HITL Monitor

Generated 2026-07-26T15:40:56.099619Z • 1231 questions • 1104 human labels

Difficulty distribution

Declared
Estimated
Human

Calibration map (b̂ vs human label)

Y-axis: Easy = 0, Medium = 0.5, Hard = 1. Few labels = expected early on.
b̂ is the difficulty estimator's continuous score — roughly −3 (very easy) to +3 (very hard). Each dot is one human-labelled question at its b̂ (x) versus the human's label (y). Good calibration shows the dots rising left-to-right: low b̂ gets "Easy", high b̂ gets "Hard". Isotonic regression is the monotonic (never-decreasing) curve fitted through these dots — the function that turns a raw b̂ into a calibrated difficulty, re-fit every 30 labels. A dot that breaks the rising trend (high b̂ but labelled "Easy") flags a question the model mis-rated. The κ panel below is the single-number summary of that agreement.

Cohen's κ — model vs human

Threshold for healthy: κ ≥ 0.5. With <30 labels, expect noisy/below-threshold values.
Cohen's κ measures how often the model's difficulty bucket matches the human's, corrected for the agreement you would get by random chance. κ = 1 is perfect agreement, κ = 0 is no better than chance, negative κ is systematic disagreement. It is recomputed every 30 ingested labels.

Recent warnings & errors (60)

Problems the run loop would otherwise only print to the terminal — a generation slot dropped on a JSON parse failure, a rubric-mining reply that would not parse, a generation/ingest that errored. They pile up here (newest first) so you can spot and debug them async.
2026-07-26 15:36:47 [warning] generator.pattern_preset — t03_general-functions/P023: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.828 vs q_t03_067). This pattern may be running out of distinct instances.
2026-07-26 15:33:54 [warning] generator.pattern_preset — t03_general-functions/P013: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.856 vs q_t03_051). This pattern may be running out of distinct instances.
2026-07-26 15:31:16 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)
2026-07-26 15:25:27 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)
2026-07-26 15:25:27 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)
2026-07-26 15:22:12 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)
2026-07-26 15:18:25 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)
2026-06-22 12:53:35 [warning] generator.pattern_preset — t03_general-functions/P023: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.917 vs q_t03_068). This pattern may be running out of distinct instances.
2026-06-22 12:50:13 [warning] generator.pattern_preset — t03_general-functions/P023: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.837 vs q_t03_053). This pattern may be running out of distinct instances.
2026-06-21 12:12:31 [warning] generator.pattern_preset — t06_rational-functions/P039: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.894 vs q_t06_027). This pattern may be running out of distinct instances.
2026-06-21 12:10:44 [warning] generator.pattern_preset — t06_rational-functions/P038: dedup exhausted after 2 attempts; accepting best fallback (cosine=0.926 vs q_t06_016). This pattern may be running out of distinct instances.
2026-06-21 12:08:59 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)
2026-06-21 12:07:45 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)
2026-06-21 12:07:45 [warning] generator.pattern_preset — t06_rational-functions/P037: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.818 vs q_t06_039). This pattern may be running out of distinct instances.
2026-06-21 12:06:38 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)
2026-06-21 11:59:42 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)
2026-06-21 11:59:42 [warning] generator.pattern_preset — t06_rational-functions/P038: CIRCUIT OPEN — 3 best-of fallbacks in this batch is draining Claude $ for ~0 marginal diversity. Remaining slots scheduled for this pattern will be SKIPPED. Consider authoring a sibling pattern or expanding the canonical examples.
2026-06-21 11:59:42 [warning] generator.pattern_preset — t06_rational-functions/P038: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.867 vs q_t06_041). This pattern may be running out of distinct instances.
2026-06-21 11:57:36 [warning] generator.pattern_preset — t06_rational-functions/P038: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.924 vs q_t06_065). This pattern may be running out of distinct instances.
2026-06-21 11:55:50 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)
2026-06-21 11:55:50 [warning] generator.pattern_preset — t06_rational-functions/P038: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.907 vs q_t06_066). This pattern may be running out of distinct instances.
2026-06-21 11:48:26 [warning] generator.pattern_preset — t06_rational-functions/P038: CIRCUIT OPEN — 3 best-of fallbacks in this batch is draining Claude $ for ~0 marginal diversity. Remaining slots scheduled for this pattern will be SKIPPED. Consider authoring a sibling pattern or expanding the canonical examples.
2026-06-21 11:48:26 [warning] generator.pattern_preset — t06_rational-functions/P038: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.897 vs q_t06_064). This pattern may be running out of distinct instances.
2026-06-21 11:46:25 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)
2026-06-21 11:46:25 [warning] generator.pattern_preset — t06_rational-functions/P038: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.926 vs q_t06_046). This pattern may be running out of distinct instances.
2026-06-21 11:43:26 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)
2026-06-21 11:43:26 [warning] generator.pattern_preset — t06_rational-functions/P038: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.949 vs q_t06_031). This pattern may be running out of distinct instances.
2026-06-21 05:30:50 [warning] generator.pattern_preset — t16_vectors/P157: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.848 vs q_t16_017). This pattern may be running out of distinct instances.
2026-06-21 04:54:01 [warning] visual.llm_svg — LLM-SVG produced no figure — sanitiser stripped everything. intent='Shows the line y = x − 2 with an open circle at the removable discontinuity (−2,' detail='<svg xmlns="http://www.w3.org/2000/svg" viewBox="-80 -80 360 300" width="540" height="450">\n <defs>\n <marker id="arrowhead" markerWidth="8" markerHeight="6"'
2026-06-21 04:42:41 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)
2026-06-21 04:42:41 [warning] generator.pattern_preset — t03_general-functions/P022: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.800 vs q_t03_060). This pattern may be running out of distinct instances.
2026-06-21 04:38:29 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)
2026-06-21 04:34:28 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)
2026-06-21 04:32:30 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)
2026-06-21 04:24:47 [warning] visual.matplotlib — matplotlib sandbox — rejected unsafe code — imports are not allowed (np / plt / math are pre-provided)
2026-06-21 04:07:23 [warning] generator.pattern_preset — t15_complex-number/P119: CIRCUIT OPEN — 3 best-of fallbacks in this batch is draining Claude $ for ~0 marginal diversity. Remaining slots scheduled for this pattern will be SKIPPED. Consider authoring a sibling pattern or expanding the canonical examples.
2026-06-21 04:07:23 [warning] generator.pattern_preset — t15_complex-number/P119: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.848 vs q_t15_030). This pattern may be running out of distinct instances.
2026-06-21 04:03:28 [warning] generator.pattern_preset — t15_complex-number/P119: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.960 vs q_t15_029). This pattern may be running out of distinct instances.
2026-06-21 04:01:54 [warning] generator.pattern_preset — t15_complex-number/P119: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.807 vs q_t15_026). This pattern may be running out of distinct instances.
2026-06-17 04:21:53 [warning] generator.pattern_preset — t21_factor-theorem-remainder-theorem-vietas-formula/P148: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.834 vs q_t21_031). This pattern may be running out of distinct instances.
2026-06-17 04:18:34 [warning] generator.pattern_preset — t21_factor-theorem-remainder-theorem-vietas-formula/P149: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.824 vs q_t21_021). This pattern may be running out of distinct instances.
2026-06-17 04:17:20 [warning] generator.pattern_preset — t21_factor-theorem-remainder-theorem-vietas-formula/P148: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.844 vs q_t21_019). This pattern may be running out of distinct instances.
2026-06-17 04:16:13 [warning] generator.pattern_preset — t21_factor-theorem-remainder-theorem-vietas-formula/P147: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.896 vs q_t21_029). This pattern may be running out of distinct instances.
2026-06-17 04:14:55 [warning] generator.pattern_preset — t21_factor-theorem-remainder-theorem-vietas-formula/P146: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.814 vs q_t21_010). This pattern may be running out of distinct instances.
2026-06-16 11:44:46 [warning] generator.pattern_preset — t21_factor-theorem-remainder-theorem-vietas-formula/P149: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.911 vs q_t21_016). This pattern may be running out of distinct instances.
2026-06-16 11:38:56 [warning] generator.pattern_preset — t21_factor-theorem-remainder-theorem-vietas-formula/P148: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.802 vs q_t21_009). This pattern may be running out of distinct instances.
2026-06-16 10:54:32 [warning] generator.pattern_preset — t21_factor-theorem-remainder-theorem-vietas-formula/P149: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.884 vs q_t21_011). This pattern may be running out of distinct instances.
2026-06-16 10:50:35 [warning] generator.pattern_preset — t21_factor-theorem-remainder-theorem-vietas-formula/P149: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.865 vs q_t21_006). This pattern may be running out of distinct instances.
2026-06-16 06:24:25 [warning] generator.pattern_preset — t20_binomial-expansion/P145: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.858 vs q_t20_041). This pattern may be running out of distinct instances.
2026-06-16 06:22:51 [warning] generator.pattern_preset — t20_binomial-expansion/P141: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.934 vs q_t20_020). This pattern may be running out of distinct instances.
2026-06-16 06:20:46 [warning] generator.pattern_preset — t20_binomial-expansion/P140: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.826 vs q_t20_037). This pattern may be running out of distinct instances.
2026-06-16 06:18:53 [warning] generator.pattern_preset — t20_binomial-expansion/P145: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.813 vs q_t20_027). This pattern may be running out of distinct instances.
2026-06-16 06:17:18 [warning] generator.pattern_preset — t20_binomial-expansion/P144: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.856 vs q_t20_036). This pattern may be running out of distinct instances.
2026-06-16 06:01:21 [warning] generator.pattern_preset — t20_binomial-expansion/P143: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.822 vs q_t20_022). This pattern may be running out of distinct instances.
2026-06-16 05:58:31 [warning] generator.pattern_preset — t20_binomial-expansion/P140: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.887 vs q_t20_021). This pattern may be running out of distinct instances.
2026-06-16 05:57:48 [warning] generator.pattern_preset — t20_binomial-expansion/P144: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.855 vs q_t20_023). This pattern may be running out of distinct instances.
2026-06-16 05:56:35 [warning] generator.pattern_preset — t20_binomial-expansion/P141: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.864 vs q_t20_020). This pattern may be running out of distinct instances.
2026-06-16 05:55:01 [warning] generator.pattern_preset — t20_binomial-expansion/P143: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.843 vs q_t20_006). This pattern may be running out of distinct instances.
2026-06-16 05:52:24 [warning] generator.pattern_preset — t20_binomial-expansion/P142: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.839 vs q_t20_019). This pattern may be running out of distinct instances.
2026-06-16 05:51:46 [warning] generator.pattern_preset — t20_binomial-expansion/P140: dedup exhausted after 3 attempts; accepting best fallback (cosine=0.850 vs q_t20_021). This pattern may be running out of distinct instances.

Constitution — v111

Core principles are hand-authored or curriculum-seeded rules, always applied. Mined principles are learned from reviewer notes; each carries a Beta(α, β) belief, where α = 1 + supporting reviews and β = 1 + contradicting reviews (so α=3, β=1 means 2 reviews backed it and 0 went against it). Its estimated support rate is α/(α+β). A mined principle is promoted to active — and only then fed into generation prompts — once it has at least 5 supporting reviews and the Wilson 95% lower bound on its support rate clears the promotion threshold (about 5 clean supports). An active principle is retired only if it later collects 2 or more contradictions. "Inactive" below means simply not-yet-promoted.
Core principles (77)
Active mined principles (239)
Inactive / retired mined principles (127)

Pending review queue (13)

High-score items are highest information-gain. Walk top-down.

Per-question log (1231)

Click any row to expand the human feedback that was submitted for it.
H — the model's uncertainty about the question's difficulty: the Shannon entropy (in bits, 0 to about 1.58) of its Easy/Medium/Hard posterior. H near 0 means the model is confident the question sits in one bucket; a high H means it is torn between buckets.
Score — the active-learning priority: how much the system expects to learn from a human review of this question. It blends H, Δ (the gap between the model's estimate and the difficulty the slot asked for) and novelty (how unlike the existing canonical examples the question is). Higher = review sooner — the pending queue is sorted by it.
IDPattern / SectionDeclaredEstimatedReviewedHScoreOutcomeFeedbackPreview
q_t02_001P002EasyMediumEasy0.00.527human_labelledreviewedEvaluate the following limit: $$\lim_{x\to\infty} \frac{x^3 + 5x}{e^x + x^2}$$
q_t02_002P011HardHardHard0.00.278human_labelledreviewedEvaluate the following limit by utilising logarithms: $$\lim_{x\to 0^+} \left(\sin x\right)^{\tan x}$$
q_t02_003P010HardMediumHard0.00.588human_labelledreviewedEvaluate the limit: $$\lim_{x \to 0} \frac{\ln(1+x^2) - x^2}{x^2(e^{x^2}-1)}$$
q_t02_004P002EasyMediumEasy0.00.608human_labelledreviewedEvaluate the following limit: $$\lim_{x\to\infty} \frac{\ln x + x^2}{3x^2 - 5x + 1}$$
q_t02_005P010MediumMediumMedium0.00.44human_labelledreviewedEvaluate the limit: $$\lim_{x \to 0} \frac{\sin(3x) - 3x\cos(x)}{x^3}$$
q_t02_006P001MediumMediumEasy0.00.407human_labelledreviewedEvaluate $\lim_{x \to 2} \left(3x^3 - 5x^2 + 2\cos(\pi x) + e^x\right)$.
q_t02_007P003MediumHardMedium0.00.485human_labelledreviewedEvaluate $\lim_{x \to 3} \dfrac{x^2 - x - 6}{\sqrt{2x + 3} - 3}$.
q_t02_008P009EasyEasyEasy0.00.054human_labelledreviewedProve, using the squeeze theorem, that $\lim_{x \to 0} x^2 \cos\!\left(\dfrac{1}{x^2}\right) = 0$.
q_t02_009P008EasyMediumEasy0.00.56human_labelledreviewedEvaluate the limit $\lim_{x\to 0} \dfrac{\log((1+x)^4)}{2x}$.
q_t02_010P005EasyMediumEasy0.00.558human_labelledreviewedEvaluate $\displaystyle\lim_{n\to\infty}\left(1+\dfrac{1}{n}\right)^{5n}$.
q_t02_011P004HardMediumHard0.00.527human_labelledreviewedEvaluate the limit $$\lim_{x\to\infty}\left(\sqrt{4x^2+6x-1} - 2x - 5\right).$$
q_t02_012P011HardHardHard0.00.352human_labelledreviewedEvaluate the following limit by utilising logarithms: $$\lim_{x\to 0^+} \left(1 + \sin 3x\right)^{\cot x}$$
q_t02_013P007MediumMediumMedium0.00.368human_labelledreviewedEvaluate $\displaystyle\lim_{x\to 0} \dfrac{e^{6x} - e^{2x}}{4x^2 + 5x}$.
q_t02_014P006MediumHardMedium0.00.552human_labelledreviewedEvaluate the limit $$\lim_{x \to 0} \frac{\sin 5x \cdot \tan 2x}{x \cdot \sin 4x}.$$
q_t02_015P002HardHardHard0.00.341human_labelledreviewedEvaluate the following limit: $$\lim_{x\to -\infty} \frac{\sqrt{9x^2 + 2x} + 3x}{\ln(-x) - x}$$
q_t02_016P010MediumHardHard0.00.532human_labelledreviewedEvaluate the limit: $$\lim_{x \to 0} \frac{e^{x^2} - 1 - x^2}{\sin^4 x}$$
q_t02_017P001EasyMediumEasy0.00.584human_labelledreviewedEvaluate $\lim_{x \to -1} \left(2x^4 - 3x^2 + 5x - 4\right)$.
q_t02_018P003MediumHardMedium0.00.464human_labelledreviewedEvaluate $\lim_{x \to 4} \dfrac{\sqrt{3x+4} - 4}{x^2 - 7x + 12}$.
q_t02_019P009EasyMediumEasy0.00.539human_labelledreviewedProve, using the squeeze theorem, that $\lim_{x \to 0} x^4 \sin\!\left(\dfrac{1}{x}\right) = 0$.
q_t02_020P008MediumMediumEasy0.00.387human_labelledreviewedEvaluate the limit $\lim_{x\to 0} \dfrac{\log(1+5x)}{3x}$.
q_t02_021P005EasyMediumMedium0.00.578human_labelledreviewedEvaluate $\displaystyle\lim_{n\to\infty}\left(\dfrac{n+3}{n}\right)^{2n}$.
q_t02_022P004MediumHardMedium0.00.472human_labelledreviewedEvaluate the limit $$\lim_{x\to\infty}\left(\sqrt{9x^2 - 12x + 5} - 3x + 4\right).$$
q_t02_023P011MediumHardMedium0.00.432human_labelledreviewedEvaluate the following limit by utilising logarithms: $$\lim_{x\to\infty} \left(1 + \frac{3}{x}\right)^{x^2}$$
q_t02_024P007MediumMediumMedium0.00.362human_labelledreviewedEvaluate $\displaystyle\lim_{x\to 0} \dfrac{e^{5x} - e^{3x}}{2x(e^x + 1)}$.
q_t02_025P006HardMediumMedium0.00.534human_labelledreviewedEvaluate the limit $$\lim_{x \to 0} \frac{\sin(3x^2) \cdot \tan(5x)}{x \cdot \sin(2x) \cdot \tan(-3x)}.$$
q_t02_026P002MediumMediumMedium0.00.425human_labelledreviewedEvaluate the limit $\lim_{x\to\infty} \dfrac{5x^3 - \sqrt{9x^6 + 2x^4}}{4x^3 - 7x^2 + 1}$.
q_t02_027P010MediumHardMedium0.00.528human_labelledreviewedEvaluate the limit: $$\lim_{x \to 0} \frac{x - \sin x}{x^2(e^x - 1)}$$
q_t02_028P001EasyMediumEasy0.00.63human_labelledreviewedEvaluate $\lim_{x \to 3} \left(x^3 - 4x^2 + \sin\!\left(\dfrac{\pi x}{6}\right)\right)$.
q_t02_029P003EasyMediumEasy0.00.573human_labelledreviewedEvaluate $\lim_{x \to 3} \dfrac{x^2 - 2x - 3}{2x^2 - 5x - 3}$.
q_t02_030P009MediumMediumMedium0.00.387human_labelledreviewedEvaluate the following limit using the squeeze theorem: $$\lim_{x \to 0^+} \sqrt{x}\, \cos\!\left(\frac{\pi}{x}\right)$$
q_t02_031P008HardMediumMedium0.00.571human_labelledreviewedEvaluate $\lim_{x\to 0} \dfrac{\log\left(\dfrac{1}{2}+x\right) + \log\left(\dfrac{1}{2}+x\right)^{-1}\cdot\log\left(1+4x+4x^2\right)}{x}$.
q_t02_032P005MediumHardHard0.00.513human_labelledreviewedEvaluate $\displaystyle\lim_{x\to\infty}\left(\dfrac{3x+7}{3x+1}\right)^{2x}$.
q_t02_033P004HardHardMedium0.00.295human_labelledreviewedEvaluate the limit $$\lim_{x\to\infty}\left(2x - 3 - \sqrt{4x^2 - 10x + 7}\right).$$
q_t02_034P002EasyMediumEasy0.00.543human_labelledreviewedEvaluate the following limit: $$\lim_{x\to\infty} \frac{e^x + 3x^2}{2e^x - 5x + 1}$$
q_t02_035P010MediumHardMedium0.00.535human_labelledreviewedEvaluate the limit: $\lim_{x \to 0} \dfrac{\sin(3x) - 3x}{x^3}$
q_t02_036P001MediumMediumMedium0.00.129human_labelledreviewedEvaluate $\displaystyle\lim_{x \to -2} \left(2x^4 - x^3 + 3\sin\!\left(\frac{\pi x}{2}\right) + e^{x+2}\right)$.
q_t02_037P003MediumHardMedium0.00.514human_labelledreviewedEvaluate $\lim_{x \to 5} \dfrac{x^2 - 3x - 10}{\sqrt{x + 4} - 3}$.
q_t02_038P009EasyMediumEasy0.00.54human_labelledreviewedEvaluate the following limit using the squeeze theorem: $$\lim_{x \to 0^+} x^3 \cos\!\left(\frac{1}{\sqrt{x}}\right)$$
q_t02_039P008EasyMediumEasy0.00.547human_labelledreviewedEvaluate the limit $\lim_{x\to 0} \dfrac{\log((1+x)^4)}{2x}$.
q_t02_040P005EasyMediumEasy0.00.543human_labelledreviewedEvaluate $\displaystyle\lim_{x\to\infty}\left(1+\dfrac{3}{x}\right)^{x}$.
q_t02_041P004HardMediumMedium0.00.54human_labelledreviewedEvaluate the limit $$\lim_{x\to\infty}\left(\sqrt{9x^2 + 6x - 2} - 3x - 5\right).$$
q_t02_042P011HardHardHard0.00.096human_labelledreviewedEvaluate the following limit by utilising logarithms: $$\lim_{x\to 0^+} x^{\sin x}$$
q_t02_043P007MediumMediumMedium0.00.128human_labelledreviewedEvaluate $\displaystyle\lim_{x\to 0} \dfrac{e^{7x} - e^{4x} - e^{3x} + 1}{x^2}$.
q_t02_044P006MediumMediumMedium0.00.128human_labelledreviewedEvaluate the limit $\lim_{x \to 0} \dfrac{\sin 4x \tan 3x}{6x^2}$.
q_t02_045P002HardHardHard0.00.092human_labelledreviewedEvaluate the limit: $$\lim_{x\to\infty} \frac{\sqrt{9x^4 + 2x^3} - 3x^2}{x + \ln x}$$
q_t02_046P010MediumHardMedium0.00.529human_labelledreviewedEvaluate the limit: $$\lim_{x \to 0} \frac{e^{2x} - 1 - 2x}{\sin^2 x}$$
q_t02_047P001EasyMediumEasy0.00.545human_labelledreviewedEvaluate $\displaystyle\lim_{x \to 2} \left(x^4 - 3x^3 + 2x - 5\right)$.
q_t02_048P003MediumHardMedium0.00.514human_labelledreviewedEvaluate $\lim_{x \to 2} \dfrac{\sqrt{3x - 2} - 2}{x^2 - 5x + 6}$.
q_t02_049P009EasyMediumEasy0.00.54human_labelledreviewedEvaluate the following limit using the squeeze theorem: $$\lim_{x \to 0^+} x \sin\!\left(\frac{1}{x^2}\right)$$
q_t02_050P008MediumHardMedium0.00.535human_labelledreviewedEvaluate the limit: $\lim_{x\to 0} \dfrac{\log(1+5x) + \log(1+3x)}{4x}$
q_t02_051P005EasyEasyMedium0.00.001human_labelledreviewedEvaluate $\displaystyle\lim_{x\to\infty}\left(\dfrac{2x+9}{2x+3}\right)^{x}$.
q_t02_052P004MediumHardMedium0.00.508human_labelledreviewedEvaluate the limit $$\lim_{x\to\infty}\left(\sqrt{25x^2 - 10x + 3} - 5x + 2\right).$$
q_t02_053P011MediumMediumEasy0.00.128human_labelledreviewedEvaluate the following limit by utilising logarithms: $$\lim_{x\to\infty} x^{\frac{1}{\ln x}}$$
q_t02_054P007MediumMediumMedium0.00.128human_labelledreviewedEvaluate $\displaystyle\lim_{x\to 0} \dfrac{e^{6x} - e^{2x}}{e^{3x} - e^{x}}$.
q_t02_055P006HardMediumMedium0.00.543human_labelledreviewedEvaluate the limit: $\lim_{x \to 0} \dfrac{\sin(5x)\tan(3x)}{\sin(2x)\tan(4x)}$
q_t02_056P002MediumMediumMedium0.00.131human_labelledreviewedEvaluate the limit: $\lim_{x\to\infty} \dfrac{5x^3 - \sqrt{9x^6 + 2x^4}}{4x^3 - 7x + 1}$
q_t02_057P010MediumMediumMedium0.00.124human_labelledreviewedEvaluate the limit: $\lim_{x \to 0} \dfrac{\sin(3x) - 3x}{x^3}$
q_t02_058P001EasyMediumEasy0.00.548human_labelledreviewedEvaluate $\displaystyle\lim_{x \to -3} \left(x^3 + 2x^2 - 4x + 7\right)$.
q_t02_059P003EasyEasyMedium0.00.001human_labelledreviewedEvaluate $\lim_{x \to 4} \dfrac{x^2 - 5x + 4}{\sqrt{x + 5} - 3}$.
q_t02_060P009MediumMediumMedium0.00.124human_labelledreviewedEvaluate the following limit using the squeeze theorem: $$\lim_{x \to \infty} \frac{\sin(x^3)}{x^2 + 1}$$
q_t02_061P008HardHardHard0.00.098human_labelledreviewedEvaluate $\lim_{x\to 0} \dfrac{\log\left(\dfrac{1}{2}+2x\right) + \log\left(\dfrac{1}{4}+x\right) + \log 8}{x}$.
q_t02_062P005MediumHardMedium0.00.516human_labelledreviewedEvaluate $\displaystyle\lim_{x\to\infty}\left(\dfrac{4x+11}{4x+3}\right)^{3x}$.
q_t02_063P004HardMediumMedium0.00.543human_labelledreviewedEvaluate the limit: $\lim_{x\to\infty}\left(\sqrt{9x^2+6x+5}-\sqrt{9x^2-12x+1}\right)$.
q_t02_064P009MediumMediumMedium0.00.0human_labelledreviewedEvaluate the following limit using the squeeze theorem: $$\lim_{x \to \infty} \frac{\cos\!\left(e^x\right)}{x^2 + 3x + 1}$$
q_t02_065P002MediumMediumMedium0.00.0human_labelledreviewedEvaluate the limit: $\lim_{x\to\infty} \dfrac{5x^3 - \sqrt{9x^6 + 2x^5}}{4x^3 + 3x - 1}$
q_t02_066P010HardMediumHard0.00.5human_labelledreviewedEvaluate the limit: $\lim_{x \to 0} \dfrac{e^{x^2} - \cos x - x^2}{x^4}$.
q_t02_067P009MediumMediumMedium0.00.0human_labelledreviewedEvaluate the following limit using the squeeze theorem: $$\lim_{x \to 0^+} \sqrt{x}\,\cos\!\left(\frac{\pi}{x^2}\right)$$
q_t02_068P002EasyMediumEasy0.00.5human_labelledreviewedEvaluate the limit: $$\lim_{x\to\infty} \frac{\ln x + x^3}{2x^3 - 5x}$$
q_t02_069P010MediumHardHard0.00.5human_labelledreviewedEvaluate the limit: $$\lim_{x \to 0} \frac{e^{\sin x} - e^x}{\tan x - x}.$$
q_t02_070P001MediumMediumEasy0.00.0human_labelledreviewedEvaluate $\displaystyle\lim_{x \to -1} \left(2x^4 - 3x^3 + x^2 - 5\cos(\pi x) + e^{2x}\right)$.
q_t02_071P003MediumMediumEasy0.00.0human_labelledreviewedEvaluate $\lim_{x \to 5} \dfrac{\sqrt{3x - 6} - 3}{x^2 - 4x - 5}$.
q_t02_072P009EasyMediumEasy0.00.5human_labelledreviewedEvaluate the following limit using the squeeze theorem: $$\lim_{x \to 0} x^2 \cos\!\left(\frac{1}{x^3}\right)$$
q_t02_073P008EasyMediumEasy0.00.5human_labelledreviewedEvaluate the limit $\lim_{x\to 0} \dfrac{\log((1+x)^4)}{2x}$.
q_t02_074P005EasyMediumMedium0.00.5human_labelledreviewedEvaluate $\displaystyle\lim_{x\to\infty}\left(1+\dfrac{3}{x}\right)^{2x}$.
q_t02_075P004HardMediumHard0.00.5human_labelledreviewedEvaluate the limit $$\lim_{x\to\infty}\left(\sqrt{5x^2 + 3x - 2} - \sqrt{5}\,x + 7\right).$$
q_t02_076P011HardMediumMedium0.00.5human_labelledreviewedEvaluate the following limit by utilising logarithms: $$\lim_{x\to\infty} \left(\ln x\right)^{\frac{1}{x}}$$
q_t02_077P007MediumMediumMedium0.00.0human_labelledreviewedEvaluate $\displaystyle\lim_{x\to 0} \dfrac{e^{5x} - e^{2x}}{x(e^{3x}+2)}$.
q_t02_078P006MediumMediumMedium0.00.0human_labelledreviewedEvaluate the limit $$\lim_{x \to 0} \frac{\tan(3x) \cdot \sin(x^2)}{x^2 \cdot \sin(6x)}.$$
q_t02_079P002HardMediumHard0.00.5human_labelledreviewedEvaluate $\lim_{x\to\infty} \dfrac{\sqrt{9x^4 + 2x^3} - 3x^2}{\ln x + x - e^{-x}}$.
q_t02_080P010MediumMediumEasy0.00.0human_labelledreviewedEvaluate the limit: $$\lim_{x \to 0} \frac{\sin(x^2) - x^2 \cos x}{x^4}.$$
q_t02_081P001EasyEasyEasy0.00.0human_labelledreviewedFind $\displaystyle\lim_{x \to 2}\left(x^3 - 4x^2 + 3x + 7\right)$.
q_t02_082P003MediumMediumMedium0.00.0human_labelledreviewedEvaluate $\lim_{x \to 2} \dfrac{x^3 - 8}{\sqrt{5x - 1} - 3}$.
q_t02_083P009EasyMediumEasy0.00.5human_labelledreviewedEvaluate the following limit using the squeeze theorem: $$\lim_{x \to 0^+} x^3 \sin\!\left(\frac{1}{\sqrt{x}}\right)$$
q_t02_084P008MediumMediumEasy0.00.0human_labelledreviewedEvaluate the limit $\lim_{x \to 0} \dfrac{\log(1 + 5x)}{3x}$.
q_t02_085P005EasyEasyMedium0.00.0human_labelledreviewedEvaluate $\displaystyle\lim_{x\to\infty}\left(\dfrac{2x+9}{2x+3}\right)^{x}$.
q_t02_086P004MediumMediumMedium0.00.0human_labelledreviewedEvaluate the limit $$\lim_{x\to\infty}\left(2x - 3 - \sqrt{4x^2 - 10x + 1}\right).$$
q_t02_087P011MediumMediumMedium0.00.0human_labelledreviewedEvaluate the following limit by utilising logarithms: $$\lim_{x\to 0^+} \left(1 + 3x\right)^{\frac{1}{\ln x}}$$
q_t02_088P007MediumMediumEasy0.00.0human_labelledreviewedEvaluate $\displaystyle\lim_{x\to 0} \dfrac{e^{6x} - e^{2x} - e^{4x} + 1}{3x^2}$.
q_t02_089P006HardMediumHard0.00.5human_labelledreviewedEvaluate the limit $$\lim_{x \to 0} \frac{\sin(2x^2) \cdot \tan(5x)}{x \cdot \sin(3x) \cdot \tan(-4x)}.$$
q_t02_090P002MediumMediumEasy0.00.0human_labelledreviewedEvaluate the limit: $\lim_{x\to\infty} \dfrac{5x^3 - \sqrt{9x^6 + 2x^4}}{4x^3 - 7x^2 + 1}$.
q_t02_091P010MediumHardMedium0.00.5human_labelledreviewedEvaluate the limit: $$\lim_{x \to 0} \frac{\ln(\cos x)}{x \sin x}.$$
q_t02_092P001EasyMediumEasy0.00.5human_labelledreviewedEvaluate $\displaystyle\lim_{x \to 2} \left(3\sin\!\left(\frac{\pi x}{6}\right) - x^2 + e^{x-2}\right)$.
q_t02_093P003EasyMediumEasy0.00.5human_labelledreviewedEvaluate $\lim_{x \to 4} \dfrac{x^2 - 3x - 4}{x^2 - 16}$.
q_t02_094P009MediumMediumMedium0.00.0human_labelledreviewedEvaluate the following limit using the squeeze theorem: $$\lim_{x\to\infty} \frac{\sin(x^3)}{x^2 + 1}$$
q_t02_095P008HardMediumMedium0.00.5human_labelledreviewedEvaluate $\lim_{x\to 0} \dfrac{\log\left(\dfrac{(1+2x)^3}{1+x}\right)}{x}$.
q_t02_096P005MediumMediumMedium0.00.0human_labelledreviewedEvaluate $\displaystyle\lim_{x\to\infty}\left(\dfrac{3x+7}{3x+1}\right)^{4x+3}$.
q_t02_097P004HardMediumHard0.00.5human_labelledreviewedEvaluate the limit $$\lim_{x\to\infty}\left(\sqrt{9x^2 + 6x - 4} - 3x - \frac{1}{x+1}\right).$$
q_t02_098P002EasyMediumEasy0.00.494human_labelledreviewedEvaluate the limit: $$\lim_{x\to\infty} \frac{e^x + x^4}{3e^x - x^2}.$$
q_t02_099P011MediumMedium—0.00.007discarded—Evaluate the following limit by utilising logarithms: $$\lim_{x\to 0^+} x^{\sin x}$$
q_t02_100P001MediumMedium—0.00.02discarded—Evaluate $\displaystyle\lim_{x \to \pi} \left(2x^2 - \cos(2x) + e^{x - \pi}\right)$.
q_t02_101P006EasyMedium—0.00.49discarded—Evaluate the limit $$\lim_{x \to 0} \frac{\sin 5x}{\tan 2x}.$$
q_t02_102P002HardHard—0.00.013discarded—Evaluate the limit: $$\lim_{x\to -\infty} \frac{\sqrt{4x^2 + 3x} + 2x}{x^3 + e^x}.$$
q_t02_103P002EasyMediumEasy0.00.486human_labelledreviewedEvaluate the limit: $$\lim_{x\to\infty} \frac{x^3 + \ln x}{x^3 + e^x}.$$
q_t02_104P006MediumMediumMedium0.00.017human_labelledreviewedEvaluate the limit $$\lim_{x \to 0} \frac{\sin(5x) \cdot \tan(x^2)}{x^2 \cdot \tan(2x)}.$$
q_t02_105P004MediumMediumMedium0.00.021human_labelledreviewedEvaluate the limit $$\lim_{x\to\infty}\left(\sqrt{4x^2 + 12x + 5} - 2x\right).$$ Hint: try completing the square inside the radical.
q_t02_106P002EasyMediumEasy0.00.5human_labelledreviewedEvaluate the limit: $$\lim_{x\to\infty} \frac{\ln x + x^2}{4x^2 - 3x + 5}$$
q_t02_107P010MediumMediumMedium0.00.0human_labelledreviewedEvaluate the limit: $$\lim_{x \to 0} \frac{e^{2x} - 1 - 2x}{x(e^x - 1)}.$$
q_t02_108P006EasyMedium—0.00.5needs_human—Find $\displaystyle\lim_{x \to 0} \dfrac{x^2}{\tan(4x)\cdot\sin(x)}$, showing clearly how the standard limit forms $\displaystyle\lim_{u \to
q_t02_109P005EasyMedium—0.00.5needs_human—Evaluate $\displaystyle\lim_{x \to 0}(1+5x)^{\frac{1}{x}}$.
q_t02_110P011EasyEasy—0.00.0needs_human—Evaluate the following limit by utilising logarithms: $$\lim_{x\to\infty} \left(1 + \frac{2}{x}\right)^{\!\frac{x}{3}}$$
q_t02_111P007EasyMedium—0.00.5needs_human—Evaluate $\displaystyle\lim_{t\to 0} \dfrac{3e^{4t} - 3e^{t}}{2t}$.
q_t02_112P001EasyMedium—0.00.5needs_human—Evaluate $\displaystyle\lim_{x \to 3}\left(2x^2 - 7x + e^{x-3} + \cos(\pi x)\right)$.
q_t02_113P002EasyMedium—0.00.5needs_human—Evaluate $\displaystyle\lim_{x\to\infty} \frac{2x^3 + \ln x}{\sqrt[3]{8x^9 + x^6}}$.
q_t02_114P008EasyMedium—0.00.5needs_human—Find $\lim_{x \to 0} \dfrac{\ln\left((1+x)^4\right)}{2x}$.
q_t02_115P003EasyMedium—0.00.5needs_human—Evaluate $\displaystyle\lim_{x \to 9} \dfrac{\sqrt{x} - 3}{x - 9}$.
q_t02_116P009EasyMedium—0.00.5needs_human—Evaluate the following limit using the squeeze theorem: $$\lim_{x \to \infty} \frac{3\cos(e^x)}{x + 2}$$
q_t02_117P004EasyMedium—0.00.5needs_human—Evaluate the limit $$\lim_{x\to\infty} \left(x - \sqrt{x^2 - 6x + 2}\right).$$
q_t02_118P010EasyMedium—0.00.5needs_human—Evaluate the following limit using L'Hôpital's rule: $$\lim_{x \to 0} \frac{\sin(4x)}{e^{2x} - 1}.$$
q_t03_001P012MediumMediumMedium0.00.0human_labelledreviewedA relation $R$ is defined by $x^2 + 4x - y^2 + 2y = -1$. (a) Show that $R$ represents a hyperbola by writing the equation in standard form.
q_t03_002P021EasyMediumEasy0.00.5human_labelledreviewedConsider the function $f(x) = x^2 - 6x + (2k + 1)$. Find all real values of $k$ such that $f(x) > 0$ for all $x \in \mathbb{R}$.
q_t03_003P016HardHardHard0.00.0human_labelledreviewedLet $h(x) = \dfrac{\sqrt{4x^2 + 9}}{e^{2x} - 5e^x + 4}$. (a) State the domain of $h$. (b) Find all vertical asymptotes of $h$, supporting
q_t03_004P020MediumMediumMedium0.00.0human_labelledreviewedFind all real values of $k$ such that $\dfrac{x^2 + 9}{x} \geq k$ for all $x > 0$.
q_t03_005P018EasyEasyMedium0.00.0human_labelledreviewedSolve exactly $\sqrt{3x + 4} = x - 2$, finding all values of $x$ that satisfy this equation.
q_t03_006P015EasyMediumEasy0.00.5human_labelledreviewedThe graph of $y = f(x)$ is transformed to the graph $y = 2f(x - 3) + 1$. (a) Describe this transformation as a sequence of simple transform
q_t03_007P014EasyEasyEasy0.00.0human_labelledreviewedConsider the functions $f$ and $g$ defined by $f(x) = \ln x$ and $g(x) = \ln(3x - 6)$, where each function has the largest possible domain.
q_t03_008P024MediumMediumHard0.00.0human_labelledreviewedThe graph of $y = f(x)$ is shown, where $f(x) = x^2 - 4x + 3 = (x-1)(x-3)$. The parabola opens upward with vertex at $(2, -1)$, crossing the
q_t03_009P019HardMediumEasy0.00.5human_labelledreviewedUse a graphic display calculator or numerical methods to solve the equation ln(x² + 1) = 2sin(πx) − 0.5x, correct to three decimal places.
q_t03_010P017HardMediumMedium0.00.5human_labelledreviewedDetermine whether each of the following functions is odd, even, or neither. Justify your answer fully by computing $f(-x)$, $g(-x)$, and $h(
q_t03_011P022EasyEasyMedium0.00.0human_labelledreviewedSketch the graph of $f(x) = x e^{-x}$, showing clearly all intercepts, asymptotes, and any stationary points.
q_t03_012P013EasyEasyMedium0.00.0human_labelledreviewedLet $f(x) = \sqrt{x - 1}$ and $g(x) = \dfrac{1}{x + 2}$. (a) Find $(f \circ g)(x)$. (b) Find $(g \circ f)(x)$. (c) State the domain of ea
q_t03_013P023HardMediumHard0.00.5human_labelledreviewedLet $f(x) = x^4 - 8x^2 + 7$. **(a)** Show that $f$ is **not** one-to-one on $\mathbb{R}$. **(b)** Find the smallest real value of $k$ such
q_t03_014P012EasyMediumEasy0.00.5human_labelledreviewedA relation $R$ is defined by $x^2 + y^2 + 6x - 4y = 3$. (a) Show that $R$ represents a circle by writing the equation in standard form. (b
q_t03_015P021MediumMediumMedium0.00.0human_labelledreviewedConsider the equation $x^2 + (k-2)x + (k^2 - 3k + 1) = 0$, where $k \in \mathbb{R}$. Determine all values of $k$ such that the equation has
q_t03_016P016MediumHardMedium0.00.5human_labelledreviewedLet $f(x) = \dfrac{\sqrt{9x^2 + 16}}{e^{2x} - 4e^x + 3}$. (a) State the domain of $f$. (b) Find all vertical asymptotes of $f$, supporting
q_t03_017P020MediumMediumMedium0.00.0human_labelledreviewedFind all real values of $k$ such that $2x + \dfrac{8}{x^2} > k$ for all $x > 0$.
q_t03_018P018MediumMediumMedium0.00.0human_labelledreviewedFind the exact value(s) of x that satisfy the equation $\sqrt{3x + 4} = x - 2$.
q_t03_019P015HardMediumHard0.00.5human_labelledreviewedThe graph of $y = h(x)$ is transformed to the graph of $y = -3h(-2x - 8) + 5$. (a) Describe this transformation as an ordered sequence of s
q_t03_020P014MediumMediumMedium0.00.0human_labelledreviewedConsider the functions f and g defined by f(x) = ln x and g(x) = ln(3x − 6), where each function has the largest possible domain. (a) Write
q_t03_021P024MediumMediumMedium0.00.0human_labelledreviewedThe graph of y = f(x) = x^2 - 4x + 3 is shown for -1 ≤ x ≤ 5. Note that f(x) = (x-1)(x-3), so the graph is an upward-opening parabola with z
q_t03_022P019MediumMediumEasy0.00.0human_labelledreviewedUse a graphic display calculator or numerical methods to solve $\ln(x+2) = \sin(2x)$, correct to three decimal places.
q_t03_023P017EasyMediumEasy0.00.5human_labelledreviewedDetermine whether each of the following functions is odd, even, or neither. Justify your answer by computing $f(-x)$, $g(-x)$, and $h(-x)$ a
q_t03_024P022MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $f(x) = \dfrac{x^2 + x - 6}{x + 1}$, showing clearly all intercepts, asymptotes, and any stationary points.
q_t03_025P013MediumMediumMedium0.00.0human_labelledreviewedLet $f(x) = \ln(x + 4)$ and $g(x) = \dfrac{3}{x - 1}$. (a) Find $(f \circ g)(x)$. (b) Find $(g \circ f)(x)$. (c) State the domain of each
q_t03_026P012MediumMediumMedium0.00.0human_labelledreviewedA relation $R$ is defined by $2x - y^2 + 6y = 7$. (a) Show that $R$ represents a sideways (horizontal) parabola by writing the equation in
q_t03_027P021EasyMediumEasy0.00.5human_labelledreviewedConsider the function $h(x) = \sqrt{x^2 + 2kx + (3k - 2)}$. Find all real values of $k$ such that $h(x)$ is defined for all $x \in \mathbb{R
q_t03_028P016HardHardHard0.00.0human_labelledreviewedLet $h(x) = \dfrac{\ln(x^2 - 4)}{\sqrt{x^2 + 3x} - 6}$. (a) State the domain of $h$. (b) Find all vertical asymptotes of $h$, if any, and j
q_t03_029P020MediumMediumMedium0.00.0human_labelledreviewedFind all real values of $k$ such that $3x^2 - 12x + 16 > k$ for all real $x$.
q_t03_030P018EasyMediumMedium0.00.5human_labelledreviewedFind the exact value(s) of $x$ that satisfy the equation $\sqrt{5x + 6} = x$.
q_t03_031P015EasyMediumEasy0.00.5human_labelledreviewedThe graph of $y = f(x)$ is transformed to the graph $y = 2f(x - 3) + 1$. (a) Describe this transformation as a sequence of simple transform
q_t03_032P014EasyMedium—0.00.5discarded—Consider the functions f and g defined by f(x) = ln x and g(x) = ln(3x - 6), where each function has the largest possible domain. (a) Write
q_t03_033P024MediumMedium—0.00.0discarded—The graph of $y = f(x)$ is shown, where $f(x) = x^2 - 4x + 3 = (x-1)(x-3)$. The parabola opens upward, crossing the $x$-axis at $x = 1$ and
q_t03_034P019HardMediumEasy0.00.5human_labelledreviewedUse a graphic display calculator or numerical methods to solve the equation ln(x² + 1) = 2sin(πx) − 0.5x, giving all solutions correct to th
q_t03_035P017HardMediumHard0.00.5human_labelledreviewedDetermine whether each of the following functions is odd, even, or neither. Justify your answer fully by computing $f(-x)$, $g(-x)$, and $h(
q_t03_036P022EasyEasyMedium0.00.0human_labelledreviewedSketch the graph of $f(x) = \frac{x^2 - 4}{x + 1}$, showing clearly all intercepts, asymptotes, and any stationary points.
q_t03_037P013EasyEasyMedium0.00.0human_labelledreviewedLet $f(x) = \sqrt{x + 3}$ and $g(x) = \dfrac{2}{x - 1}$. (a) Find $(f \circ g)(x)$. (b) Find $(g \circ f)(x)$. (c) State the domain of ea
q_t03_038P021EasyMediumEasy0.00.5human_labelledreviewedConsider the equation $x^2 - (2k+1)x + (k^2 - k - 6) = 0$, where $k \in \mathbb{R}$. Determine all values of $k$ such that the equation has
q_t03_039P012HardHardHard0.00.0human_labelledreviewedA relation $R$ is defined by $9x^2 - 4y^2 - 18x + 16y = 43$. (a) Show that $R$ represents a hyperbola by writing the equation in the standa
q_t03_040P015MediumMediumMedium0.00.0human_labelledreviewedThe graph of y = h(x) is transformed to the graph y = 2h(4x - 8) - 3. (a) Describe this transformation as a sequence of simple transformati
q_t03_041P024MediumHardMedium0.00.5human_labelledreviewedThe graph of $y = f(x)$ is shown, where $f(x) = (x+2)(x-1)(x-3)$. The curve is a cubic with simple zeros at $x = -2$, $x = 1$, and $x = 3$,
q_t03_042P018HardMediumMedium0.00.5human_labelledreviewedSolve exactly $\sqrt{4x^2 - 7} = 2x - 1$, finding all values of $x$ that satisfy this equation.
q_t03_043P017EasyMediumEasy0.00.5human_labelledreviewedDetermine whether each of the following functions is odd, even, or neither. Justify your answer by computing $f(-x)$, $g(-x)$, and $h(-x)$ a
q_t03_044P019EasyMediumEasy0.00.5human_labelledreviewedUse a graphic display calculator or numerical method to solve ln(x) = 2 - x, correct to three decimal places.
q_t03_045P022MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $f(x) = \dfrac{x^2 + 2x + 3}{x - 1}$, showing clearly all intercepts, asymptotes, and any stationary points.
q_t03_046P013MediumMediumMedium0.00.0human_labelledreviewedLet $f(x) = \dfrac{x}{x+2}$ and $g(x) = \sqrt{x - 3}$. (a) Find $(f \circ g)(x)$. (b) Find $(g \circ f)(x)$. (c) State the domain of each
q_t03_047P023MediumHardMedium0.00.5human_labelledreviewedLet $f(x) = x^3 - 6x^2 + 9x + 1$. **(a)** Show that $f$ is **not** one-to-one on $\mathbb{R}$. **(b)** Find the smallest real value of $k$
q_t03_048P016EasyMediumMedium0.00.5human_labelledreviewedLet $f(x) = \dfrac{\sqrt{x^2 + 9}}{x}$. (a) State the domain of $f$. (b) Find all vertical asymptotes of $f$, if any, justifying your answ
q_t03_049P014HardHardMedium0.00.0human_labelledreviewedConsider the functions f and g defined by f(x) = ln x and g(x) = ln(3x − 12), where each function has the largest possible domain. (a) Writ
q_t03_050P020MediumMediumMedium0.00.0human_labelledreviewedFind all real values of $k$ such that $x^2 + \dfrac{16}{x^2} \geq k$ for all $x \neq 0$.
q_t03_051P013EasyEasyEasy0.00.004human_labelledreviewedLet $f(x) = \log_2(x)$ and $g(x) = \sqrt{x - 4}$. (a) Find $(f \circ g)(x)$. (b) Find $(g \circ f)(x)$. (c) State the domain of each comp
q_t03_052P013EasyEasyEasy0.00.01human_labelledreviewedLet $f(x) = \ln(x)$ and $g(x) = x^2 + 1$. (a) Find $(f \circ g)(x)$. (b) Find $(g \circ f)(x)$. (c) State the domain of each composite fu
q_t03_053P023MediumHardMedium0.00.472human_labelledreviewedLet $f(x) = x^3 - 3x^2 - 9x + 5$. **(a)** Show that $f$ is **not** one-to-one on $\mathbb{R}$. **(b)** Find the smallest real value of $k$
q_t03_054P022MediumHardMedium0.00.477human_labelledreviewedSketch the graph of $f(x) = x^2 e^{-x}$, showing clearly all intercepts, asymptotes, and any stationary points.
q_t03_055P012MediumHardMedium0.00.48human_labelledreviewedA relation $R$ is defined by $4x^2 - y^2 - 8x + 4y = 16$. (a) Show that $R$ represents a hyperbola by writing the equation in the standard
q_t03_056P013EasyEasyEasy0.00.004human_labelledreviewedLet $f(x) = \dfrac{1}{x + 3}$ and $g(x) = \sqrt{2x - 1}$. (a) Find $(f \circ g)(x)$, simplifying your answer. (b) Find $(g \circ f)(x)$, s
q_t03_057P013EasyEasyEasy0.00.007human_labelledreviewedLet $f(x) = \sqrt{4 - x}$ and $g(x) = \dfrac{2}{x+1}$. (a) Find $(f \circ g)(x)$, simplifying your answer. (b) Find $(g \circ f)(x)$, simp
q_t03_058P022MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $f(x) = \dfrac{x^2 + 2x - 3}{x - 2}$, showing clearly all intercepts with the axes, any asymptotes, and any stationary p
q_t03_059P022EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $g(t) = \dfrac{t^2 - 9}{t + 1}$, showing clearly all intercepts with the axes, any asymptotes, and any stationary points
q_t03_060P022EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $h(x) = \dfrac{x^2 - 4}{x + 2}$, showing clearly all intercepts with the axes, any asymptotes, and any stationary points
q_t03_061P022HardHardHard0.00.0human_labelledreviewedSketch the graph of $$f(x) = \frac{x^3 - 4x}{x^2 - 1},$$ showing clearly all intercepts with the coordinate axes, all asymptotes, and any st
q_t03_062P022MediumHardHard0.00.5human_labelledreviewedSketch the graph of $f(x) = \dfrac{x^3 + x^2 - 4}{x^2 - x - 2}$, showing clearly all intercepts with the coordinate axes, all asymptotes, an
q_t03_063P024EasyEasyEasy0.00.0human_labelledreviewedThe graph of $y = h(x)$ is shown, where $h(x) = (x + 1)(x - 2)$. The parabola opens upward with zeros at $x = -1$ and $x = 2$, vertex at $\l
q_t03_064P024EasyEasyEasy0.00.0human_labelledreviewedThe diagram below shows the graph of $y = p(t)$, where $p(t) = \sin t$ for $0 \leq t \leq 2\pi$. The function has zeros at $t = 0$, $t = \pi
q_t03_065P024HardHardMedium0.00.0human_labelledreviewedThe diagram below shows the graph of $y = f(x)$, where $$f(x) = x(x-3)(x+2).$$ The cubic has zeros at $x = -2$, $x = 0$, and $x = 3$, a lo
q_t03_066P024MediumHardEasy0.00.5human_labelledreviewedThe diagram below shows the graph of $y = g(u)$, where $$g(u) = (2u + 1)(u - 2)(u - 4).$$ The cubic has zeros at $u = -\tfrac{1}{2}$, $u =
q_t03_067P023EasyEasyMedium0.00.0human_labelledreviewedLet $g(t) = -t^3 + 3t^2 + 4$. **(a)** Show that $g$ is **not** one-to-one on $\mathbb{R}$. **(b)** Find the largest real value of $k$ such
q_t03_068P023HardHardMedium0.00.0human_labelledreviewedLet $f(x) = 2x^3 - 9x^2 + 12x - 4$. (a) Show that $f$ is not one-to-one on $\mathbb{R}$. (b) Find the smallest value of $k$ such that $f$
q_t03_069P023MediumHardMedium0.00.5human_labelledreviewedLet $f(x) = x^3 - 6x^2 + 9x + 2$. (a) Show that $f$ is not one-to-one on $\mathbb{R}$. (b) Find the smallest real value of $k$ such that $
q_t03_070P021EasyEasy—0.00.0needs_human—Consider the function $f(x) = (k-1)x^2 + 2x + (k-1)$, where $k \in \mathbb{R}$. Find all real values of $k$ such that $f(x) > 0$ for all $x
q_t03_071P012EasyEasy—0.00.0needs_human—A relation $R$ is defined by $x^2 + 4y^2 - 2x = 3$. **(a)** Show that $R$ represents an ellipse by writing the equation in the standard for
q_t03_072P015EasyEasy—0.00.0needs_human—The graph of $y = p(x)$ is transformed to give the graph of $y = -p(2x - 6) + 4$. **(a)** Describe this transformation as an ordered sequen
q_t03_073P024EasyEasy—0.00.0needs_human—The diagram below shows the graph of $y = f(s)$, where $$f(s) = -(s - 1)(s + 3).$$ The parabola opens downward, has zeros at $s = -3$ and
q_t03_074P018EasyEasy—0.00.0needs_human—Solve exactly $\sqrt{x + 6} = x + 4$, finding all values of $x$ that satisfy the equation.
q_t03_075P017EasyEasy—0.00.0needs_human—Determine whether each of the following functions is odd, even, or neither. Justify your answer by computing $p(-x)$, $q(-x)$, and $r(-x)$ a
q_t03_076P019EasyMedium—0.00.5needs_human—Use a graphic display calculator to solve ln(x) = 2 − x, correct to three decimal places.
q_t03_077P022EasyEasy—0.00.0needs_human—Sketch the graph of $f(x) = \dfrac{2x + 6}{x - 1}$, showing clearly all intercepts with the coordinate axes, all asymptotes, and any station
q_t03_078P013EasyEasy—0.00.0needs_human—Let $f(x) = \ln(x + 4)$ and $g(x) = \dfrac{x + 1}{x - 2}$. **(a)** Find $(f \circ g)(x)$, simplifying your answer. **(b)** Find $(g \circ
q_t03_079P023EasyEasy—0.00.0needs_human—Let $g(x) = x^2 - 6x + 5$. **(a)** Show that $g$ is **not** one-to-one on $\mathbb{R}$. **(b)** Find the smallest real value of $k$ such t
q_t03_080P016EasyEasy—0.00.0needs_human—Let $f(x) = \dfrac{2x + 1}{\sqrt{x^2 + 3}}$. **(a)** State the domain of $f$. **(b)** Find all vertical asymptotes of $f$, if any. Justify
q_t03_081P014EasyMedium—0.00.5needs_human—Consider the functions $f$ and $g$ defined by $f(x) = e^{x}$ and $g(x) = e^{2x+4}$, where each function has domain $\mathbb{R}$. The graph
q_t03_082P020EasyMedium—0.00.5needs_human—Find all real values of $k$ such that $e^x + e^{-x} > k$ for all real $x$.
q_t04_001P029EasyMediumEasy0.00.506human_labelledreviewedFind the minimum value of $y = x^2 - 4x + 9$.
q_t04_002P026MediumMediumEasy0.00.378human_labelledreviewedSolve the equation $3x^2 + 5x - 1 = 0$, giving your answers in surd form.
q_t04_003P028MediumMediumMedium0.00.0human_labelledreviewedFind the number of real solutions of the equation $\left(x - \dfrac{1}{x}\right)^2 + 2\left(x - \dfrac{1}{x}\right) - 8 = 0$.
q_t04_004P029EasyMediumEasy0.00.5human_labelledreviewedFind the maximum value of $y = -x^2 + 8x - 7$.
q_t04_005P026HardMediumMedium0.00.5human_labelledreviewedSolve the equation $\dfrac{3x}{x-2} + \dfrac{4}{x+1} = 5$, giving your answers in surd form.
q_t04_006P027MediumMediumEasy0.00.0human_labelledreviewedDetermine the number of real solutions of the equation $2x^2 - 5x + 4 = 0$.
q_t04_007P025EasyMediumEasy0.00.5human_labelledreviewedSketch the graph of $y = 2x^2 - 8x + 6$, showing clearly the coordinates of the vertex, the $x$-intercepts, and the $y$-intercept.
q_t04_008P030EasyMediumEasy0.00.5human_labelledreviewedIn triangle $PQR$, point $S$ lies on $PQ$ and point $T$ lies on $PR$. It is given that $PS = x$ cm, $SQ = 5$ cm, $PT = 3$ cm, and $TR = x$ c
q_t04_009P028EasyMediumEasy0.00.5human_labelledreviewedFind the number of real solutions of the equation $e^{2x} - 6e^x + 9 = 0$.
q_t04_010P029MediumMediumMedium0.00.0human_labelledreviewedA farmer has 40 metres of fencing and wishes to enclose a rectangular plot against a long straight wall, so that the wall forms one side of
q_t04_011P026HardMediumMedium0.00.5human_labelledreviewedSolve the equation $\dfrac{3}{x-2} + \dfrac{5}{x+1} = 4$. Give your answers in surd form.
q_t04_012P027HardMediumMedium0.00.5human_labelledreviewedDetermine the number of real solutions of the equation $-\dfrac{3}{4}x^2 + \dfrac{5}{2}x - \dfrac{25}{12} = 0$.
q_t04_013P025EasyMediumEasy0.00.5human_labelledreviewedSketch the graph of $y = -2x^2 - 4x + 6$, showing clearly the coordinates of the vertex, the $x$-intercepts, and the $y$-intercept.
q_t04_014P030EasyMedium—0.00.5discarded—In triangle PQR, point S lies on PQ and point T lies on PR such that ST is parallel to QR. It is given that PS = x cm, SQ = 5 cm, PT = 3 cm,
q_t04_015P028HardMediumMedium0.00.5human_labelledreviewedFind the number of real solutions of the equation $$\left(\sin x\right)^2 - \sin x - 2 = 0, \quad x \in [0, 4\pi].$$
q_t04_016P029EasyMediumEasy0.00.5human_labelledreviewedFind the minimum value of $y = x^2 - 10x + 29$.
q_t04_017P026MediumMediumMedium0.00.0human_labelledreviewedSolve the equation $2x^2 + 3x - 7 = 0$. Give your answers in surd form.
q_t04_018P027MediumMediumEasy0.00.0human_labelledreviewedDetermine the number of real solutions of the equation $3x^2 - 7x + 5 = 0$.
q_t04_019P025MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = -2x^2 + 6x + 8$, showing clearly the coordinates of the vertex, the $x$-intercepts, and the $y$-intercept.
q_t04_020P030MediumMediumEasy0.00.0human_labelledreviewedIn triangle PQR, point S lies on PQ and point T lies on PR such that ST is parallel to QR. It is given that PS = x cm, SQ = 5 cm, PT = (x +
q_t04_021P028HardMediumMedium0.00.5human_labelledreviewedFind the number of real solutions of the equation $$\left(\ln x\right)^2 - \ln\left(x^3\right) - 10 = 0.$$
q_t04_022P029MediumMediumEasy0.00.0human_labelledreviewedA ball is thrown vertically upward. Its height $h$ metres above the ground at time $t$ seconds is given by $$h = -5t^2 + 30t + 2.$$ Find the
q_t04_023P026MediumMediumMedium0.00.0human_labelledreviewedSolve the equation $5x^2 - 4kx - k^2 = 0$, giving your answers in terms of $k$.
q_t04_024P027MediumMediumEasy0.00.0human_labelledreviewedDetermine the number of real solutions of the equation $-2x^2 + 6x - 5 = 0$.
q_t04_025P025EasyMediumEasy0.00.5human_labelledreviewedSketch the graph of $y = 3x^2 - 12x + 9$, showing clearly the coordinates of the vertex, the $x$-intercepts, and the $y$-intercept.
q_t04_026P030MediumMedium—0.00.0discarded—In triangle PQR, point S lies on PQ and point T lies on PR such that ST is parallel to QR. It is given that PS = x cm, SQ = 5 cm, PT = (x +
q_t04_027P028MediumMediumMedium0.00.0human_labelledreviewedFind the number of real solutions of the equation $$\left(x + \frac{1}{x}\right)^2 - 2\left(x + \frac{1}{x}\right) - 8 = 0.$$
q_t04_028P028MediumMediumMedium0.00.0human_labelledreviewedFind the number of real solutions of the equation $$\left(\ln x\right)^2 - \ln x - 6 = 0.$$
q_t04_029P029EasyMediumEasy0.00.5human_labelledreviewedFind the minimum value of $y = x^2 - 10x + 29$.
q_t04_030P026HardMediumMedium0.00.5human_labelledreviewedSolve the equation $\dfrac{3}{x-2} + \dfrac{x}{x+1} = 4$, giving your answers in surd form.
q_t04_031P027MediumMediumMedium0.00.0human_labelledreviewedDetermine the number of real solutions of the equation $\dfrac{5}{2}x^2 - 3x + \dfrac{9}{10} = 0$.
q_t04_032P025EasyMediumEasy0.00.5human_labelledreviewedSketch the graph of $y = -2x^2 + 4x + 6$, showing clearly the coordinates of the vertex, the $x$-intercepts, and the $y$-intercept.
q_t04_033P030EasyMediumEasy0.00.5human_labelledreviewedIn triangle $ABC$, point $D$ lies on $AB$ and point $E$ lies on $AC$ such that $DE \parallel BC$. It is given that $AD = 4$ cm, $DB = x$ cm,
q_t04_034P028EasyMediumEasy0.00.5human_labelledreviewedFind the number of real solutions of the equation $$e^{2x} - 3e^x + 2 = 0.$$
q_t04_035P029MediumMediumEasy0.00.0human_labelledreviewedFind the minimum value of y = 2x² - 12x + 23.
q_t04_036P026HardMediumHard0.00.5human_labelledreviewedSolve the equation $\sqrt{2x+5} = x - 1$, giving your answers in surd form where necessary, and clearly stating any roots that must be rejec
q_t04_037P027HardMediumMedium0.00.5human_labelledreviewedDetermine the number of real solutions of the equation $$\frac{3}{4}x^2 - \frac{\sqrt{3}}{2}x + \frac{1}{3} = 0.$$
q_t04_038P025EasyMediumEasy0.00.5human_labelledreviewedSketch the graph of $y = 3x^2 - 12x + 9$, showing clearly the coordinates of the vertex, the $x$-intercepts, and the $y$-intercept.
q_t04_039P030EasyMediumEasy0.00.5human_labelledreviewedIn triangle $LMN$, point $P$ lies on $LM$ and point $Q$ lies on $LN$ such that $PQ \parallel MN$. It is given that $LP = 6$ cm, $PM = x$ cm,
q_t04_040P028HardMediumHard0.00.5human_labelledreviewedFind the number of real solutions of the equation $$\left(x + \frac{1}{x}\right)^2 - 3\left|x + \frac{1}{x}\right| - 4 = 0.$$
q_t04_041P029EasyMediumEasy0.00.5human_labelledreviewedFind the maximum value of $y = -x^2 + 4x + 1$.
q_t04_042P026MediumMediumEasy0.00.0human_labelledreviewedSolve the equation $2x^2 + 5x - 1 = 0$. Give your answers in surd form.
q_t04_043P027MediumMediumEasy0.00.0human_labelledreviewedDetermine the number of real solutions of the equation $2x^2 - 5x + 4 = 0$.
q_t04_044P025MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = -2x^2 + 3x + 5$, showing clearly the coordinates of the vertex, the $x$-intercepts, and the $y$-intercept. State th
q_t04_045P030MediumMediumMedium0.00.0human_labelledreviewedIn triangle $PQR$, point $S$ lies on $PQ$ and point $T$ lies on $PR$ such that $ST \parallel QR$. It is given that $PS = x$ cm, $SQ = (x + 2
q_t04_046P028HardMediumMedium0.00.5human_labelledreviewedFind the number of real solutions of the equation $$\sin^2 x - 3\sin x + 2 = 0, \quad x \in [-2\pi, 2\pi].$$
q_t04_047P029MediumMediumEasy0.00.0human_labelledreviewedFind the minimum value of $y = 2x^2 - 12x + 23$.
q_t04_048P026MediumMediumMedium0.00.0human_labelledreviewedSolve the equation $3x^2 - 2x(x + 1) = 5(x + 2)$, giving your answers in surd form.
q_t04_049P027MediumMediumMedium0.00.0human_labelledreviewedDetermine the number of real solutions of the equation $y = -\dfrac{3}{4}x^2 + \dfrac{5}{2}x - \dfrac{25}{12}$.
q_t04_050P025EasyMediumEasy0.00.5human_labelledreviewedSketch the graph of $y = -x^2 + 2x + 8$, showing clearly the coordinates of the vertex, the $x$-intercepts, and the $y$-intercept.
q_t04_051P030MediumMediumMedium0.00.0human_labelledreviewedIn triangle $PQR$, point $S$ lies on $PQ$ and point $T$ lies on $PR$ such that $ST \parallel QR$. It is given that $PS = x$ cm, $SQ = (x + 2
q_t04_052P028MediumMediumHard0.00.0human_labelledreviewedFind the number of real solutions of the equation $$4\cosh^2 x - 8\cosh x + 3 = 0,$$ where $\cosh x = \dfrac{e^x + e^{-x}}{2}$ denotes the h
q_t04_053P026EasyMediumEasy0.00.477human_labelledreviewedSolve the equation $3x^2 - 7x + 2 = 0$.
q_t04_054P025MediumMediumEasy0.00.009human_labelledreviewedSketch the graph of $y = 2x^2 + 4x - 6$, showing clearly the coordinates of the vertex, the $x$-intercepts, and the $y$-intercept.
q_t04_055P029MediumMediumMedium0.00.049human_labelledreviewedA farmer has $80$ metres of fencing to enclose a rectangular vegetable plot. One side of the plot lies along an existing wall, so fencing is
q_t04_056P030MediumHardMedium0.00.475human_labelledreviewedIn the diagram below, triangle $ABC$ has a point $D$ on side $AB$ and a point $E$ on side $BC$ such that $DE \parallel AC$. It is given that
q_t04_057P028EasyMediumEasy0.00.513human_labelledreviewedFind the number of real solutions of the equation $$9^t - 4 \cdot 3^t - 45 = 0.$$
q_t04_058P027EasyEasyEasy0.00.027human_labelledreviewedDetermine the number of real solutions of the equation $3x^2 + 6x + 3 = 0$.
q_t04_059P026EasyMediumEasy0.00.494human_labelledreviewedSolve the equation $4x^2 + 11x - 3 = 0$.
q_t04_060P025HardMediumMedium0.00.483human_labelledreviewedThe function $g(t) = -2t^2 + 6t + k$ has its vertex on the line $y = \frac{11}{2}$. (a) Find the value of $k$. (b) Sketch the graph of $g$
q_t04_061P029HardMediumMedium0.00.507human_labelledreviewedA farmer has 120 metres of fencing to enclose a rectangular pen against a long straight barn wall. The barn wall forms one side of the pen,
q_t04_062P030MediumMediumMedium0.00.035human_labelledreviewedIn the diagram below, trapezium $ABCD$ has $AB \parallel DC$. The diagonals $AC$ and $BD$ intersect at point $P$. It is given that $AP = x$
q_t04_063P028MediumMediumEasy0.00.02human_labelledreviewedFind the number of real solutions of the equation $$\left(x - \frac{2}{x}\right)^2 + \left(x - \frac{2}{x}\right) - 6 = 0.$$
q_t04_064P027HardMediumMedium0.00.489human_labelledreviewedA curve is defined by $y = \dfrac{2}{3}x^2 - \sqrt{5}\,x + \dfrac{15}{8}$. Determine the number of real solutions of the equation $y = 0$.
q_t04_065P026MediumMediumEasy0.00.052human_labelledreviewedSolve the equation $3t^2 - 4t - 6 = 0$, giving your answers in surd form.
q_t04_066P025EasyEasyEasy0.00.017human_labelledreviewedSketch the graph of $y = x^2 - 6x + 8$, showing clearly the coordinates of the vertex, the $x$-intercepts, and the $y$-intercept.
q_t04_067P029MediumMediumMedium0.00.052human_labelledreviewedA farmer has 60 m of fencing to enclose a rectangular plot against a straight wall. The wall forms one side of the rectangle, so fencing is
q_t04_068P030EasyEasyEasy0.00.041human_labelledreviewedIn triangle $PQR$, point $S$ lies on $PQ$ and point $T$ lies on $PR$ such that $ST \parallel QR$. It is given that $PS = x$ cm, $SQ = 5$ cm,
q_t04_069P028MediumMediumEasy0.00.023human_labelledreviewedFind the number of real solutions of the equation $$\left(\ln x\right)^2 - \ln x^3 - 10 = 0.$$
q_t04_070P027EasyEasyEasy0.00.033human_labelledreviewedDetermine the number of real solutions of the equation $2x^2 - 8x + 8 = 0$.
q_t04_071P026MediumMediumMedium0.00.061human_labelledreviewedSolve the equation $\dfrac{3}{m+1} + \dfrac{2}{m-2} = 1$, giving your answers in surd form.
q_t04_072P025MediumMediumMedium0.00.014human_labelledreviewedSketch the graph of $h(x) = -3x^2 - 6x + 24$, showing clearly the coordinates of the vertex, any $x$-intercepts, and the $y$-intercept.
q_t04_073P029MediumHardMedium0.00.504human_labelledreviewedA rectangular garden is to be divided into three equal sections by two internal fences running parallel to one of the shorter sides, as well
q_t04_074P030HardHardHard0.00.052human_labelledreviewedIn triangle $PQR$, point $S$ lies on $PQ$ and point $T$ lies on $PR$ such that $ST \parallel QR$. It is given that $PS = x$ cm, $SQ = (x + 2
q_t04_075P028MediumMediumMedium0.00.022human_labelledreviewedFind the number of real solutions of the equation $\left(x + \frac{1}{x}\right)^2 - 2\left(x + \frac{1}{x}\right) - 8 = 0$.
q_t04_076P027MediumMediumEasy0.00.044human_labelledreviewedDetermine the number of real solutions of the equation $-\dfrac{2}{3}t^2 + 3t - \dfrac{27}{8} = 0$.
q_t04_077P026EasyMediumEasy0.00.484human_labelledreviewedSolve the equation $3x^2 - 10x + 8 = 0$.
q_t04_078P025EasyMediumEasy0.00.468human_labelledreviewedSketch the graph of $p(x) = 2x^2 + 4x - 6$, showing clearly the coordinates of the vertex, the $x$-intercepts, and the $y$-intercept.
q_t04_079P029MediumMediumEasy0.00.051human_labelledreviewedA farmer has $80$ metres of fencing to enclose a rectangular pen against a long straight barn wall. The wall forms one side of the rectangle
q_t04_080P030HardHardHard0.00.05human_labelledreviewedIn the diagram below, a circle has centre $O$ and radius $r$ cm. A chord $AB$ is drawn, and from a point $C$ on the circle (on the major arc
q_t04_081P028MediumMediumMedium0.00.04human_labelledreviewedFind the number of real solutions of the equation $$\left(x^2 - 3x\right)^2 - 2\left(x^2 - 3x\right) - 8 = 0.$$
q_t04_082P027HardMediumHard0.00.505human_labelledreviewedA curve is defined by the equation $h(u) = \dfrac{3}{5}u^2 - \dfrac{7}{4}u + \dfrac{245}{192}$. Determine the number of real values of $u$ f
q_t04_083P028MediumMediumMedium0.00.029human_labelledreviewedFind the number of real solutions of the equation $$\sin^2\theta - 3\sin\theta + 2 = 0, \quad 0 \leq \theta \leq 2\pi.$$
q_t04_084P026HardMediumMedium0.00.501human_labelledreviewedThe polynomial $P(x) = 2x^3 - 3x^2 - 11x + 6$ has a factor $(x - 3)$. By first dividing $P(x)$ by $(x - 3)$, solve the equation $P(x) = 0$,
q_t04_085P030HardHardMedium0.00.033human_labelledreviewedIn the diagram below, two chords $AC$ and $BD$ of a circle intersect at an interior point $P$. It is given that $AP = x$ cm, $PC = (x + 3)$
q_t04_086P029MediumMediumEasy0.00.032human_labelledreviewedA manufacturer produces cylindrical tins. For a particular product line, the sum of the height and the radius of each tin is fixed at $12$ c
q_t04_087P025EasyMediumEasy0.00.492human_labelledreviewedSketch the graph of $h(x) = -3x^2 - 12x - 9$, showing clearly the coordinates of the vertex, the $x$-intercepts, and the $y$-intercept. Stat
q_t04_088P027HardMediumEasy0.00.498human_labelledreviewedA function $f$ is defined by $f(t) = \left(k^2 - 3k + 2\right)t^2 - \left(2k^2 - 5k + 3\right)t + \left(k^2 - 2k + 1\right)$, where $k$ is a
q_t04_089P028MediumMediumMedium0.00.017human_labelledreviewedFind the number of real solutions of the equation $$\left(x - \frac{2}{x}\right)^2 + 3\left(x - \frac{2}{x}\right) - 10 = 0.$$
q_t04_090P026EasyMediumEasy0.00.507human_labelledreviewedSolve the equation $6t^2 + t - 12 = 0$.
q_t04_091P030EasyEasyEasy0.00.027human_labelledreviewedIn the diagram below, a ladder $PQ$ of length $15$ m leans against a vertical wall $QR$. A horizontal brace $ST$ is attached to the ladder a
q_t04_092P029MediumMediumMedium0.00.029human_labelledreviewedA graphic designer is creating a rectangular banner. The banner must have a total area of $200 \text{ cm}^2$. The printed region inside the
q_t04_093P025EasyEasyEasy0.00.009human_labelledreviewedSketch the graph of $y = x^2 - 6x + 8$, showing clearly the coordinates of the vertex, the $x$-intercepts, and the $y$-intercept.
q_t04_094P027MediumMediumEasy0.00.032human_labelledreviewedA physicist models the trajectory of a projectile using the equation $p(x) = -\dfrac{2}{3}x^2 + \dfrac{5}{4}x - \dfrac{75}{96}$, where $x$ r
q_t04_095P028EasyEasyEasy0.00.013human_labelledreviewedFind the number of real solutions of the equation $$\left(x + \frac{1}{x}\right)^2 - 2\left(x + \frac{1}{x}\right) - 8 = 0.$$
q_t04_096P026HardMediumHard0.00.494human_labelledreviewedThe function $f(x) = 3x^2 - (k+5)x + 2k$ has roots $\alpha$ and $\beta$. Given that $\alpha^2 + \beta^2 = \dfrac{13}{9}$, find the possible
q_t04_097P030MediumMediumMedium0.00.035human_labelledreviewedIn the diagram below, a flagpole $MN$ stands vertically on horizontal ground. At a point $P$ on the ground, a taut wire runs from $P$ to the
q_t04_098P029MediumHardMedium0.00.5human_labelledreviewedA farmer wants to build a rectangular chicken coop with an internal dividing wall parallel to one pair of sides, as shown in the diagram bel
q_t04_099P025MediumMediumMedium0.00.015human_labelledreviewedSketch the graph of $h(x) = -2x^2 - 8x - 3$, showing clearly the coordinates of the vertex, any $x$-intercepts, and the $y$-intercept.
q_t04_100P027EasyMediumEasy0.00.503human_labelledreviewedA ball is thrown upward and its height above the ground (in metres) is modelled by $h(t) = -5t^2 + 4t - 1$, where $t$ is the time in seconds
q_t04_101P028MediumMediumMedium0.00.014human_labelledreviewedFind the number of real solutions of the equation $$(\ln x)^2 - \ln(x^3) - 10 = 0.$$
q_t04_102P026MediumMediumEasy0.00.021human_labelledreviewedSolve the equation $3x^2 - 4x - 6 = 0$. Give your answers in surd form.
q_t05_001P034MediumMediumMedium0.00.0human_labelledreviewedFind the number of real solutions of the equation $3e^{4x} - 10e^{2x} - 8 = 0$.
q_t05_002P032EasyMediumMedium0.00.5human_labelledreviewedDetermine the number of zeroes of the function $f(x) = x^3 - 3x^2 - 9x + 5$.
q_t05_003P033HardMediumMedium0.00.5human_labelledreviewedSketch the graph of $y = -2x^4 + 10x^2 - 8$, showing clearly the coordinates of all $x$-intercepts, the $y$-intercept, and all turning point
q_t05_004P031MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of y = x^3 - 6x^2 + 9x + 2, showing clearly the coordinates of the y-intercept and any turning points.
q_t05_005P034EasyMediumEasy0.00.5human_labelledreviewedFind the number of real solutions of the equation $x^4 - 13x^2 + 36 = 0$.
q_t05_006P032EasyMedium—0.00.5discarded—Determine the number of real zeroes of the function $f(x) = x^3 - 3x^2 - 9x + 5$.
q_t05_007P033EasyMediumMedium0.00.5human_labelledreviewedSketch the graph of $y = x^4 - 5x^2 + 4$, showing clearly the coordinates of any intercepts and turning points.
q_t05_008P031MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = x^3 - 6x^2 + 9x + 1$, showing clearly the coordinates of the $y$-intercept and any turning points.
q_t05_009P034HardMediumHard0.00.5human_labelledreviewedDetermine the number of real solutions of the equation $$\ln^2(x^2) - 5\ln(x^2) + 4 = 0,$$ where $x \neq 0$.
q_t05_010P032HardMediumMedium0.00.5human_labelledreviewedDetermine the number of real zeroes of the function $f(x) = 2x^3 - 9x^2 + 12x - 7$.
q_t05_011P033EasyMediumEasy0.00.5human_labelledreviewedSketch the graph of $y = x^4 - 10x^2 + 9$, showing clearly the coordinates of any intercepts and turning points.
q_t05_012P031EasyMediumMedium0.00.5human_labelledreviewedSketch the graph of $y = -x^3 + 3x^2 + 9x - 2$, showing clearly the coordinates of the $y$-intercept and any turning points.
q_t05_013P034HardMediumMedium0.00.5human_labelledreviewedDetermine the number of real solutions of the equation $$\sin^4 x - \frac{5}{4}\sin^2 x + \frac{1}{4} = 0$$ on the interval $0 \leq x \leq 2
q_t05_014P032EasyMediumMedium0.00.5human_labelledreviewedDetermine the number of real zeroes of the function $f(x) = x^3 - 3x^2 - 9x + 5$.
q_t05_015P033MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = -x^4 + 13x^2 - 36$, showing clearly the coordinates of all $x$-intercepts, the $y$-intercept, and all turning point
q_t05_016P031MediumMediumEasy0.00.0human_labelledreviewedSketch the graph of $y = 2x^3 + 3x^2 - 12x + 1$, showing clearly the coordinates of the $y$-intercept and any turning points.
q_t05_017P034MediumMediumMedium0.00.0human_labelledreviewedDetermine the number of real solutions of the equation $$2\tan^4\theta - 7\tan^2\theta + 3 = 0$$ on the interval $0 \leq \theta < \pi$, wher
q_t05_018P032MediumMedium—0.00.0discarded—Determine the number of real zeroes of the function $f(x) = x^3 - 3x^2 - 9x + 5$.
q_t05_019P033HardMediumHard0.00.5human_labelledreviewedSketch the graph of $y = 2x^4 - 7x^2 - 4$, showing clearly the coordinates of all $x$-intercepts, the $y$-intercept, and all turning points.
q_t05_020P031MediumMedium—0.00.0discarded—Sketch the graph of $y = x^3 - 6x^2 + 9x + 2$, showing clearly the coordinates of any turning points and the $y$-intercept.
q_t05_021P034MediumMediumEasy0.00.0human_labelledreviewedFind the number of real solutions of the equation $$2\left(\ln x\right)^2 - 7\ln x + 3 = 0,$$ where $x > 0$.
q_t05_022P032MediumMedium—0.00.0discarded—Determine the number of real zeroes of the function $f(x) = x^3 - 3x^2 - 9x + 5$.
q_t05_023P033EasyMediumMedium0.00.5human_labelledreviewedSketch the graph of $y = -x^4 + 5x^2 - 4$, showing clearly the coordinates of all $x$-intercepts, the $y$-intercept, and all turning points.
q_t05_024P031MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = -2x^3 - 3x^2 + 12x + 4$, showing clearly the coordinates of the $y$-intercept and any turning points. Your sketch m
q_t05_025P034MediumMediumMedium0.00.0human_labelledreviewedFind the number of real solutions of the equation $$4\cosh^2 t - 9\cosh t + 5 = 0,$$ where $\cosh t = \dfrac{e^t + e^{-t}}{2}$ denotes the h
q_t05_026P034MediumMediumMedium0.00.0human_labelledreviewedFind the number of real solutions of the equation $$9^x - 4 \cdot 3^x - 45 = 0.$$
q_t05_027P032EasyMedium—0.00.5discarded—Determine the number of real zeroes of the function $f(x) = x^3 - 3x^2 - 9x + 5$.
q_t05_028P033HardMediumHard0.00.5human_labelledreviewedSketch the graph of $y = 2x^4 - 7x^2 - 4$, showing clearly the coordinates of any $x$-intercepts, $y$-intercept, and turning points.
q_t05_029P031MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = 3x^3 - 20x^2 + 36x - 16$, showing clearly the coordinates of the $y$-intercept and any turning points. Your sketch
q_t05_030P034EasyMediumEasy0.00.5human_labelledreviewedFind the number of real solutions of the equation $$e^{2x} - 3e^{x} + 2 = 0.$$
q_t05_031P032EasyMediumMedium0.00.5human_labelledreviewedDetermine the number of real zeroes of the function $f(x) = x^3 + 3x^2 - 24x + 5$.
q_t05_032P033EasyMedium—0.00.5discarded—Sketch the graph of $y = -x^4 + 5x^2 - 4$, showing clearly the coordinates of all $x$-intercepts, the $y$-intercept, and all turning points.
q_t05_033P031MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = x^3 + \dfrac{3}{2}x^2 - 6x + 1$, showing clearly the coordinates of the $y$-intercept and any turning points. Your
q_t05_034P034HardMediumHard0.00.5human_labelledreviewedDetermine the number of real solutions of the equation $$\ln^2(x^2+1) - 3\ln(x^2+1) + 2 = 0,$$ where $\ln$ denotes the natural logarithm.
q_t05_035P032HardMediumMedium0.00.5human_labelledreviewedDetermine the number of real zeroes of the function $f(x) = 2x^3 + 3x^2 - 36x + 11$.
q_t05_036P033EasyMediumEasy0.00.5human_labelledreviewedSketch the graph of $y = 3x^4 - 12x^2 + 9$, showing clearly the coordinates of all $x$-intercepts, the $y$-intercept, and all turning points
q_t05_037P031EasyMediumEasy0.00.5human_labelledreviewedSketch the graph of $y = 2x^3 + 9x^2 - 60x + 5$, showing clearly the coordinates of the $y$-intercept and any turning points. Your sketch mu
q_t05_038P034HardMediumHard0.00.5human_labelledreviewedDetermine the number of real solutions of the equation $$2\left(\tan^2 x + 1\right)^2 - 7\left(\tan^2 x + 1\right) + 3 = 0$$ on the interval
q_t05_039P032EasyMediumMedium0.00.5human_labelledreviewedDetermine the number of real zeroes of the function $f(x) = x^3 - 3x^2 - 9x + 5$.
q_t05_040P033MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = x^4 - 5x^2 + 4$, showing clearly the coordinates of any intercepts and turning points.
q_t05_041P031MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = 3x^3 - 5x^2 - 4x + 2$, showing clearly the coordinates of the $y$-intercept and any turning points. Your sketch mus
q_t05_042P034MediumMediumMedium0.00.0human_labelledreviewedFind the number of real solutions of the equation $$3\sin^4\theta - 10\sin^2\theta + 3 = 0$$ for $\theta \in [0, 2\pi)$.
q_t05_043P032MediumMedium—0.00.0discarded—Determine the number of real zeroes of the function $f(x) = x^3 - 3x^2 - 9x + 5$.
q_t05_044P033HardMediumMedium0.00.5human_labelledreviewedSketch the graph of $y = 3x^4 - 16x^2 + 5$, showing clearly the coordinates of all $x$-intercepts, the $y$-intercept, and all turning points
q_t05_045P031MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = x^3 - \dfrac{3}{2}x^2 - 6x + 8$, showing clearly the coordinates of the $y$-intercept and any turning points. Your
q_t05_046P034MediumMediumMedium0.00.0human_labelledreviewedFind the number of real solutions of the equation $$\cosh^2(x) - \frac{7}{2}\cosh(x) + \frac{3}{2} = 0,$$ where $\cosh(x) = \dfrac{e^x + e^{
q_t05_047P032MediumMedium—0.00.0discarded—Determine the number of real zeroes of the function $f(x) = x^3 - 3x^2 - 9x + 5$.
q_t05_048P033EasyMediumMedium0.00.5human_labelledreviewedSketch the graph of $y = x^4 - 13x^2 + 36$, showing clearly the coordinates of any intercepts and turning points.
q_t05_049P031MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = x^3 + \dfrac{3}{2}x^2 - 6x - 5$, showing clearly the coordinates of the $y$-intercept and any turning points. Your
q_t05_050P034MediumMediumMedium0.00.0human_labelledreviewedDetermine the number of real solutions of the equation $$4\tan^4\theta - 17\tan^2\theta + 4 = 0$$ for $\theta \in \left(-\dfrac{\pi}{2},\, \
q_t05_051P033MediumHardHard0.00.484human_labelledreviewedSketch the graph of $y = 2x^4 - 7x^2 - 4$, showing clearly the coordinates of all $x$-intercepts, the $y$-intercept, and all turning points.
q_t05_052P031EasyEasyEasy0.00.032human_labelledreviewedSketch the graph of $f(t) = t^3 - 6t^2 + 8$, showing clearly the coordinates of the $y$-intercept and any turning points. Your sketch must i
q_t05_053P034MediumMediumEasy0.00.03human_labelledreviewedFind the number of real solutions of the equation $e^{4x} - 10e^{2x} + 9 = 0$.
q_t05_054P032MediumMediumEasy0.00.048human_labelledreviewedDetermine the number of real zeroes of the function $p(t) = 2t^3 + 9t^2 - 60t + 4$.
q_t05_055P033MediumHardEasy0.00.479human_labelledreviewedSketch the graph of $y = -x^4 + 10x^2 - 9$, showing clearly the coordinates of all $x$-intercepts, the $y$-intercept, and all turning points
q_t05_056P031EasyEasyEasy0.00.04human_labelledreviewedSketch the graph of $g(u) = -2u^3 + 3u^2 + 12u - 4$, showing clearly the coordinates of the $y$-intercept and any turning points. Your sketc
q_t05_057P034EasyMediumEasy0.00.48human_labelledreviewedFind the number of real solutions of the equation $x^4 - 10x^2 + 9 = 0$.
q_t05_058P032EasyEasyEasy0.00.037human_labelledreviewedDetermine the number of real zeroes of the cubic function $q(s) = s^3 + 3s^2 - 4$.
q_t05_059P033HardHardMedium0.00.01human_labelledreviewedSketch the graph of $y = 2x^4 - 7x^2 - 4$, showing clearly the coordinates of all intercepts and turning points.
q_t05_060P031HardHardMedium0.00.043human_labelledreviewedSketch the graph of $h(s) = 2s^3 - 3s^2 - 36s + 10$, showing clearly the coordinates of the $y$-intercept and all turning points. Your sketc
q_t05_061P034MediumMediumEasy0.00.025human_labelledreviewedDetermine the number of real solutions of the equation $$e^{4x} - 5e^{2x} + 4 = 0.$$
q_t05_062P032MediumMediumMedium0.00.039human_labelledreviewedDetermine the number of real zeroes of the cubic function $P(u) = u^3 + \frac{3}{2}u^2 - 6u + 20$.
q_t05_063P033HardHardMedium0.00.031human_labelledreviewedSketch the graph of $y = -2x^4 + 9x^2 - 4$, showing clearly the coordinates of all $x$-intercepts, the $y$-intercept, and all turning points
q_t05_064P031MediumHardMedium0.00.482human_labelledreviewedSketch the graph of $p(w) = -w^3 + \dfrac{3}{2}w^2 + 6w - 4$, showing clearly the $p$-intercept and the coordinates of any turning points. Y
q_t05_065P034EasyMediumEasy0.00.468human_labelledreviewedFind the number of real solutions of the equation $$x^4 - 8x^2 + 12 = 0.$$
q_t05_066P032MediumMediumMedium0.00.036human_labelledreviewedDetermine the number of real zeroes of the cubic function $r(w) = 3w^3 - 16w^2 + 12w + 40$.
q_t05_067P033EasyEasyEasy0.00.009human_labelledreviewedSketch the graph of $y = x^4 - 5x^2 + 4$, showing clearly the coordinates of any intercepts and turning points.
q_t05_068P031MediumHardMedium0.00.477human_labelledreviewedSketch the graph of $f(x) = -2x^3 - 3x^2 + 12x + 5$, showing clearly the coordinates of the $y$-intercept and any turning points. Your sketc
q_t05_069P034EasyEasyEasy0.00.073human_labelledreviewedFind the number of real solutions of the equation $$\sin^2(x) - \frac{3}{2}\sin(x) + \frac{1}{2} = 0$$ for $x \in [0, 2\pi]$.
q_t05_070P032MediumMediumEasy0.00.033human_labelledreviewedDetermine the number of real zeroes of the cubic function $q(x) = x^3 - \frac{3}{2}x^2 - 18x + 5$.
q_t05_072P031MediumHardMedium0.00.473human_labelledreviewedSketch the graph of $g(x) = x^3 + 3x^2 - 9x - 2$, showing clearly the coordinates of the $y$-intercept and any turning points. Your sketch m
q_t05_073P034HardMediumMedium0.00.482human_labelledreviewedDetermine the number of real solutions of the equation $$\left(x^2 - 3x\right)^2 - 2\left(x^2 - 3x\right) - 8 = 0.$$
q_t05_074P032MediumMediumMedium0.00.028human_labelledreviewedDetermine the number of real zeroes of the cubic function $f(u) = 2u^3 - 3u^2 - 12u + 7$.
q_t05_076P031EasyEasyEasy0.00.041human_labelledreviewedSketch the graph of $h(u) = u^3 - 3u + 2$, showing clearly the coordinates of the $h$-intercept and any turning points. Your sketch must ind
q_t05_077P034EasyMediumEasy0.00.465human_labelledreviewedFind the number of real solutions of the equation $$e^{2x} - 5e^{x} + 6 = 0.$$
q_t05_078P032MediumMediumEasy0.00.028human_labelledreviewedDetermine the number of real zeroes of the cubic function $h(s) = s^3 + \frac{3}{2}s^2 - 6s + 10$.
q_t05_080P031MediumHardEasy0.00.454human_labelledreviewedSketch the graph of $y = x^3 - 6x^2 + 9x + 1$, showing clearly the coordinates of the $y$-intercept and any turning points.
q_t05_081P034HardMediumMedium0.00.481human_labelledreviewedDetermine the number of real solutions of the equation $$2\cos^4\theta - 5\cos^2\theta + 2 = 0$$ for $\theta \in [0, 2\pi)$.
q_t05_082P032MediumMediumMedium0.00.024human_labelledreviewedDetermine the number of real zeroes of the cubic function $q(t) = t^3 - \frac{3}{2}t^2 - 18t + 40$.
q_t05_083P034HardMediumMedium0.00.504human_labelledreviewedDetermine the number of real solutions of the equation $$\left(\ln x\right)^4 - 13\left(\ln x\right)^2 + 36 = 0,$$where $\ln x$ denotes the
q_t05_084P031HardHardMedium0.00.022human_labelledreviewedSketch the graph of $p(x) = -2x^3 + 3x^2 + 12x - 5$, showing clearly the coordinates of the $y$-intercept and all turning points. Your sketc
q_t05_085P033MediumHardEasy0.00.474human_labelledreviewedSketch the graph of $y = x^4 - 5x^2 + 4$, showing clearly the coordinates of any intercepts and turning points.
q_t05_086P032EasyEasyEasy0.00.021human_labelledreviewedDetermine the number of real zeroes of the cubic function $p(s) = 2s^3 + 3s^2 - 12s + 20$.
q_t05_087P034HardMediumMedium0.00.498human_labelledreviewedDetermine the number of real solutions of the equation $$2\left(x^2 - 3\right)^2 - 9\left(x^2 - 3\right) + 9 = 0.$$
q_t05_088P031MediumHardMedium0.00.491human_labelledreviewedSketch the graph of $C(w) = -w^3 + w^2 + 8w - 4$, showing clearly the coordinates of the $C$-intercept and any turning points. Your sketch m
q_t05_089P033EasyEasyEasy0.00.006human_labelledreviewedSketch the graph of $y = -x^4 + 5x^2 - 4$, showing clearly the coordinates of all intercepts and turning points.
q_t05_090P032EasyEasyEasy0.00.019human_labelledreviewedDetermine the number of real zeroes of the cubic function $g(v) = v^3 + 3v^2 - 9v - 10$.
q_t05_091P034MediumMediumMedium0.00.024human_labelledreviewedDetermine the number of real solutions of the equation $$\left(x^2 - 2x\right)^2 - 7\left(x^2 - 2x\right) + 12 = 0.$$
q_t05_092P031EasyEasyEasy0.00.019human_labelledreviewedSketch the graph of $f(t) = -t^3 + 3t^2 + 9t - 2$, showing clearly the coordinates of the $f$-intercept and any turning points. Your sketch
q_t05_093P033MediumHardEasy0.00.474human_labelledreviewedSketch the graph of $y = x^4 - 5x^2 + 4$, showing clearly the coordinates of any intercepts and turning points.
q_t05_094P032EasyEasyEasy0.00.018human_labelledreviewedDetermine the number of real zeroes of the cubic function $h(u) = u^3 - 3u^2 - 9u + 5$.
q_t05_095P034HardMediumHard0.00.506human_labelledreviewedDetermine the number of real solutions of the equation $$2\left(\arctan x\right)^2 - 5\left(\arctan x\right) + 2 = 0,$$ where $\arctan x$ de
q_t05_096P031MediumHardEasy0.00.489human_labelledreviewedSketch the graph of $h(u) = u^3 + u^2 - 8u + 6$, showing clearly the coordinates of the $h$-intercept, any turning points, and the correct e
q_t05_098P032MediumMediumEasy0.00.02human_labelledreviewedDetermine the number of real zeroes of the cubic function $q(w) = w^3 + \frac{3}{2}w^2 - 6w + 10$.
q_t05_099P034EasyMediumEasy0.00.504human_labelledreviewedFind the number of real solutions of the equation $$\sin^4 t - 10\sin^2 t + 9 = 0.$$
q_t05_100P031MediumHardEasy0.00.491human_labelledreviewedSketch the graph of $g(s) = 3s^3 - 3s^2 - 36s + 4$, showing clearly the coordinates of the $g$-intercept and all turning points. Your sketch
q_t06_001P038MediumHardHard0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{x^2 - x - 6}{x^2 - 5x + 6}$, by first expressing it in the form $y = a + \dfrac{b}{x-2} + \dfrac{c}{x-3}$. Y
q_t06_002P039EasyMediumEasy0.00.5human_labelledreviewedSolve the equation $$\frac{3x+1}{x+2} = 4.$$
q_t06_003P036HardHardHard0.00.0human_labelledreviewedSketch the graph of $$y = \frac{3x - 6}{x^2 + x - 12},$$ showing clearly all intercepts, asymptotes, and the behaviour of the function on ea
q_t06_004P037MediumHardHard0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{x^2 + x - 6}{x - 1}$, showing clearly all intercepts, asymptotes, and the coordinates of any stationary poin
q_t06_005P035EasyMediumEasy0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{3x - 6}{x + 2}$, showing clearly all intercepts and asymptotes.
q_t06_006P038EasyMediumMedium0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{x^2 + x - 2}{x^2 - x - 2}$, by first expressing it in the form $y = a + \dfrac{b}{x-2} + \dfrac{c}{x+1}$. Cl
q_t06_007P039EasyMediumEasy0.00.5human_labelledreviewedSolve the equation $$\frac{5}{x-3} + 2 = \frac{1}{x-3}.$$
q_t06_008P036MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $$y = \frac{2x - 6}{x^2 - x - 6},$$ showing clearly all intercepts, asymptotes (stating their equations), any removable
q_t06_009P037HardHardHard0.00.0human_labelledreviewedSketch the graph of $y = \dfrac{3x^2 - 4x + 5}{x - 3}$, showing clearly all intercepts, asymptotes, and the coordinates of any stationary po
q_t06_010P035HardMediumMedium0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{5x + 3}{2x - 7}$, showing clearly all intercepts and asymptotes. On your sketch, indicate which branch lies
q_t06_011P038EasyMedium—0.00.5discarded—Sketch the graph of $y = \dfrac{x^2 + x - 2}{x^2 - x - 2}$, by first expressing it in the form $y = a + \dfrac{b}{x-2} + \dfrac{c}{x+1}$. In
q_t06_012P039EasyMediumEasy0.00.5human_labelledreviewedSolve the equation $$\frac{4}{x+3} + \frac{x}{2} = 1.$$
q_t06_013P036HardHardHard0.00.0human_labelledreviewedSketch the graph of $$y = \frac{3x + 9}{x^2 - x - 6},$$ showing clearly all intercepts, asymptotes, and the behaviour of $y$ on each side of
q_t06_014P037EasyMediumMedium0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{x^2 - 2x - 8}{x + 3}$, showing clearly all intercepts and asymptotes.
q_t06_015P035MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = \dfrac{4 - 3x}{2x + 5}$, showing clearly all intercepts and asymptotes. On your sketch, label the exact coordinates
q_t06_016P038MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = \dfrac{x^2 - x - 6}{x^2 - x - 2}$, by first expressing it in the form $y = a + \dfrac{b}{x-2} + \dfrac{c}{x+1}$. Yo
q_t06_017P039MediumMediumEasy0.00.0human_labelledreviewedSolve the equation $$\frac{3}{x+2} - \frac{1}{x-1} = \frac{2}{x^2+x-2}.$$
q_t06_018P036MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $$y = \frac{2x - 1}{x^2 - 3x - 10},$$ showing clearly all intercepts, asymptotes, and the behaviour of $y$ on each side
q_t06_019P037HardHardHard0.00.0human_labelledreviewedSketch the graph of $y = \dfrac{2x^2 - 5x - 12}{2x + 1}$, showing clearly all intercepts, asymptotes, and the coordinates of any stationary
q_t06_020P035MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = \dfrac{3x + 8}{2x - 6}$, showing clearly the equations of both asymptotes and the exact coordinates of all intercep
q_t06_021P038MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{3x^2 - x - 2}{x^2 - x - 2}$, by first expressing it in the form $y = a + \dfrac{b}{x-2} + \dfrac{c}{x+1}$. S
q_t06_022P039MediumMediumMedium0.00.0human_labelledreviewedSolve the equation $$\frac{2}{x+3} + \frac{x}{x-2} = \frac{3x-1}{x^2+x-6}.$$
q_t06_023P036EasyMediumMedium0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{x-1}{x^2-x-6}$, showing clearly all intercepts, asymptotes, and the behaviour of the function near each vert
q_t06_024P037MediumHardHard0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{x^2 - 2x - 3}{x + 2}$, showing clearly all intercepts, asymptotes, and the coordinates of any stationary poi
q_t06_025P035MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = \dfrac{5 - 2x}{3x + 4}$, showing clearly the equations of both asymptotes and the exact coordinates of all intercep
q_t06_026P038MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = \dfrac{x^2 - x - 6}{x^2 - x - 2}$, by first expressing it in the form $y = a + \dfrac{b}{x-2} + \dfrac{c}{x+1}$. Yo
q_t06_027P039EasyMediumEasy0.00.5human_labelledreviewedSolve the equation $$\frac{3}{x-4} = \frac{x-2}{x-4} + 1.$$
q_t06_028P036HardHardMedium0.00.0human_labelledreviewedSketch the graph of $$y = \frac{3x + 6}{x^2 - x - 12},$$ showing clearly all intercepts, asymptotes, and the behaviour of $y$ on each side o
q_t06_029P037MediumHardHard0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{3x^2 - x - 4}{x - 3}$, showing clearly all intercepts, asymptotes, and the coordinates of any stationary poi
q_t06_030P035EasyMediumEasy0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{2x + 9}{x - 4}$, clearly labelling the equations of both asymptotes and the exact coordinates of all axis in
q_t06_031P038EasyMediumMedium0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{x^2 - x - 2}{x^2 - 3x + 2}$, by first expressing it in the form $y = a + \dfrac{b}{x-1} + \dfrac{c}{x-2}$. I
q_t06_032P039EasyMediumEasy0.00.5human_labelledreviewedSolve the equation $$\frac{4}{x+1} = \frac{x+7}{3x+3}.$$
q_t06_033P036MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $$y = \frac{4x - 8}{x^2 + x - 6},$$ showing clearly all intercepts, asymptotes (stating their equations), any removable
q_t06_034P037HardHardMedium0.00.0human_labelledreviewedSketch the graph of $y = \dfrac{3x^2 - 4x - 4}{x - 2}$, showing clearly all intercepts, asymptotes, and the coordinates of any stationary po
q_t06_035P035HardHardMedium0.00.0human_labelledreviewedSketch the graph of $y = \dfrac{4 - 7x}{2x + 3}$, showing clearly the equations of both asymptotes and the exact coordinates of all axis int
q_t06_036P038EasyMediumMedium0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{x^2 + 2x - 3}{x^2 + 4x + 3}$, by first expressing it in the form $y = a + \dfrac{b}{x+1} + \dfrac{c}{x+3}$.
q_t06_037P039EasyMediumEasy0.00.5human_labelledreviewedSolve the equation $$\frac{x}{x+3} + 1 = \frac{5}{x+3}.$$
q_t06_038P036HardHardHard0.00.0human_labelledreviewedSketch the graph of $y = \dfrac{3x - 6}{x^2 - x - 6}$, showing clearly all intercepts, asymptotes, any removable discontinuities, and the be
q_t06_039P037EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $y = \dfrac{x^2 + 4x + 3}{x - 1}$, showing clearly all intercepts and asymptotes.
q_t06_040P035MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{6 - 5x}{3x + 2}$, showing clearly the equations of both asymptotes and the exact coordinates of all axis int
q_t06_041P038MediumHardHard0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{x^3 - 4x^2 + x + 6}{x^2 - 3x - 10}$, by first expressing it in the form $y = ax + b + \dfrac{c}{x-5} + \dfra
q_t06_042P039MediumMediumMedium0.00.0human_labelledreviewedSolve the equation $$\frac{x+1}{x-3} - \frac{2x}{x+2} = 1.$$
q_t06_043P036MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $$y = \frac{3x + 9}{x^2 - 2x - 8},$$ showing clearly all intercepts, asymptotes, and the behaviour of $y$ near each vert
q_t06_044P037HardHard—0.00.0discarded—Sketch the graph of $y = \dfrac{3x^2 - 4x - 4}{x - 2}$, showing clearly all intercepts, asymptotes, and the coordinates of any stationary po
q_t06_045P035MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{3x + 8}{4 - x}$, clearly labelling the equations of both asymptotes and the exact coordinates of all axis in
q_t06_046P038MediumHardHard0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{x^3 - 2x^2 - 9x + 18}{x^2 - x - 6}$, by first expressing it in the form $y = ax + b + \dfrac{c}{x-3} + \dfra
q_t06_047P039MediumMediumMedium0.00.0human_labelledreviewedSolve the equation $$\frac{x}{x+2} - \frac{6}{x^2-4} = \frac{1}{x-2}.$$
q_t06_048P036EasyEasyMedium0.00.0human_labelledreviewedSketch the graph of $$y = \frac{2x - 10}{x^2 + 2x - 15},$$ showing clearly all intercepts, asymptotes, any removable discontinuities, and th
q_t06_049P037MediumHardHard0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{2x^2 - x + 6}{x + 2}$, showing clearly all intercepts, asymptotes, and the coordinates of any stationary poi
q_t06_050P035MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = \dfrac{3x - 4}{2x + 6}$, showing clearly all intercepts and asymptotes.
q_t06_051P039EasyMedium—0.00.494discarded—Solve the equation $$\frac{3}{t-4} = \frac{t}{t-4} - 2,$$ where $t$ is a real number.
q_t06_052P037HardHard—0.00.015discarded—A curve $C$ is defined by $$f(t) = \frac{3t^2 + 10t - 8}{2t - 1}, \quad t \neq \frac{1}{2}.$$ Sketch the graph of $C$, showing clearly: - al
q_t06_053P036MediumHardMedium0.00.493human_labelledreviewedSketch the graph of $$h(t) = \frac{3t - 12}{t^2 - t - 12},$$ showing clearly all intercepts, asymptotes, any removable discontinuities, and
q_t06_054P036EasyEasyMedium0.00.034human_labelledreviewedSketch the graph of $$f(u) = \frac{u + 3}{u^2 - u - 6},$$ showing clearly all intercepts, asymptotes, and the behaviour of $f$ near each ver
q_t06_055P036EasyEasyMedium0.00.018human_labelledreviewedSketch the graph of $g(x) = \dfrac{x - 1}{x^2 - x - 6}$, showing clearly all intercepts, asymptotes, and the behaviour of the graph near eac
q_t06_056P036HardHardHard0.00.043human_labelledreviewedSketch the graph of $$R(s) = \frac{2s + 8}{3s^2 - 3s - 36},$$ showing clearly all intercepts, asymptotes, any removable discontinuities, and
q_t06_057P036MediumHardMedium0.00.467human_labelledreviewedConsider the function $f(x) = \dfrac{x - 3}{x^2 - x - 6}$. (a) Factorise the denominator $x^2 - x - 6$. (b) Hence write down the equations
q_t06_058P039EasyMediumEasy0.00.496human_labelledreviewedSolve the equation $$\frac{3}{t+1} = \frac{t+5}{2t+2}.$$
q_t06_059P035MediumHardMedium0.00.493human_labelledreviewedSketch the graph of $y = \dfrac{2x + 9}{3 - x}$, clearly labelling the equations of both asymptotes and the exact coordinates of all axis in
q_t06_060P036MediumHardMedium0.00.497human_labelledreviewedSketch the graph of $$h(t) = \frac{3t - 9}{t^2 + t - 12},$$ showing clearly all intercepts, asymptotes, any removable discontinuities, and t
q_t06_061P039EasyMediumEasy0.00.496human_labelledreviewedSolve the equation $$\frac{2}{n+3} = \frac{n-1}{2n+6}.$$
q_t06_062P035MediumMediumMedium0.00.014human_labelledreviewedSketch the graph of $y = \dfrac{4t + 3}{2t - 6}$, for $t \in \mathbb{R}$, $t \neq 3$. Clearly label the equations of both asymptotes and the
q_t06_063P036MediumHardMedium0.00.488human_labelledreviewedSketch the graph of $$f(x) = \frac{3x + 6}{x^2 - x - 6},$$ showing clearly all intercepts, asymptotes, and the behaviour of $f$ on each side
q_t06_064P038EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $y = \dfrac{x^2 - 4x + 3}{x^2 - 5x + 6}$, by first expressing it in the form $y = a + \dfrac{b}{x - 2} + \dfrac{c}{x - 3
q_t06_065P038HardHardMedium0.00.0human_labelledreviewedSketch the graph of $y = \dfrac{2x^3 - 3x^2 - 8x + 12}{x^2 + x - 6}$, by first expressing it in the form $y = ax + b + \dfrac{c}{x - 2} + \d
q_t06_066P038MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{3x^2 - x - 4}{x^2 - 2x - 3}$, by first expressing it in the form $y = a + \dfrac{b}{x-3} + \dfrac{c}{x+1}$.
q_t06_067P038MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $y = \dfrac{2x^3 + 3x^2 - 2x}{2x^2 - x - 1}$, by first expressing it in the form $y = ax + b + \dfrac{c}{x-1} + \dfrac{d
q_t06_068P038EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $y = \dfrac{x^2 + 2x - 8}{x^2 + x - 12}$, by first expressing it in the form $y = a + \dfrac{b}{x-3} + \dfrac{c}{x+4}$.
q_t06_069P038EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $y = \dfrac{x^2+2x-3}{x^2+x-6}$, by first expressing it in the form $y = a + \dfrac{b}{x-2} + \dfrac{c}{x+3}$.
q_t06_070P035MediumMediumEasy0.00.0human_labelledreviewedSketch the graph of $g(u) = \dfrac{6 - 2u}{3u + 9}$, for $u \in \mathbb{R}$, $u \neq -3$. Clearly label on your sketch: - the equations of
q_t06_071P037EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $y = \dfrac{x^2 - x - 6}{x + 1}$, showing clearly all intercepts and asymptotes.
q_t06_072P036EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $$y = \frac{x - 3}{x^2 + x - 12},$$ showing clearly all intercepts, asymptotes, and the behaviour of $y$ on each side of
q_t06_073P038HardHardHard0.00.0human_labelledreviewedSketch the graph of $y = \dfrac{x^3 - 7x + 6}{x^3 - 2x^2 - 5x + 6}$, by first expressing it in the form $y = a + \dfrac{b}{x+2} + \dfrac{c}{
q_t06_074P039MediumMediumMedium0.00.0human_labelledreviewedSolve the equation $$\frac{3}{x^2 - 4} = \frac{1}{x - 2} - \frac{1}{x + 3}.$$
q_t07_001P041MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = ||x + 1| - 4|$, clearly showing all x-intercepts, y-intercept, and the coordinates of any vertices.
q_t07_002P040EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $y = |x^2 - x - 6|$ for $-3 \le x \le 5$, clearly labelling any x-intercepts, y-intercepts, and turning points.
q_t07_003P042HardHardHard0.00.0human_labelledreviewedSketch the graph of $y = |3x - 6| - |2x + 4| + |x + 1|$, clearly showing all critical values, the piecewise formula on each interval, and th
q_t07_004P041MediumHard—0.00.5discarded—Sketch the graph of $y = ||x + 1| - 4|$, clearly showing all x-intercepts, y-intercept, and the coordinates of any vertices.
q_t07_005P040EasyMediumEasy0.00.5human_labelledreviewedSketch the graph of $y = |\sin x|$ for $0 \le x \le 2\pi$, clearly labelling any x-intercepts and the coordinates of any turning points.
q_t07_006P042EasyEasyMedium0.00.0human_labelledreviewedSketch the graph of $y = |x + 5| - |2x - 2| + |x - 3|$, clearly identifying all critical values, writing the piecewise formula on each inter
q_t07_007P041EasyEasyMedium0.00.0human_labelledreviewedSketch the graph of $y = ||x + 1| - 2|$, clearly showing all x-intercepts, y-intercept, and the coordinates of any vertices.
q_t07_008P040MediumMediumEasy0.00.0human_labelledreviewedSketch the graph of $y = |2x^2 - 5x - 3|$, clearly showing the x-intercepts, the vertex, and the behaviour of the graph.
q_t07_009P042HardHardHard0.00.0human_labelledreviewedSketch the graph of $y = |3x + 9| - |4x - 8| + |2x - 2|$, clearly identifying all critical values, deriving the piecewise formula on each in
q_t07_010P041HardHardHard0.00.0human_labelledreviewedSketch the graph of $y = ||x^2 - 4x - 5| - 4|$, clearly showing all x-intercepts, the coordinates of all local minima and maxima, and any ax
q_t07_011P041EasyEasyMedium0.00.0human_labelledreviewedSketch the graph of $y = ||2x - 6| - 4|$, clearly showing all $x$-intercepts, the $y$-intercept, and the coordinates of any vertices.
q_t07_012P040MediumHardEasy0.00.5human_labelledreviewedSketch the graph of $y = |x^2 - 2x - 3|$ for $-2 \le x \le 5$. On your sketch, clearly label the coordinates of any x-intercepts, y-intercep
q_t07_013P042MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = |2x + 6| - |3x - 3| + |x - 5|$, clearly identifying all critical values, deriving the piecewise formula on each int
q_t07_014P041MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $y = ||2x + 4| - 3|$, clearly indicating all x-intercepts, y-intercept, and the coordinates of any vertices.
q_t07_015P040MediumHard—0.00.5discarded—Sketch the graph of $y = |x^2 - 2x - 3|$ for $-2 \le x \le 5$. On your sketch, clearly label the x-intercepts, the y-intercept, and the coor
q_t07_016P042EasyEasyMedium0.00.0human_labelledreviewedSketch the graph of $y = |x - 2| + |x + 4| - |2x - 6|$, clearly identifying all critical values, writing the piecewise formula on each inter
q_t07_017P041EasyEasyMedium0.00.0human_labelledreviewedSketch the graph of $y = ||x + 1| - 2|$, clearly indicating all x-intercepts, y-intercept, and the coordinates of any vertices.
q_t07_018P040MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $y = |2x^2 - 2x - 12|$, clearly showing the coordinates of any x-intercepts, the y-intercept, and the vertex of the unde
q_t07_019P042MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $y = |3x - 6| - |x + 2| + |2x + 8|$, clearly identifying all critical values, deriving the piecewise formula on each int
q_t07_020P041MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $y = ||x + 1| - 4|$, clearly showing all x-intercepts, y-intercepts, and the coordinates of any vertices (turning points
q_t07_021P040EasyEasyMedium0.00.0human_labelledreviewedSketch the graph of $y = |x^2 - 5x + 4|$ for $0 \le x \le 5$, clearly labelling the $x$-intercepts, the $y$-intercept, and the coordinates o
q_t07_022P042HardHardHard0.00.0human_labelledreviewedSketch the graph of $y = |3x + 6| - |2x - 4| - |x - 1|$ for $x \in \mathbb{R}$, clearly identifying all critical values, deriving the piecew
q_t07_023P041HardHardHard0.00.0human_labelledreviewedSketch the graph of $y = ||x^2 + 2x - 8| - 5|$, clearly showing all x-intercepts, vertices, and any axes of symmetry.
q_t07_024P040MediumMedium—0.00.0discarded—Sketch the graph of $y = |x^2 - 2x - 3|$, clearly showing all x-intercepts, the y-intercept, and the turning point of the graph.
q_t07_025P042HardHardHard0.00.0human_labelledreviewedSketch the graph of $y = |3x - 9| - |2x + 4| + |4x + 8|$ for $x \in \mathbb{R}$, clearly identifying all critical values, deriving the piece
q_t07_026P041MediumHardHard0.00.5human_labelledreviewedSketch the graph of $y = ||x^2 - 4x| - 3|$, clearly showing all $x$-intercepts, the $y$-intercept, any axes of symmetry, and the coordinates
q_t07_027P040EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $y = |2x - x^2|$ for $-1 \le x \le 3$, clearly labelling the $x$-intercepts, the $y$-intercept, and the coordinates of t
q_t07_028P042HardMediumHard0.00.5human_labelledreviewedSketch the graph of $y = |3x - 6| + |2x + 4| - |x - 1| + |x + 2|$, clearly showing all critical (breakpoint) values, the piecewise-linear fo
q_t07_029P041MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $y = ||3x - 6| - 4|$, clearly indicating all $x$-intercepts, the $y$-intercept, and the coordinates of any vertices (tur
q_t07_030P040EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $y = |x^2 - 4|$ for $-3 \le x \le 3$. On your sketch, clearly label the coordinates of any $x$-intercepts, the $y$-inter
q_t07_031P042EasyEasyMedium0.00.0human_labelledreviewedSketch the graph of $y = |x - 1| + |x + 2| + |x - 3|$ for $-5 \leq x \leq 6$. Identify all critical (breakpoint) values, write down the piec
q_t07_032P041EasyEasyMedium0.00.0human_labelledreviewedSketch the graph of $y = ||x^2 - 9| - 4|$, clearly showing all $x$-intercepts, the $y$-intercept, and the coordinates of any vertices. State
q_t07_033P040MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = |2\sin x - 1|$ for $0 \le x \le 2\pi$. On your sketch, clearly label the x-intercepts, any local maximum points, an
q_t07_034P042HardHardHard0.00.0human_labelledreviewedSketch the graph of $y = |3x - 9| - |2x + 4| + |x + 7|$ for $x \in \mathbb{R}$. (a) Identify all critical values, derive the piecewise form
q_t07_035P041HardHardHard0.00.0human_labelledreviewedSketch the graph of $y = ||x^2 - 4x - 5| - 4|$, clearly showing all x-intercepts, the coordinates of any local minima and maxima, and any ax
q_t07_036P040EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $y = |x^2 - 2x - 3|$ for $-2 \le x \le 5$, clearly labelling the $x$-intercepts, the $y$-intercept, and the coordinates
q_t07_037P042EasyMediumMedium0.00.5human_labelledreviewedSketch the graph of $y = |x + 4| - |2x - 2| + |x - 6|$ for $x \in \mathbb{R}$. (a) Find all critical values, write down the piecewise formu
q_t07_038P041HardHardHard0.00.0human_labelledreviewedSketch the graph of $y = ||x^2 - 4x + 3| - 2|$, clearly showing all x-intercepts, vertices, and any axes of symmetry.
q_t07_039P040EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $y = |x(x - 4)|$ for $-1 \le x \le 5$. On your sketch, clearly label the $x$-intercepts, the $y$-intercept, and the coor
q_t07_040P042MediumHardHard0.00.5human_labelledreviewedSketch the graph of $y = |2x + 6| - |3x - 3| + |x - 5|$ for $x \in \mathbb{R}$. (a) Find all critical values, derive the piecewise formula
q_t07_041P041MediumHardHard0.00.5human_labelledreviewedSketch the graph of $y = ||2x^2 + 4x - 6| - 8|$, clearly showing all $x$-intercepts, the coordinates of all vertices (turning points), and a
q_t07_042P040MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = |2x^2 - 5x - 3|$, clearly showing all x-intercepts, the y-intercept, and the vertex of the underlying parabola.
q_t07_043P042MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $y = |x - 4| + |3x + 3| - |2x - 2|$ for $x \in \mathbb{R}$. (a) Identify all critical values, derive the piecewise form
q_t07_044P041HardHardMedium0.00.0human_labelledreviewedSketch the graph of $y = ||3x^2 - 12| - 6|$, clearly showing all $x$-intercepts, the coordinates of all vertices (local minima and maxima),
q_t07_045P040MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $y = |2x^2 - 8x + 6|$ for $-1 \le x \le 5$, clearly showing all x-intercepts, the y-intercept, and any turning points.
q_t07_046P042MediumHardHard0.00.5human_labelledreviewedSketch the graph of $y = |3x + 6| - |2x - 4| + |x - 7|$ for $x \in \mathbb{R}$. (a) Identify all critical values, derive the piecewise form
q_t07_047P041MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $y = ||2x^2 - 8| - 6|$, clearly showing all $x$-intercepts, the $y$-intercept, any axes of symmetry, and the coordinates
q_t07_048P040EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $y = |x^2 + x - 6|$ for $-4 \le x \le 3$, clearly labelling the $x$-intercepts, the $y$-intercept, and the coordinates o
q_t07_049P042MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $y = |2x + 4| - |3x - 3| + |x - 5|$, clearly identifying all critical values, deriving the piecewise formula on each int
q_t07_050P041MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $y = ||x + 1| - 4|$, clearly showing all x-intercepts, y-intercept, and the coordinates of any vertices.
q_t07_051P041MediumMediumEasy0.00.0human_labelledreviewedSketch the graph of $y = ||x - 3| - 2|$, clearly showing all x-intercepts, the y-intercept, and the coordinates of any vertices.
q_t07_052P040EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $y = |x^2 - 1|$ for $-3 \le x \le 3$, clearly labelling the $x$-intercepts, the $y$-intercept, and the coordinates of th
q_t07_053P042HardMediumMedium0.00.5human_labelledreviewedSketch the graph of $y = |3x - 6| + |2x + 4| - |x - 1| + |x + 2|$, clearly showing all critical points, the slope of each linear piece, and
q_t07_054P041MediumHardEasy0.00.5human_labelledreviewedSketch the graph of $y = ||x + 1| - 4|$, clearly indicating all x-intercepts, y-intercept, and the coordinates of any vertices (turning poin
q_t07_055P040EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $y = |x^2 - 2x - 3|$, clearly showing all x-intercepts, the y-intercept, and the vertex.
q_t07_056P042EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $y = |x + 1| - |2x - 4| + |x - 7|$, clearly identifying all critical values, deriving the piecewise formula on each inte
q_t07_057P041EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $y = ||x - 1| - 2|$, clearly indicating all x-intercepts, y-intercept, and the coordinates of any vertices (turning poin
q_t07_058P040MediumMediumEasy0.00.0human_labelledreviewedSketch the graph of $y = |x^2 - 2x - 3|$ for $-2 \le x \le 5$. On your sketch, clearly label the x-intercepts, the y-intercept, and the coor
q_t07_059P042HardHardMedium0.00.0human_labelledreviewedSketch the graph of $y = |3x - 9| - |2x + 6| + |x + 1|$ for $x \in \mathbb{R}$, clearly identifying all critical values, deriving the piecew
q_t07_060P041HardHardMedium0.00.0human_labelledreviewedSketch the graph of $y = ||x^2 - 4x + 3| - 2|$, clearly showing all x-intercepts, vertices, and any axis of symmetry.
q_t07_061P040EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $y = |x^2 - 2x - 3|$, clearly showing all x-intercepts, the vertex of the underlying parabola, and the y-intercept.
q_t07_062P042EasyMediumEasy0.00.5human_labelledreviewedSketch the graph of $y = |x - 2| + |x + 4| - |x - 1|$. Identify all critical values, derive the piecewise formula on each interval, and lab
q_t07_063P041HardHardMedium0.00.0human_labelledreviewedSketch the graph of $y = ||x^2 - 4x + 3| - 2|$, clearly showing all x-intercepts, vertices, and any axis of symmetry.
q_t07_064P040EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $y = |x(x-4)|$ for $-1 \le x \le 5$, clearly labelling the coordinates of the $x$-intercepts, the $y$-intercept, and the
q_t07_065P042MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $y = |2x - 6| - |x + 2| + |3x + 3|$, clearly identifying all critical values, deriving the piecewise formula on each int
q_t07_066P041MediumHardEasy0.00.5human_labelledreviewedSketch the graph of $y = ||2x - 6| - 4|$, clearly indicating all x-intercepts, the y-intercept, the coordinates of all vertices, and any axi
q_t07_067P040MediumHardEasy0.00.5human_labelledreviewedSketch the graph of $y = |2x^2 - 5x - 3|$, clearly showing the coordinates of any x-intercepts, the y-intercept, and the vertex of the parab
q_t07_068P042MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = |2x + 4| - |3x - 3| + |x - 5|$, clearly identifying all critical values, deriving the piecewise formula on each int
q_t07_069P041HardHardMedium0.00.0human_labelledreviewedSketch the graph of $y = ||x^2 - 4x + 3| - 2|$, clearly showing all x-intercepts, vertices, and any axis of symmetry.
q_t07_070P040MediumHardEasy0.00.5human_labelledreviewedSketch the graph of $y = |2x^2 - 5x - 3|$, clearly showing the coordinates of any x-intercepts, the y-intercept, and the vertex of the under
q_t07_071P042MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $y = |2x + 6| - |3x - 3| + |x + 1|$ for $x \in \mathbb{R}$. **(a)** Find all critical values, derive the piecewise form
q_t07_072P041MediumHardEasy0.00.5human_labelledreviewedSketch the graph of $y = ||2x + 4| - 3|$, clearly indicating all x-intercepts, the y-intercept, and the coordinates of any vertices.
q_t07_073P040EasyEasyEasy0.00.0human_labelledreviewedSketch the graph of $y = |x^2 - 2x - 8|$, clearly labelling the coordinates of the $x$-intercepts, the $y$-intercept, and the turning point
q_t07_074P042MediumHardMedium0.00.5human_labelledreviewedSketch the graph of $y = |x - 2| + |3x + 3| - |2x - 8|$, clearly identifying all critical values, deriving the piecewise formula on each int
q_t07_075P041MediumHardEasy0.00.5human_labelledreviewedSketch the graph of $y = \bigl||x + 1| - 4\bigr|$, clearly showing all x-intercepts, y-intercept, and the coordinates of any vertices.
q_t08_001P046MediumMediumEasy0.00.0human_labelledreviewedThe number of registered users (in millions) on a social media platform is modelled by $$U(t) = 5 + 3\log_2(4t + 2),$$ where $t$ is the numb
q_t08_002P047EasyEasyMedium0.00.0human_labelledreviewedSolve each of the following equations for $x$. **(a)** $2^{x+3} = 8^{x-1}$ **(b)** $3 \cdot 4^x - 10 \cdot 2^x + 8 = 0$ **(c)** $x^{\log_
q_t08_003P044HardMediumHard0.00.5human_labelledreviewedSketch the graph of $y = \log_2(3x + 5) - 3$, clearly identifying the vertical asymptote, the end behaviours, and any intercepts with the co
q_t08_004P045MediumMedium—0.00.0discarded—A printing press was purchased for $\$24\,500$ when new. After $7$ years, its resale value had fallen to $\$13\,800$. Assuming the press dep
q_t08_005P043EasyMediumEasy0.00.5human_labelledreviewedSketch the graph of $y = 2e^{x-1} - 4$, clearly showing the horizontal asymptote and any intercepts with the axes.
q_t08_006P048EasyEasyEasy0.00.0human_labelledreviewedSolve each of the following equations for $x$: (a) $\log(5x+3) - \log(x-1) = \log 8$ (b) $\log_4 x + \log_2 x = 6$ (c) $\log_6 x + \log_x
q_t08_007P046EasyMediumEasy0.00.5human_labelledreviewedThe depth of sediment (in centimetres) at the bottom of a reservoir is modelled by $$D(t) = 3 + 5\ln(t + 1),$$ where $t$ is the number of ye
q_t08_008P047MediumHardMedium0.00.5human_labelledreviewedSolve each of the following equations for $x$. **(a)** $4^{x+1} = 32^{x-2}$ **(b)** $5 \cdot 25^x - 26 \cdot 5^x + 5 = 0$ **(c)** $x^{3 -
q_t08_009P044HardMediumHard0.00.5human_labelledreviewedSketch the graph of $y = \log_3(4 - 2x) + 1$, clearly identifying the vertical asymptote, the end behaviours, and all intercepts with the co
q_t08_010P045HardMediumMedium0.00.5human_labelledreviewedA vineyard purchases a specialised harvesting robot for $\$87\,500$. Due to rapid advances in agricultural technology, the robot's resale va
q_t08_011P043EasyMediumMedium0.00.5human_labelledreviewedSketch the graph of $y = -2e^{x+1} + 6$, clearly showing the horizontal asymptote and any intercepts with the axes.
q_t08_012P048EasyEasyEasy0.00.0human_labelledreviewedSolve each of the following equations for $x$: **(a)** $\ln(4x + 3) - \ln(2x - 1) = \ln 3$ **(b)** $\log_{27} x + \log_9 x + \log_3 x = \f
q_t08_013P046HardMediumEasy0.00.5human_labelledreviewedThe cumulative amount of crude oil extracted from a maturing oil field, measured in millions of barrels, is modelled by $$V(t) = 120 + 45\lo
q_t08_014P047EasyEasyMedium0.00.0human_labelledreviewedSolve each of the following equations for $x$. (a) $2^{x+3} = 4^{2x-1}$ (b) $3 \cdot 9^x - 10 \cdot 3^x + 3 = 0$ (c) $x^{1 + \log_{10} x}
q_t08_015P044MediumMediumHard0.00.0human_labelledreviewedSketch the graph of $y = \log_3(5 - 2x) - 2$, clearly identifying the domain, the vertical asymptote, the end behaviours, and all intercepts
q_t08_016P045MediumMediumEasy0.00.0human_labelledreviewedA boat was purchased in 2010 for $24000. By 2019, its value had fallen to $14500. Assuming the value depreciates exponentially at a constant
q_t08_017P043MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = -3e^{2x-1} + 5$, clearly showing the horizontal asymptote and any intercepts with the axes.
q_t08_018P048MediumHardEasy0.00.5human_labelledreviewedSolve for $x$ in each of the following equations: (a) $\log(5x+3) - \log(x-1) = \log 8$ (b) $\log_{25} x + \log_5 x + \log_{125} x = \frac
q_t08_019P046HardHardMedium0.00.0human_labelledreviewedThe concentration of a pharmaceutical compound in a patient's bloodstream, measured in micrograms per litre (μg/L), is modelled by $$C(t) =
q_t08_020P047MediumHardMedium0.00.5human_labelledreviewedSolve each of the following equations for $x$. (a) $4^{x+3} = 8^{2x-1}$ (b) $3 \cdot 25^x - 16 \cdot 5^x + 5 = 0$ (c) $x^{3 + \log_{10} x
q_t08_021P044MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = \log_2(3 - 6x) + 2$, clearly identifying the domain, the vertical asymptote, the end behaviours, and all intercepts
q_t08_022P045MediumMediumMedium0.00.0human_labelledreviewedA commercial espresso machine was purchased by a café for $\$31\,200$ in January 2009. By January 2021, the same machine had a resale value
q_t08_023P043EasyMediumMedium0.00.5human_labelledreviewedSketch the graph of $y = 4 - 2e^{-x+3}$, clearly showing the horizontal asymptote and any intercepts with the axes.
q_t08_024P048MediumHard—0.00.5discarded—Solve for $x$ in each of the following equations: (a) $\log(5x+3) - \log(x-1) = \log 8$ (b) $\log_{25} x + \log_5 x + \log_{125} x = \frac
q_t08_025P046MediumMediumEasy0.00.0human_labelledreviewedThe noise level inside a concert venue, measured in decibels (dB), is modelled by $$L(t) = 85 + 9\log_3(2t + 3),$$ where $t$ is the number o
q_t08_026P046MediumMediumMedium0.00.0human_labelledreviewedThe brightness of a star, measured in apparent magnitude units, is observed to change as dust slowly clears from a nebula. The apparent magn
q_t08_027P047EasyEasyMedium0.00.0human_labelledreviewedSolve each of the following equations for $x$. (a) $2^{x+3} = 16^{x-1}$ (b) $3 \cdot 4^x - 10 \cdot 2^x + 8 = 0$ (c) $x^{1 + \log_{10} x}
q_t08_028P044HardMediumMedium0.00.5human_labelledreviewedSketch the graph of $y = \log_2(4 - 3x) - 2$, clearly identifying the domain, the vertical asymptote, the end behaviours, and all intercepts
q_t08_029P045MediumMediumMedium0.00.0human_labelledreviewedA photography studio purchases a high-end digital camera system for $\$42\,600$ in 2011. By 2018, the resale value of the same system has fa
q_t08_030P043EasyMediumMedium0.00.5human_labelledreviewedSketch the graph of $y = 3e^{-x-2} - 6$, clearly showing the horizontal asymptote and any intercepts with the axes.
q_t08_031P048EasyEasyEasy0.00.0human_labelledreviewedSolve each of the following equations for $x$: **(a)** $\ln(x+6) - \ln(2x-1) = \ln 2$ **(b)** $\log_8 x + \log_4 x + \log_2 x = 11$ **(c)
q_t08_032P046EasyMediumEasy0.00.5human_labelledreviewedThe total distance (in kilometres) that a migrating bird has travelled is modelled by $$D(t) = 20 + 15\ln(3t + 2),$$ where $t$ is the number
q_t08_033P047MediumHardMedium0.00.5human_labelledreviewedSolve each of the following equations for $x$. **(a)** $27^{x-1} = 9^{2x+3}$ **(b)** $2 \cdot 36^x - 13 \cdot 6^x + 6 = 0$ **(c)** $x^{\l
q_t08_034P044HardHardHard0.00.0human_labelledreviewedSketch the graph of $y = \log_{1/2}(3x - 6) + 4$. Your sketch must include: the equation of the vertical asymptote, the behaviour of $y$ as
q_t08_035P045HardMediumMedium0.00.5human_labelledreviewedA piece of industrial equipment was purchased in January 2008 for $47 500. By January 2019, its resale value had fallen to $18 200. Assuming
q_t08_036P043EasyMediumEasy0.00.5human_labelledreviewedSketch the graph of $y = 3e^{x+2} - 12$, clearly showing the horizontal asymptote and any intercepts with the axes.
q_t08_037P048EasyEasyEasy0.00.0human_labelledreviewedSolve each of the following equations for $x$: **(a)** $\log(4x + 9) - \log(2x - 1) = \log 3$ **(b)** $\log_{81} x + \log_{27} x + \log_9
q_t08_038P046HardHardMedium0.00.0human_labelledreviewedThe accumulated light exposure (in lux-hours) received by a deep-sea coral colony is modelled by $$E(t) = 2.4 + 6.5\log_4(3t + 4),$$ where $
q_t08_039P047EasyEasyEasy0.00.0human_labelledreviewedSolve each of the following equations for $x$. (a) $2^{x+3} = 16^{x-1}$ (b) $3 \cdot 9^x - 10 \cdot 3^x + 3 = 0$ (c) $x^{1 + \log_{10} x}
q_t08_040P044MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = \log_3(2x + 7) - 2$, clearly identifying the domain, the vertical asymptote, the end behaviours, and all intercepts
q_t08_041P045MediumMediumMedium0.00.0human_labelledreviewedA vineyard purchases a new oak barrel for wine ageing at a cost of $\$3\,750$ in the year 2007. Due to gradual degradation, the barrel's res
q_t08_042P043MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = -4e^{3x+2} + 7$, clearly showing the horizontal asymptote and any intercepts with the axes.
q_t08_043P048MediumHardEasy0.00.5human_labelledreviewedSolve each equation for $x$: (a) $\log(5x+2) - \log(x-3) = \log 8$ (b) $\log_{25} x + \log_5 x + \log_{125} x = \frac{11}{6}$ (c) $\log_4
q_t08_044P046HardHardHard0.00.0human_labelledreviewedThe cumulative rainfall (in millimetres) recorded at a weather station during a prolonged dry season is modelled by $$R(t) = 12 + 8\log_5(2t
q_t08_045P047MediumHardMedium0.00.5human_labelledreviewedSolve each of the following equations for $x$. (a) $4^{x+3} = 8^{2x-1}$ (b) $3 \cdot 25^x - 8 \cdot 5^x + 4 = 0$ (c) $x^{3 + \log_{10} x}
q_t08_046P044MediumMediumMedium0.00.0human_labelledreviewedSketch the graph of $y = \log_2(5x - 3) + 1$, clearly identifying the domain, the vertical asymptote, the end behaviours, and all intercepts
q_t08_047P045MediumMediumMedium0.00.0human_labelledreviewedA luxury wristwatch was purchased at auction for $\$6\,850$ in 2006. By 2019, the same watch had appreciated in value to $\$11\,340$. Assumi
q_t08_048P043EasyMediumMedium0.00.5human_labelledreviewedSketch the graph of $y = 4e^{-x+2} - 8$, clearly showing the horizontal asymptote and any intercepts with the axes.
q_t08_049P048MediumHard—0.00.5discarded—Solve for $x$ in each of the following equations: (a) $\log(5x+3) - \log(x-1) = \log 8$ (b) $\log_{25} x + \log_5 x + \log_{125} x = \frac
q_t08_050P046MediumMediumEasy0.00.0human_labelledreviewedThe number of subscribers (in thousands) to an online streaming service is modelled by $S(t) = 30 + 15\ln(3t + 1)$, where $t$ is the number
q_t08_051P044EasyEasy—0.00.014discarded—Sketch the graph of $y = \log_2(x + 4) - 1$. Your sketch should clearly show the vertical asymptote, the $x$-intercept, and the $y$-intercep
q_t08_052P043MediumMedium—0.00.014discarded—Sketch the graph of $y = -2e^{3x+6} + 10$, clearly showing the horizontal asymptote and any intercepts with the axes.
q_t08_053P047MediumHard—0.00.49discarded—Solve each of the following equations for $x$. **(a)** $5^{3x-1} = 25^{x+4}$ **(b)** $2 \cdot 49^x - 15 \cdot 7^x + 7 = 0$ **(c)** $x^{\l
q_t08_054P048MediumHard—0.00.49discarded—Solve each of the following equations for $x$: **(a)** $\log_3(4x - 5) - \log_3(x + 7) = \log_3 3$ **(b)** $\log_{64} x + \log_{16} x + \l
q_t08_055P046EasyMedium—0.00.495discarded—The number of subscribers to an online channel is modelled by $N(t) = 200 + 50\ln(t + 1)$, where $N$ is the number of subscribers and $t$ is
q_t08_056P045EasyMedium—0.00.492discarded—A laptop computer was purchased new for $2500. After 6 years it is worth $1100. Assuming the laptop depreciates exponentially at a constant
q_t08_057P044EasyMedium—0.00.492discarded—Sketch the graph of $y = \log_3(2x - 1) + 2$, clearly identifying the domain, the vertical asymptote, the end behaviours, and all intercepts
q_t08_058P043HardMedium—0.00.486discarded—Sketch the graph of $y = 4 - 3e^{-2x+5}$, clearly showing the horizontal asymptote, the $y$-intercept, and the $x$-intercept.
q_t08_059P047HardHard—0.00.011discarded—Solve each of the following equations for $x$. **(a)** $8^{2x+1} = 4^{3x-5}$ **(b)** $6 \cdot 25^x - 31 \cdot 5^x + 5 = 0$ **(c)** $x^{3\
q_t08_060P048MediumHard—0.00.49discarded—Solve each of the following equations for $x$: **(a)** $\log_2(5x + 4) - \log_2(x - 2) = \log_2 9$ **(b)** $\log_{125} x + \log_{25} x + \
q_t08_061P046MediumMedium—0.00.018discarded—The number of subscribers (in thousands) to an online magazine is modelled by $S(t) = 5 + 8\ln(3t + 1)$, where $t$ is the number of months a
q_t08_062P045HardMedium—0.00.488discarded—A commercial property was purchased in 2008 for $\$850\,000$. By 2019, its value had fallen to $\$610\,000$ due to exponential depreciation
q_t08_063P044MediumMedium—0.00.012discarded—Sketch the graph of $y = \log_2(4 - 3x) + 1$, clearly identifying the domain, the vertical asymptote, the end behaviours, and all intercepts
q_t08_064P043EasyMedium—0.00.492discarded—Sketch the graph of $y = 5e^{-x+2} - 15$, clearly showing the horizontal asymptote and any intercepts with the axes.
q_t08_065P047MediumHard—0.00.492discarded—Solve each of the following equations for $x$. **(a)** $243^{x-2} = 9^{3x+1}$ **(b)** $3 \cdot 4^x - 14 \cdot 2^{x+1} + 16 = 0$ **(c)** $
q_t08_066P048EasyEasy—0.00.009discarded—Solve each of the following equations for $x$: **(a)** $\log_5(3x + 10) - \log_5(x + 2) = \log_5(x + 1)$ **(b)** $\log_{64} x + \log_8 x +
q_t08_067P046MediumMedium—0.00.02discarded—The cumulative distance (in kilometres) cycled by a participant in a charity ride is modelled by $$D(n) = 20 + 11\log_4(5n + 4),$$ where $n$
q_t08_068P045EasyMedium—0.00.488discarded—A new laptop computer is purchased for $\$1200$. After $4$ years, its resale value has fallen to $\$430$. Assuming the value depreciates exp
q_t08_073P046MediumMedium—0.00.018discarded—The number of subscribers (in thousands) to an online newsletter is modelled by $N(t) = 5 + 3\ln(2t + 1)$, where $t$ is the number of months
q_t08_074P045MediumMedium—0.00.007discarded—A piece of industrial equipment was purchased in 2008 for $24000. By 2020, its value had fallen to $10500. Assuming the value depreciates ex
q_t08_079P046MediumMedium—0.00.017discarded—The number of subscribers (in thousands) to an online magazine is modelled by $N(t) = 5 + 3\ln(4t + 1)$, where $t$ is the time in months aft
q_t08_080P045HardMedium—0.00.492discarded—A commercial property was purchased at the start of 2008 for $\$420\,000$. By the start of 2021, its value had risen to $\$695\,000$. Assumi
q_t09_001P056MediumMediumMedium0.00.0human_labelledreviewedPoints $A$ and $B$ lie on the circumference of a circle with centre $O$ and radius $7$ cm. The chord $AB$ has length $9$ cm. Find the centra
q_t09_002P055EasyMediumMedium0.00.5human_labelledreviewedIn triangle ABC, AB = 6, BC = 9, and angle BAC = 50°. Find the length of AC and the area of triangle ABC.
q_t09_003P057HardHardHard0.00.0human_labelledreviewedLet $n$ be a positive integer and let $x$ be a real number with $\sin x \neq 0$. (a) By multiplying by $2\sin x$ and applying the product-t
q_t09_004P051MediumMediumMedium0.00.0human_labelledreviewedShow that $$\frac{\sin 2x}{1 - \cos 2x} = \frac{1}{\tan x}.$$
q_t09_005P049EasyMediumEasy0.00.5human_labelledreviewedFind all values of $x$ such that $2\sin x + 1 = 0$ \quad $(0 \le x < 2\pi)$.
q_t09_006P053EasyMediumEasy0.00.5human_labelledreviewedExpand and simplify $(\sin x + \cos x)^2$, expressing your answer in terms of $\sin 2x$.
q_t09_007P052EasyEasyMedium0.00.0human_labelledreviewedShow that $\cos\!\left(\theta + \dfrac{\pi}{4}\right) = \dfrac{1}{\sqrt{2}}(\cos\theta - \sin\theta)$. Hence, show that $\dfrac{\cos\!\left
q_t09_008P054MediumMediumMedium0.00.0human_labelledreviewedFor all values of $x$ such that $3\tan x + \sqrt{5} = 0$, find all possible values of $\sin x$.
q_t09_009P050HardMediumHard0.00.5human_labelledreviewedFind all values of x such that $2\sqrt{3}\sin\!\left(3x - \dfrac{\pi}{3}\right) = -3$ for $0 \le x < \dfrac{4\pi}{3}$.
q_t09_010P056HardHardHard0.00.0human_labelledreviewedPoints $P$ and $Q$ lie on the circumference of a circle with centre $O$ and radius $r$ cm. The chord $PQ$ makes an angle of $28^\circ$ with
q_t09_011P055EasyMediumMedium0.00.5human_labelledreviewedIn triangle ABC, AB = 6, BC = 9, and angle BAC = 55°. Find the length of AC and the area of triangle ABC.
q_t09_012P057EasyEasyHard0.00.0human_labelledreviewedShow that $\displaystyle\sum_{k=1}^{n} \sin(kx) = \dfrac{\sin\!\left(\dfrac{nx}{2}\right)\sin\!\left(\dfrac{(n+1)x}{2}\right)}{\sin\!\left(\
q_t09_013P051HardMediumMedium0.00.5human_labelledreviewedShow that $$\frac{1 + \sin 2x - \cos 2x}{1 + \sin 2x + \cos 2x} = \tan x.$$
q_t09_014P049EasyMediumEasy0.00.5human_labelledreviewedFind all values of $x$ such that $\sqrt{3}\tan x - 1 = 0$ \quad $(0 \le x < 2\pi)$.
q_t09_015P053MediumMediumMedium0.00.0human_labelledreviewedExpand and simplify $\sin\!\left(x + \dfrac{\pi}{3}\right)\cos\!\left(x + \dfrac{\pi}{6}\right)$, expressing your answer in the form $a\sin
q_t09_016P052MediumHard—0.00.5discarded—Show that $\sin\!\left(\theta + \dfrac{\pi}{3}\right) = \dfrac{1}{2}\sin\theta + \dfrac{\sqrt{3}}{2}\cos\theta$. Hence, show that $\dfrac{\
q_t09_017P054MediumMediumMedium0.00.0human_labelledreviewedFor all values of $\theta$ such that $5\sin\theta + 3\cos\theta = 0$, find all possible values of $\cos\theta$.
q_t09_018P050MediumMediumMedium0.00.0human_labelledreviewedFind all values of x such that $2\sin\!\left(3x - \dfrac{\pi}{6}\right) = \sqrt{2}$, where $0 \le x < \pi$.
q_t09_019P056HardHardHard0.00.0human_labelledreviewedPoints $A$ and $B$ lie on the circumference of a circle with centre $O$ and radius $r$ cm. The chord $AB$ makes an angle of $42°$ with the r
q_t09_020P055MediumMediumMedium0.00.0human_labelledreviewedIn triangle ABC, AB = 9 cm, BC = 13 cm, and angle BAC = 52°. Find the two possible values of angle ACB, and for each case find the correspon
q_t09_021P057MediumHardHard0.00.5human_labelledreviewedLet $n$ be a positive integer and let $\theta$ be a real number with $\sin\theta \neq 0$. **(a)** By multiplying both sides by $2\sin\theta
q_t09_022P051MediumMediumEasy0.00.0human_labelledreviewedShow that $\dfrac{\sin 2x}{1 - \cos 2x} = \cot x$.
q_t09_023P049EasyMediumEasy0.00.5human_labelledreviewedFind all values of $x$ such that $2\cos x + 1 = 0$ $(0 \le x < 2\pi)$.
q_t09_024P053MediumMediumMedium0.00.0human_labelledreviewedExpand and simplify $(\sin x + \cos x)^2(\sin x - \cos x)^2$.
q_t09_025P052MediumMedium—0.00.0discarded—Show that $\cos\!\left(\theta - \dfrac{\pi}{6}\right) = \dfrac{\sqrt{3}}{2}\cos\theta + \dfrac{1}{2}\sin\theta$. Hence, show that $\dfrac{\
q_t09_026P056MediumMediumMedium0.00.0human_labelledreviewedLet A and B be points on the circumference of a circle with centre O. Chord AB has length 9 cm and the radius of the circle is 7 cm. Find th
q_t09_027P055EasyMedium—0.00.5discarded—In triangle ABC, AB = 6 cm, BC = 9 cm, and angle BAC = 55°. Find the length of AC and the area of triangle ABC.
q_t09_028P057HardHardMedium0.00.0human_labelledreviewedLet $n$ be a positive integer and let $\alpha$ be a real number with $\sin\left(\frac{3\alpha}{2}\right) \neq 0$. **(a)** By multiplying bo
q_t09_029P051MediumMediumEasy0.00.0human_labelledreviewedShow that $$\frac{1 + \cos x}{\sin x} = \cot\frac{x}{2}.$$ *You may use the identities $\cos x = 2\cos^2\dfrac{x}{2} - 1$ and $\sin x = 2\s
q_t09_030P049EasyMediumMedium0.00.5human_labelledreviewedFind all values of $\theta$ such that $2\tan\theta + 2\sqrt{3} = 0$ $\quad(0 \le \theta < 2\pi)$.
q_t09_031P053EasyEasyEasy0.00.0human_labelledreviewedExpand and simplify $\cos\left(x + \dfrac{\pi}{4}\right)\cos\left(x - \dfrac{\pi}{4}\right)$, expressing your answer in terms of $\cos 2x$.
q_t09_032P052EasyMedium—0.00.5discarded—Show that $\sin\left(\theta + \frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\sin\theta + \frac{1}{2}\cos\theta$. Hence, show that $\dfrac{\sin\le
q_t09_033P054MediumHardMedium0.00.5human_labelledreviewedFind all possible values of $\sin\theta$ for all values of $\theta$ satisfying the equation $4\cos^2\theta - 3\sin\theta\cos\theta = 0$, whe
q_t09_034P050HardMediumHard0.00.5human_labelledreviewedFind all values of $x$ such that $\sqrt{2}\,\tan\!\left(2x + \dfrac{\pi}{3}\right) = \sqrt{6}$, where $0 \le x < \pi$.
q_t09_035P056HardMediumMedium0.00.5human_labelledreviewedLet A and B be points on the circumference of a circle with centre O and radius r cm. The chord AB has length 9 cm, and the perpendicular di
q_t09_036P055EasyEasyEasy0.00.0human_labelledreviewedIn triangle ABC, AB = 6 cm, BC = 9 cm, and angle BAC = 35°. Find the length of AC and the area of triangle ABC.
q_t09_037P057EasyMediumHard0.00.5human_labelledreviewedLet $n$ be a positive integer and let $\theta$ be a real number with $\sin\theta \neq 0$. Show that $$\cos\theta + \cos 3\theta + \cos 5\th
q_t09_038P051HardMediumHard0.00.5human_labelledreviewedShow that $\dfrac{\sin 2x}{1 - \cos 2x} - \dfrac{1 - \cos 2x}{\sin 2x} = 2\cot 2x$.
q_t09_039P049EasyMediumEasy0.00.5human_labelledreviewedFind all values of $\theta$ such that $2\cos\theta - \sqrt{2} = 0$ $\quad(0 \le \theta < 2\pi)$.
q_t09_040P053MediumMediumMedium0.00.0human_labelledreviewedExpand and simplify $\left(\sin x + \sqrt{3}\cos x\right)^2$.
q_t09_041P052MediumMediumMedium0.00.0human_labelledreviewedShow that $\cos\left(\theta + \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}(\cos\theta - \sin\theta)$. Hence, show that $\dfrac{\cos\left(\theta
q_t09_042P054MediumMediumMedium0.00.0human_labelledreviewedFind all possible values of $\tan\theta$ for all values of $\theta$ satisfying the equation $3\sin^2\theta - 7\sin\theta\cos\theta + 2\cos^2
q_t09_043P050MediumMediumMedium0.00.0human_labelledreviewedFind all values of $x$ such that $2\cos\!\left(4x + \dfrac{\pi}{6}\right) = -\sqrt{2}$, where $0 \le x < \dfrac{\pi}{2}$.
q_t09_044P056HardMediumMedium0.00.5human_labelledreviewedLet A and B be points on the circumference of a circle with centre O and radius r cm. The chord AB has length 9 cm and forms an angle of 28°
q_t09_045P055MediumMediumMedium0.00.0human_labelledreviewedIn triangle ABC, AB = 9 cm, BC = 13 cm, and angle BAC = 52°. Find the two possible values of angle BCA, and for each case find the correspon
q_t09_046P057MediumHardMedium0.00.5human_labelledreviewedLet $n$ be a positive integer and let $\phi$ be a real number with $\sin\!\left(\frac{\phi}{2}\right) \neq 0$. **(a)** By multiplying both
q_t09_047P051MediumMediumMedium0.00.0human_labelledreviewedShow that $$\frac{\cos 2x}{1 + \sin 2x} = \frac{1 - \tan x}{1 + \tan x}.$$ You may use the identities $\cos 2x = \cos^2 x - \sin^2 x$ and $
q_t09_048P049EasyMediumEasy0.00.5human_labelledreviewedFind all values of $\alpha$ such that $2\sin\alpha - \sqrt{3} = 0$ $\quad(0 \le \alpha < 2\pi)$.
q_t09_049P053MediumMedium—0.00.0discarded—Expand and simplify $(\sin x + \cos x)^2(\sin x - \cos x)^2$.
q_t09_050P052MediumMediumMedium0.00.0human_labelledreviewedShow that $\cos\left(\theta + \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}(\cos\theta - \sin\theta)$. Hence, show that $\dfrac{\cos\left(\theta
q_t09_051P053EasyMedium—0.00.481discarded—Expand and simplify $(\sin x + \cos x)^2$.
q_t09_052P056MediumMedium—0.00.008discarded—Let A and B be points on the circumference of a circle with centre O. Chord AB has length 9 cm, and the radius of the circle is 7 cm. Find t
q_t09_053P056MediumHard—0.00.5discarded—Points $P$ and $Q$ lie on the circumference of a circle with centre $O$ and radius $4 \text{ cm}$. The chord $PQ$ forms an angle of $42°$ wi
q_t10_001P059EasyMediumEasy0.00.5human_labelledreviewedDifferentiate the following functions: (a) $y = x^4 + e^x - 3x + 7$ (b) $y = \sqrt{x} + \frac{1}{x^2}$
q_t10_002P070MediumMediumMedium0.00.0human_labelledreviewedA conical paper cup is being filled with water at a constant rate of $3$ cm$^3$ s$^{-1}$. The cone has a fixed half-angle of $30°$, so that
q_t10_003P069MediumHardHard0.00.5human_labelledreviewedConsider the function $f(x) = x^2 e^{-0.4x}$ defined on the domain $0 \leq x \leq 10$. (a) Find the $x$-intercepts of $f(x)$ in the given d
q_t10_004P058MediumMediumMedium0.00.0human_labelledreviewedDetermine whether the function $$f(x) = \begin{cases} x^2 + 3\sin\!\left(\tfrac{\pi}{2}(x-2)\right) & x \leq 2 \\ e^{2(x-2)} + 3x - 7 & x >
q_t10_005P060EasyEasyMedium0.00.0human_labelledreviewedDifferentiate $f(x) = \ln(x^3 + \sin x)$ and find the exact value of $f'\!\left(\dfrac{\pi}{2}\right)$.
q_t10_006P062EasyEasyEasy0.00.0human_labelledreviewedDifferentiate the function $f(x) = \dfrac{e^x}{x^2}$ using the quotient rule. Hence, find the gradient of the curve at $x = 1$.
q_t10_007P068EasyEasyMedium0.00.0human_labelledreviewedFind the $x$-coordinates of the critical points and the inflection points of the function $f(x) = \dfrac{x^2 - 4}{e^x}$. For the inflection
q_t10_008P065HardHardHard0.00.0human_labelledreviewedLet $f(x) = x^3 - 3x^2 + 2$. Find all equations of tangent lines to the curve $y = f(x)$ that pass through the point $(3, -4)$.
q_t10_009P066HardMediumMedium0.00.5human_labelledreviewedFind the values of the real constants $a$ and $b$ such that the curve $f(x) = a\ln x + bx^2$ has a tangent line at the point where $x = e$ t
q_t10_010P061MediumMediumEasy0.00.0human_labelledreviewedFind the derivative of $f(x) = x^3 \ln x$ and hence find the exact value of $f'(e)$.
q_t10_011P067MediumMediumMedium0.00.0human_labelledreviewedConsider the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$. Find the coordinates of the points on this ellipse at which the tangent has a g
q_t10_012P071HardHard—0.00.0discarded—A manufacturer is designing a closed cylindrical tin can that must hold a fixed volume of 500π cm³. The cost of material for the circular to
q_t10_013P064MediumMediumMedium0.00.0human_labelledreviewedFind the equation of the tangent to the curve $y = x^2 e^{x-2}$ at $x = 2$.
q_t10_014P063EasyMediumEasy0.00.5human_labelledreviewedDifferentiate the function $y = e^{x^2 + 3x}$ and find the exact value of $\dfrac{dy}{dx}$ at $x = 0$.
q_t10_015P059MediumMediumEasy0.00.0human_labelledreviewedFind the derivative of the function $$g(x) = \cos x - \frac{1}{\sqrt{x}} + 2e^x - \ln x,$$ and hence find the exact value of $g'(1)$.
q_t10_016P070EasyMediumEasy0.00.5human_labelledreviewedA spherical snowball is melting so that its volume decreases at a constant rate of $12$ cm$^3$ s$^{-1}$. Find the rate at which the radius o
q_t10_017P069MediumHardMedium0.00.5human_labelledreviewedConsider the function $f(x) = x^2 e^{-0.4x}$ defined on the domain $0 \leq x \leq 10$. (a) Find the $x$-intercepts of $f(x)$ in the given d
q_t10_018P058EasyEasyEasy0.00.0human_labelledreviewedDetermine whether the function $$f(x) = \begin{cases} 3x^2 - 5x + 1 & x \leq 2 \\ \ln(x - 1) + 3x - 5 & x > 2 \end{cases}$$ is differentiabl
q_t10_019P060MediumMediumMedium0.00.0human_labelledreviewedDifferentiate $f(x) = \sin(e^x + x^2)$ and find the exact value of $f'(0)$.
q_t10_020P062MediumMediumMedium0.00.0human_labelledreviewedDifferentiate the function $f(x) = \dfrac{x^2}{\ln x}$ using the quotient rule. Hence, find the gradient of the curve $y = \dfrac{x^2}{\ln x
q_t10_021P068MediumHardMedium0.00.5human_labelledreviewedConsider the function $f(x) = \dfrac{x^2 + 2x}{e^x}$, defined for all $x \in \mathbb{R}$. (a) Find $f'(x)$ and hence determine the $x$-coor
q_t10_022P065HardHardHard0.00.0human_labelledreviewedLet $f(x) = \ln x - x$ for $x > 0$. Find all equations of tangent lines to the curve $y = f(x)$ that pass through the point $(0, -3)$.
q_t10_023P066MediumMediumMedium0.00.0human_labelledreviewedThe function $f(x) = ax^3 + bx$ passes through the point $(2, 10)$ and has a tangent with gradient $9$ at $x = 1$. Find the values of $a$ an
q_t10_024P061MediumMediumMedium0.00.0human_labelledreviewedFind the derivative of $g(x) = x^2 \sin x$ and hence find the exact value of $g'\!\left(\dfrac{\pi}{2}\right)$.
q_t10_025P067EasyEasyMedium0.00.0human_labelledreviewedConsider the circle $x^2 + y^2 = 25$. Find the coordinates of the points on this circle at which the tangent has a gradient of $\dfrac{3}{4}
q_t10_026P071EasyEasyMedium0.00.0human_labelledreviewedA gardener wants to build a rectangular vegetable patch with a total area of $36$ m². Three sides of the patch will be enclosed by wooden fe
q_t10_027P064MediumMediumMedium0.00.0human_labelledreviewedFind the equation of the tangent to the curve $y = x^2 \sin x$ at $x = \dfrac{\pi}{2}$.
q_t10_028P063HardMediumHard0.00.5human_labelledreviewedDifferentiate the function $h(x) = e^{\sin(x)\ln(x)}$ and find the exact value of $h'\!\left(\dfrac{\pi}{2}\right)$.
q_t10_029P059MediumMediumMedium0.00.0human_labelledreviewedFind the derivative of the function $$h(x) = \sin x + \frac{1}{\sqrt[3]{x}} - e^x + 5x^3,$$ and hence find the exact value of $h'\!\left(\df
q_t10_030P070HardHardHard0.00.0human_labelledreviewedA trough has a cross-section in the shape of an isosceles trapezoid. The bottom edge of the trapezoid has length $1$ m, the two slanted side
q_t10_031P065MediumMediumHard0.00.005human_labelledreviewedFind all equations of tangent lines to the curve $y = x^2 e^x$ that pass through the point $(1, 0)$.
q_t10_032P058MediumHardMedium0.00.492human_labelledreviewedDetermine whether the function $$p(x) = \begin{cases} \sin(\pi x) + 2x^2 - 8 & x \leq 2 \\ x\ln(x-1) - 3x + 6 & x > 2 \end{cases}$$ is diffe
q_t10_033P070EasyEasyEasy0.00.01human_labelledreviewedA cube is being inflated so that its volume increases at a constant rate of $12$ cm$^3$ s$^{-1}$. Find the rate at which the surface area of
q_t10_034P061HardMediumMedium0.00.5human_labelledreviewedDifferentiate the function $h(x) = x^2 \sin x \cdot e^x$ by first writing it as a product of two sub-functions $u(x) = x^2 \sin x$ and $v(x)
q_t10_035P068EasyEasyMedium0.00.006human_labelledreviewedFind the coordinates of the critical points and the inflection points of the function $f(x) = \ln(x^2 - 4x + 8)$, defined for all $x \in \ma
q_t10_036P063HardMediumHard0.00.497human_labelledreviewedLet $f(x) = e^{\sin^2(x) - \cos(x)}$. (a) Find $f'(x)$, showing all working including a substitution $u = g(x)$. (b) Find the exact value o
q_t10_037P064MediumMediumMedium0.00.007human_labelledreviewedFind the equation of the tangent to the curve $y = e^x \cos x$ at $x = \dfrac{\pi}{2}$.
q_t10_038P071EasyEasyMedium0.00.007human_labelledreviewedA cylindrical tin can with an open top must have a volume of $250\pi$ cm³. The can has a circular base and a curved side wall. The material
q_t10_039P059MediumMediumMedium0.00.012human_labelledreviewedFind the derivative of the function $$f(x) = 3x^4 - \ln x + \frac{1}{\sqrt[4]{x}} + \cos x,$$ and hence find the exact value of $f'\!\left(\
q_t10_040P062HardMediumHard0.00.496human_labelledreviewedDifferentiate the function $f(x) = \dfrac{x^2 e^x}{\sin x}$ using the quotient rule. Hence, find the gradient of the curve $y = f(x)$ at $x
q_t10_041P060HardMediumHard0.00.498human_labelledreviewedLet $f(x) = \ln\!\left(e^{2x} + \sin^2 x\right)$. Find $f'(x)$ and hence find the exact value of $f'(0)$.
q_t10_042P067MediumHardMedium0.00.494human_labelledreviewedConsider the hyperbola $\dfrac{x^2}{25} - \dfrac{y^2}{9} = 1$. Find the coordinates of the points on this hyperbola at which the tangent has
q_t10_043P069MediumHardEasy0.00.495needs_humanreviewedConsider the function $f(x) = x^2 e^{-0.4x}$ defined on the domain $0 \leq x \leq 10$. (a) Write down the $x$-intercept(s) of $f$. (b) Fin
q_t10_044P066MediumMediumEasy0.00.012human_labelledreviewedFind the values of the real constants $a$ and $b$ such that the function $f(x) = ae^x + b\sin x$ satisfies $f(0) = 3$ and has a tangent with
q_t10_045P065EasyEasyMedium0.00.011human_labelledreviewedFind the equation(s) of the tangent line(s) to the curve $f(x) = x^2 + 2x$ that pass through the point $(1, -2)$.
q_t10_046P058EasyMediumEasy0.00.493human_labelledreviewedDetermine whether the function $$f(x) = \begin{cases} 2x^2 - 5x + 3 & x \leq 3 \\ e^{x-3} + x - 4 & x > 3 \end{cases}$$ is differentiable at
q_t10_047P070MediumMediumMedium0.00.011human_labelledreviewedA particle of sand is dropped into a still pond, creating a ripple that spreads outward as a circle. The area of the disturbed surface incre
q_t10_048P061MediumMediumMedium0.00.014human_labelledreviewedFind the derivative of $f(x) = \sin x \cdot \ln x$ and hence find the exact value of $f'\!\left(\dfrac{\pi}{2}\right)$.
q_t11_001P088MediumMediumMedium0.00.012human_labelledreviewedFind the area of the finite region enclosed by the curves $y = x^3 - 3x^2$ and $y = x^2 - 3x$.
q_t11_002P073EasyEasyEasy0.00.022human_labelledreviewedEvaluate $\displaystyle\int e^{4x+3}\,dx$.
q_t11_003P073HardMediumMedium0.00.497human_labelledreviewedEvaluate $\displaystyle\int \frac{3}{(5-2x)^4}\,dx$.
q_t11_004P089HardMediumHard0.00.489human_labelledreviewedThe region bounded by the curve $y = (x+1)\sqrt{3-x}$, the $x$-axis, and the lines $x = -1$ and $x = 3$ is rotated about the $x$-axis. Find
q_t11_005P085MediumMediumMedium0.00.012human_labelledreviewedGiven that $\displaystyle\int_1^5 f(x)\,dx = 10$ and $\displaystyle\int_1^5 g(x)\,dx = 3$, find the value of $$I = \int_5^1 \bigl[2f(x) - 3g
q_t11_006P072MediumMediumMedium0.00.018human_labelledreviewedEvaluate $\displaystyle\int \frac{3x^4 - 2x^2 + 5\sqrt{x}}{x^2}\,dx$.
q_t11_007P084MediumMediumMedium0.00.006human_labelledreviewedLet $f(x)$ be a differentiable function with a differentiable derivative. Some values of $f$ and $f'$ are given in the table below: | $x$ |
q_t11_008P087MediumMediumMedium0.00.003human_labelledreviewedFind the total finite area enclosed between the curve $y = x^3 - x^2 - 6x$ and the $x$-axis.
q_t11_009P075MediumMediumEasy0.00.007human_labelledreviewedEvaluate $\displaystyle\int x^2 e^{x^3 + 1}\,dx$. Use the substitution $u = x^3 + 1$.
q_t11_010P086EasyMediumEasy0.00.497human_labelledreviewedThe function $h$ is even and $\displaystyle\int_0^5 h(x)\,dx = 9$. Find $\displaystyle\int_{-5}^{5} \bigl(h(x) + x^3\bigr)\,dx$.
q_t11_011P082EasyMediumMedium0.00.497human_labelledreviewedThe area under the curve $y = 3x + 2$ from $x = 0$ to $x = 4$ is approximated using a right Riemann sum with $6$ equal subintervals. Calcula
q_t11_012P089HardMediumHard0.00.496human_labelledreviewedThe region bounded by the curve $y = \sin x + \cos x$, the $x$-axis, and the lines $x = 0$ and $x = \dfrac{\pi}{2}$ is rotated about the $x$
q_t11_013P077MediumMediumHard0.00.006human_labelledreviewedEvaluate $\displaystyle\int \frac{x^2}{\sqrt{4 - x^2}}\,dx$.
q_t11_014P072EasyMediumEasy0.00.499human_labelledreviewedEvaluate $\displaystyle\int (2x - 3)^2\,dx$.
q_t11_015P075MediumMediumEasy0.00.02human_labelledreviewedEvaluate $\displaystyle\int \frac{\sin(\ln x)}{x}\,dx$, using the substitution $u = \ln x$.
q_t11_016P079HardMediumHard0.00.492human_labelledreviewedEvaluate $\displaystyle\int e^{2x}\sin(3x)\,dx$.
q_t11_017P085EasyMediumEasy0.00.492human_labelledreviewedGiven that $\displaystyle\int_0^4 h(x)\,dx = 6$, find the value of $$J = \int_0^2 h(2x)\,dx - \int_4^0 h(x)\,dx.$$
q_t11_018P082MediumMediumMedium0.00.037human_labelledreviewedEvaluate $\displaystyle\lim_{n\to\infty}\sum_{r=1}^{n}\frac{3}{n}\left(1+\frac{3r}{n}\right)^{4}$ by identifying it as a definite integral.
q_t11_019P075MediumMediumMedium0.00.026human_labelledreviewedEvaluate $\displaystyle\int \tan^5(x)\sec^2(x)\,dx$, using the substitution $u = \tan(x)$.
q_t11_020P086EasyMediumEasy0.00.481human_labelledreviewedThe function $p$ is odd and $\displaystyle\int_0^4 p(x)\,dx = 6$. Find $\displaystyle\int_{-4}^{4} \bigl(2p(x) - 5\bigr)\,dx$.
q_t11_021P082EasyMediumEasy0.00.48human_labelledreviewedThe area under the curve $y = e^x$ from $x = 0$ to $x = 1$ is approximated using a left Riemann sum with $n = 4$ equal subintervals. Calcula
q_t11_022P089HardHardHard0.00.016human_labelledreviewedThe region bounded by the curve $y = \dfrac{x^2 + 2}{\sqrt{x}}$, the $x$-axis, and the lines $x = 1$ and $x = 4$ is rotated about the $x$-ax
q_t11_023P077MediumHardHard0.00.479human_labelledreviewedEvaluate $\displaystyle\int \frac{x^2}{\sqrt{x^2 + 9}}\,dx$ using an appropriate trigonometric substitution.
q_t11_024P074EasyMediumEasy0.00.489human_labelledreviewedEvaluate $\displaystyle\int \frac{3x^2}{x^3 + 5}\,dx$.
q_t11_025P084MediumMediumMedium0.00.049human_labelledreviewedLet $h(x)$ be a differentiable function with a differentiable derivative. Some values of $h$ and $h'$ are given in the table below: | $x$ |
q_t11_026P091MediumMediumHard0.00.015human_labelledreviewedThe region bounded by the curve $y = \sqrt{x}$, the line $x = 9$, and the $x$-axis is rotated about the line $y = 4$. Find the volume of the
q_t11_027P075MediumMediumEasy0.00.053human_labelledreviewedEvaluate $\displaystyle\int \frac{\cos(\sqrt{x})}{\sqrt{x}}\,dx$, using the substitution $u = \sqrt{x}$.
q_t11_028P081EasyMediumMedium0.00.456human_labelledreviewedExpress $\dfrac{5x+1}{(x+1)(x-2)}$ in partial fractions. Hence evaluate $\displaystyle\int \frac{5x+1}{(x+1)(x-2)}\,dx$.
q_t11_029P079EasyEasyMedium0.00.043human_labelledreviewedEvaluate $\displaystyle\int e^{-t}\cos(2t)\,dt$ by letting $I = \int e^{-t}\cos(2t)\,dt$ and applying integration by parts twice.
q_t11_030P077EasyEasyEasy0.00.053human_labelledreviewedEvaluate $\displaystyle\int \frac{1}{\sqrt{x^2 + 16}}\,dx$ using the trigonometric substitution $x = 4\tan\theta$.
q_t11_031P090HardMediumHard0.00.484human_labelledreviewedThe region $R$ is bounded by the curve $y = \sqrt{x - 2}$, the line $y = 3$, and the $y$-axis. Find the volume of the solid formed when $R$
q_t11_032P073HardMediumHard0.00.487human_labelledreviewedEvaluate $\displaystyle\int \frac{\sin(\ln 3 - 2x)}{\sqrt[3]{\cos(\ln 3 - 2x)}}\,dx$.
q_t11_033P093MediumMediumMedium0.00.034human_labelledreviewedA company manufactures custom phone cases. Its marginal revenue from selling $x$ units per day is modelled by $$\frac{dR}{dx} = 12 - 0.6x +
q_t11_034P088MediumHardMedium0.00.481human_labelledreviewedFind the area of the finite region enclosed by the curves $y = x^3 - x$ and $y = 2x^2 - 2$. (a) Find the $x$-coordinates of all intersectio
q_t11_035P072HardMediumMedium0.00.494human_labelledreviewedEvaluate $\displaystyle\int \frac{(x^2 + 3)^2 - 9x^2}{x^{3/2}}\,dx$.
q_t11_036P076MediumMediumMedium0.00.0human_labelledreviewedEvaluate $\displaystyle\int \sin^3(x)\cos^4(x)\,dx$.
q_t11_037P080EasyEasyMedium0.00.03human_labelledreviewedLet $I_n = \displaystyle\int_0^1 x^n e^{2x}\,dx$ for $n \in \mathbb{N}$. **(a)** Show that $I_n = \dfrac{e^2}{2} - \dfrac{n}{2}\,I_{n-1}$.
q_t11_038P092MediumMediumMedium0.00.01human_labelledreviewedThe region enclosed by $y = \sqrt{x}$, $y = 0$, $x = 1$, and $x = 4$ is rotated about the $x$-axis. Separately, the region enclosed by $y =
q_t11_039P086EasyMediumEasy0.00.483human_labelledreviewedThe function $q$ is even and $\displaystyle\int_0^3 q(x)\,dx = 5$. Find $\displaystyle\int_{-3}^{3} \bigl(4q(x) - x^5\bigr)\,dx$.
q_t11_040P087MediumHardHard0.00.48human_labelledreviewedFind the total finite area enclosed between the curve $y = x^4 - 8x^2 + 12$ and the $x$-axis, given that the curve crosses the $x$-axis at $
q_t11_041P089EasyEasyMedium0.00.023human_labelledreviewedFind the volume of the solid formed when the region bounded by the curve $y = 3x - x^2$, the $x$-axis, and the lines $x = 1$ and $x = 3$ is
q_t11_042P078MediumMediumMedium0.00.025human_labelledreviewedEvaluate $\displaystyle\int x^2 \ln(3x)\,dx$.
q_t11_043P082MediumHardMedium0.00.499human_labelledreviewedEvaluate $\displaystyle\lim_{n\to\infty}\sum_{r=1}^{n}\frac{2}{n}\sin\!\left(\frac{\pi r}{n}\right)$ by identifying it as a definite integra
q_t11_044P085MediumMediumMedium0.00.036human_labelledreviewedGiven that $\displaystyle\int_0^4 g(x)\,dx = 7$ and $\displaystyle\int_0^2 g(x)\,dx = 3$, find $\displaystyle\int_1^3 g(2x-2)\,dx$.
q_t11_045P083HardHardHard0.00.067human_labelledreviewedConsider the function $$G(t) = 5 + \int_{\ln 2}^{e^t - 1} \frac{\sqrt{1 + \ln(1+u)}}{(1+u)} \, du, \quad t \ge 0.$$ (a) Find $G(0)$. (b) R
q_t11_046P074MediumMediumMedium0.00.068human_labelledreviewedEvaluate $\displaystyle\int \frac{\ln x}{x(1 + \ln x)^2}\,dx$.
q_t11_047P084MediumMediumMedium0.00.048human_labelledreviewedDefine the function $F$ by $$F(x) = \int_1^{x^3} \frac{e^t}{t^2 + 1}\,dt.$$ (a) Write down $F'(x)$, the derivative of $F$ with respect to $
q_t11_048P091EasyEasyMedium0.00.025human_labelledreviewedThe region bounded by the curve $y = x^2 + 1$, the line $x = 2$, and the $y$-axis is rotated about the line $y = 5$. Find the volume of the
q_t11_049P075EasyMediumEasy0.00.5human_labelledreviewedEvaluate $\displaystyle\int 3x^2 \cos(x^3)\,dx$, using the substitution $u = x^3$.
q_t11_050P081MediumHardHard0.00.482human_labelledreviewedExpress $\dfrac{4x^2 - x + 3}{x(x^2 + 3)}$ in partial fractions. Hence evaluate $\displaystyle\int \frac{4x^2 - x + 3}{x(x^2 + 3)}\,dx$.
q_t11_051P079HardHardHard0.00.038human_labelledreviewedEvaluate $\displaystyle\int e^{3x}\cos(2x)\,dx$ by letting $I = \displaystyle\int e^{3x}\cos(2x)\,dx$ and applying integration by parts twic
q_t11_052P077MediumHardMedium0.00.491human_labelledreviewedEvaluate $\displaystyle\int \frac{\sqrt{x^2 - 25}}{x}\,dx$ for $x > 5$, using the trigonometric substitution $x = 5\sec\theta$.
q_t11_053P090HardHardHard0.00.028human_labelledreviewedThe region enclosed by the curve $y = \ln(x-1)$, the line $x = 3$, and the $x$-axis is rotated $2\pi$ radians about the $y$-axis. Find the e
q_t11_054P082MediumMedium—0.00.0discarded—Evaluate $\displaystyle\lim_{n\to\infty}\sum_{r=1}^{n}\frac{4}{n}\cdot\frac{1}{\left(1+\frac{4r}{n}\right)^{2}}$ by identifying it as a defi
q_t11_055P072MediumMedium—0.00.0needs_human—Find $\displaystyle\int \sqrt{x}\left(x - \frac{1}{\sqrt{x}}\right)^{\!2} dx$.
q_t11_056P072MediumMedium—0.00.0needs_human—Find $\displaystyle\int \frac{x^3 + 4x - 3}{\sqrt{x}}\,dx$.
q_t11_057P072MediumMedium—0.00.0needs_human—Find $\displaystyle\int \frac{(x^2-3)^2}{\sqrt[3]{x^2}}\,dx$.
q_t11_058P072MediumMedium—0.00.0needs_human—Find $\displaystyle\int \frac{x^2 - 3\sqrt[4]{x^3} + 1}{\sqrt[4]{x}}\,dx$.
q_t11_059P072EasyMedium—0.00.5needs_human—Find $\displaystyle\int \frac{3x^4 - 6x^2 + 2}{x^2}\,dx$.
q_t11_060P072HardMedium—0.00.5needs_human—Find $\displaystyle\int \frac{\left(\sqrt{x}+x\right)^3}{x\sqrt{x}}\,dx$.
q_t11_061P072MediumMedium—0.00.0needs_human—Find $\displaystyle\int \frac{(x^2 + 1)(\sqrt{x^3} - \sqrt{x})}{\sqrt{x}}\,dx$.
q_t11_062P072MediumMedium—0.00.0needs_human—Find $\displaystyle\int (3\sqrt{t}-1)(t+\sqrt{t})\,dt$.
q_t11_063P072MediumMedium—0.00.0needs_human—Find $\displaystyle\int \bigl(2\sqrt[3]{x} - x\bigr)\!\bigl(x + x^{-1/3}\bigr)\,dx$.
q_t11_064P072HardMedium—0.00.5needs_human—Find $\displaystyle\int \frac{x^4 - 1}{\sqrt{x}\,(x+1)}\,dx$.
q_t11_065P072HardMedium—0.00.5needs_human—Find $\displaystyle\int \frac{\left(\sqrt{x}+x^{-1/3}\right)^2}{\sqrt[6]{x}}\,dx$.
q_t11_066P072HardMedium—0.00.5needs_human—Find $\displaystyle\int \frac{x^3 + 2x^2 - 3\sqrt{x}}{x^{2/3}}\,dx$.
q_t11_067P072MediumMedium—0.00.0needs_human—Find $\displaystyle\int \frac{3x^2 + 5\sqrt{x} - 2}{x\sqrt[4]{x}}\,dx$.
q_t11_068P072HardMedium—0.00.5needs_human—Find $\displaystyle\int \frac{\left(\sqrt[3]{x^2} - x^{-1}\right)^2}{\sqrt[4]{x}}\,dx$.
q_t11_069P072HardMedium—0.00.5needs_human—Find $\displaystyle\int x^{-3/4}\!\left(x^{5/4}+x^{-1/4}\right)^{\!3}\,dx$.
q_t12_001P095EasyMedium—1.250.507discarded—The acceleration of a cyclist moving along a straight path is given by $$a(t) = \begin{cases} 3 & 0 \le t < 2 \\ 7 - 2t & 2 \le t < 6 \\ 2\c
q_t12_002P094MediumHard—0.7220.344discarded—The velocity of a particle $P$ (in m/s) at time $t$ seconds is given by $$v(t) = \begin{cases} 3t^2 & 0 \le t < 2 \\ 12 & 2 \le t < 6 \\ 12
q_t12_003P096MediumHard—0.3930.312discarded—The following graph shows the displacement (in metres) of a remote-controlled car C moving along a straight track, where t is measured in se
q_t12_005P094MediumHard—0.7220.366discarded—The velocity of a particle $R$ (in m/s) at time $t$ seconds is given by $$v(t) = \begin{cases} 6t - t^2 & 0 \le t < 3 \\ 9 & 3 \le t < 7 \\
q_t12_006P096EasyMedium—1.250.505discarded—The following graph shows the displacement (in metres) of a remote-controlled car moving along a straight track, where t is time in seconds.
q_t12_007P095EasyEasy—0.9430.281discarded—The acceleration of a cyclist moving along a straight path is given by $$a(t) = \begin{cases} 3 & 0 \le t < 2 \\ 7 - 2t & 2 \le t < 5 \\ -\s
q_t12_008P094MediumHard—0.7220.357discarded—The velocity of a particle $M$ (in m/s) at time $t$ seconds is given by $$v(t) = \begin{cases} 5t - t^2 & 0 \le t < 4 \\ 4 & 4 \le t < 9 \\
q_t12_009P096MediumHard—0.3670.311discarded—The following graph shows the displacement (in metres) of a remote-controlled car moving along a straight track, where t is time in seconds.
q_t12_010P095MediumHard—0.5870.356discarded—The acceleration of a particle moving along a straight track is given by $$a(t) = \begin{cases} 3t & 0 \le t < 2 \\ 10 - 2t & 2 \le t < 6 \\
q_t12_011P094EasyEasy—0.0110.042discarded—The velocity of a particle $N$ (in m/s) at time $t$ seconds is given by $$v(t) = \begin{cases} 4t & 0 \le t < 3 \\ 12 & 3 \le t < 8 \\ 12 -
q_t12_012P096HardHard—0.3670.156discarded—The following graph shows the displacement (in metres) of a remote-controlled car moving along a straight track, where t is time in seconds.
q_t12_013P095HardHard—0.3670.115discarded—The acceleration of a particle moving along a straight line is given by $$a(t) = \begin{cases} 2t - 1 & 0 \le t < 4 \\ 7 - \tfrac{3}{2}(t-4)
q_t12_014P094MediumHard—0.6990.367discarded—The velocity of a particle P (in m/s) is given by $$v(t) = \begin{cases} 3t^2 & 0 \le t < 2 \\ 12 & 2 \le t < 6 \\ 12 - 2(t-6)^2 & 6 \le t \
q_t12_015P096HardHard—0.3670.157discarded—The following graph shows the displacement (in metres) of a remote-controlled car moving along a straight track, where t is measured in seco
q_t12_016P094MediumHardMedium0.00.486human_labelledreviewedThe velocity of a particle P (in m/s) is given by $$v(t) = \begin{cases} 3t & 0 \le t < 3 \\ 9 & 3 \le t < 7 \\ 9 - (t-7)^2 & 7 \le t \le 10
q_t12_017P095MediumHardHard0.00.482human_labelledreviewedThe acceleration of a cyclist moving along a straight track is given by $$a(t) = \begin{cases} 3t & 0 \le t < 2 \\ 10 - 2t & 2 \le t < 7 \\
q_t12_018P094EasyEasyEasy0.00.01human_labelledreviewedThe velocity of a particle $R$ (in $\text{m/s}$) is given by $$v(t) = \begin{cases} 4t & 0 \le t < 3 \\ 12 & 3 \le t < 6 \\ 12 - 2(t-6)^2 &
q_t12_019P096HardHard—0.00.017discarded—The displacement (in metres) of a particle P moving along a straight line is given by $$s(t) = \begin{cases} \dfrac{1}{3}t^3 & 0 \le t < 3 \
q_t12_020P095MediumHardHard0.00.482human_labelledreviewedThe acceleration of a rocket sled moving along a straight test track is given by $$a(t) = \begin{cases} 2t - 1 & 0 \le t < 3 \\ 7 - \dfrac{3
q_t12_021P094EasyEasyMedium0.00.023human_labelledreviewedThe velocity of a boat $B$ (in $\text{m/s}$) moving along a straight canal is given by $$v(t) = \begin{cases} 6 - 2t & 0 \le t < 2 \\ 2 & 2
q_t12_022P096EasyEasyEasy0.00.016human_labelledreviewedThe displacement (in metres) of a remote-controlled car moving along a straight track is given by $$s(t) = \begin{cases} 2t^2 & 0 \le t < 3
q_t12_023P095EasyMediumMedium0.00.489human_labelledreviewedThe acceleration of a skateboarder moving along a straight ramp is given by $$a(t) = \begin{cases} 4 - t & 0 \le t < 4 \\ -\dfrac{1}{2}(t -
q_t12_024P094MediumHardMedium0.00.491human_labelledreviewedThe velocity of a drone $D$ (in $\text{m/s}$) flying along a straight horizontal path is given by $$v(t) = \begin{cases} 2\sin\!\left(\dfrac
q_t12_025P096HardHardHard0.00.023human_labelledreviewedThe displacement (in metres) of a submarine moving along a straight course is given by $$s(t) = \begin{cases} \dfrac{1}{4}t^4 - 3t^2 & 0 \l
q_t12_026P095HardHardHard0.00.021human_labelledreviewedThe acceleration of a submarine moving along a straight course is given by $$a(t) = \begin{cases} t^2 - 4 & 0 \le t < 3 \\ \dfrac{3}{2}(t -
q_t12_027P094EasyEasyEasy0.00.019human_labelledreviewedThe velocity of a cyclist $C$ (in $\text{m/s}$) travelling along a straight road is given by $$v(t) = \begin{cases} 5t & 0 \le t < 4 \\ 20 &
q_t12_028P096EasyEasy—0.00.02discarded—The displacement (in metres) of a cyclist moving along a straight road is given by $$s(t) = \begin{cases} \dfrac{1}{4}t^2 & 0 \le t < 6 \\[
q_t12_029P095HardHard—0.00.016discarded—The acceleration of a spacecraft moving along a straight launch corridor is given by $$a(t) = \begin{cases} t^2 - 2t & 0 \le t < 3 \\ \dfrac
q_t12_030P094EasyEasy—0.00.019discarded—The velocity of a train $T$ (in $\text{m/s}$) moving along a straight track is given by $$v(t) = \begin{cases} 3t^2 & 0 \le t < 2 \\ 12 & 2
q_t12_031P096MediumHard—0.00.491discarded—The displacement (in metres) of a kayak moving along a straight river channel is given by $$s(t) = \begin{cases} \dfrac{1}{2}t^2 + t & 0 \l
q_t12_032P095MediumHard—0.00.488discarded—The acceleration of a surfer moving along a straight wave channel is given by $$a(t) = \begin{cases} 3 - \dfrac{3}{2}t & 0 \le t < 2 \\ t -
q_t12_033P094MediumHard—0.00.487discarded—The velocity of a rocket sled $R$ (in $\text{m/s}$) moving along a straight test track is given by $$v(t) = \begin{cases} 2t^2 & 0 \le t < 3
q_t12_034P096MediumHard—0.00.487discarded—The displacement (in metres) of a hot-air balloon rising and drifting along a straight vertical path is given by $$s(t) = \begin{cases} 3t^
q_t12_035P095HardHard—0.00.012discarded—The acceleration of a particle moving along a straight line is given by $$a(t) = \begin{cases} t + 2 & 0 \le t < 3 \\ 8 - \frac{3}{2}(t-3) &
q_t12_036P094MediumHard—0.00.488discarded—The velocity of a cable car $C$ (in $\text{m/s}$) travelling along a straight mountain track is given by $$v(t) = \begin{cases} \dfrac{t^2}{
q_t12_037P096MediumHard—0.00.483discarded—The following graph shows the displacement (in metres) of a remote-controlled car P moving along a track, where $t$ is time in seconds. $$s(
q_t12_038P095MediumHard—0.00.485discarded—The acceleration of a rowing boat moving along a straight canal is given by $$a(t) = \begin{cases} 3 - \dfrac{t}{2} & 0 \le t < 6 \\ t - 9 &
q_t12_039P094EasyEasy—0.00.024discarded—The velocity of a skateboarder $S$ (in $\text{m/s}$) along a straight path is given by $$v(t) = \begin{cases} \sqrt{4t} & 0 \le t < 4 \\ 4 &
q_t12_040P096MediumHard—0.00.489discarded—The displacement (in metres) of a marble rolling along a curved track is given by $$s(t) = \begin{cases} \dfrac{1}{3}t^3 - 2t & 0 \le t < 3
q_t12_041P095MediumHard—0.00.484discarded—The acceleration of a cable car moving along a straight mountain track is given by $$a(t) = \begin{cases} 2\cos\!\left(\dfrac{\pi t}{6}\righ
q_t13_001P097MediumMediumEasy0.00.023human_labelledreviewedFind $w$ as an explicit function of $t$, given that $w = 2 + \ln 3$ when $t = 3$: $$\frac{dw}{dt} = \frac{1}{t} + e^{2t} - 4t.$$
q_t13_002P103EasyEasyMedium0.00.021human_labelledreviewedFind the first four non-zero terms of the power series solution to the differential equation $$\frac{du}{dt} = 1 - u,$$ given that $u = 3$ w
q_t13_003P098EasyEasyEasy0.00.014human_labelledreviewedSolve the differential equation below, giving the answer in the form $y = f(t)$. It is given that $y = 3$ when $t = 0$: $$\frac{dy}{dt} = \f
q_t13_004P099HardHardHard0.00.012human_labelledreviewedSolve the differential equation using the substitution $v = \dfrac{y}{x}$: $$\frac{dy}{dx} = \frac{x^3 + 3xy^2}{x^2 y + y^3}.$$
q_t13_005P102HardMediumMedium0.00.496human_labelledreviewedSolve the following differential equation: $$\frac{d^2u}{dt^2} = 3e^{2t} - 4t^3 + 6t$$ given that $u = 2$ and $\dfrac{du}{dt} = -\dfrac{1}{2
q_t13_006P100MediumHardMedium0.00.489human_labelledreviewedFind an integrating factor for the differential equation below and hence solve it, given that $y = 2$ when $x = 1$: $$\frac{dy}{dx} + \frac{
q_t13_007P101HardMediumMedium0.00.494human_labelledreviewedThe concentration $C$ (in mol/L) of a chemical in a reactor is modelled by the differential equation $$\frac{dC}{dt} = 0.05C\ln\!\left(\frac
q_t13_008P098HardMediumMedium0.00.474human_labelledreviewedSolve the differential equation below, giving the answer in the form $y = f(x)$. It is given that $y = 1$ when $x = 1$: $$\frac{dy}{dx} = \f
q_t13_009P103MediumMediumMedium0.00.023human_labelledreviewedFind the power series solution to the differential equation $$\frac{dy}{dx} = x^2 - 2y,$$ given that $y = 3$ when $x = 0$, expressing your a
q_t13_010P102EasyMediumEasy0.00.483human_labelledreviewedSolve the following differential equation: $$\frac{d^2y}{dx^2} = 4x + e^{2x}$$ given that $y = 3$ and $\dfrac{dy}{dx} = 2$ when $x = 0$.
q_t13_011P097HardHardMedium0.00.027human_labelledreviewedFind $y$ as an explicit function of $x$, given that $y = \dfrac{\pi}{4} - \ln 2$ when $x = 1$: $$\frac{dy}{dx} = \frac{1}{x^2 + 1} - \frac{2
q_t13_012P099MediumHardMedium0.00.473human_labelledreviewedSolve the differential equation using the substitution $v = \dfrac{y}{x}$: $$\frac{dy}{dx} = \frac{2xy + y^2}{x^2}.$$
q_t13_013P100EasyMediumEasy0.00.483human_labelledreviewedFind an integrating factor for the differential equation below and hence solve it, given that $y = 3$ when $x = 0$: $$\frac{dy}{dx} + 2y = 4
q_t13_014P101EasyEasyMedium0.00.027human_labelledreviewedThe height $h$ (in metres) of water in a draining tank is modelled by the differential equation $$\frac{dh}{dt} = -0.3\sqrt{h} + 0.05t,$$ wh
q_t13_015P098MediumMediumMedium0.00.013human_labelledreviewedSolve the differential equation below, giving the answer in the form $y = f(x)$. It is given that $y = 2$ when $x = 0$: $$\frac{dy}{dx} = \f
q_t13_016P103EasyMediumEasy0.00.483human_labelledreviewedFind the power series solution to the differential equation $$\frac{dy}{dx} = y + 2x,$$ given that $y = 1$ when $x = 0$, expressing your ans
q_t13_017P102MediumMediumEasy0.00.024human_labelledreviewedSolve the following differential equation: $$\frac{d^2y}{dx^2} = \sin(2x) + 3x^2$$ given that $y = 4$ and $\dfrac{dy}{dx} = -1$ when $x = 0$
q_t13_018P097EasyMediumEasy0.00.506human_labelledreviewedFind $s$ as an explicit function of $t$, given that $s = 5 + e^{2}$ when $t = 2$: $$\frac{ds}{dt} = 3t^2 - e^t.$$
q_t13_019P099HardHardHard0.00.025human_labelledreviewedSolve the differential equation using the substitution $v = \dfrac{y}{x}$: $$\frac{dy}{dx} = \frac{3x^2 y - y^3}{x^3 - 3xy^2}.$$
q_t13_020P100MediumHardMedium0.00.484human_labelledreviewedFind an integrating factor for the differential equation below and hence solve it, given that $y = 0$ when $x = 0$: $$\frac{dy}{dx} + y\cot
q_t13_021P101MediumMediumMedium0.00.046human_labelledreviewedThe velocity $v$ (in m/s) of a skydiver during free fall is modelled by the differential equation $$\frac{dv}{dt} = 9.8 - 0.005v^2 \ln(1 + v
q_t13_022P098MediumMediumMedium0.00.026human_labelledreviewedSolve the differential equation below, giving the answer in the form $y = f(x)$. It is given that $y = 0$ when $x = 1$: $$\frac{dy}{dx} = \f
q_t13_023P103EasyMediumEasy0.00.483human_labelledreviewedFind the power series solution to the differential equation $$\frac{dy}{dx} = xy + 1,$$ given that $y = 2$ when $x = 0$, expressing your ans
q_t13_024P102MediumMediumEasy0.00.022human_labelledreviewedSolve the following differential equation: $$\frac{d^2y}{dx^2} = 2\cos(x) + 4x$$ given that $y = 3$ and $\dfrac{dy}{dx} = 1$ when $x = 0$.
q_t13_025P097MediumMediumMedium0.00.045human_labelledreviewedFind $p$ as an explicit function of $x$, given that $p = 3 - \dfrac{\pi}{6}$ when $x = 0$: $$\frac{dp}{dx} = \frac{2}{x^2 + 4} - \sin x + 3x
q_t13_026P098MediumMediumMedium0.00.022human_labelledreviewedSolve the differential equation below, giving the answer in the form $y = f(x)$. It is given that $y = 1$ when $x = 0$: $$\frac{dy}{dx} = \f
q_t13_027P103EasyMediumEasy0.00.489human_labelledreviewedFind the power series solution to the differential equation $$\frac{dy}{dx} = x - 3y,$$ given that $y = 2$ when $x = 0$, expressing your ans
q_t13_028P101HardMediumHard0.00.499human_labelledreviewedThe voltage $V$ (in volts) across a capacitor in a nonlinear circuit is modelled by the differential equation $$\frac{dV}{dt} = \frac{0.08}{
q_t13_029P100MediumHardMedium0.00.487human_labelledreviewedFind an integrating factor for the differential equation below and hence solve it, given that $y = 0$ when $x = 0$: $$\frac{dy}{dx} + \frac{
q_t13_030P099EasyEasyEasy0.00.002human_labelledreviewedSolve the differential equation using the substitution $v = \dfrac{y}{x}$: $$\frac{dy}{dx} = \frac{x^2 + y^2}{xy}.$$
q_t13_031P097EasyMediumEasy0.00.496human_labelledreviewedFind $u$ as an explicit function of $x$, given that $u = 4 - e^{3}$ when $x = 3$: $$\frac{du}{dx} = e^{x} - 2x.$$
q_t13_032P102EasyEasyEasy0.00.015human_labelledreviewedSolve the following differential equation: $$\frac{d^2y}{dx^2} = 6x^2 - e^{3x}$$ given that $y = 0$ and $\dfrac{dy}{dx} = 3$ when $x = 0$.
q_t13_033P098MediumMediumEasy0.00.007human_labelledreviewedSolve the differential equation giving the answer in the form $y = f(x)$. It is given that $y = 2$ when $x = 0$: $$\frac{dy}{dx} = \frac{x}{
q_t13_034P103HardMediumMedium0.00.493human_labelledreviewedFind the power series solution to the differential equation $$\frac{dy}{dx} = x^2 y + \sin x,$$ given that $y = 0$ when $x = 0$, expressing
q_t13_035P101HardMediumHard0.00.505human_labelledreviewedThe angular displacement $\theta$ (in radians) of a pendulum with nonlinear damping is modelled by the differential equation $$\frac{d\theta
q_t13_036P100EasyMediumEasy0.00.496human_labelledreviewedFind an integrating factor for the differential equation below and hence solve it, given that $y = 5$ when $x = 0$: $$\frac{dy}{dx} - 3y = 6
q_t13_037P099EasyEasyMedium0.00.015human_labelledreviewedSolve the differential equation using the substitution $v = \dfrac{y}{x}$: $$\frac{dy}{dx} = \frac{x^2 + 3y^2}{2xy}.$$
q_t13_038P097HardHardMedium0.00.036human_labelledreviewedFind $w$ as an explicit function of $x$, given that $w = 2 + \ln 3$ when $x = -1$: $$\frac{dw}{dx} = \frac{3}{x^2 + 9} - \frac{2}{x} + e^{2x
q_t13_039P102EasyEasyEasy0.00.027human_labelledreviewedSolve the following differential equation: $$\frac{d^2s}{dt^2} = 12t - e^{2t}$$ given that $s = 5$ and $\dfrac{ds}{dt} = -\dfrac{1}{2}$ when
q_t13_040P098MediumMedium—0.00.007discarded—Solve the differential equation giving the answer in the form $y = f(x)$. It is given that $y = 2$ when $x = 0$: $$\frac{dy}{dx} = \frac{x}{
q_t13_041P103MediumMediumMedium0.00.023human_labelledreviewedFind the power series solution to the differential equation $$\frac{dy}{dx} = x^2 + xy,$$ given that $y = 3$ when $x = 0$, expressing your a
q_t13_042P101MediumMediumHard0.00.03human_labelledreviewedThe depth $y$ (in metres) of water in a tidal estuary is modelled by the differential equation $$\frac{dy}{dt} = 0.3\sqrt{y}\ln(y + 1) - 0.1
q_t13_043P100MediumHardMedium0.00.487human_labelledreviewedFind an integrating factor for the differential equation below and hence solve it, given that $y = 3$ when $x = 1$: $$\frac{dy}{dx} + \frac{
q_t13_044P099HardHardMedium0.00.023human_labelledreviewedSolve the differential equation using the substitution $v = \dfrac{y}{x}$: $$\frac{dy}{dx} = \frac{2x^2 + xy - y^2}{x^2 + xy}.$$ Express you
q_t13_045P097MediumMediumMedium0.00.039human_labelledreviewedFind $q$ as an explicit function of $t$, given that $q = 2 + \ln 2$ when $t = 1$: $$\frac{dq}{dt} = \frac{1}{t} + \frac{4}{t^2 + 4} - 2t.$$
q_t13_046P102MediumMediumEasy0.00.032human_labelledreviewedSolve the following differential equation: $$\frac{d^2\theta}{dt^2} = t^2 - 2e^{t/2}$$ given that $\theta = 3$ and $\dfrac{d\theta}{dt} = 6$
q_t13_047P098MediumMediumMedium0.00.013human_labelledreviewedSolve the differential equation below, giving the answer in the form $y = f(x)$. It is given that $y = 1$ when $x = 1$: $$\frac{dy}{dx} = \f
q_t13_048P103EasyMediumEasy0.00.495human_labelledreviewedFind the power series solution to the differential equation $$\frac{dv}{dt} = t - 2v,$$ given that $v = 1$ when $t = 0$, expressing your ans
q_t13_049P101MediumMediumMedium0.00.022human_labelledreviewedA population of insects $N$ (in thousands) is modelled by the differential equation $$\frac{dN}{dt} = 0.05N\ln(N+1) - 0.02N,$$ where $t$ is
q_t13_050P100MediumMediumMedium0.00.031human_labelledreviewedFind an integrating factor for the differential equation below and hence solve it, given that $y = 2$ when $x = 0$: $$\frac{dy}{dx} + \frac{
q_t14_001P109EasyMediumEasy0.00.5human_labelledreviewedThe function $\sin x$ is approximated by the Maclaurin series truncated after two terms: $\sin x \approx x - \dfrac{x^3}{3!}$. Using the nex
q_t14_002P108MediumHardEasy0.00.5human_labelledreviewedUse the Maclaurin expansion of $\cos x$ to find an approximation for $\cos(0.3)$, giving your answer correct to 4 decimal places. Hence, fin
q_t14_003P107MediumHardMedium0.00.5human_labelledreviewedFind the first four non-zero terms of the binomial expansion of $\dfrac{1}{1+x^2}$. Hence, by integrating term-by-term and using the fact t
q_t14_004P110MediumMediumEasy0.00.0human_labelledreviewedUse the Maclaurin series of $\cos x$ to evaluate $$\lim_{x \to 0} \frac{1 - \cos x}{2x^2}.$$
q_t14_005P104MediumMediumMedium0.00.0human_labelledreviewedUsing the definition of the Maclaurin series, find the first three non-zero terms of the expansion of $\cos 2x$.
q_t14_006P105EasyMediumEasy0.00.5human_labelledreviewedFind the first four non-zero terms of the Maclaurin series for $\cos(2x)$ by substituting into a known Maclaurin series.
q_t14_007P111EasyMediumMedium0.00.5human_labelledreviewedGiven that $f(x) = \frac{1 + ax}{1 - 2x}$, where $a$ is a constant, and that the Maclaurin expansion of $f(x)$ begins $1 + 5x + bx^2 + \cdot
q_t14_008P106MediumMediumMedium0.00.0human_labelledreviewedBy first writing down the Maclaurin series for $\dfrac{1}{1+x^3}$, find the first four non-zero terms of the Maclaurin expansion of $\ln(1+x
q_t14_009P109MediumMediumEasy0.00.0human_labelledreviewedThe function $f(x) = \ln(1-x)$ is approximated by the Maclaurin series truncated after three terms: $$\ln(1-x) \approx -x - \frac{x^2}{2} -
q_t14_010P108MediumMediumMedium0.00.0human_labelledreviewedUse the Maclaurin expansion of $\ln(1+x)$ to find an approximation for $\ln(1.4)$, giving your answer correct to 3 decimal places. Hence, fi
q_t14_011P107EasyEasyEasy0.00.0human_labelledreviewedFind the first three non-zero terms of the geometric expansion of $\dfrac{1}{1+x}$, valid for $|x| < 1$. Given that $\dfrac{d}{dx}\bigl(\ln
q_t14_012P110HardMediumMedium0.00.5human_labelledreviewedUse the Maclaurin series of $\cos x$ to evaluate $\displaystyle\lim_{x \to 0} \frac{1 - \cos x - \tfrac{1}{2}x^2}{x^4}$.
q_t14_013P104HardHardHard0.00.0human_labelledreviewedUsing the definition of the Maclaurin series, find the first three non-zero terms of the expansion of $f(x) = \ln(\cos x)$.
q_t14_014P105MediumMediumMedium0.00.0human_labelledreviewedUse a known Maclaurin series and substitution to find the first four non-zero terms of the expansion of $\cos(2x^2)$.
q_t14_015P111HardMediumHard0.00.5human_labelledreviewedGiven that $f(x) = \frac{(a + bx)^2}{(1-x)^3}$, where $a$ and $b$ are constants, and that the Maclaurin expansion of $f(x)$ begins $4 - 4x +
q_t14_016P109EasyMediumEasy0.00.5human_labelledreviewedThe function $f(x) = e^x$ is approximated by the Maclaurin series truncated after three terms: $$e^x \approx 1 + x + \frac{x^2}{2!}.$$ Using
q_t14_017P108HardHardMedium0.00.0human_labelledreviewedThe Maclaurin series for $\ln(1+x)$ is given by $\ln(1+x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4} + \cdots$, valid for $-1 <
q_t14_018P107HardHard—0.00.0discarded—Define $f(x) = \arctan\!\left(\dfrac{x}{2}\right)$. (a) Show that $f'(x) = \dfrac{1}{2}\cdot\dfrac{1}{1+(x/2)^2}$, and hence write down a b
q_t14_019P110MediumMediumMedium0.00.0human_labelledreviewedUse the Maclaurin series of $\cos x$ to evaluate $\displaystyle\lim_{x \to 0} \frac{1 - \cos x}{x^2}$.
q_t14_020P104HardHardHard0.00.0human_labelledreviewedUsing the definition of the Maclaurin series, find the first three non-zero terms of the expansion of $f(x) = x e^{x^2}$.
q_t14_021P105MediumMediumMedium0.00.0human_labelledreviewedFind the first four non-zero terms of the Maclaurin series for $\ln(3 + 9x)$ by first expressing it in a form that matches a known standard
q_t14_022P111MediumMediumMedium0.00.0human_labelledreviewedLet $f(x) = \frac{a + bx}{\sqrt{1-2x}}$, where $a$ and $b$ are constants. The Maclaurin series for $f(x)$ begins $2 + 7x + Cx^2 + \cdots$. F
q_t14_023P106HardHardHard0.00.0human_labelledreviewedBy first expressing $\dfrac{x}{1-x^2}$ as a sum of partial fractions, and hence writing down a Maclaurin series for $\dfrac{x}{1-x^2}$, find
q_t14_024P109MediumMediumEasy0.00.0human_labelledreviewedThe function $\sin x$ is approximated by the truncated Maclaurin series $\sin x \approx x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!}$. Use the next
q_t14_025P108EasyMediumEasy0.00.5human_labelledreviewedUse the Maclaurin expansion of $\sin x$ to find an approximation for $\sin(0.5)$, giving your answer correct to 3 decimal places. The Macla
q_t14_026P107HardHardMedium0.00.0human_labelledreviewedDefine $f(x) = \arctan\!\left(\dfrac{x}{2}\right)$. (a) Show that $f'(x) = \dfrac{2}{4+x^2}$. (b) Write $\dfrac{2}{4+x^2}$ in the form $\d
q_t14_027P110HardMediumMedium0.00.5human_labelledreviewedEvaluate the limit $$\lim_{x \to 0} \frac{e^x - 1 - x - \dfrac{x^2}{2}}{x^2 \ln(1+x)}$$ by substituting the Maclaurin series of $e^x$ and $\
q_t14_028P109EasyMediumEasy0.00.5human_labelledreviewedThe function $f(x) = \cos x$ is approximated by the Maclaurin series truncated after two terms: $$\cos x \approx 1 - \frac{x^2}{2!}.$$ Using
q_t14_029P108HardHardHard0.00.0human_labelledreviewedThe Maclaurin series for $\arctan x$ is given by $$\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} - \cdots, \
q_t14_030P107HardHardHard0.00.0human_labelledreviewedLet $f(x) = \arctan\left(\dfrac{x}{2}\right)$. (a) Show that $f'(x) = \dfrac{1}{2} \cdot \dfrac{1}{1 + \left(\frac{x}{2}\right)^2} = \dfrac
q_t14_031P110MediumMediumMedium0.00.0human_labelledreviewedEvaluate the limit $$\lim_{x \to 0} \frac{\sin x - x\cos x}{x^3}$$ by substituting the Maclaurin series of $\sin x$ and $\cos x$ into the ex
q_t14_032P104HardHardHard0.00.0human_labelledreviewedUsing the definition of the Maclaurin series, find the first three non-zero terms of the expansion of $f(x) = \tan x$.
q_t14_033P105MediumMediumMedium0.00.0human_labelledreviewedFind the first four non-zero terms of the Maclaurin series for $\sin(2x^3)$ by substituting into a known standard Maclaurin series.
q_t14_034P111MediumMediumMedium0.00.0human_labelledreviewedLet $f(x) = \dfrac{p + qx}{(1-x)^2}$, where $p$ and $q$ are constants. The Maclaurin series for $f(x)$ begins $2 + 9x + 16x^2 + \cdots$. Fin
q_t14_035P106HardHardHard0.00.0human_labelledreviewedBy first using the geometric series formula to write down the Maclaurin series for $\dfrac{1}{1+4x^2}$, find the first four non-zero terms o
q_t14_036P109MediumMediumMedium0.00.0human_labelledreviewedThe function $f(x) = \sin(3x)$ is approximated by the Maclaurin series truncated after two terms: $$\sin(3x) \approx 3x - \frac{(3x)^3}{3!}.
q_t14_037P108EasyMediumMedium0.00.5human_labelledreviewedUse the Maclaurin expansion of $e^x$ to find an approximation for $e^{0.3}$, giving your answer correct to 3 decimal places. The Maclaurin
q_t14_038P107HardHardMedium0.00.0human_labelledreviewedDefine $g(x) = \ln\!\left(\sqrt{\dfrac{1+x}{1-x}}\right)$ for $|x| < 1$. **(a)** Show that $g(x) = \tfrac{1}{2}\ln(1+x) - \tfrac{1}{2}\ln(1
q_t14_039P110HardMediumMedium0.00.5human_labelledreviewedEvaluate the limit $$\lim_{x \to 0} \frac{\sin x - \tan x}{x^2 \ln(1+x)}$$ by substituting the Maclaurin series of $\sin x$, $\tan x$, and $
q_t14_040P105MediumMediumEasy0.00.02human_labelledreviewedFind the first four non-zero terms of the Maclaurin series for $f(x) = \arctan(2x)$ by substituting an appropriate expression into the known
q_t14_041P104MediumMediumMedium0.00.011human_labelledreviewedUsing the definition of the Maclaurin series, find the first three non-zero terms of the expansion of $f(x) = e^x \sin x$.
q_t14_042P108EasyEasyEasy0.00.017human_labelledreviewedThe Maclaurin series for $\cos x$ is given by $$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots, \quad x \in \mathbb{
q_t14_043P109HardMediumHard0.00.491human_labelledreviewedThe function $g(x) = x^2 e^{-x^2}$ is approximated near $x = 0$ by truncating its Maclaurin series after the term in $x^6$: $$g(x) \approx x
q_t14_044P110HardMediumMedium0.00.497human_labelledreviewedEvaluate the limit $$\lim_{x \to 0} \frac{\cos x - e^{-x^2/2}}{x^4}$$ by substituting the Maclaurin series of $\cos x$ and $e^u$ (with $u =
q_t14_045P105MediumMediumEasy0.00.017human_labelledreviewedFind the first four non-zero terms of the Maclaurin series for $g(x) = \ln(3 + 6x)$ by first writing $3 + 6x$ in a suitable form and then su
q_t14_046P104MediumMediumMedium0.00.017human_labelledreviewedUsing the definition of the Maclaurin series, find the first three non-zero terms of the expansion of $f(x) = x^2 e^{-3x}$.
q_t14_047P108EasyEasyEasy0.00.021human_labelledreviewedThe Maclaurin series for $e^x$ is given by $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots, \quad x \in \mathbb{R}
q_t14_048P109EasyMediumEasy0.00.483human_labelledreviewedThe function $h(t) = \sin t$ is approximated near $t = 0$ by the Maclaurin series truncated after two terms: $$\sin t \approx t - \frac{t^3}
q_t14_049P107HardHardHard0.00.036human_labelledreviewedLet $f(x) = \arctan\left(\frac{x}{2}\right)$. (a) Show that $f'(x) = \dfrac{1}{2} \cdot \dfrac{1}{1 + \left(\frac{x}{2}\right)^2} = \dfrac{
q_t14_050P106MediumMediumMedium0.00.017human_labelledreviewedBy first using the geometric series formula to write down the Maclaurin series for $\dfrac{1}{1+9x^2}$, find the first four non-zero terms o
q_t15_001P113EasyMediumEasy0.00.477human_labelledreviewedThe complex number $z$ is given by $z = 3e^{i\frac{\pi}{4}}$. Find the value of $\text{Re}(z^4)$.
q_t15_002P122HardMediumMedium0.00.482human_labelledreviewedFind $z = \dfrac{3 - 5i}{(2 + 3i)^2}$ in the form $a + bi$, where $a, b \in \mathbb{R}$.
q_t15_003P112MediumMediumMedium0.00.012human_labelledreviewedExpress $z = 8e^{i\frac{7\pi}{6}}$ in the form $a + bi$, where $a, b \in \mathbb{R}$, giving exact values.
q_t15_004P112MediumMediumMedium0.00.017human_labelledreviewedExpress $z = 10e^{-i\frac{\pi}{3}}$ in the form $a + bi$, where $a, b \in \mathbb{R}$, giving exact values.
q_t15_005P121HardHardHard0.00.009human_labelledreviewedLet $z = \sqrt{3}\left(\cos\dfrac{\pi}{6} + i\sin\dfrac{\pi}{6}\right)$, where $n \in \mathbb{Z}^+$ and $n \geq 2$. The points represented o
q_t15_006P124HardHardHard0.00.023human_labelledreviewedLet $n$ be a positive integer and let $x$ be a real number with $x \neq 2k\pi$ for any integer $k$. **(a)** Show that $$\sum_{r=0}^{n} e^{i
q_t15_007P120MediumMediumMedium0.00.004human_labelledreviewedIn an Argand diagram, the point $A$ represents the complex number $3 + i$ and the point $B$ represents the complex number $1 + 4i$. The shap
q_t15_008P112MediumMediumMedium0.00.015human_labelledreviewedExpress $z = 5\sqrt{2}\, e^{\,i\frac{3\pi}{4}}$ in the form $a + bi$, where $a, b \in \mathbb{R}$, giving exact values.
q_t15_009P121HardHardHard0.00.018human_labelledreviewedLet $z = 2\left(\cos\dfrac{\pi}{6} + i\sin\dfrac{\pi}{6}\right)$, where $n \in \mathbb{Z}^+$. The points represented on an Argand diagram by
q_t15_010P114MediumMediumMedium0.00.018human_labelledreviewedIf $z = 3 - 2i$, determine the stretch factor and the angle of anticlockwise rotation achieved by multiplying $z$ by $\alpha = -1 + \sqrt{3}
q_t15_011P114EasyMediumEasy0.00.477human_labelledreviewedLet $z = 3 + 4i$. Describe the geometric effect on $z$ in the Argand plane when $z$ is multiplied by $\alpha = \sqrt{3} + i$. State the stre
q_t15_012P114MediumMediumMedium0.00.033human_labelledreviewedLet $w = -2 + 5i$. Determine the stretch factor and the angle of anticlockwise rotation (in radians) achieved when $w$ is multiplied by $\mu
q_t15_013P114HardMediumMedium0.00.482human_labelledreviewedLet $z = -3 + \sqrt{5}\,i$. Determine the stretch factor and the angle of anticlockwise rotation (in radians) achieved when $z$ is multiplie
q_t15_014P115MediumMediumMedium0.00.021human_labelledreviewedFind all solutions to $z^5 = -4\sqrt{2} + 4\sqrt{2}\,i$ and plot them on the Argand diagram.
q_t15_015P115EasyEasyEasy0.00.013human_labelledreviewedFind all solutions to $z^6 = -8i$ and plot them on the Argand diagram.
q_t15_016P115MediumMediumMedium0.00.008human_labelledreviewedFind all solutions to $z^4 = -\sqrt{3} - i$ and plot them on the Argand diagram.
q_t15_017P115HardMediumMedium0.00.47human_labelledreviewedFind all solutions to $z^6 = -27 + 27\sqrt{3}\,i$ and plot them on the Argand diagram.
q_t15_018P116MediumHardMedium0.00.476human_labelledreviewedFind the area of the polygon formed by the solutions of $z^4 = -8 + 8\sqrt{3}\,i$.
q_t15_019P116EasyEasyEasy0.00.043human_labelledreviewedFind the area of the triangle formed by the solutions of $z^3 = 27$.
q_t15_020P116MediumMediumMedium0.00.041human_labelledreviewedFind the area of the pentagon formed by the solutions of $z^5 = -32i$.
q_t15_021P116HardHardHard0.00.036human_labelledreviewedFind the area of the octagon formed by the solutions of $z^8 = -128 + 128i$.
q_t15_022P117MediumMediumMedium0.00.02human_labelledreviewedLet $\omega$ be a primitive $7$th root of unity, so that $\omega = e^{2\pi i/7}$ and $\omega \neq 1$. Show that $$( 1 - \omega)(1 - \omega^2
q_t15_023P117EasyMediumEasy0.00.456human_labelledreviewedLet $\omega = e^{2\pi i/3}$ be a primitive cube root of unity. Show that $(1 - \omega)(1 - \omega^2) = 3$.
q_t15_024P117MediumMedium—0.00.022discarded—Let $\omega = e^{2\pi i/7}$, a primitive 7th root of unity. Show that $(1 - \omega)(1 - \omega^2)(1 - \omega^3)(1 - \omega^4)(1 - \omega^5)(
q_t15_025P117HardMediumMedium0.00.475human_labelledreviewedLet $\omega = e^{2\pi i/9}$ be a primitive 9th root of unity. Show that $(1 - \omega)(1 - \omega^2)(1 - \omega^3)(1 - \omega^4)(1 - \omega^5
q_t15_026P119MediumHardHard0.00.454human_labelledreviewedGiven that $z = e^{i\theta}$, find the modulus and argument of $z + i$.
q_t15_027P118MediumHardMedium0.00.49human_labelledreviewedLet $\omega$ be the first complex root of $z^5 = 1$, that is, $\omega = e^{2\pi i/5}$. (a) Show that $(\omega^*)^3 + (\omega^*)^4 = \omega^
q_t15_028P118EasyEasyMedium0.00.007human_labelledreviewedLet $w$ be the first imaginary solution to $z^6 = 1$, that is, $w = e^{i\pi/3}$. (a) Show that $(w^*)^2 + (w^*)^4 = w^4 + w^2$. (b) Using
q_t15_029P119MediumHardMedium0.00.5human_labelledreviewedGiven that $z = e^{i\theta}$, where $0 < \theta < \pi$, find the modulus and argument of $w = z - i$, giving your answers in terms of $\thet
q_t15_030P119EasyEasyMedium0.00.0human_labelledreviewedGiven that $z = e^{i\theta}$, where $-\pi < \theta < 0$, find the modulus and argument of $w = z + i$, giving your answers in terms of $\the
q_t15_031P119EasyEasyEasy0.00.0human_labelledreviewedGiven that $z = e^{i\theta}$, where $0 < \theta < \pi$, find the modulus and argument of $w = z + i\sqrt{3}$, giving your answers in terms o
q_t16_001P151MediumMediumMedium0.00.012human_labelledreviewedFind all values of $k$ such that the vectors $\mathbf{u} = \begin{pmatrix} k^2 \\ 3 \\ -2 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 2
q_t16_002P171MediumMediumMedium0.00.02human_labelledreviewedThe points $P$, $Q$, $R$ and $S$ have position vectors $\mathbf{p} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$, $\mathbf{q} = \begin{pmatrix} 7
q_t16_003P152MediumMediumMedium0.00.04human_labelledreviewedLet $\mathbf{p} = \begin{pmatrix} 3 \\ m \\ m^2 - 4 \end{pmatrix}$ and $\mathbf{q} = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}$. Find all va
q_t16_004P169HardHardMedium0.00.022human_labelledreviewedFind the acute angle between the line $$\mathbf{r} = \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix} + t\begin{pmatrix} 4 \\ -2 \\ 1 \end{pmatrix
q_t16_005P164HardMediumMedium0.00.464human_labelledreviewedThe plane $\Pi$ has vector equation $$\mathbf{r} = \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix} + s\begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}
q_t16_006P151MediumMediumMedium0.00.041human_labelledreviewedFind all values of $m$ such that the vectors $\mathbf{p} = \begin{pmatrix} m \\ 4 \\ m^2 \end{pmatrix}$ and $\mathbf{q} = \begin{pmatrix} 3
q_t16_007P171EasyEasyEasy0.00.011human_labelledreviewedThe points $A$, $B$, $C$ and $D$ have position vectors $\mathbf{a} = \begin{pmatrix} 0 \\ 2 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 5
q_t16_008P155HardMediumMedium0.00.473human_labelledreviewedShow that the lines $L_1$ and $L_2$, defined by $$L_1: \quad x = 3 + 2t, \quad y = -1 + 3t, \quad z = 2 - t$$ $$L_2: \quad x = 1 + 3s, \qu
q_t16_009P165MediumMediumHard0.00.032human_labelledreviewedThe plane $\Pi$ has equation $3x - 4y + 5z = 20$, and the point $Q(2, -1, 3)$ does not lie on $\Pi$. Find the shortest distance from $Q$ to
q_t16_010P163EasyEasyMedium0.00.007human_labelledreviewedThe line $l$ has vector equation $\mathbf{r} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \end{pmatrix}$. The poi
q_t16_011P158EasyMediumMedium0.00.445human_labelledreviewedFind the shortest distance between the lines $l_1$ and $l_2$ with vector equations $$l_1: \mathbf{r} = \begin{pmatrix} 0 \\ 1 \\ 3 \end{pmat
q_t16_012P159MediumHardMedium0.00.445human_labelledreviewedFind the shortest distance between the lines $l_1$ and $l_2$, which have vector equations $$l_1: \mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ -1 \
q_t16_013P167EasyMediumEasy0.00.463human_labelledreviewedDetermine whether the three planes $x + y + z = 6$, $2x + y - z = 3$, and $x - y + 2z = 5$ intersect in a unique point. If they do, find the
q_t16_014P170MediumMediumMedium0.00.043human_labelledreviewedLet $\mathbf{u}$ and $\mathbf{v}$ be any two vectors in $\mathbb{R}^3$. **(a)** By first expanding $|\mathbf{u} + \mathbf{v}|^2$ using the
q_t16_015P173EasyEasyEasy0.00.038human_labelledreviewedLet $\mathbf{p}$ and $\mathbf{q}$ be non-zero vectors such that $|\mathbf{p} + 2\mathbf{q}| = |\mathbf{p} - 2\mathbf{q}|$. By expanding both
q_t16_016P156HardMediumMedium0.00.455human_labelledreviewedThe line $l$ has vector equation $$\mathbf{r} = \begin{pmatrix} -3 \\ 0 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} 4 \\ -3 \\ -2 \end{pmat
q_t16_017P157MediumMediumMedium0.00.004human_labelledreviewedThe line $l$ has vector equation $$\mathbf{r} = \begin{pmatrix} 1 \\ 4 \\ -3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatr
q_t16_018P162MediumMediumMedium0.00.018human_labelledreviewedTwo particles $A$ and $B$ move in three-dimensional space. At time $t$ seconds ($t \geq 0$), their position vectors $\mathbf{r}_A$ and $\mat
q_t16_019P172MediumMediumEasy0.00.028human_labelledreviewedFind the coordinates of the point $P$ that divides the line segment joining $A(-3, 5)$ and $B(9, -1)$ in the ratio $3:1$.
q_t16_020P168EasyEasyMedium0.00.028human_labelledreviewedThree planes are defined by the equations Π₁: x + y + z = 4, Π₂: x − y + 2z = 2, Π₃: 2x + 3z = 6. (a) Show that the three planes do not me
q_t16_021P166MediumMediumMedium0.00.035human_labelledreviewedFind the line of intersection of the planes $3x - y + 2z = 7$ and $x + y - z + 1 = 0$. Express your answer as a vector equation of a line in
q_t16_022P161MediumHardMedium0.00.454human_labelledreviewedA particle is projected from the origin with initial speed $26 \text{ m s}^{-1}$ at an angle of elevation $\theta$ above the horizontal, whe
q_t16_023P151EasyEasyEasy0.00.021human_labelledreviewedFind the value of $t$ such that the vectors $\mathbf{a} = \begin{pmatrix} 5 \\ -2 \\ t \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 3 \\
q_t16_024P164MediumMediumMedium0.00.027human_labelledreviewedA plane $\Pi$ has vector equation $$\mathbf{r} = \begin{pmatrix} 2 \\ 0 \\ -1 \end{pmatrix} + s\begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix} + t
q_t16_025P173MediumMediumMedium0.00.056human_labelledreviewedLet $\mathbf{u}$ and $\mathbf{v}$ be non-zero vectors. Given that $|\mathbf{u} + \mathbf{v}|^2 + |\mathbf{u} - \mathbf{v}|^2 = 4|\mathbf{u}|
q_t16_026P166MediumMediumMedium0.00.046human_labelledreviewedFind the line of intersection of the planes $\Pi_1: 2x - 3y + z = 5$ and $\Pi_2: x + y = 3 - 2z$. Express your answer as a vector equation o
q_t16_027P158MediumMediumMedium0.00.004human_labelledreviewedFind the shortest distance between the lines $l_1$ and $l_2$, which have vector equations $$l_1: \mathbf{r} = \begin{pmatrix} 2 \\ -1 \\ 4 \
q_t16_028P171HardMediumMedium0.00.483human_labelledreviewedThe points $P$, $Q$, $R$ and $S$ have position vectors $\mathbf{p} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$, $\mathbf{q} = \begin{pmatrix} 7
q_t16_029P173EasyMediumEasy0.00.457human_labelledreviewedLet $\mathbf{a}$ and $\mathbf{b}$ be non-zero vectors such that $|\mathbf{a} + \mathbf{b}|^2 = |\mathbf{a}|^2 + |\mathbf{b}|^2$. By expandin
q_t16_030P166MediumMediumMedium0.00.046human_labelledreviewedFind the line of intersection of the planes $\Pi_1: 4x - y + 3z = 10$ and $\Pi_2: 2y - z = 3x - 2$. Express your answer as a vector equation
q_t16_031P161MediumMediumMedium0.00.034human_labelledreviewedA ball is kicked from a point on flat ground with an initial speed of $65 \text{ m s}^{-1}$. The ball leaves the ground at an angle of eleva
q_t16_032P151MediumHardMedium0.00.494human_labelledreviewedFind all values of $n$ such that the vectors $\mathbf{u} = \begin{pmatrix} n \\ 2n \\ -3 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} n^
q_t16_033P156EasyMediumEasy0.00.465human_labelledreviewedFind the Cartesian equation of the line $m$ with vector equation $$\mathbf{r} = \begin{pmatrix} 4 \\ -3 \\ 0 \end{pmatrix} + \lambda \begin{
q_t16_034P152EasyMediumEasy0.00.47human_labelledreviewedFind the values of $a$ and $b$ such that the vectors $\mathbf{u} = \begin{pmatrix} 4 \\ a \\ -6 \end{pmatrix}$ and $\mathbf{v} = \begin{pmat
q_t16_035P164HardHardMedium0.00.038human_labelledreviewedThe plane Π has vector equation $$\mathbf{r} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix} + s\begin{pmatrix} 1 \\ 4 \\ -2 \end{pmatrix} + t
q_t16_036P153MediumMediumEasy0.00.029human_labelledreviewedFind the acute and obtuse angles between the vectors $\mathbf{u} = \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}$ and $\mathbf{v} = \begin{pmat
q_t16_037P151EasyMediumEasy0.00.479human_labelledreviewedFind the value of $p$ such that the vectors $\mathbf{u} = \begin{pmatrix} 6 \\ p \\ -4 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 2 \\
q_t16_038P164MediumHardMedium0.00.487human_labelledreviewedA shipping company models the floor of a cargo hold as a plane $\Pi$. The plane has vector equation $$\mathbf{r} = \begin{pmatrix} 1 \\ 3 \\
q_t16_039P173MediumMediumMedium0.00.032human_labelledreviewedLet $\mathbf{u}$ and $\mathbf{v}$ be non-zero vectors such that $|\mathbf{u}| = |\mathbf{v}|$. By expanding using the identity $|\mathbf{w}|
q_t16_040P166MediumMediumMedium0.00.039human_labelledreviewedFind the line of intersection of the planes $\Pi_1: 3x + y - 2z = 8$ and $\Pi_2: x - 3y + z + 4 = 2x$. Express your answer as a vector equat
q_t16_041P158MediumMediumMedium0.00.003human_labelledreviewedThe lines $l_1$ and $l_2$ have vector equations $$l_1: \mathbf{r} = \begin{pmatrix} 5 \\ 0 \\ -3 \end{pmatrix} + \lambda \begin{pmatrix} 1 \
q_t16_042P158MediumMediumMedium0.00.004human_labelledreviewedFind the shortest distance between the lines $l_1$ and $l_2$, which have vector equations $$l_1: \mathbf{r} = \begin{pmatrix} 1 \\ -2 \\ 5 \
q_t16_043P171HardMediumMedium0.00.481human_labelledreviewedThe points $P$, $Q$, $R$ and $S$ have position vectors $\mathbf{p} = \begin{pmatrix} -3 \\ 1 \end{pmatrix}$, $\mathbf{q} = \begin{pmatrix} 4
q_t16_044P173EasyEasyMedium0.00.055human_labelledreviewedLet $\mathbf{m}$ and $\mathbf{n}$ be non-zero vectors such that $|2\mathbf{m} + \mathbf{n}| = |\mathbf{n}|$. By expanding both sides using t
q_t16_045P166MediumMediumMedium0.00.032human_labelledreviewedFind the line of intersection of the planes $\Pi_1: 2x + y - 3z = 1$ and $\Pi_2: y - z = 4 - 3x$. Express your answer as a vector equation o
q_t16_046P161MediumHardMedium0.00.473human_labelledreviewedA stone is launched from the top of a cliff with initial speed $52 \text{ m s}^{-1}$ at an angle of elevation $\phi$ above the horizontal, w
q_t16_047P151MediumMediumEasy0.00.038human_labelledreviewedTwo vectors are defined as $\mathbf{p} = \begin{pmatrix} 3 \\ c^2 \\ -2 \end{pmatrix}$ and $\mathbf{q} = \begin{pmatrix} c \\ -4 \\ 5c \end{
q_t16_048P156EasyMediumEasy0.00.467human_labelledreviewedThe line $k$ passes through the point with position vector $\begin{pmatrix} 0 \\ 3 \\ -2 \end{pmatrix}$ and has direction vector $\begin{pma
q_t16_049P152EasyMediumEasy0.00.486human_labelledreviewedFind the values of $c$ and $d$ such that the vectors $\mathbf{u} = \begin{pmatrix} 5 \\ -3 \\ d \end{pmatrix}$ and $\mathbf{v} = \begin{pmat
q_t16_050P164HardHardHard0.00.05human_labelledreviewedA surveyor models a sloped terrain surface as a plane $\Pi$. The plane has vector equation $$\mathbf{r} = \begin{pmatrix} -2 \\ 3 \\ 1 \end{
q_t16_051P153MediumMediumMedium0.00.047human_labelledreviewedTwo displacement vectors are defined as $\mathbf{p} = \begin{pmatrix} 5 \\ -2 \\ 3 \end{pmatrix}$ and $\mathbf{q} = \begin{pmatrix} -1 \\ 4
q_t16_052P157EasyEasyMedium0.00.0human_labelledreviewedFind the shortest distance from the point $Q(0, 3, 5)$ to the line $l$, where $l$ has vector equation $$\mathbf{r} = \begin{pmatrix} 2 \\ 1
q_t16_053P157HardMediumMedium0.00.5human_labelledreviewedThe line $l$ has vector equation $$\mathbf{r} = \begin{pmatrix} -1 \\ 3 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmat
q_t16_054P157MediumMediumMedium0.00.0human_labelledreviewedA particle is observed at position $P(7, 0, -4)$. A beam of light travels along the line $l$ with vector equation $$\mathbf{r} = \begin{pma
q_t16_055P172EasyMediumEasy0.00.5human_labelledreviewedFind the coordinates of the point $Q$ that divides the line segment joining $M(2, -5)$ and $N(10, 7)$ in the ratio $2:3$.
q_t16_056P172HardMediumMedium0.00.5human_labelledreviewedThe points $A$ and $B$ have position vectors $\vec{OA} = \begin{pmatrix} -5 \\ 3 \\ 2 \end{pmatrix}$ and $\vec{OB} = \begin{pmatrix} 7 \\ -3
q_t16_057P172MediumMediumMedium0.00.0human_labelledreviewedThe vertices of a triangle have position vectors $\vec{OA} = \begin{pmatrix} 4 \\ -1 \\ 6 \end{pmatrix}$, $\vec{OB} = \begin{pmatrix} -2 \\
q_t17_001P206EasyMediumEasy0.00.482human_labelledreviewedGiven that $P(A' \cup B) = 0.75$ and $P(A') = 0.55$, find the value of $P(B \mid A)$.
q_t17_002P199MediumMediumMedium0.00.061human_labelledreviewedFind the number of distinct arrangements of all the letters in the word $\textbf{REARRANGED}$.
q_t17_003P226HardMediumMedium0.00.488human_labelledreviewedThe box-and-whisker plots below summarise the daily screen time (in minutes) recorded over one month for two groups of university students:
q_t17_004P211EasyMediumEasy0.00.453human_labelledreviewedThe probability distribution of a discrete random variable $X$ is given by the function $$P(X = x) = kx^2$$ for $x \in \{1, 2, 3, 4\}$. Find
q_t17_005P220MediumMediumMedium0.00.042human_labelledreviewedFor $T \sim N(52, 36)$, find the value of $a$, correct to two decimal places, such that $P(a < T < 61) = 0.73$.
q_t17_006P227MediumMediumMedium0.00.016human_labelledreviewedThe following frequency table shows the distances (in km) that students in a class travel to school each day. | Distance (km) | Frequency |
q_t17_007P214MediumMediumMedium0.00.074human_labelledreviewedA biased coin has a probability of $\frac{3}{8}$ of landing on heads. Determine the minimum number of times the coin must be tossed so that
q_t17_008P225EasyEasyEasy0.00.077human_labelledreviewedA student recorded the number of hours spent studying (h) and the resulting test score (s, out of 100) for seven different tests. | Hours s
q_t17_009P228EasyMediumEasy0.00.46human_labelledreviewedA game is played with two fair six-sided dice. The player wins $4 for each die that shows a 6. If both dice show a 6, the player wins an add
q_t17_010P200HardMediumMedium0.00.476human_labelledreviewedA school committee consists of $3$ teachers, $4$ senior students, and $2$ junior students. All $9$ members are to be seated in a row of $9$
q_t17_011P219MediumMedium—0.00.017discarded—A random variable X is distributed normally with a mean of 15. Given that P(X < 21) = 0.91, find the exact value of P(12 < X < 21).
q_t17_012P211EasyMediumEasy0.00.487human_labelledreviewedThe probability distribution of a discrete random variable $W$ is given by the function $$P(W = w) = k(w + 3)$$ for $w \in \{0, 1, 2, 3\}$.
q_t17_013P214MediumMediumMedium0.00.022human_labelledreviewedA quality-control inspector examines items from a production line. Each item independently has a probability of $0.35$ of being defective. D
q_t17_014P217MediumMediumMedium0.00.009human_labelledreviewedA continuous random variable $T$ has probability density function defined by $$f(t) = \begin{cases} kt^3 & 0 \le t \le 2 \\ 0 & \text{other
q_t17_015P199MediumMediumMedium0.00.022human_labelledreviewedFind the number of distinct arrangements of all the letters in the word $\textbf{STATISTICS}$.
q_t17_016P228HardMediumHard0.00.491human_labelledreviewedA carnival game uses a standard six-sided die and a bag containing 4 red balls and 6 blue balls. A player rolls the die once. If the result
q_t17_017P219EasyMediumEasy0.00.493human_labelledreviewedA random variable $T$ is distributed normally with a mean of $40$. Given that $P(T > 46) = 0.09$, find the exact value of $P(34 \leq T \leq
q_t17_018P212EasyMediumEasy0.00.498human_labelledreviewedA drawer contains $4$ black socks and $3$ white socks. Two socks are drawn at random without replacement. Let $X$ be the number of white soc
q_t17_019P213MediumMediumMedium0.00.017human_labelledreviewedA factory quality-control inspector examines items coming off a production line. The production line is known to produce $8\%$ defective ite
q_t17_020P220MediumMediumMedium0.00.011human_labelledreviewedThe daily water consumption (in litres) at a small office is modelled by $X \sim N(85, 64)$. Find the value of $a$, correct to two decimal p
q_t17_021P210MediumHardHard0.00.489human_labelledreviewedThe following is a probability distribution table for the discrete random variable $W$. | $w$ | $1$ | $2$ | $3$ | $4$ | |---|---|---|---|--
q_t17_022P205EasyMediumEasy0.00.488human_labelledreviewedLet $A$ and $B$ be two events in a sample space $S$, where $P(S) = 1$. The following conditions are given: $$P(A \cap B) = 0.15$$ $$P(A' \
q_t17_023P204HardMediumHard0.00.496human_labelledreviewedProve the identity: $$\sum_{r=2}^{n} r(r-1)\binom{n}{r} = n(n-1)\cdot 2^{n-2}$$ where $n$ is an integer with $n \geq 2$.
q_t17_024P200MediumMediumEasy0.00.025human_labelledreviewedA theatre director is arranging $10$ performers in a single row on stage. The cast consists of $2$ lead actors, $3$ ensemble singers, and $5
q_t17_025P221HardMediumMedium0.00.488human_labelledreviewedThe random variable $T$ follows a normal distribution with unknown mean $\mu$ and unknown standard deviation $\sigma$, so that $T \sim N(\mu
q_t18_001P128MediumMediumMedium0.00.019human_labelledreviewedLet $n \in \mathbb{Z}$. Prove by contrapositive that if $n^2$ is not divisible by 3, then $n$ is not divisible by 3.
q_t18_002P129EasyEasyMedium0.00.027human_labelledreviewedLet $n$ be a positive integer. Prove that $n$ is odd $\iff$ $n^2$ is odd.
q_t18_003P126HardHardHard0.00.015human_labelledreviewedProve by exhaustion that for every integer $n$, the expression $n^5 - 5n^3 + 4n$ is divisible by $120$.
q_t18_004P127MediumMediumMedium0.00.02human_labelledreviewedLet $a$ and $b$ be positive real numbers such that $a + b = 1$. Prove by contradiction that $\dfrac{1}{a} + \dfrac{1}{b} \geq 4$.
q_t18_005P125EasyMediumEasy0.00.498human_labelledreviewedProve that the product of two consecutive even integers is always divisible by $8$.
q_t18_006P130EasyEasyMedium0.00.013human_labelledreviewedProve by mathematical induction that $5^n + 3 \geq 8n$ for all integers $n \geq 1$.
q_t18_007P128EasyMediumEasy0.00.485human_labelledreviewedLet $n \in \mathbb{Z}$. Prove by contrapositive that if $n^2$ is even, then $n$ is even.
q_t18_008P129MediumMediumMedium0.00.027human_labelledreviewedLet $a$ and $b$ be integers. Prove that $ab$ is odd $\iff$ both $a$ and $b$ are odd.
q_t18_009P126HardHardHard0.00.022human_labelledreviewedProve by exhaustion that for every integer $n$, the expression $n^4 + 2n^3 - n^2 - 2n$ is divisible by $24$.
q_t18_010P127HardHardHard0.00.019human_labelledreviewedLet $f : \mathbb{R} \to \mathbb{R}$ be a twice-differentiable function satisfying $f''(x) > 0$ for all $x \in \mathbb{R}$. Suppose that $x_0
q_t18_011P125EasyMediumEasy0.00.489human_labelledreviewedProve that the sum of two odd integers is always even.
q_t18_012P130EasyMediumMedium0.00.492human_labelledreviewedProve by mathematical induction that $2^n \geq 2n$ for all integers $n \geq 1$.
q_t18_013P128HardMediumHard0.00.497human_labelledreviewedLet $f : \mathbb{R} \to \mathbb{R}$ be a twice-differentiable function satisfying $f''(x) > 0$ for all $x \in \mathbb{R}$. Prove by contrapo
q_t18_014P129EasyEasyEasy0.00.023human_labelledreviewedLet $n$ be an integer. Prove that $n$ is divisible by $2$ $\iff$ $n^2$ is divisible by $4$. *Hint: For the forward direction, write $n = 2k
q_t18_015P126MediumHardMedium0.00.504human_labelledreviewedProve by exhaustion that for every integer $n$, the value of $n^2(n^2 - 1)$ is divisible by $12$.
q_t18_016P127MediumMediumMedium0.00.037human_labelledreviewedLet $n$ be a positive integer. Prove by contradiction that if $n^2$ is divisible by $3$, then $n$ is divisible by $3$.
q_t18_017P125MediumMediumMedium0.00.027human_labelledreviewedProve that for any integer $n$, the expression $n^3 - n$ is always divisible by $6$. You may use the fact that a product of two consecutive
q_t18_018P130MediumMediumMedium0.00.021human_labelledreviewedProve by mathematical induction that $n^3 + 2n$ is divisible by $3$ for all integers $n \geq 1$.
q_t18_019P128HardMediumHard0.00.501human_labelledreviewedLet $a, b \in \mathbb{R}$ with $a < b$. Suppose $f : [a, b] \to \mathbb{R}$ is a continuous function that is differentiable on $(a, b)$. Pro
q_t18_020P129MediumHardMedium0.00.486human_labelledreviewedLet $f : \mathbb{R} \to \mathbb{R}$ be a differentiable function. Prove that: $$f(x) = f(-x) \text{ for all } x \in \mathbb{R} \iff f'(x) =
q_t18_021P126MediumMedium—0.00.025discarded—Prove by exhaustion that for every integer $n$, the expression $n^3 + 2n$ is divisible by $3$.
q_t18_022P127MediumMediumMedium0.00.022human_labelledreviewedLet $p$ and $q$ be real numbers satisfying $p > 0$ and $q > 0$. Prove by contradiction that $$\frac{p}{q} + \frac{q}{p} \geq 2.$$
q_t18_023P125EasyMediumEasy0.00.509human_labelledreviewedProve that if $a$ divides $b$ and $a$ divides $c$, then $a$ divides $b + c$, where $a, b, c$ are integers. (Here, $a \mid b$ means there exi
q_t18_024P130MediumMediumMedium0.00.024human_labelledreviewedProve by mathematical induction that $\displaystyle\sum_{r=1}^{n} r(r+1) = \frac{n(n+1)(n+2)}{3}$ for all integers $n \geq 1$.
q_t18_025P128MediumMediumEasy0.00.015human_labelledreviewedLet $a, b \in \mathbb{Z}$. Prove by contrapositive that if $a \cdot b$ is odd, then both $a$ and $b$ are odd.
q_t18_026P130EasyMediumMedium0.00.49human_labelledreviewedProve by mathematical induction that $4^n - 1$ is divisible by $3$ for all integers $n \geq 1$.
q_t18_027P128MediumMediumMedium0.00.012human_labelledreviewedLet $a, b \in \mathbb{Z}$. Prove by contrapositive that if $a^2 + b^2$ is divisible by $4$, then both $a$ and $b$ are even. *(Hint: begin b
q_t18_028P125MediumMediumMedium0.00.018human_labelledreviewedProve that for any two real numbers $a$ and $b$, the inequality $a^2 + b^2 \geq 2ab$ holds. Hence, by writing $a = \sqrt{x}$ and $b = \sqrt{
q_t18_029P127MediumEasyEasy0.00.502human_labelledreviewedLet $a$, $b$, and $c$ be real numbers satisfying $a + b + c = 0$. Prove by contradiction that it is impossible for all three of the inequali
q_t18_030P126MediumMediumMedium0.00.019human_labelledreviewedProve by exhaustion that for every integer $n$, the expression $n^3 + 2n$ is divisible by $3$.
q_t18_031P129EasyEasyEasy0.00.013human_labelledreviewedLet $n$ be an integer. Prove that $n$ is divisible by $3$ $\iff$ $n^2$ is divisible by $9$. *Hint: For the forward direction, write $n = 3k
q_t18_032P130EasyMediumMedium0.00.503human_labelledreviewedProve by mathematical induction that $\displaystyle\sum_{r=1}^{n} (2r - 1) = n^2$ for all integers $n \geq 1$.
q_t18_033P128MediumMediumEasy0.00.014human_labelledreviewedLet $n \in \mathbb{Z}$. Prove by contrapositive that if $3n^2 + 2$ is odd, then $n$ is odd. *(Hint: begin by writing down the contrapositiv
q_t18_034P125MediumMediumEasy0.00.022human_labelledreviewedLet $p$ and $q$ be rational numbers, so that $p = \dfrac{a}{b}$ and $q = \dfrac{c}{d}$ for some integers $a, b, c, d$ with $b \neq 0$ and $d
q_t18_035P127MediumMediumMedium0.00.024human_labelledreviewedLet $p$ and $q$ be positive real numbers. Prove by contradiction that if $pq > 9$, then at least one of $p$ or $q$ is greater than $3$.
q_t18_036P126EasyMediumEasy0.00.501human_labelledreviewedProve by exhaustion that for every integer $n$, the expression $n^2 + n$ is divisible by $2$.
q_t18_037P129HardHardHard0.00.02human_labelledreviewedLet $f : \mathbb{R} \to \mathbb{R}$ be a twice-differentiable function. Prove that: $$f(x+y) = f(x) + f(y) + xy \quad \text{for all } x, y \
q_t18_038P130HardMediumMedium0.00.501human_labelledreviewedProve by mathematical induction that for all integers $n \geq 1$, $$\sum_{r=1}^{n} r \cdot 2^r = (n-1) \cdot 2^{n+1} + 2.$$
q_t18_039P128MediumMediumMedium0.00.013human_labelledreviewedLet $p$ and $q$ be integers. Prove by contrapositive that if $p^2 + q^2$ is odd, then exactly one of $p$ or $q$ is odd. *(Hint: begin by wr
q_t18_040P125HardMediumMedium0.00.503human_labelledreviewedLet $f : \mathbb{R} \to \mathbb{R}$ and $g : \mathbb{R} \to \mathbb{R}$ be functions satisfying the following two conditions: - **Condition
q_t18_041P128MediumMediumEasy0.00.0human_labelledreviewedLet $n \in \mathbb{Z}$. Prove by contrapositive that if $n^2$ is not divisible by $3$, then $n$ is not divisible by $3$.
q_t18_042P125HardHardHard0.00.0human_labelledreviewedLet $f : \mathbb{R} \to \mathbb{R}$ be a function satisfying the following two properties: - **Property 1 (Subadditivity of differences):**
q_t18_043P130EasyMediumMedium0.00.5human_labelledreviewedProve by mathematical induction that $7^n - 1$ is divisible by $6$ for all integers $n \geq 1$.
q_t18_044P126MediumHardHard0.00.5human_labelledreviewedProve by exhaustion that for every integer $n$, the expression $n^4 - n^2$ is divisible by $12$. (Hint: consider the residues of $n$ modulo
q_t18_045P129MediumMediumEasy0.00.0human_labelledreviewedLet $n$ be a positive integer. Prove that $n$ leaves a remainder of $1$ or $5$ when divided by $6$ $\iff$ $n$ is not divisible by $2$ and no
q_t18_046P127MediumMediumEasy0.00.0human_labelledreviewedLet $n$ be an integer. Prove by contradiction that if $n^2$ is odd, then $n$ is odd.
q_t18_047P128EasyMediumEasy0.00.5human_labelledreviewedLet $a$ and $b$ be integers. Prove by contrapositive that if $a + b$ is odd, then at least one of $a$ or $b$ is odd. (You may use without p
q_t18_048P125EasyMediumEasy0.00.5human_labelledreviewedProve that the product of an even integer and any integer is always even. You may use the fact that an integer $n$ is even if and only if $
q_t18_049P130HardMediumMedium0.00.5human_labelledreviewedProve by mathematical induction that for all integers $n \geq 1$, $$\sum_{r=1}^{n} r \cdot 2^r = (n-1)\cdot 2^{n+1} + 2.$$
q_t18_050P126MediumMediumEasy0.00.0human_labelledreviewedProve by exhaustion that for every integer $n$, the value of $n^2 + 3n + 2$ is always even. (Hint: consider the two cases $n$ even and $n$
q_t19_001P135MediumHardMedium0.00.485human_labelledreviewedThe expressions $3m - 1$, $m + 5$, and $4m - 3$ are three consecutive terms of a sequence. (a) Find the value of $m$ that makes the sequenc
q_t19_002P137EasyEasyEasy0.00.001human_labelledreviewedA colony of bacteria numbers $500$ at the start of an experiment. Every hour, the number of bacteria increases to $120\%$ of the previous ho
q_t19_003P134MediumMediumMedium0.00.0human_labelledreviewedConsider the geometric sequence 24, 12, 6, 3, …. Find the least integer n such that |S∞ − Sn| < 0.05.
q_t19_004P131MediumMediumMedium0.00.0human_labelledreviewedA geometric sequence has its $3$rd term equal to $36$ and its $6$th term equal to $\ \dfrac{4}{3}$. Find an expression for the general term
q_t19_005P136MediumHardMedium0.00.5human_labelledreviewedAn infinite geometric series has first term $a$ and common ratio $r = \dfrac{3}{2x - 1}$, where $x$ is a real number. **(a)** Find the rang
q_t19_006P133MediumHardMedium0.00.5human_labelledreviewedThe sum of the first $n$ terms of a sequence $(u_n)$ is given by $$S_n = 4 \cdot 2^n + 3n - 4.$$ (a) Find the general term $u_n$ for $n \ge
q_t19_007P132EasyEasyEasy0.00.0human_labelledreviewedA sequence is defined by the recurrence relation $u_{n+1} = 3u_n - 4$, with $u_1 = 3$. (a) Calculate the first four terms of the sequence.
q_t19_008P135EasyMediumEasy0.00.5human_labelledreviewedThe expressions $3k + 1$, $5k - 1$, and $8k - 4$ are three consecutive terms of an arithmetic sequence. Find the value of $k$.
q_t19_009P137MediumHardMedium0.00.5human_labelledreviewedA colony of bacteria is observed in a laboratory. At the start of the first hour, there are $800$ bacteria. The number of bacteria increases
q_t19_010P134MediumMediumMedium0.00.0human_labelledreviewedConsider the geometric sequence $3, 6, 12, 24, \ldots$ Find the least integer $n$ such that $S_n > 750$.
q_t19_011P131MediumMediumEasy0.00.0human_labelledreviewedAn arithmetic sequence has its $5$th term equal to $3$ and its $12$th term equal to $-25$. Find an expression for the general term $u_n$, an
q_t19_012P136EasyEasyEasy0.00.0human_labelledreviewedAn infinite geometric series has first term $a = 5$ and common ratio $r = \dfrac{2}{k+1}$, where $k$ is a real number. **(a)** Find the ran
q_t19_013P133HardHardMedium0.00.001human_labelledreviewedThe sum of the first $n$ terms of a sequence $(u_n)$ is given by $$S_n = \frac{2}{3}\left(1 - \left(\frac{1}{2}\right)^n\right) + n^2 - n.$$
q_t19_014P132HardHardHard0.00.0human_labelledreviewedA sequence $(u_n)_{n \geq 1}$ is defined by the recurrence relation $$u_{n+1} = 5u_n - 3^n, \quad u_1 = 1.$$ (a) Calculate the first four t
q_t19_015P135MediumHardMedium0.00.5human_labelledreviewedThe expressions $3k - 1$, $k + 3$, and $2k - 1$ are three consecutive terms of a sequence. (a) Find the values of $k$ for which the sequenc
q_t19_016P137EasyEasyEasy0.00.0human_labelledreviewedA scientist places a colony of 500 bacteria in a dish. The number of bacteria doubles every hour. (a) Write down the number of bacteria in
q_t19_017P134MediumHardMedium0.00.5human_labelledreviewedA geometric sequence has first three terms $100, 70, 49, \ldots$ Find the least integer $n$ such that $|S_{\infty} - S_n| < 1$.
q_t19_018P131MediumMediumEasy0.00.0human_labelledreviewedA geometric sequence $\{b_n\}$ has its $2$nd term equal to $20$ and its $5$th term equal to $\dfrac{5}{16}$. Find an expression for the gene
q_t19_019P136MediumMediumMedium0.00.0human_labelledreviewedAn infinite geometric series has first term $a$ and common ratio $r = \dfrac{2}{3-2t}$, where $t$ is a real number. **(a)** Find the range
q_t19_020P133MediumHardMedium0.00.5human_labelledreviewedThe sum of the first $n$ terms of a sequence $(v_n)$ is given by $$S_n = 3 \cdot 5^n - 2n^2 - 3.$$ (a) Find the general term $v_n$ for $n \
q_t19_021P132EasyEasyEasy0.00.0human_labelledreviewedA sequence $(a_n)_{n \geq 1}$ is defined by the recurrence relation $$a_{n+1} = 4a_n + 6,$$ with $a_2 = 22$. (a) Find $a_1$, and then calcu
q_t19_022P135EasyMediumMedium0.00.5human_labelledreviewedThe expressions $n^2 - 3$, $2n + 1$, and $n + 10$ are three consecutive terms of an arithmetic sequence. Find all possible values of $n$.
q_t19_023P137MediumHard—0.00.5discarded—A car is purchased for $24,000. Its value depreciates by 15% each year. (a) Write down the value of the car after $n$ years. (b) Find the
q_t19_024P131HardMediumMedium0.00.5human_labelledreviewedA geometric sequence has all positive terms. The 3rd term is 12 and the 6th term is 324. Find the general term $a_n$, and hence find the val
q_t19_025P134EasyEasyMedium0.00.0human_labelledreviewedA geometric sequence has first three terms $162, 54, 18, \ldots$ Find the least integer $n$ such that $S_n > 242$.
q_t19_026P136MediumMediumMedium0.00.0human_labelledreviewedAn infinite geometric series has first term $a$ and common ratio $r$. The sum to infinity of the series is 15. A second infinite geometric s
q_t19_027P132MediumMediumMedium0.00.0human_labelledreviewedA sequence is defined by the recurrence relation $u_{n+1} = 3u_n - 4$, with $u_1 = 3$. (a) Calculate the first four terms of the sequence.
q_t19_028P135MediumMediumMedium0.00.0human_labelledreviewedThe expressions $3m + 1$, $5m - 3$, and $9m - 11$ are three consecutive terms of a sequence. (a) Find the value of $m$ for which the three
q_t19_029P133EasyEasyEasy0.00.0human_labelledreviewedThe sum of the first $n$ terms of a sequence $(u_n)$ is given by $$S_n = 3n^2 + 7n.$$ (a) Find the general term $u_n$ using the identity $u
q_t19_030P137EasyEasyMedium0.00.0human_labelledreviewedA cup of tea is made at a temperature of $95°\text{C}$. Each minute, the temperature of the tea drops to $80\%$ of its temperature at the st
q_t19_031P131HardMediumMedium0.00.5human_labelledreviewedA geometric sequence has its 3rd term equal to 12 and its 6th term equal to 96. Find the first term and common ratio, write down the general
q_t19_032P134MediumMediumMedium0.00.0human_labelledreviewedConsider the geometric sequence 2, 6, 18, …. Find the least integer n such that S_n > 500.
q_t19_033P133MediumHardMedium0.00.5human_labelledreviewedThe sum of the first $n$ terms of a sequence $(w_n)$ is given by $$S_n = 6\cdot 4^n - 2n^2 - 6.$$ (a) Find the general term $w_n$ for $n \g
q_t19_034P131MediumMediumMedium0.00.0human_labelledreviewedAn arithmetic sequence $\{v_n\}$ satisfies $v_3 + v_7 = 20$ and $v_3 \cdot v_7 = 96$. Determine the two possible expressions for the general
q_t19_035P134EasyEasyEasy0.00.0human_labelledreviewedConsider the geometric sequence $5, 10, 20, 40, \ldots$ Find the least integer $n$ such that $S_n > 1000$.
q_t19_036P132HardHardHard0.00.0human_labelledreviewedA sequence $(u_n)_{n \geq 1}$ is defined by the recurrence relation $$u_{n+1} = 3u_n + 2^n,$$ with $u_3 = 21$. (a) Find $u_2$ and $u_1$, th
q_t19_037P137EasyEasyEasy0.00.0human_labelledreviewedA colony of bacteria doubles in size every hour. At the start of the experiment (hour 0), there are 50 bacteria. (a) Write down the number
q_t19_038P137EasyEasyEasy0.00.0human_labelledreviewedA new social media post is shared by $3$ people on the first day. Each day after that, every person who shared it the previous day shares it
q_t19_039P137EasyEasyEasy0.00.0human_labelledreviewedA radioactive substance has a mass of $200$ grams at the start of an experiment. Every hour, the mass of the substance decreases to $75\%$ o
q_t20_001P145MediumMediumMedium0.00.0human_labelledreviewedUsing the expansions of $(1+x)^{2n}$ and the product $(1+x)^n(1+x)^n$, show that $$\sum_{k=0}^{n}\binom{n}{k}^2 = \binom{2n}{n}.$$
q_t20_002P138EasyMediumEasy0.00.5human_labelledreviewedFind the coefficient of $x^3$ in the expansion of $\left(x + \frac{1}{3}\right)^6$.
q_t20_003P142HardMediumMedium0.00.5human_labelledreviewedGiven that the coefficient of $x^4$ in the expansion of $\left(3 - \dfrac{x}{k}\right)^9$ is $\dfrac{5376}{k}$, find the possible values of
q_t20_004P141MediumMediumMedium0.00.0human_labelledreviewedFind the coefficient of $x^4$ in the expansion of $(1 - 2x + 3x^2)^6$.
q_t20_005P140EasyMediumMedium0.00.5human_labelledreviewedFind the coefficient of $x^3$ in the expansion of $(1 + 3x)^4(1 + x)^3$.
q_t20_006P143EasyMediumEasy0.00.5human_labelledreviewedGiven that the expanded expression is $32 + 240x + 720x^2 + 1080x^3 + 810x^4 + 243x^5$, find the single powered expression of the form $(a +
q_t20_007P144EasyMediumEasy0.00.5human_labelledreviewedFind the first four terms in ascending powers of $x$ of $\left(1 + 2x\right)^{-3}$.
q_t20_008P139MediumMediumEasy0.00.0human_labelledreviewedFind the coefficient of $x^3$ in the binomial series expansion of $(9 - 2x)^{1/2}$, valid for $|x| < \dfrac{9}{2}$.
q_t20_009P145HardMedium—0.00.5discarded—Show that, for any positive integer $n$, $$\sum_{k=0}^{n} (-1)^k \binom{n}{k} \binom{n}{n-k} \cdot \frac{1}{k+1} = \frac{1}{n+1},$$ by con
q_t20_010P138HardMediumMedium0.00.5human_labelledreviewedFind the coefficient of $x^5$ in the expansion of $\left(3x^2 - \dfrac{2}{\sqrt{x}}\right)^7$.
q_t20_011P142EasyMediumEasy0.00.5human_labelledreviewedGiven that the coefficient of $x^2$ in the expansion of $(1 + kx)^8$ is $112$, find the possible values of $k$.
q_t20_012P141EasyMediumMedium0.00.5human_labelledreviewedFind the coefficient of $x^3$ in the expansion of $(2 + x - x^2)^5$.
q_t20_013P140HardMediumMedium0.00.5human_labelledreviewedFind the coefficient of $x^4$ in the expansion of $\left(2 + x - 3x^2\right)\left(1 - 2x\right)^7$.
q_t20_014P143EasyMediumEasy0.00.5human_labelledreviewedGiven that the expanded expression is $16t^4 - 96t^3 + 216t^2 - 216t + 81$, find the single powered expression of the form $(a + bt)^n$ from
q_t20_015P144MediumMediumMedium0.00.0human_labelledreviewedFind the first four terms in ascending powers of $x$ of $\left(1 - \dfrac{x}{2}\right)^{-\frac{1}{2}}$.
q_t20_016P139MediumMediumMedium0.00.0human_labelledreviewedFind the coefficient of $x^2$ in the binomial series expansion of $(8 - 6x)^{-\frac{2}{3}}$, valid for $|x| < \dfrac{4}{3}$.
q_t20_017P145MediumMediumMedium0.00.0human_labelledreviewedShow that, for any positive integer $n$, $$\sum_{k=0}^{n} k\binom{n}{k} = n \cdot 2^{n-1}.$$ by substituting a suitable value of $x$ into
q_t20_018P138MediumMediumMedium0.00.0human_labelledreviewedFind the coefficient of $x^2$ in the expansion of $\left(\dfrac{x^3}{2} - \dfrac{3}{x}\right)^8$.
q_t20_019P142HardMediumMedium0.00.5human_labelledreviewedGiven that the coefficient of $x^6$ in the expansion of $(k + 2x)^8$ is $448$, find the possible values of $k$.
q_t20_020P141MediumMediumMedium0.00.0human_labelledreviewedFind the coefficient of $x^5$ in the expansion of $(1 + 3x - 2x^3)^4$.
q_t20_021P140MediumMediumMedium0.00.0human_labelledreviewedFind the coefficient of $x^4$ in the expansion of $(3 - x + 2x^2)(1 + 3x)^5$.
q_t20_022P143MediumMediumEasy0.00.0human_labelledreviewedGiven that the expanded expression is $$16 + 96x + 216x^2 + 216x^3 + 81x^4,$$ find the single powered expression $(a + bx)^n$ from which it
q_t20_023P144EasyMediumMedium0.00.5human_labelledreviewedFind the first four terms in ascending powers of $x$ of $\left(1 + \dfrac{x}{3}\right)^{\frac{1}{3}}$.
q_t20_024P143EasyMediumEasy0.00.5human_labelledreviewedGiven that the expanded expression is $$81m^4 + 108m^3n + 54m^2n^2 + 12mn^3 + n^4,$$ find the single powered expression of the form $(am + b
q_t20_025P138HardMediumMedium0.00.5human_labelledreviewedFind the coefficient of $x^2$ in the expansion of $\left(2x^3 - \dfrac{1}{x^2}\right)^{10}$.
q_t20_026P141EasyMediumMedium0.00.5human_labelledreviewedFind the coefficient of $x^4$ in the expansion of $(2 + x - x^2)^4$.
q_t20_027P145HardMediumHard0.00.5human_labelledreviewedShow that, for any positive integer $n$, $$\sum_{k=0}^{n} (-1)^k \binom{n}{k} \frac{1}{k+1} = \frac{1}{n+1}.$$ *Hint: consider the expansi
q_t20_028P144MediumMediumMedium0.00.0human_labelledreviewedFind the first four terms in ascending powers of $x$ of $\left(1 - \dfrac{3x}{2}\right)^{\frac{2}{3}}$.
q_t20_029P140EasyMediumMedium0.00.5human_labelledreviewedFind the coefficient of $x^3$ in the expansion of $(1 + 2x)^5(2 - x)^3$.
q_t20_030P142MediumMediumMedium0.00.0human_labelledreviewedGiven that the coefficient of $x^3$ in the expansion of $\left(2x + \dfrac{3}{k}\right)^7$ is $\,15120$, find the possible values of $k$.
q_t20_031P139MediumMediumMedium0.00.0human_labelledreviewedFind the coefficient of $x^3$ in the binomial series expansion of $\left(27 - 4x\right)^{-\frac{1}{3}}$, valid for $\left|x\right| < \dfrac{
q_t20_032P143MediumMediumEasy0.00.0human_labelledreviewedGiven that the expanded expression is $32 + 240x + 720x^2 + 1080x^3 + 810x^4 + 243x^5$, write it as a single expression of the form $(a + bx
q_t20_033P138HardMediumMedium0.00.5human_labelledreviewedFind the coefficient of $x^0$ (the constant term) in the expansion of $\left(\dfrac{x^2}{3} - \dfrac{\sqrt{3}}{x^3}\right)^{10}$.
q_t20_034P141MediumMediumMedium0.00.0human_labelledreviewedFind the coefficient of $x^6$ in the expansion of $(3 - x^2 + 2x^3)^5$.
q_t20_035P145MediumMediumMedium0.00.0human_labelledreviewedShow that, for any positive integer $n$, $$\binom{n}{1} - 2\binom{n}{2} + 3\binom{n}{3} - \cdots + (-1)^{n-1}\,n\binom{n}{n} = 0,$$ that i
q_t20_036P144MediumMediumMedium0.00.0human_labelledreviewedFind the first four terms in ascending powers of $x$ of $\left(1 + \dfrac{2x}{3}\right)^{-\frac{3}{2}}$, stating the values of $x$ for which
q_t20_037P140EasyMediumMedium0.00.5human_labelledreviewedFind the coefficient of $x^2$ in the expansion of $(1 + 3x)^4(1 + x)^3$.
q_t20_038P142EasyMediumEasy0.00.5human_labelledreviewedGiven that the coefficient of $x^2$ in the expansion of $\left(3 + kx\right)^5$ is $270$, find the possible values of $k$.
q_t20_039P139MediumMediumMedium0.00.0human_labelledreviewedFind the coefficient of $x^2$ in the binomial series expansion of $\left(16 + 5x\right)^{-\frac{3}{4}}$, valid for $\left|x\right| < \dfrac{
q_t20_040P144MediumMediumMedium0.00.002human_labelledreviewedFind the first four terms in ascending powers of $x$ of $\left(1 - \dfrac{x}{3}\right)^{-\frac{5}{2}}$, stating the values of $x$ for which
q_t20_041P145HardHardHard0.00.003human_labelledreviewedShow that, for any positive integer $n$, $$\sum_{k=0}^{n} (-1)^k \binom{n}{k} \frac{1}{2k+1} = \frac{(n!)^2 \cdot 4^n}{(2n+1)!}.$$ *You ma
q_t20_042P138EasyMediumEasy0.00.498human_labelledreviewedFind the coefficient of $x^4$ in the expansion of $\left(x + \dfrac{1}{2}\right)^8$.
q_t20_043P143EasyMediumEasy0.00.498human_labelledreviewedGiven that the expanded expression is $$243k^5 - 405k^4 + 270k^3 - 90k^2 + 15k - 1,$$ find the single powered expression of the form $(ak +
q_t20_044P140MediumMediumMedium0.00.001human_labelledreviewedFind the coefficient of $x^4$ in the expansion of $\left(1 + 2x^2 - x^3\right)(2 + x)^6$.
q_t20_045P139MediumMediumMedium0.00.002human_labelledreviewedFind the coefficient of $x^3$ in the binomial series expansion of $\left(32 + 5x\right)^{-\frac{3}{5}}$, valid for $\left|x\right| < \dfrac{
q_t20_046P142MediumMediumEasy0.00.002human_labelledreviewedGiven that the coefficient of $x^4$ in the expansion of $\left(kx^2 - \dfrac{1}{x}\right)^5$ is $-10$, find the possible values of $k$.
q_t20_047P141MediumMediumMedium0.00.001human_labelledreviewedFind the coefficient of $x^5$ in the expansion of $(2 - x^2 + 3x^3)^4$.
q_t20_048P144EasyMediumEasy0.00.498human_labelledreviewedFind the first four terms in ascending powers of $x$ of $\left(1 + \dfrac{x}{2}\right)^{-4}$.
q_t20_049P145HardMediumHard0.00.498human_labelledreviewedLet $n$ be a positive integer. By considering the product of the expansions of $(1 + x)^n$ and $(1 + x)^n$, show that $$\sum_{k=0}^{n}\binom
q_t20_050P138MediumMediumEasy0.00.002human_labelledreviewedFind the coefficient of $x^3$ in the expansion of $\left(2x + \dfrac{1}{3}\right)^9$.
q_t21_001P149MediumHardMedium0.00.497human_labelledreviewedObtain the values of $a$, $b$, $c$, and $p$ if $$x^3 - x^2 + x - 6 = a(x-p)^3 + b(x-p)^2 + c(x-p).$$
q_t21_002P150EasyMediumEasy0.00.499human_labelledreviewedThe equation $x^2 - 8x + 12 = 0$ has roots $\alpha$ and $\beta$. Find the value of $\alpha^2 + \beta^2$.
q_t21_003P147HardMediumMedium0.00.499human_labelledreviewedThe polynomial $f(x) = 2x^3 + ax^2 + bx + c$ satisfies the following conditions: - When $f(x)$ is divided by $(x - 1)$, the remainder is $6
q_t21_004P148MediumHardMedium0.00.497human_labelledreviewedFor some polynomial $f(x)$, the remainder when $f(x)$ is divided by $(x - 3)^2$ is $5x - 4$, and the remainder when $f(x)$ is divided by $(x
q_t21_005P146EasyMediumEasy0.00.498human_labelledreviewedFind the polynomial $g(x) = x^3 + ax^2 + bx - 6$, where $a, b \in \mathbb{R}$, given that $x - 1$ is a factor of $g(x)$ and that $g(-1) = -8
q_t21_006P149EasyMediumMedium0.00.497human_labelledreviewedObtain the values of $a, b, c, p$ if $$x^3 + x^2 + x - 3 = a(x - p)^3 + b(x - p)^2 + c(x - p).$$
q_t21_007P150EasyMediumEasy0.00.498human_labelledreviewedThe equation $x^2 - 7x + 10 = 0$ has roots $\alpha$ and $\beta$. Find the value of $\dfrac{1}{\alpha} + \dfrac{1}{\beta}$.
q_t21_008P147MediumMediumEasy0.00.003human_labelledreviewedThe polynomial $g(x) = x^3 + ax^2 + bx - 4$ satisfies the following conditions: - When $g(x)$ is divided by $(x - 2)$, the remainder is $10
q_t21_009P148HardHardHard0.00.002human_labelledreviewedFor a polynomial $g(t)$, the remainder when $g(t)$ is divided by $(t + 2)^2$ is $6t - 1$, and the remainder when $g(t)$ is divided by $(t -
q_t21_010P146HardMediumMedium0.00.498human_labelledreviewedLet $p(x) = x^4 + ax^3 + bx^2 + cx + d$, where $a, b, c, d \in \mathbb{R}$. It is given that $(x + 2)$ and $(x - 3)$ are factors of $p(x)$,
q_t21_011P149EasyEasyMedium0.00.002human_labelledreviewedFind the values of $a$, $b$, $c$, and $p$ such that $$x^3 + 3x^2 + 3x - 7 = a(x-p)^3 + b(x-p)^2 + c(x-p).$$
q_t21_012P150EasyMediumEasy0.00.499human_labelledreviewedThe equation $2x^2 - 6x + 3 = 0$ has roots $p$ and $q$. Find the value of $\dfrac{p}{q} + \dfrac{q}{p}$.
q_t21_013P147HardMediumMedium0.00.499human_labelledreviewedThe polynomial $h(t) = t^4 + pt^3 + qt^2 - 8t + r$ satisfies the following conditions: - When $h(t)$ is divided by $(t + 1)$, the remainder
q_t21_014P148EasyEasyEasy0.00.002human_labelledreviewedFor a polynomial $h(u)$, the remainder when $h(u)$ is divided by $(u - 4)^2$ is $3u - 5$, and the remainder when $h(u)$ is divided by $(u -
q_t21_015P146MediumMediumEasy0.00.002human_labelledreviewedFind the polynomial $h(x) = x^3 + ax^2 + bx + c$, where $a, b, c \in \mathbb{R}$, given that $(x + 3)$ is a factor of $h(x)$, that $h(2) = 2
q_t21_016P149MediumHardMedium0.00.499human_labelledreviewedDetermine the values of $a$, $b$, $c$, and $p$ such that $$2t^3 - 7t^2 + 2t + 3 = a(t - p)^3 + b(t - p)^2 + c(t - p).$$
q_t21_017P150MediumHardHard0.00.5human_labelledreviewedThe cubic equation $2x^3 - 3x^2 - 11x + 6 = 0$ has roots $\alpha$, $\beta$, and $\gamma$. Find the value of $\alpha^2\beta + \alpha^2\gamma
q_t21_018P147MediumMediumEasy0.00.003human_labelledreviewedThe polynomial $f(x) = x^3 + ax^2 - 3x + b$ satisfies the following conditions: - When $f(x)$ is divided by $(x - 2)$, the remainder is $6$
q_t21_019P148HardMediumHard0.00.498human_labelledreviewedA polynomial $p(x)$ satisfies the following two conditions: - the remainder when $p(x)$ is divided by $(2x - 1)^2$ is $8x - 3$, - the remain
q_t21_020P146MediumMediumMedium0.00.001human_labelledreviewedFind the polynomial $f(x) = x^3 + ax^2 + bx + c$, where $a, b, c \in \mathbb{R}$, given that $(x - 2)$ and $(x + 3)$ are factors of $f(x)$,
q_t21_021P149MediumMediumMedium0.00.001human_labelledreviewedFind the values of $a$, $b$, $c$, and $p$, where $p$ is a positive integer, such that $$x^3 - x^2 - 5x - 3 = a(x-p)^3 + b(x-p)^2 + c(x-p).$$
q_t21_022P150MediumMediumMedium0.00.004human_labelledreviewedThe cubic equation $x^3 - 6x^2 + kx - 10 = 0$, where $k$ is a real constant, has roots $\alpha$, $\beta$, and $\gamma$. Given that $\alpha^2
q_t21_023P147EasyMediumEasy0.00.499human_labelledreviewedThe polynomial $f(x) = x^2 + 6x + c$ is divided by $x - 2$. The remainder is 5. Find the value of $c$.
q_t21_024P148MediumHardMedium0.00.498human_labelledreviewedA polynomial $w(s)$ satisfies the following two conditions: - the remainder when $w(s)$ is divided by $(s - 5)^2$ is $7s - 3$, - the remaind
q_t21_025P146MediumMediumMedium0.00.001human_labelledreviewedLet $$f(x) = x^4 + ax^3 + bx^2 + cx - 6,$$ where $a, b, c \in \mathbb{R}$. It is given that $x - 1$ and $x + 2$ are factors of $f(x)$, and t
q_t21_026P148EasyEasyEasy0.00.003human_labelledreviewedFor a polynomial $m(y)$, the remainder when $m(y)$ is divided by $(y - 3)^2$ is $5y - 2$, and the remainder when $m(y)$ is divided by $(y +
q_t21_027P146MediumMediumMedium0.00.002human_labelledreviewedLet $q(x) = x^3 + ax^2 + bx + c$, where $a, b, c \in \mathbb{R}$. It is given that $(x - 3)$ is a factor of $q(x)$, that $(x + 1)$ is a fact
q_t21_028P149MediumMediumMedium0.00.001human_labelledreviewedFind the values of $a$, $b$, $c$, and $p$ such that $$x^3 - x^2 - x - 2 = a(x-p)^3 + b(x-p)^2 + c(x-p).$$
q_t21_029P147MediumMediumMedium0.00.005human_labelledreviewedThe polynomial $p(t) = 2t^3 - 5t^2 + at + b$ satisfies the following conditions: - When $p(t)$ is divided by $(t + 1)$, the remainder is $-
q_t21_030P150MediumMediumMedium0.00.006human_labelledreviewedThe polynomial $P(t) = 3t^3 + bt^2 + ct - 8$, where $b$ and $c$ are real constants, has roots $\alpha$, $\beta$, and $\gamma$. It is known t
q_t21_031P148EasyEasyEasy0.00.004human_labelledreviewedA polynomial $h(t)$ satisfies the following two conditions: - the remainder when $h(t)$ is divided by $(t - 3)^2$ is $2t + 7$, - the remain
q_t21_032P146EasyEasyEasy0.00.004human_labelledreviewedFind the polynomial $h(x) = x^3 + ax^2 + bx - 10$, where $a, b \in \mathbb{R}$, given that $(x - 2)$ is a factor of $h(x)$ and that $h(-1) =
q_t21_033P149MediumMediumMedium0.00.004human_labelledreviewedDetermine the values of $a$, $b$, $c$, and $p$ such that $$s^3 - 3s^2 - 10s + 24 = a(s - p)^3 + b(s - p)^2 + c(s - p),$$ where $p$ is a ne
q_t21_034P147MediumMediumMedium0.00.004human_labelledreviewedThe polynomial $g(u) = 2u^3 - u^2 + pu + q$ satisfies the following conditions: - When $g(u)$ is divided by $(u + 2)$, the remainder is $-3
q_t21_035P150MediumMediumMedium0.00.004human_labelledreviewedThe cubic polynomial $P(x) = x^3 + ax^2 - 7x + b$, where $a$ and $b$ are real constants, has roots $\alpha$, $\beta$, and $\gamma$. Given th
q_t21_036P148EasyEasyEasy0.00.003human_labelledreviewedA polynomial $g(n)$ satisfies the following two conditions: - the remainder when $g(n)$ is divided by $(n + 2)^2$ is $3n - 1$, - the remain
q_t21_037P146HardHardMedium0.00.003human_labelledreviewedLet $g(x) = x^4 + ax^3 + bx^2 + cx + d$, where $a, b, c, d \in \mathbb{R}$. It is given that $(x - 2)$ is a repeated factor of $g(x)$, that
q_t21_038P149HardMediumMedium0.00.498human_labelledreviewedDetermine the values of $a$, $b$, $c$, and $p$ such that $$2t^3 - 7t^2 - 17t + 10 = a(t - p)^3 + b(t - p)^2 + c(t - p),$$ where $p$ is an
q_t21_039P147MediumMediumMedium0.00.006human_labelledreviewedThe polynomial $h(s) = s^3 - 4s^2 + ms + n$ satisfies the following conditions: - When $h(s)$ is divided by $(s + 2)$, the remainder is $-2
q_t21_040P150HardHardHard0.00.004human_labelledreviewedThe polynomial $P(x) = 3x^4 + ax^3 + bx^2 + 39x - 30$, where $a, b \in \mathbb{R}$, has a root $z = 2 + i$. (a) Write down another root of
q_t21_041P149MediumMediumMedium0.00.0human_labelledreviewedDetermine the values of $a$, $b$, $c$, and $p$ such that $$2u^3 - 9u^2 + 7u + 6 = a(u - p)^3 + b(u - p)^2 + c(u - p),$$ where $p$ is an in
q_t21_042P146HardMediumHard0.00.5human_labelledreviewedLet $p(x) = x^4 + ax^3 + bx^2 + cx + d$, where $a, b, c, d \in \mathbb{R}$. It is given that $(x + 1)$ is a repeated factor of $p(x)$, that
q_t21_043P150EasyMediumEasy0.00.5human_labelledreviewedThe equation $2t^2 - 8t + 5 = 0$ has roots $m$ and $n$. Find the value of $(m - n)^2$.
q_t21_044P147MediumMediumMedium0.00.0human_labelledreviewedThe polynomial $p(t) = 2t^3 + at^2 + bt - 6$ satisfies the following conditions: - When $p(t)$ is divided by $(t - 3)$, the remainder is $2
q_t21_045P148MediumMediumMedium0.00.0human_labelledreviewedA polynomial $p(x)$ is known to satisfy two conditions: - when $p(x)$ is divided by $(2x - 1)^2$, the remainder is $6x - 4$, - when $p(x)$
q_t21_046P149MediumMediumMedium0.00.0human_labelledreviewedFind the values of $a$, $b$, $c$, and $p$ such that $$x^3 - x^2 - x - 2 = a(x-p)^3 + b(x-p)^2 + c(x-p).$$
q_t21_047P146EasyEasyEasy0.00.0human_labelledreviewedFind the polynomial $p(x) = x^3 + ax^2 + bx + c$, where $a, b, c \in \mathbb{R}$, given that $(x - 2)$ is a factor of $p(x)$, that $p(0) = 6
q_t21_048P150EasyMediumEasy0.00.5human_labelledreviewedThe equation $3x^2 - 7x + 1 = 0$ has roots $\alpha$ and $\beta$. The equation $x^2 + px + q = 0$ has roots $2\alpha$ and $2\beta$. Find the
q_t21_049P147HardMediumMedium0.00.5human_labelledreviewedThe polynomial $f(x) = x^4 - 3x^3 + ax^2 + bx + c$ satisfies the following conditions: - When $f(x)$ is divided by $(x - 2)$, the remainder
q_t21_050P148MediumHardMedium0.00.5human_labelledreviewedA polynomial $k(t)$ satisfies the following two conditions: - the remainder when $k(t)$ is divided by $(t + 3)^2$ is $5t - 7$, - the remain