Mathematics 题库

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历年真题风格题库（原始题库，可參考）

Paper 1 常规题 (65%, 2h15m) · Paper 2 MCQ (35%, 1h15m, 45题)。所有计算题附完整步骤与答案。 商科重点：复利、百分比链、统计图表英文描述。完成后交 模考阅卷/批改指令。

Part 1 Number & Algebra（Q1–Q30）

Q1 · Basic · Number

Evaluate: 3² + 4 × 2 − 5. Answer: 3²=9; 4×2=8; 9+8−5=12. Marking: order of operations (BIDMAS): 12.

Q2 · Basic · Fraction

Simplify: 2/3 + 1/6. Answer: 4/6 + 1/6 = 5/6. Marking: common denominator 6.

Q3 · Drill · Percentage

A shirt costs $400, now 25% off. Sale price? Answer: 400 × (1−0.25) = $300. Marking: 400×0.75=300.

Q4 · Drill · Percentage change

Price rose from $80 to $100. % increase? Answer: (100−80)/80 × 100% = 25%. Marking: change/original.

Q5 · Drill · Ratio

Divide $900 in ratio 2:3:4. Answer: Total 9 parts; 1 part=$100; → $200, $300, $400. Marking: 2+3+4=9.

Q6 · Basic · Indices

Simplify: x³ · x⁴. Answer: x⁷. Marking: add exponents.

Q7 · Drill · Indices

Evaluate: (2³)². Answer: 2⁶ = 64. Marking: multiply exponents.

Q8 · Basic · Standard form

Write 0.00045 in standard form. Answer: 4.5 × 10⁻⁴. Marking: move decimal 4 places.

Q9 · Drill · Scientific calc

Evaluate 1.2 × 10⁵ ÷ (3 × 10²). Answer: (1.2/3) × 10³ = 0.4 × 10³ = 4 × 10². Marking: 400.

Q10 · Basic · Algebraic expansion

Expand: 3(x + 2) − 2(x − 1). Answer: 3x+6−2x+2 = x+8. Marking: distribute, collect.

Q11 · Drill · Factorise

Factorise: x² − 9. Answer: (x−3)(x+3) [difference of squares]. Marking: a²−b².

Q12 · Drill · Solve linear

Solve: 2x + 5 = 17. Answer: 2x=12 → x=6. Marking: isolate x.

Q13 · Drill · Simultaneous

Solve: x + y = 10, x − y = 4. Answer: Add: 2x=14 → x=7; y=3. Marking: elimination.

Q14 · Basic · Sequence

Arithmetic sequence: 3, 7, 11, … Find 10th term. Answer: a=3, d=4; T₁₀ = 3 + 9×4 = 39. Marking: Tₙ = a+(n−1)d.

Q15 · Drill · Sequence sum

Sum first 20 terms of 2,5,8,…? Answer: a=2,d=3; S₂₀ = 20/2[2×2 + 19×3] = 10[4+57]=610. Marking: Sₙ=n/2[2a+(n−1)d].

Q16 · Basic · Interest (simple)

$5000 at 4% p.a. simple for 3 years. Interest? Answer: I = Prt = 5000×0.04×3 = $600. Marking: simple interest formula.

Q17 · Drill · Compound interest

$8000 at 5% p.a. compounded annually for 2 years. Amount? Answer: A = 8000(1.05)² = 8000×1.1025 = $8820. Marking: A=P(1+r)ⁿ.

Q18 · Drill · Compound (more periods)

$10,000 at 6% p.a. compounded monthly for 1 year. Answer: A = 10000(1+0.06/12)¹² = 10000(1.005)¹² ≈ $10,616.78. Marking: r/n, nt.

Q19 · Basic · Profit/Loss

CP $120, SP $150. Profit %? Answer: 30/120 ×100% = 25%. Marking: profit/cost.

Q20 · Drill · Discount chain

A $200 item gets 10% off, then extra 5% off. Final price? Answer: 200×0.9=180; 180×0.95=$171. Marking: successive multipliers.

Q21 · Basic · Surds

Simplify: √50. Answer: √(25×2) = 5√2. Marking: factor perfect square.

Q22 · Drill · Logarithm

Solve: 2ˣ = 32. Answer: 2ˣ = 2⁵ → x=5. Marking: same base.

Q23 · Basic · Polynomial

If x=2 is a root of x² + ax − 6 = 0, find a. Answer: 4 + 2a − 6 = 0 → 2a=2 → a=1. Marking: substitute root.

Q24 · Drill · Inequality

Solve: 3x − 4 > 8. Answer: 3x > 12 → x > 4. Marking: keep inequality direction.

Q25 · Basic · Variation

y ∝ x². When x=3, y=18. Find y when x=5. Answer: y=kx²; 18=k×9 → k=2; y=2×25=50. Marking: direct square variation.

Q26 · Drill · Inverse variation

y ∝ 1/x. x=4 → y=3. Find y when x=6. Answer: y=k/x; 3=k/4 → k=12; y=12/6=2. Marking: inverse.

Q27 · Basic · Rationalise

Simplify 1/√2. Answer: √2/2. Marking: multiply numerator/denominator by √2.

Q28 · Drill · Significant figures

Round 0.04738 to 3 s.f. Answer: 0.0474. Marking: first non-zero digit counts.

Q29 · Basic · Sets

If n(A)=12, n(B)=8, n(A∩B)=3, find n(A∪B). Answer: 12+8−3 = 17. Marking: inclusion-exclusion.

Q30 · Drill · Remainder

Find remainder when x³−2x+1 divided by (x−1). Answer: f(1)=1−2+1=0 → remainder 0 (x−1 is factor). Marking: remainder theorem.

Part 2 Measures, Finance & Functions（Q31–Q60）

Q31 · Basic · Area

Rectangle 8 cm by 5 cm. Area? Answer: 40 cm². Marking: length×width.

Q32 · Drill · Circle

Circle radius 7 cm. Circumference (π=22/7)? Answer: 2πr = 2×(22/7)×7 = 44 cm. Marking: C=2πr.

Q33 · Drill · Circle area

Area of circle radius 7 cm (π=22/7)? Answer: πr² = (22/7)×49 = 154 cm². Marking: A=πr².

Q34 · Basic · Volume

Cuboid 3×4×5 cm. Volume? Answer: 60 cm³. Marking: l×w×h.

Q35 · Drill · Cylinder

Cylinder r=3 cm, h=10 cm. Volume (π=3.14)? Answer: πr²h = 3.14×9×10 = 282.6 cm³. Marking: V=πr²h.

Q36 · Basic · Speed

Distance 240 km, time 3 h. Average speed? Answer: 240/3 = 80 km/h. Marking: speed = distance/time.

Q37 · Drill · Distance-time

A train travels 90 km in 1.5 h, then 60 km in 1 h. Avg speed whole journey? Answer: total 150 km / 2.5 h = 60 km/h. Marking: total dist/total time.

Q38 · Basic · Exchange rate

HK$7.8 = US$1. How many US$ for HK$780? Answer: 780/7.8 = US$100. Marking: divide by rate.

Q39 · Drill · Tax

Income $300,000; first $100,000 tax-free, rest at 15%. Tax? Answer: 0.15 × 200,000 = $30,000. Marking: progressive slice.

Q40 · Basic · Gradient

Line through (1,2) and (4,8). Gradient? Answer: (8−2)/(4−1) = 6/3 = 2. Marking: rise/run.

Q41 · Drill · Midpoint

Midpoint of (2,3) and (8,7)? Answer: ((2+8)/2, (3+7)/2) = (5,5). Marking: average coordinates.

Q42 · Basic · Function

If f(x) = 2x + 1, find f(3). Answer: 2×3+1 = 7. Marking: substitute.

Q43 · Drill · Function inverse

f(x)=3x−2. Find f⁻¹(x). Answer: y=3x−2 → x=(y+2)/3 → f⁻¹(x)=(x+2)/3. Marking: swap, solve.

Q44 · Basic · Linear graph

Line y = 2x + 3. y-intercept? Answer: 3 (at x=0). Marking: c in y=mx+c.

Q45 · Drill · Quadratic vertex

y = x² − 4x + 3. Vertex x-coordinate? Answer: −b/2a = 4/2 = 2. Marking: vertex formula.

Q46 · Basic · Solve quadratic

Solve x² − 5x + 6 = 0. Answer: (x−2)(x−3)=0 → x=2,3. Marking: factorise.

Q47 · Drill · Quadratic formula

Solve 2x² + 3x − 2 = 0. Answer: x = [−3 ± √(9+16)]/4 = [−3 ±5]/4 → x=0.5 or −2. Marking: discriminant 25.

Q48 · Basic · Exponential graph

y = 2ˣ. As x→−∞, y→ ___. Answer: 0 (x-axis asymptote). Marking: decays to 0.

Q49 · Drill · Log equation

Solve log₂(x) = 5. Answer: x = 2⁵ = 32. Marking: exponentiate.

Q50 · Basic · Distance formula

Distance between (0,0) and (3,4)? Answer: √(3²+4²)=5. Marking: Pythagoras.

Q51 · Drill · Angle sum

Interior angle sum of a pentagon? Answer: (5−2)×180 = 540°. Marking: (n−2)×180.

Q52 · Basic · Exterior angle

Regular hexagon exterior angle? Answer: 360/6 = 60°. Marking: 360/n.

Q53 · Drill · Similar triangles

Two similar triangles, sides ratio 1:3. Area ratio? Answer: 1²:3² = 1:9. Marking: area ∝ scale².

Q54 · Basic · Pythagoras

Right triangle legs 5,12. Hypotenuse? Answer: √(25+144)=13. Marking: 13-12-5 triple.

Q55 · Drill · Trigonometry

In right triangle, sin θ = 3/5. cos θ = ? Answer: cos θ = 4/5 (3-4-5 triangle). Marking: sin²+cos²=1.

Q56 · Basic · Trig ratio

tan 45° = ? Answer: 1. Marking: standard value.

Q57 · Drill · Sine rule

Triangle: a/sin A = b/sin B. Given a=10, A=30°, B=45°, find b. Answer: b = 10·sin45/sin30 = 10×(√2/2)/(0.5) = 10√2 ≈14.14. Marking: sine rule.

Q58 · Basic · Bearings

A point is on bearing 090° from O. Direction? Answer: East. Marking: 090 = east.

Q59 · Drill · Symmetry

A square has ___ lines of symmetry. Answer: 4. Marking: 2 diagonals + 2 midlines.

Q60 · Basic · Transformations

Translation (x,y)→(x+3,y−2) moves point right 3 and ___ 2. Answer: down 2. Marking: y decrease = down.

Part 3 Statistics & Probability（Q61–Q90）

Q61 · Basic · Mean

Data: 4,6,8,10. Mean? Answer: 28/4 = 7. Marking: sum/n.

Q62 · Drill · Median

Data: 3,7,9,11,15. Median? Answer: middle = 9. Marking: ordered middle.

Q63 · Drill · Mode

Data: 2,3,3,5,7,3. Mode? Answer: 3. Marking: most frequent.

Q64 · Basic · Range

Data: 5,9,2,11. Range? Answer: 11−2 = 9. Marking: max−min.

Q65 · Drill · Standard deviation concept

Which dataset is more spread? A:{5,5,5,5} or B:{1,5,5,9}? Answer: B (has variation; A sd=0). Marking: spread = variability.

Q66 · Basic · Probability

Fair die: P(rolling a 4)? Answer: 1/6. Marking: one of six.

Q67 · Drill · Probability OR

P(A)=0.3, P(B)=0.4, mutually exclusive. P(A or B)? Answer: 0.3+0.4 = 0.7. Marking: add for ME.

Q68 · Drill · Probability AND

P(A)=0.5, P(B|A)=0.2. P(A and B)? Answer: 0.5×0.2 = 0.1. Marking: P(A∩B)=P(A)P(B|A).

Q69 · Basic · Tree diagram

Bag: 2 red, 3 blue. Draw one, replace, draw again. P(red then blue)? Answer: (2/5)×(3/5) = 6/25 = 0.24. Marking: with replacement.

Q70 · Drill · Without replacement

Bag 2 red, 3 blue. Draw 2 without replacement. P(both red)? Answer: (2/5)×(1/4) = 2/20 = 1/10. Marking: dependence.

Q71 · Basic · Expected value

Game: win $10 with prob 0.2, lose $2 with prob 0.8. Expected gain? Answer: 10×0.2 + (−2)×0.8 = 2 − 1.6 = $0.40. Marking: Σ x·P(x).

Q72 · Drill · Histogram

A histogram bar for class 10–20 has frequency density 3 with width 10. Frequency? Answer: fd × width = 3×10 = 30. Marking: freq = fd × class width.

Q73 · Basic · Pie chart

A category is 25% of total. Angle in pie? Answer: 0.25×360 = 90°. Marking: percentage × 360.

Q74 · Drill · Cumulative frequency

CF at upper boundary gives count ___ that value. Answer: less than or equal to. Marking: cumulative = "up to".

Q75 · Basic · Quartiles

For n=20 ordered data, position of lower quartile? Answer: (n+1)/4 = 21/4 = 5.25th value. Marking: quartile position.

Q76 · Drill · Correlation

Scatter shows points rising left→right. Correlation? Answer: Positive. Marking: direction of trend.

Q77 · Basic · Outlier

In {2,3,4,5,100}, the outlier is: Answer: 100. Marking: extreme value.

Q78 · Drill · Stem-leaf

Stem 2 | leaves 3,5,7 represents values ___. Answer: 23,25,27. Marking: stem= tens.

Q79 · Basic · Permutation

Arrange 3 different books in a row: ways? Answer: 3! = 6. Marking: factorial.

Q80 · Drill · Combination

Choose 2 from 5: C(5,2)? Answer: 5!/(2!3!) = 10. Marking: combination.

Q81 · Basic · Box plot

Box plot: median inside box, whiskers to min/max. It shows ___. Answer: five-number summary (min, Q1, median, Q3, max). Marking: distribution shape.

Q82 · Drill · Percentile

If you are 80th percentile in height, you are taller than ___% of peers. Answer: 80%. Marking: percentile definition.

Q83 · Basic · Independent events

P(A)=0.5, P(B)=0.3, independent. P(A∩B)? Answer: 0.5×0.3 = 0.15. Marking: independent multiply.

Q84 · Drill · Conditional

P(A|B) = P(A∩B)/P(B). If P(A∩B)=0.1, P(B)=0.25, P(A|B)=? Answer: 0.1/0.25 = 0.4 ×... = 0.4. (0.1/0.25=0.4) Marking: conditional formula.

Q85 · Basic · Biased die

Die lands 6 with prob 0.3 (others equal). P(6)? Answer: 0.3. Marking: given.

Q86 · Drill · Normal approx (concept)

In a normal distribution, about ___% lie within 1 sd of mean. Answer: ~68%. Marking: empirical rule.

Q87 · Basic · Sampling

A sample of every 10th student is ___ sampling. Answer: systematic. Marking: fixed interval.

Q88 · Drill · Bias

A survey only online excludes non-internet users → ___ bias. Answer: selection bias. Marking: non-representative sample.

Q89 · Basic · Two-way table

Table: 30 boys, 20 like sport (12 boys). Girls who like sport? Answer: total like=20; boys like=12 → girls like=8 (of 30−? girls total? if 50 students: girls=20, boys=30; girls like=8). Marking: subtract.

Q90 · Drill · Index numbers

Price index 2010=100, 2020=140. Price rose ___%. Answer: 40%. Marking: (140−100)/100.

Part 4 Calculus & Applied（Q91–Q110）

Q91 · Basic · Derivative

If y = x³, dy/dx = ? Answer: 3x². Marking: power rule.

Q92 · Drill · Derivative

y = 2x² + 5x − 3. dy/dx at x=1? Answer: dy/dx = 4x+5; at x=1 = 9. Marking: differentiate, substitute.

Q93 · Basic · Stationary point

y = x² − 4x. Stationary at x=? Answer: dy/dx = 2x−4 =0 → x=2. Marking: set derivative 0.

Q94 · Drill · Second derivative

y = x³. d²y/dx² = ? Answer: 6x. Marking: differentiate twice.

Q95 · Basic · Integral

∫ 3x² dx = ? Answer: x³ + C. Marking: reverse power rule.

Q96 · Drill · Definite integral

∫₀² 2x dx = ? Answer: [x²]₀² = 4 − 0 = 4. Marking: evaluate bounds.

Q97 · Basic · Economics application

TR = 100q − 2q². MR = d(TR)/dq. MR = ? Answer: 100 − 4q. Marking: marginal revenue.

Q98 · Drill · Max revenue

With TR = 100q − 2q², max revenue at q=? Answer: MR=0 → 100−4q=0 → q=25. Marking: revenue max where MR=0.

Q99 · Basic · Gradient of tangent

Curve y=x² at x=3. Gradient of tangent? Answer: dy/dx=2x=6. Marking: derivative gives tangent slope.

Q100 · Drill · Area under curve

∫₀¹ x dx = ? Answer: [x²/2]₀¹ = 1/2. Marking: area triangle = 0.5.

Q101 · Structured · Finance word problem

David invests $20,000 at 4% p.a. compounded quarterly for 3 years. Find the final amount and the interest earned. Answer: A = 20000(1+0.04/4)^(4×3) = 20000(1.01)^12 ≈ 20000×1.126825 = $22,536.50. Interest = 22,536.50 − 20,000 = $2,536.50. Marking: identify r/n and nt (1); compute amount (2); subtract principal (1).

Q102 · Structured · Statistics report (English)

Given dataset {10,12,14,14,16,18,20}, find mean and comment on distribution in one English sentence. Answer: Mean = 104/7 ≈ 14.86. "The data are slightly right-skewed with a mean of about 14.9 and a mode of 14." Marking: mean (1); correct English description with skew/centre (2-3).

Q103 · Structured · Geometry

A cone has base radius 3 cm and height 4 cm. Find slant height and volume (π=3.14). Answer: l = √(3²+4²)=5 cm. V = (1/3)πr²h = (1/3)×3.14×9×4 = 37.68 cm³. Marking: slant (1); volume formula (1); compute (1).

Q104 · Structured · Probability tree

A box has 4 red, 6 green. Two drawn with replacement. Find P(exactly one red). Answer: P(RG)+P(GR) = (4/10)(6/10)+(6/10)(4/10) = 24/100+24/100 = 48/100 = 0.48. Marking: two branches (2); sum (1).

Q105 · Structured · Simultaneous (word)

Tickets: adult $x, child $y. 2 adults + 3 children cost $110; 1 adult + 2 children cost $65. Find x,y. Answer: 2x+3y=110; x+2y=65 → x=65−2y; 2(65−2y)+3y=110 → 130−y=110 → y=20; x=25. Marking: form equations (1); solve (2).

Q106 · Structured · Trigonometry (3D)

A 10 m ladder leans on a wall, base 6 m from wall. Height reached? Answer: h = √(10²−6²)=√64=8 m. Marking: Pythagoras.

Q107 · Structured · Linear programming (concept)

Objective z=3x+2y, constraints x≥0,y≥0,x+y≤10. Max z? Answer: Max at corner (10,0): z=30; or (0,10): z=20 → max 30. Marking: corners (2); evaluate (1).

Q108 · Structured · Graph interpretation (English)

Describe the trend shown: sales rose from 100 to 250 units over 5 years. Answer: "Sales increased steadily from 100 units in Year 1 to 250 units in Year 5, more than doubling over the period." Marking: accurate trend + formal English (2-3).

Q109 · Structured · Variation application

Cost C varies jointly as workers w and hours h: C = kwh. C=$600 for 3 workers, 4 hours. Cost for 5 workers, 6 hours? Answer: 600=k×3×4 → k=50. C=50×5×6 = $1500. Marking: find k (1); apply (2).

Q110 · Integrated · Mock-style long question

A company's profit P (in $1000) is modelled by P = −2q² + 40q − 100, where q is output in hundreds. (a) Find output for max profit. (b) Max profit. (c) Outputs giving zero profit. Answer: (a) dP/dq = −4q+40=0 → q=10 (hundred). (b) P(10)=−200+400−100=100 → $100,000. (c) −2q²+40q−100=0 → divide −2: q²−20q+50=0 → q = [20±√(400−200)]/2 = [20±√200]/2 = 10±7.07 → q≈2.93 or 17.07. Marking: derivative/max (2); substitute (1); solve quadratic (2); interpret (1).

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商科方向重点练 Q16–Q20（利息/折扣）、Q91–Q100（边际/最优化）。完成后用 模考阅卷指令 生成失分报告。