題目 1 · 選擇題
1 分Differentiate \(y=\ln(3x)\). Circle your answer.
- A.\(\tfrac{3}{x}\)
- B.\(\tfrac{1}{x}\)
- C.\(\tfrac{1}{3x}\)
- D.\(\ln 3\)
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解題
\(\tfrac{d}{dx}\ln(3x)=\tfrac{3}{3x}=\tfrac1x\).
評分準則
B1 for \(\tfrac1x\).
AQA A-Level · Thinka 原創模擬試題
2022 AQA A-Level Mathematics 7357 模擬試題連答案詳解
An original Thinka practice paper modelled on the structure and difficulty of the Jun 2022 AQA A Level Mathematics 7357 paper. Not affiliated with or reproduced from AQA.
卷一: Pure Mathematics
Answer all questions. Entirely pure content; calculator and AQA formulae booklet permitted.
11 題目 · 100 分
題目 2 · 選擇題
1 分Find \(\displaystyle\int e^{2x}\,dx\). Circle your answer.
- A.\(2e^{2x}+c\)
- B.\(\tfrac12 e^{2x}+c\)
- C.\(e^{2x}+c\)
- D.\(2e^{x}+c\)
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解題
\(\int e^{2x}dx=\tfrac12 e^{2x}+c\).
評分準則
B1 for \(\tfrac12 e^{2x}+c\).
題目 3 · Short
5 分Find the first three terms, in ascending powers of \(x\), of the binomial expansion of \((2-3x)^7\).
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解題
\((2-3x)^7=2^7+\binom71 2^6(-3x)+\binom72 2^5(-3x)^2+\dots=128-7(64)(3)x+21(32)(9)x^2=128-1344x+6048x^2\).
評分準則
M1 binomial structure; A1 128; A1 -1344x; M1 third-term values; A1 6048x².
題目 4 · Structured
12 分A circle \(C\) has equation \(x^2+y^2-6x+4y-12=0\). (a) Find the centre and radius. (b) Show that the point \((7,1)\) lies on \(C\). (c) Find the equation of the tangent to \(C\) at \((7,1)\). (d) Find the length of the tangent to \(C\) drawn from the external point \((10,6)\).
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解題
(a) \((x-3)^2+(y+2)^2=25\): centre \((3,-2)\), radius \(5\). (b) \(49+1-42+4-12=0\) ✓. (c) Radius to \((7,1)\) has gradient \(\tfrac{1-(-2)}{7-3}=\tfrac34\), so tangent gradient \(-\tfrac43\): \(y-1=-\tfrac43(x-7)\Rightarrow 4x+3y=31\). (d) Tangent length \(=\sqrt{(10-3)^2+(6+2)^2-5^2}=\sqrt{49+64-25}=\sqrt{88}=9.38\).
評分準則
M1A1 centre & radius; M1A1 (b); M1 radius gradient; M1 tangent gradient; A1 tangent equation; M1A1 tangent length.
題目 5 · Structured
10 分(a) An arithmetic series has first term 5 and common difference 3. Find (i) the 20th term and (ii) the sum of the first 20 terms. (b) A geometric series has first term 5 and common ratio 1.1. Find (i) the 20th term and (ii) the sum of the first 20 terms. (c) State, with a reason, whether the geometric series has a sum to infinity.
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解題
(a)(i) \(5+19(3)=62\); (ii) \(S_{20}=\tfrac{20}{2}(10+57)=670\). (b)(i) \(5(1.1)^{19}=30.6\); (ii) \(S_{20}=5\cdot\tfrac{1.1^{20}-1}{0.1}=286\). (c) No: \(|r|=1.1>1\), so the series diverges.
評分準則
M1A1 (a)(i,ii); M1A1 (b)(i,ii); M1A1 (b)(ii); B1 (c) with reason.
題目 6 · Structured
12 分\(f(x)=x^3-5x+1\). (a) Show that a root of \(f(x)=0\) lies between 2 and 3. (b) Apply the Newton–Raphson method with \(x_0=2\) to find \(x_1\) and \(x_2\). (c) The iteration \(x_{n+1}=\sqrt[3]{5x_n-1}\) is used with \(x_0=2\). Find the root to 3 decimal places.
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解題
(a) \(f(2)=-1<0\), \(f(3)=13>0\): a sign change implies a root in \((2,3)\). (b) \(f'(x)=3x^2-5\); \(x_1=2-\tfrac{-1}{7}=2.143\); \(x_2=2.143-\tfrac{f(2.143)}{f'(2.143)}=2.129\). (c) \(x_1=\sqrt[3]{9}=2.080,\;x_2=2.111,\;x_3=2.123,\;x_4=2.128,\dots\to 2.128\).
評分準則
M1 evaluate ends; A1 sign-change conclusion; M1 NR formula; A1 x₁; A1 x₂; M1 iterate; A1 2.128.
題目 7 · Structured
10 分\(p(x)=2x^3+ax^2-7x+b\). Given that \((x-1)\) and \((x+2)\) are factors of \(p(x)\): (a) find \(a\) and \(b\); (b) factorise \(p(x)\) completely; (c) solve \(p(x)=0\).
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解題
(a) \(p(1)=2+a-7+b=0\Rightarrow a+b=5\); \(p(-2)=-16+4a+14+b=0\Rightarrow4a+b=2\). Subtracting, \(3a=-3\Rightarrow a=-1,\,b=6\). (b) \(p(x)=2x^3-x^2-7x+6=(x-1)(x+2)(2x-3)\). (c) \(x=1,-2,\tfrac32\).
評分準則
M1 p(1)=0; M1 p(-2)=0; A1 a,b; M1 divide; A1 full factorisation; A1 three roots.
題目 8 · Structured
8 分(a) A geometric series has first term 8 and common ratio \(\tfrac34\). Find its sum to infinity. (b) Solve \(\tan 2x=\sqrt3\) for \(0\le x\le\pi\). (c) Solve \(3\sin x=2\cos x\) for \(0\le x\le2\pi\), giving answers to 2 decimal places.
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解題
(a) \(\tfrac{8}{1-3/4}=32\). (b) \(2x=\tfrac{\pi}{3},\tfrac{4\pi}{3}\Rightarrow x=\tfrac{\pi}{6},\tfrac{2\pi}{3}\). (c) \(\tan x=\tfrac23\Rightarrow x=0.588\) or \(0.588+\pi=3.73\).
評分準則
B1 (a); M1 2x values; A1 both x; M1 tan x; A1 0.59; A1 3.73.
題目 9 · Structured
10 分(a) Prove the identity \(\dfrac{\sin\theta}{1+\cos\theta}+\dfrac{1+\cos\theta}{\sin\theta}\equiv\dfrac{2}{\sin\theta}\). (b) Hence solve \(\dfrac{\sin\theta}{1+\cos\theta}+\dfrac{1+\cos\theta}{\sin\theta}=4\) for \(0<\theta<2\pi\). (c) Show that \(\dfrac{1-\cos2x}{\sin2x}\equiv\tan x\).
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解題
(a) Common denominator: \(\tfrac{\sin^2\theta+(1+\cos\theta)^2}{\sin\theta(1+\cos\theta)}=\tfrac{\sin^2\theta+1+2\cos\theta+\cos^2\theta}{\sin\theta(1+\cos\theta)}=\tfrac{2+2\cos\theta}{\sin\theta(1+\cos\theta)}=\tfrac{2}{\sin\theta}\). (b) \(\tfrac{2}{\sin\theta}=4\Rightarrow\sin\theta=\tfrac12\Rightarrow\theta=\tfrac{\pi}{6},\tfrac{5\pi}{6}\). (c) \(\tfrac{1-\cos2x}{\sin2x}=\tfrac{2\sin^2x}{2\sin x\cos x}=\tan x\).
評分準則
M1 combine fractions; M1 use \(\sin^2+\cos^2=1\); A1 result; M1A1 (b); M1A1 (c).
題目 10 · Structured
15 分(a) Find \(\displaystyle\int\left(4x^3-\tfrac{2}{x^2}+e^{2x}\right)dx\). (b) Use the substitution \(u=x^2+1\) to find \(\displaystyle\int 2x\sqrt{x^2+1}\,dx\). (c) Find the area enclosed between the curves \(y=x^2\) and \(y=2x\).
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解題
(a) \(x^4+\tfrac{2}{x}+\tfrac12 e^{2x}+c\). (b) \(u=x^2+1,\,du=2x\,dx\Rightarrow\int\sqrt u\,du=\tfrac23 u^{3/2}=\tfrac23(x^2+1)^{3/2}+c\). (c) Intersect at \(x=0,2\); area \(=\int_0^2(2x-x^2)dx=[x^2-\tfrac{x^3}{3}]_0^2=4-\tfrac83=\tfrac43\).
評分準則
M1A1A1 (a); M1 substitution; A1 (b); M1 limits 0,2; M1 integrate difference; A1 4/3.
題目 11 · Structured
16 分A curve has equation \(y=x^3-3x^2-9x+5\). (a) Find \(\dfrac{dy}{dx}\). (b) Find the coordinates of the stationary points and determine their nature. (c) Find the coordinates of the point of inflection. (d) Find the equation of the tangent at \(x=-1\). (e) State the set of values of \(x\) for which the curve is increasing.
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解題
(a) \(\tfrac{dy}{dx}=3x^2-6x-9=3(x-3)(x+1)\). (b) \(x=-1\Rightarrow(−1,10)\); \(x=3\Rightarrow(3,-22)\). \(\tfrac{d^2y}{dx^2}=6x-6\): at \(x=-1\) negative (max), at \(x=3\) positive (min). (c) \(6x-6=0\Rightarrow x=1,\,y=-6\). (d) \(x=-1\) is a stationary point, so the tangent is \(y=10\). (e) Increasing where \(\tfrac{dy}{dx}>0\): \(x<-1\) or \(x>3\).
評分準則
M1A1 derivative; M1 stationary; A1 both points; M1A1 natures; M1A1 inflection; B1 tangent; M1A1 increasing set.
卷二: Pure & Mechanics
Answer all questions. Section A is pure; Section B is mechanics (take g = 9.8 m s⁻²).
14 題目 · 100 分
題目 1 · 選擇題
1 分Find \(\displaystyle\int\tfrac1x\,dx\). Circle your answer.
- A.\(\ln|x|+c\)
- B.\(-\tfrac{1}{x^2}+c\)
- C.\(\tfrac{1}{2}x^2+c\)
- D.\(x\ln x+c\)
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解題
\(\int\tfrac1x dx=\ln|x|+c\).
評分準則
B1 for \(\ln|x|+c\).
題目 2 · Short
3 分Express \(\dfrac{5x-7}{(x-1)(x-3)}\) in partial fractions.
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解題
Let \(\tfrac{5x-7}{(x-1)(x-3)}=\tfrac{A}{x-1}+\tfrac{B}{x-3}\). Then \(5x-7=A(x-3)+B(x-1)\). \(x=1:-2=-2A\Rightarrow A=1\); \(x=3:8=2B\Rightarrow B=4\).
評分準則
M1 set up identity; M1 substitute values; A1 A=1, B=4.
題目 3 · Proof
6 分Prove by contradiction that \(\sqrt3\) is irrational.
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解題
Assume \(\sqrt3=\tfrac pq\) in lowest terms with \(q\ne0\). Then \(p^2=3q^2\), so \(3\mid p^2\Rightarrow3\mid p\). Write \(p=3k\): \(9k^2=3q^2\Rightarrow q^2=3k^2\), so \(3\mid q\). Then \(3\) divides both \(p\) and \(q\), contradicting lowest terms. Hence \(\sqrt3\) is irrational.
評分準則
B1 assume rational in lowest terms; M1 \(p^2=3q^2\); A1 3|p; M1 substitute; A1 3|q; A1 contradiction.
題目 4 · Structured
10 分An open-topped box has a square base of side \(x\,\text{cm}\) and volume \(500\,\text{cm}^3\). (a) Show that the external surface area is \(S=x^2+\dfrac{2000}{x}\). (b) Find the value of \(x\) that minimises \(S\). (c) Find the minimum surface area.
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解題
(a) Height \(h=\tfrac{500}{x^2}\); \(S=x^2+4xh=x^2+\tfrac{2000}{x}\). (b) \(\tfrac{dS}{dx}=2x-\tfrac{2000}{x^2}=0\Rightarrow x^3=1000\Rightarrow x=10\). (c) \(S=100+200=300\,\text{cm}^2\) (and \(\tfrac{d^2S}{dx^2}>0\) confirms a minimum).
評分準則
M1 height; A1 show S; M1 differentiate; M1 solve =0; A1 x=10; A1 S=300.
題目 5 · Structured
10 分(a) A population grows so that \(\dfrac{dP}{dt}=0.5P\). Show that \(P=Ae^{0.5t}\). (b) Given \(P=200\) when \(t=0\), find \(P\) when \(t=4\). (c) Solve the differential equation \(\dfrac{dy}{dx}=6x^2y\), giving \(y\) in terms of \(x\).
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解題
(a) \(\int\tfrac{dP}{P}=\int0.5\,dt\Rightarrow\ln P=0.5t+c\Rightarrow P=Ae^{0.5t}\). (b) \(A=200\Rightarrow P=200e^{2}=1478\). (c) \(\int\tfrac{dy}{y}=\int6x^2dx\Rightarrow\ln y=2x^3+c\Rightarrow y=Be^{2x^3}\).
評分準則
M1 separate; A1 show; M1 A=200; A1 1478; M1 separate; A1 \(y=Be^{2x^3}\).
題目 6 · Structured
10 分(a) Solve \(3^{2x}-10\cdot3^{x}+9=0\). (b) Solve \(\log_2 x+\log_2(x-2)=3\). (c) Evaluate \(\log_3 81-\log_3 9\).
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解題
(a) Let \(y=3^x\): \(y^2-10y+9=0\Rightarrow(y-1)(y-9)=0\Rightarrow3^x=1\,(x=0)\) or \(3^x=9\,(x=2)\). (b) \(\log_2 x(x-2)=3\Rightarrow x^2-2x=8\Rightarrow(x-4)(x+2)=0\Rightarrow x=4\) (reject \(x=-2\)). (c) \(\log_3 81-\log_3 9=4-2=2\).
評分準則
M1 substitution; A1 both x; M1 combine logs; M1 solve; A1 x=4; A1 (c).
題目 7 · Structured
10 分(a) Express \(3\sin x+4\cos x\) in the form \(R\sin(x+\alpha)\), \(R>0\), \(0<\alpha<90^\circ\). (b) State the maximum value and the value of \(x\) in \(0\le x\le360^\circ\) at which it occurs. (c) Solve \(3\sin x+4\cos x=2.5\) for \(0\le x\le360^\circ\).
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解題
(a) \(R=\sqrt{3^2+4^2}=5\), \(\tan\alpha=\tfrac43\Rightarrow\alpha=53.13^\circ\): \(5\sin(x+53.13^\circ)\). (b) Max \(5\) when \(x+53.13^\circ=90^\circ\Rightarrow x=36.9^\circ\). (c) \(\sin(x+53.13^\circ)=0.5\Rightarrow x+53.13^\circ=30^\circ,150^\circ\); only \(150^\circ\) is valid \(\Rightarrow x=96.9^\circ\).
評分準則
M1 R=5; M1 α; A1 form; M1A1 (b); M1 solve; A1 96.9°.
題目 8 · 選擇題
1 分Find the weight of a particle of mass \(5\,\text{kg}\) (take \(g=9.8\)). Circle your answer.
- A.\(5\,\text{N}\)
- B.\(0.51\,\text{N}\)
- C.\(49\,\text{N}\)
- D.\(9.8\,\text{N}\)
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解題
Weight \(=mg=5\times9.8=49\,\text{N}\).
評分準則
B1 for 49 N.
題目 9 · Structured
6 分A particle moves in a straight line with acceleration \(a=(6t-4)\,\text{m s}^{-2}\) and velocity \(2\,\text{m s}^{-1}\) when \(t=0\). (a) Find \(v\) in terms of \(t\). (b) Find \(v\) when \(t=3\). (c) Find the displacement during the first 3 seconds.
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解題
(a) \(v=\int(6t-4)dt=3t^2-4t+C\); \(v(0)=2\Rightarrow C=2\), so \(v=3t^2-4t+2\). (b) \(v(3)=27-12+2=17\). (c) \(s=\int_0^3(3t^2-4t+2)dt=[t^3-2t^2+2t]_0^3=27-18+6=15\,\text{m}\).
評分準則
M1 integrate a; A1 v; B1 (b); M1 integrate v; A1 15 m.
題目 10 · Structured
9 分A particle of mass \(4\,\text{kg}\) lies on a smooth plane inclined at \(30^\circ\) to the horizontal. It is connected by a light inextensible string over a smooth pulley at the top of the plane to a particle of mass \(6\,\text{kg}\) hanging freely. The system is released from rest. (a) Find the acceleration. (b) Find the tension in the string. (c) State one assumption you have used about the string.
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解題
For the hanging mass: \(6g-T=6a\). For the mass on the plane: \(T-4g\sin30^\circ=4a\). Adding: \(6g-4g(0.5)=10a\Rightarrow a=\tfrac{58.8-19.6}{10}=3.92\,\text{m s}^{-2}\). Then \(T=6g-6a=58.8-23.52=35.3\,\text{N}\). (c) The string is light and inextensible (so tension is constant and the masses share one acceleration).
評分準則
M1 equation hanging mass; M1 equation on plane; A1 add; A1 a=3.92; M1 T; A1 35.3; B1 assumption.
題目 11 · Structured
10 分A uniform beam \(AB\) of length \(8\,\text{m}\) and weight \(400\,\text{N}\) rests horizontally on supports at \(A\) and \(B\). A person of weight \(600\,\text{N}\) stands \(2\,\text{m}\) from \(A\). (a) Find the magnitude of the reaction at each support. (b) Find how far from \(A\) the person must stand for the two reactions to be equal.
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解題
(a) Moments about \(A\): \(R_B(8)=400(4)+600(2)=2800\Rightarrow R_B=350\,\text{N}\); \(R_A=400+600-350=650\,\text{N}\). (b) For equal reactions each is \(500\,\text{N}\): \(500(8)=400(4)+600d\Rightarrow600d=2400\Rightarrow d=4\,\text{m}\).
評分準則
M1 moments about A; A1 R_B; M1 resolve vertically; A1 R_A; M1 set reactions equal; A1 d=4.
題目 12 · Structured
8 分A particle moves so that its position vector is \(\mathbf{r}=(t^2-2t)\mathbf{i}+(t^2-4t)\mathbf{j}\), where \(t\) is in seconds. (a) Find its velocity when \(t=1\). (b) Find its speed when \(t=1\). (c) Find the time at which the particle is moving parallel to \(\mathbf{i}\).
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解題
\(\mathbf{v}=(2t-2)\mathbf{i}+(2t-4)\mathbf{j}\). (a) At \(t=1\): \(\mathbf{v}=0\mathbf{i}-2\mathbf{j}\). (b) Speed \(=\sqrt{0^2+(-2)^2}=2\,\text{m s}^{-1}\). (c) Parallel to \(\mathbf{i}\) when the \(\mathbf{j}\)-component is zero: \(2t-4=0\Rightarrow t=2\).
評分準則
M1 differentiate r; A1 v at t=1; M1A1 speed; M1 set j-comp =0; A1 t=2.
題目 13 · Structured
10 分A block of mass \(10\,\text{kg}\) rests on a rough horizontal surface with coefficient of friction \(0.3\). It is pulled by a force of \(50\,\text{N}\) acting at \(20^\circ\) above the horizontal (take \(g=9.8\)). (a) Find the normal reaction. (b) Find the friction force. (c) Find the acceleration of the block.
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解題
(a) Resolve vertically: \(N=10g-50\sin20^\circ=98-17.1=80.9\,\text{N}\). (b) \(F=\mu N=0.3(80.9)=24.3\,\text{N}\). (c) Horizontally: \(50\cos20^\circ-F=10a\Rightarrow46.98-24.27=10a\Rightarrow a=2.27\,\text{m s}^{-2}\).
評分準則
M1 resolve vertically; A1 N; M1A1 friction; M1 resolve horizontally; A1 a=2.27.
題目 14 · Structured
6 分A ball is thrown horizontally with speed \(15\,\text{m s}^{-1}\) from a point \(20\,\text{m}\) above horizontal ground (take \(g=9.8\), ignore air resistance). (a) Find the time taken to reach the ground. (b) Find the horizontal distance travelled before landing.
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解題
(a) Vertically \(20=\tfrac12(9.8)t^2\Rightarrow t^2=4.08\Rightarrow t=2.02\,\text{s}\). (b) Horizontally (constant speed) \(x=15(2.02)=30.3\,\text{m}\).
評分準則
M1 vertical equation; A1 t=2.02; M1 horizontal motion; A1 30.3 m; B1 method.
Paper 3: Pure & Statistics
Answer all questions. Section A is pure; Section B is statistics (Large Data Set context).
13 題目 · 100 分
題目 1 · 選擇題
1 分State the period of \(y=\sin(\pi x)\). Circle your answer.
- A.\(\pi\)
- B.\(\tfrac12\)
- C.\(2\)
- D.\(2\pi\)
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解題
Period \(=\tfrac{2\pi}{\pi}=2\).
評分準則
B1 for 2.
題目 2 · Structured
10 分Differentiate each of the following. (a) \(y=(2x+1)^5\). (b) \(y=x\ln x\). (c) \(y=\dfrac{x}{x^2+1}\). (d) Hence state the gradient of the curve in part (a) at \(x=0\).
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解題
(a) Chain rule: \(10(2x+1)^4\). (b) Product rule: \(\ln x+1\). (c) Quotient rule: \(\tfrac{(x^2+1)-x(2x)}{(x^2+1)^2}=\tfrac{1-x^2}{(x^2+1)^2}\). (d) At \(x=0\): \(10(1)^4=10\).
評分準則
M1A1 (a); M1A1 (b); M1A1 (c); M1A1 (d).
題目 3 · Structured
9 分Find each integral. (a) \(\displaystyle\int(3x+2)^4\,dx\). (b) \(\displaystyle\int\tfrac{6}{2x-1}\,dx\). (c) \(\displaystyle\int_1^2\tfrac{4}{x^2}\,dx\).
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解題
(a) \(\tfrac{(3x+2)^5}{15}+c\). (b) \(3\ln|2x-1|+c\). (c) \([-\tfrac{4}{x}]_1^2=-2-(-4)=2\).
評分準則
M1A1 (a); M1A1 (b); M1 antiderivative; A1 (c)=2.
題目 4 · Structured
10 分(a) Solve \(2\cos 2x+1=0\) for \(0\le x\le2\pi\). (b) Prove that \(\sec^2 x-1\equiv\tan^2 x\). (c) Solve \(\sin x=\cos x\) for \(0\le x\le2\pi\).
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解題
(a) \(\cos2x=-\tfrac12\Rightarrow2x=\tfrac{2\pi}{3},\tfrac{4\pi}{3},\tfrac{8\pi}{3},\tfrac{10\pi}{3}\Rightarrow x=\tfrac{\pi}{3},\tfrac{2\pi}{3},\tfrac{4\pi}{3},\tfrac{5\pi}{3}\). (b) \(\sec^2x-1=\tfrac{1}{\cos^2x}-1=\tfrac{1-\cos^2x}{\cos^2x}=\tfrac{\sin^2x}{\cos^2x}=\tan^2x\). (c) \(\tan x=1\Rightarrow x=\tfrac{\pi}{4},\tfrac{5\pi}{4}\).
評分準則
M1 cos2x=-1/2; A1 all four x; M1A1 (b); M1A1 (c).
題目 5 · Structured
10 分A radioactive substance decays so that its mass is \(m=m_0e^{-kt}\), with a half-life of 20 days. (a) Show that \(k=\dfrac{\ln2}{20}\). (b) Given \(m_0=50\,\text{g}\), find the mass after 30 days. (c) Find, to the nearest day, the time for the mass to fall to \(10\,\text{g}\). (d) Find the rate of decay (in g per day) at \(t=0\).
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解題
(a) \(\tfrac12 m_0=m_0e^{-20k}\Rightarrow e^{-20k}=\tfrac12\Rightarrow k=\tfrac{\ln2}{20}\). (b) \(k=0.03466\); \(m=50e^{-1.0397}=17.7\,\text{g}\). (c) \(10=50e^{-kt}\Rightarrow t=\tfrac{\ln5}{0.03466}=46\,\text{days}\). (d) \(\tfrac{dm}{dt}=-km_0e^{-kt}\); at \(t=0\), \(=-0.03466(50)=-1.73\,\text{g/day}\).
評分準則
M1 half-life equation; A1 show k; M1A1 (b); M1A1 (c); M1A1 (d).
題目 6 · Structured
10 分(a) Use the arithmetic-series formula to prove that the sum of the first \(n\) odd numbers is \(n^2\). (b) An arithmetic series is \(7,11,15,\dots\). Find the sum of the first 30 terms. (c) A geometric series is \(100,80,64,\dots\). Find its sum to infinity. (d) Find the first term of the series in (b) that exceeds 200.
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解題
(a) Odd numbers form an AP with \(a=1,d=2\): \(S_n=\tfrac n2(2+(n-1)2)=\tfrac n2(2n)=n^2\). (b) \(a=7,d=4\): \(S_{30}=15(14+29(4))=15(130)=1950\). (c) \(r=0.8\): \(\tfrac{100}{1-0.8}=500\). (d) \(7+(n-1)4>200\Rightarrow n>49.25\Rightarrow n=50\) (term \(=201\)).
評分準則
M1A1 (a) proof; M1A1 (b); M1A1 (c); M1A1 (d).
題目 7 · 選擇題
1 分For \(X\sim B(20,0.3)\), state the mean \(E(X)\). Circle your answer.
- A.\(0.3\)
- B.\(6\)
- C.\(4.2\)
- D.\(14\)
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解題
\(E(X)=np=20\times0.3=6\).
評分準則
B1 for 6.
題目 8 · Structured
6 分(a) Define a simple random sample. (b) A company has 500 employees, each with a payroll number. Describe how to select a simple random sample of 50 employees. (c) State one advantage of stratified sampling over simple random sampling here.
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解題
(a) A simple random sample is one in which every possible sample of the required size has an equal chance of being selected (equivalently, every member has an equal chance of being chosen). (b) Number the employees 1–500; use random numbers (or a random-number generator) to select 50 distinct payroll numbers, ignoring repeats and any number above 500. (c) Stratified sampling ensures each subgroup (e.g. department or grade) is represented in proportion, reducing sampling bias.
評分準則
B1 definition; M1 numbering; A1 random selection of 50; B1 advantage; B1 contextual reason; B1 clarity.
題目 9 · Structured
10 分Eight recorded times (in minutes) are \(12,15,15,18,20,22,25,30\). (a) Find the median. (b) Find the lower quartile, upper quartile and interquartile range. (c) Find the mean. (d) Find the standard deviation to 2 decimal places. (e) Using the rule that an outlier lies more than \(1.5\times\text{IQR}\) beyond a quartile, determine whether 30 is an outlier.
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解題
(a) Median \(=\tfrac{18+20}{2}=19\). (b) \(Q_1=15,\,Q_3=25,\,\text{IQR}=10\). (c) Mean \(=\tfrac{157}{8}=19.625\). (d) \(\sum x^2=3327\); variance \(=\tfrac{3327}{8}-19.625^2=30.73\); s.d. \(=5.54\). (e) \(Q_3+1.5(10)=40\); since \(30<40\), 30 is not an outlier.
評分準則
B1 median; B1 quartiles; B1 IQR; M1A1 mean; M1A1 s.d.; B1 outlier conclusion.
題目 10 · Structured
10 分The heights of adult plants are modelled by \(X\sim N(170,8^2)\) (in cm). (a) Find \(P(X>180)\). (b) Find \(P(165
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解題
(a) \(z=\tfrac{180-170}{8}=1.25\Rightarrow P(Z>1.25)=0.1056\). (b) \(z=\pm0.625\Rightarrow\Phi(0.625)-\Phi(-0.625)=0.7340-0.2660=0.4680\). (c) \(P(X>x)=0.10\Rightarrow z=1.2816\Rightarrow x=170+1.2816(8)=180.3\,\text{cm}\).
評分準則
M1 standardise; A1 (a); M1A1 (b); M1 z=1.2816; A1 (c).
題目 11 · Structured
10 分A spinner is claimed to land on red with probability greater than 0.4. (a) For \(X\sim B(15,0.4)\) find (i) \(P(X=6)\) and (ii) \(P(X\le4)\). (b) In 15 spins the spinner lands on red 10 times. Test, at the 5% significance level, whether the probability of red exceeds 0.4.
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解題
(a)(i) \(\binom{15}{6}0.4^6 0.6^9=0.2066\). (ii) \(P(X\le4)=0.2173\). (b) \(H_0:p=0.4\), \(H_1:p>0.4\), \(X\sim B(15,0.4)\). \(P(X\ge10)=1-P(X\le9)=0.0338<0.05\), so reject \(H_0\): evidence that the probability of red exceeds 0.4.
評分準則
M1A1 (a)(i); A1 (a)(ii); B1 hypotheses; M1 P(X≥10); A1 0.0338; A1 conclusion.
題目 12 · Structured
7 分Events \(A\) and \(B\) satisfy \(P(A)=0.5\), \(P(B)=0.4\) and \(P(A\cap B)=0.2\). (a) Find \(P(A\cup B)\). (b) Find \(P(A\mid B)\). (c) Determine, with justification, whether \(A\) and \(B\) are independent.
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解題
(a) \(P(A\cup B)=0.5+0.4-0.2=0.7\). (b) \(P(A\mid B)=\tfrac{0.2}{0.4}=0.5\). (c) \(P(A\cap B)=0.2\) and \(P(A)P(B)=0.5(0.4)=0.2\); equal, so \(A\) and \(B\) are independent.
評分準則
M1A1 (a); M1A1 (b); M1 compare products; A1 conclusion.
題目 13 · Structured
6 分The discrete random variable \(X\) has distribution \(P(X=1)=0.2,\,P(X=2)=0.3,\,P(X=3)=0.4,\,P(X=4)=0.1\). (a) Find \(E(X)\). (b) Find \(\text{Var}(X)\). (c) Find \(E(2X+1)\).
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解題
(a) \(E(X)=1(0.2)+2(0.3)+3(0.4)+4(0.1)=2.4\). (b) \(E(X^2)=1(0.2)+4(0.3)+9(0.4)+16(0.1)=6.6\); \(\text{Var}(X)=6.6-2.4^2=0.84\). (c) \(E(2X+1)=2(2.4)+1=5.8\).
評分準則
M1A1 E(X); M1 E(X²); A1 Var; A1 (c).
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