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.
Paper 1: Pure Mathematics
Answer all questions. Entirely pure content; calculator and AQA formulae booklet permitted.
11 PastPaper.question · 100 PastPaper.marks
PastPaper.question 1 · Multiple choice
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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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PastPaper.workedSolution
\(\tfrac{d}{dx}\ln(3x)=\tfrac{3}{3x}=\tfrac1x\).
PastPaper.markingScheme
B1 for \(\tfrac1x\).
PastPaper.question 2 · Multiple choice
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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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PastPaper.workedSolution
\(\int e^{2x}dx=\tfrac12 e^{2x}+c\).
PastPaper.markingScheme
B1 for \(\tfrac12 e^{2x}+c\).
PastPaper.question 3 · Short
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Find the first three terms, in ascending powers of \(x\), of the binomial expansion of \((2-3x)^7\).
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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PastPaper.workedSolution
(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 \(=\sqrt{(10-3)^2+(6+2)^2-5^2}=\sqrt{49+64-25}=\sqrt{88}=9.38\).
(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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PastPaper.workedSolution
(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.
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M1A1 (a)(i,ii); M1A1 (b)(i,ii); M1A1 (b)(ii); B1 (c) with reason.
PastPaper.question 6 · Structured
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\(f(x)=x^3-5x+1\). (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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PastPaper.workedSolution
(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) \(x_1=\sqrt[3]{9}=2.080,\;x_2=2.111,\;x_3=2.123,\;x_4=2.128,\dots\to 2.128\).
\(p(x)=2x^3+ax^2-7x+b\). Given that \((x-1)\) and \((x+2)\) are factors of \(p(x)\): (a) find \(a\) and \(b\); (b) factorise \(p(x)\) completely; (c) solve \(p(x)=0\).
M1 p(1)=0; M1 p(-2)=0; A1 a,b; M1 divide; A1 full factorisation; A1 three roots.
PastPaper.question 8 · Structured
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(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.
B1 (a); M1 2x values; A1 both x; M1 tan x; A1 0.59; A1 3.73.
PastPaper.question 9 · Structured
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(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 \(\dfrac{1-\cos2x}{\sin2x}\equiv\tan x\).
(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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PastPaper.workedSolution
(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 \(=\int_0^2(2x-x^2)dx=[x^2-\tfrac{x^3}{3}]_0^2=4-\tfrac83=\tfrac43\).
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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PastPaper.workedSolution
(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 \(x>3\).
Answer all questions. Section A is pure; Section B is mechanics (take g = 9.8 m s⁻²).
14 PastPaper.question · 100 PastPaper.marks
PastPaper.question 1 · Multiple choice
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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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PastPaper.workedSolution
\(\int\tfrac1x dx=\ln|x|+c\).
PastPaper.markingScheme
B1 for \(\ln|x|+c\).
PastPaper.question 2 · Short
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Express \(\dfrac{5x-7}{(x-1)(x-3)}\) in partial fractions.
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PastPaper.workedSolution
Let \(\tfrac{5x-7}{(x-1)(x-3)}=\tfrac{A}{x-1}+\tfrac{B}{x-3}\). Then \(5x-7=A(x-3)+B(x-1)\). \(x=1:-2=-2A\Rightarrow A=1\); \(x=3:8=2B\Rightarrow B=4\).
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M1 set up identity; M1 substitute values; A1 A=1, B=4.
PastPaper.question 3 · Proof
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Prove by contradiction that \(\sqrt3\) is irrational.
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PastPaper.workedSolution
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.
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.
(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 \(x\).
(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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PastPaper.workedSolution
(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 \(\Rightarrow x=96.9^\circ\).
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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PastPaper.workedSolution
Weight \(=mg=5\times9.8=49\,\text{N}\).
PastPaper.markingScheme
B1 for 49 N.
PastPaper.question 9 · Structured
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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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PastPaper.workedSolution
(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) \(s=\int_0^3(3t^2-4t+2)dt=[t^3-2t^2+2t]_0^3=27-18+6=15\,\text{m}\).
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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PastPaper.workedSolution
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).
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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PastPaper.workedSolution
(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 \(500\,\text{N}\): \(500(8)=400(4)+600d\Rightarrow600d=2400\Rightarrow d=4\,\text{m}\).
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M1 moments about A; A1 R_B; M1 resolve vertically; A1 R_A; M1 set reactions equal; A1 d=4.
PastPaper.question 12 · Structured
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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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\(\mathbf{v}=(2t-2)\mathbf{i}+(2t-4)\mathbf{j}\). (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: \(2t-4=0\Rightarrow t=2\).
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M1 differentiate r; A1 v at t=1; M1A1 speed; M1 set j-comp =0; A1 t=2.
PastPaper.question 13 · Structured
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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.
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.
Answer all questions. Section A is pure; Section B is statistics (Large Data Set context).
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PastPaper.question 1 · Multiple choice
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State the period of \(y=\sin(\pi x)\). Circle your answer.
A.\(\pi\)
B.\(\tfrac12\)
C.\(2\)
D.\(2\pi\)
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PastPaper.workedSolution
Period \(=\tfrac{2\pi}{\pi}=2\).
PastPaper.markingScheme
B1 for 2.
PastPaper.question 2 · Structured
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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 \(x=0\).
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 \(t=0\).
(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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PastPaper.workedSolution
(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 \(=201\)).
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M1A1 (a) proof; M1A1 (b); M1A1 (c); M1A1 (d).
PastPaper.question 7 · Multiple choice
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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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PastPaper.workedSolution
\(E(X)=np=20\times0.3=6\).
PastPaper.markingScheme
B1 for 6.
PastPaper.question 8 · Structured
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(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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PastPaper.workedSolution
(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.
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B1 definition; M1 numbering; A1 random selection of 50; B1 advantage; B1 contextual reason; B1 clarity.
PastPaper.question 9 · Structured
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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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PastPaper.workedSolution
(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.
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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(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.
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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PastPaper.workedSolution
(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.
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 \(E(2X+1)\).