2023 AQA A-Level Mathematics 7357 模擬試題連答案詳解

Differentiate \(y=\ln(5x)\). Circle your answer.

Find \(\displaystyle\int e^{3x}\,dx\). Circle your answer.

Find the first three terms, in ascending powers of \(x\), of the binomial expansion of \((3+2x)^6\).

A circle \(C\) has equation \(x^2+y^2-10x+2y+1=0\). (a) Find the centre and radius. (b) Show that the point \((5,4)\) lies on \(C\). (c) Find the equation of the tangent to \(C\) at \((5,4)\). (d) Find the length of the tangent to \(C\) from the external point \((12,3)\).

(a) An arithmetic series has first term 4 and common difference 6. Find (i) the 15th term and (ii) the sum of the first 15 terms. (b) A geometric series has first term 4 and common ratio 1.05. Find (i) the 10th term and (ii) the sum of the first 10 terms. (c) State, with a reason, whether the geometric series converges.

\(f(x)=x^3-7x+2\). (a) Show that a root of \(f(x)=0\) lies between 2 and 3. (b) Apply the Newton–Raphson method with \(x_0=2.5\) to find \(x_1\) and \(x_2\). (c) The iteration \(x_{n+1}=\sqrt[3]{7x_n-2}\) is used with \(x_0=2.5\). Find the root to 3 decimal places.

\(p(x)=2x^3+ax^2+bx-6\). Given that \((x-2)\) and \((x+1)\) are factors: (a) find \(a\) and \(b\); (b) factorise \(p(x)\) completely; (c) solve \(p(x)=0\).

(a) A geometric series has first term 6 and common ratio \(\tfrac23\). Find its sum to infinity. (b) Solve \(\tan 2x=1\) for \(0\le x\le\pi\). (c) Solve \(4\sin x=3\cos x\) for \(0\le x\le2\pi\), to 2 decimal places.

(a) Prove the identity \(\dfrac{1+\sin x}{\cos x}+\dfrac{\cos x}{1+\sin x}\equiv\dfrac{2}{\cos x}\). (b) Hence solve \(\dfrac{1+\sin x}{\cos x}+\dfrac{\cos x}{1+\sin x}=4\) for \(0

(a) Find \(\displaystyle\int\left(6x^2-\tfrac{4}{x^2}+e^{3x}\right)dx\). (b) Use the substitution \(u=x^2+4\) to find \(\displaystyle\int\tfrac{2x}{\sqrt{x^2+4}}\,dx\). (c) Find the area enclosed between the curve \(y=x^2+3\) and the line \(y=4x\).

A curve has equation \(y=2x^3-9x^2+12x-3\). (a) Find \(\dfrac{dy}{dx}\). (b) Find the stationary points and determine their nature. (c) Find the coordinates of the point of inflection. (d) Find the equation of the tangent at \(x=0\). (e) State the set of values of \(x\) for which the curve is increasing.

Find \(\displaystyle\int\tfrac1x\,dx\). Circle your answer.

Express \(\dfrac{4x+5}{(x+1)(x-2)}\) in partial fractions.

Prove by contradiction that \(\sqrt5\) is irrational.

A closed box has a square base of side \(x\,\text{cm}\) and volume \(1000\,\text{cm}^3\). (a) Show that the total surface area is \(S=2x^2+\dfrac{4000}{x}\). (b) Find the value of \(x\) that minimises \(S\). (c) Find the minimum surface area.

(a) A population grows so that \(\dfrac{dP}{dt}=0.4P\). Show that \(P=Ae^{0.4t}\). (b) Given \(P=150\) when \(t=0\), find \(P\) when \(t=5\). (c) Solve the differential equation \(\dfrac{dy}{dx}=4x^3y\), giving \(y\) in terms of \(x\).

(a) Solve \(3^{2x}-12\cdot3^{x}+27=0\). (b) Solve \(\log_3 x+\log_3(x+6)=3\). (c) Evaluate \(\log_2 32-\log_2 4\).

(a) Express \(5\sin x-12\cos x\) 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 \(5\sin x-12\cos x=6.5\) for \(0\le x\le360^\circ\).

Find the weight of a particle of mass \(8\,\text{kg}\) (take \(g=9.8\)). Circle your answer.

A particle moves in a straight line with acceleration \(a=(12t-6)\,\text{m s}^{-2}\) and velocity \(4\,\text{m s}^{-1}\) at \(t=0\). (a) Find \(v\) in terms of \(t\). (b) Find \(v\) when \(t=2\). (c) Find the displacement during the first 2 seconds.

A particle of mass \(3\,\text{kg}\) lies on a smooth plane inclined at \(30^\circ\). It is connected by a light inextensible string over a smooth pulley at the top of the plane to a mass of \(5\,\text{kg}\) hanging freely. The system is released from rest. (a) Find the acceleration. (b) Find the tension. (c) State one assumption about the string.

A uniform beam \(AB\) of length \(10\,\text{m}\) and weight \(500\,\text{N}\) rests horizontally on supports at \(A\) and \(B\). A load of \(300\,\text{N}\) is placed \(3\,\text{m}\) from \(A\). (a) Find the reaction at each support. (b) Find how far from \(A\) the load must be placed for the reactions to be equal.

A particle has position vector \(\mathbf{r}=(t^2-4t)\mathbf{i}+(t^2-2t)\mathbf{j}\) (\(t\) in seconds). (a) Find its velocity when \(t=1\). (b) Find its speed when \(t=1\). (c) Find the time at which it is moving parallel to \(\mathbf{j}\).

A ball is thrown horizontally with speed \(20\,\text{m s}^{-1}\) from a point \(25\,\text{m}\) above horizontal ground (take \(g=9.8\)). (a) Find the time to reach the ground. (b) Find the horizontal distance travelled.

A particle \(P\) of mass \(4\,\text{kg}\) lies on a smooth horizontal table. It is connected by a light inextensible string passing over a smooth pulley at the edge of the table to a particle \(Q\) 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 made about the pulley.

State the period of \(y=\sin(2\pi x)\). Circle your answer.

Differentiate each of the following. (a) \(y=(3x-2)^4\). (b) \(y=x^2\ln x\). (c) \(y=\dfrac{x}{x^2+4}\). (d) Hence state the gradient of the curve in part (a) at \(x=1\).

Find each integral. (a) \(\displaystyle\int(2x+5)^3\,dx\). (b) \(\displaystyle\int\tfrac{9}{3x+1}\,dx\). (c) \(\displaystyle\int_1^2\tfrac{6}{x^2}\,dx\).

(a) Solve \(2\cos 2x=\sqrt3\) for \(0\le x\le2\pi\). (b) Prove that \(\tan^2 x+1\equiv\sec^2 x\). (c) Solve \(2\sin x+1=0\) for \(0\le x\le2\pi\).

A radioactive sample decays as \(m=m_0e^{-kt}\), with a half-life of 12 years. (a) Find \(k\) to 3 significant figures. (b) Given \(m_0=80\,\text{g}\), find the mass after 20 years. (c) Find, to the nearest year, the time to fall to \(20\,\text{g}\). (d) Find the rate of decay at \(t=0\).

(a) An arithmetic series is \(5,9,13,\dots\). Find the sum of the first 25 terms. (b) A geometric series is \(200,150,112.5,\dots\). Find its sum to infinity. (c) Use the arithmetic-series formula to prove that \(1+3+5+\dots+(2n-1)=n^2\). (d) Find the first term of the series in (a) that exceeds 100.

For \(X\sim B(25,0.2)\), state the mean \(E(X)\). Circle your answer.

(a) Define a simple random sample. (b) A college has 600 students, each with an ID number. Describe how to take a simple random sample of 60. (c) State one advantage of stratified sampling here.

Eight values are \(10,12,14,15,18,20,24,28\). (a) Find the median. (b) Find \(Q_1\), \(Q_3\) and the IQR. (c) Find the mean. (d) Find the standard deviation to 2 decimal places. (e) Using the \(1.5\times\text{IQR}\) rule, determine whether 28 is an outlier.

The masses of items are modelled by \(X\sim N(50,6^2)\) (in grams). (a) Find \(P(X>56)\). (b) Find \(P(44

(a) For \(X\sim B(12,0.35)\) find (i) \(P(X=4)\) and (ii) \(P(X\le2)\). (b) A coin-like process is claimed to succeed with probability greater than 0.35. In 12 trials there are 9 successes. Test at the 5% level whether the success probability exceeds 0.35.

Events \(A\) and \(B\) satisfy \(P(A)=0.6\), \(P(B)=0.3\) and \(P(A\cap B)=0.18\). (a) Find \(P(A\cup B)\). (b) Find \(P(B\mid A)\). (c) Determine, with justification, whether \(A\) and \(B\) are independent.

The discrete random variable \(X\) has \(P(X=0)=0.1,\,P(X=1)=0.3,\,P(X=2)=0.4,\,P(X=3)=0.2\). (a) Find \(E(X)\). (b) Find \(\text{Var}(X)\). (c) Find \(E(3X-2)\).