2023 AQA AS-Level Mathematics 7356 模擬試題連答案詳解

The point \((2,k)\) lies on the curve \(y=3x^2-5x+1\). Find \(k\). Circle your answer.

State the period of \(y=\tan 3x\). Circle your answer.

Find the first three terms, in ascending powers of \(x\), of the binomial expansion of \((1+2x)^6\).

Solve \(2\cos^2 x+3\sin x=3\) for \(0^\circ\le x\le360^\circ\).

The points are \(A(1,3)\) and \(B(7,11)\). (a) Find the midpoint of \(AB\). (b) Find the gradient of \(AB\). (c) Find the equation of the perpendicular bisector of \(AB\).

\(f(x)=x^2-5x\). (a) Use differentiation from first principles to show that \(f'(x)=2x-5\). (b) Find the gradient at \(x=4\). (c) Find the value of \(x\) at which the gradient is 7.

A curve has equation \(y=3x^2-4x+1\). (a) Find \(\displaystyle\int y\,dx\). (b) Find the area under the curve between \(x=1\) and \(x=3\). (c) Find the x-coordinate of the point where the gradient of the curve is zero.

Prove that the product of any two consecutive even integers is divisible by 8.

The value of a car is modelled by \(V=18000e^{-kt}\), where \(V\) is in pounds and \(t\) is the age in years. When the car is 4 years old it is worth £11000. (a) Find \(k\) to 3 significant figures. (b) Find the value when the car is 7 years old. (c) Find, to the nearest year, when the value falls to £5000.

A curve has equation \(y=x^2-6x+8\). (a) Find the equation of the tangent to the curve at \(x=5\). (b) Find the coordinates of the point where this tangent crosses the x-axis.

The line \(y=2x+c\) is a tangent to the curve \(y=x^2-3x+7\). Find the value of \(c\).

A car decelerates uniformly from \(20\,\text{m s}^{-1}\) to rest in 8 seconds. Find the magnitude of its deceleration. Circle your answer.

A stone is projected vertically upwards at \(14\,\text{m s}^{-1}\) from the top of a cliff \(10\,\text{m}\) above the sea (take \(g=9.8\)). (a) Find the greatest height above the cliff top. (b) Find the speed with which it hits the sea.

Two forces \((4\mathbf{i}-3\mathbf{j})\,\text{N}\) and \((\mathbf{i}+7\mathbf{j})\,\text{N}\) act on a particle of mass \(5\,\text{kg}\). (a) Find the resultant force. (b) Find the magnitude of the acceleration.

Two particles of masses \(5\,\text{kg}\) and \(3\,\text{kg}\) are connected by a light inextensible string over a smooth pulley and released from rest. (a) Find the acceleration. (b) Find the tension.

A particle moves in a straight line with velocity \(v=2t^2-10t+8\;\text{m s}^{-1}\). (a) Find the times when it is at rest. (b) Find the acceleration when \(t=4\). (c) Find the displacement during the first 3 seconds.

A lorry of mass \(2000\,\text{kg}\) tows a trailer of mass \(800\,\text{kg}\) by a light rigid tow-bar. The engine produces a forward force of \(6000\,\text{N}\); resistances are \(500\,\text{N}\) on the lorry and \(300\,\text{N}\) on the trailer. (a) Find the acceleration. (b) Find the tension in the tow-bar.

Find \(\displaystyle\int(6x^2-4)\,dx\). Circle your answer.

Express \(\sqrt{50}+\sqrt{8}\) in the form \(k\sqrt2\). Circle your answer.

\(f(x)=2x^3-3x^2-11x+6\). (a) Show that \((x-3)\) is a factor. (b) Factorise \(f(x)\) completely.

The quadratic equation \(x^2+kx+(k+3)=0\) has no real roots. Find the set of values of \(k\).

A cup of coffee cools according to \(T=22+Ae^{-kt}\), where \(T\,^\circ\text{C}\) is the temperature \(t\) minutes after it is made. Initially \(T=78\), and after 6 minutes \(T=50\). (a) Find \(A\). (b) Find \(k\) to 3 significant figures. (c) Find the temperature after 15 minutes. (d) Find, to the nearest minute, the time to reach \(30^\circ\text{C}\).

In triangle \(PQR\), \(PQ=9\,\text{cm}\), \(QR=6\,\text{cm}\) and angle \(PQR=110^\circ\). (a) Find \(PR\). (b) Find the area of the triangle. (c) Find angle \(QPR\) to 1 decimal place.

The line \(y=2x+1\) and the curve \(y=x^2-x+1\) intersect at two points. (a) Find the coordinates of the points of intersection. (b) Write down the inequalities that define the enclosed region. (c) Find the exact area enclosed.

A curve has equation \(y=x^3-6x^2+9x+2\). (a) Find \(\dfrac{dy}{dx}\). (b) Find the coordinates of the stationary points. (c) Use the second derivative to determine the nature of each.

In an arithmetic series the 3rd term is 13 and the 8th term is 38. (a) Find the first term and common difference. (b) Find the sum of the first 20 terms. (c) Find which term of the series is equal to 98.

Express \(\dfrac{7}{3-\sqrt2}\) in the form \(a+b\sqrt2\), where \(a\) and \(b\) are integers.

Selecting every 10th person from an ordered list of customers is an example of which sampling method? Circle your answer.

Which measure of average is most affected by a single extreme outlier? Circle your answer.

Six data values are \(4, 7, 9, 10, 13, 17\). (a) Find the mean. (b) Find the standard deviation to 2 decimal places.

On each trial one of three outcomes occurs independently with \(P(X)=0.25\), \(P(Y)=0.4\), \(P(Z)=0.35\). For four trials find: (a) \(P(\text{no }Z)\); (b) \(P(\text{at least one }X)\); (c) \(P(\text{exactly two }Y)\).

The discrete random variable \(X\) has \(P(X=x)=k(5-x)\) for \(x=1,2,3,4\). (a) Find \(k\). (b) Find \(E(X)\). (c) Find \(P(X\ge3)\).

A drug company claims its treatment has a success rate greater than 0.6. In a trial of 25 patients, 20 are treated successfully. Test the claim at the 5% significance level.