A-level Sample Test
Paper 2
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State the values of |x| for which the binomial expansion of (3+2x)^{−4} is valid.

Circle your answer.

|x| < \frac{2}{3}
|x| < 1
|x| < \frac{3}{2}
|x| < 3
[1 mark] ?

A zoologist is investigating the growth of a population of red squirrels in a forest.

She uses the equation N=\frac{200}{1+9e^{-\frac{t}{5}}} as a model to predict the number of squirrels, N, in the population t weeks after the start of the investigation.

What is the size of the squirrel population at the start of the investigation?

Circle your answer.

[1 mark] ?

A curve is defined by the parametric equations

x = t^3 + 2 ,   y = t^2 − 1


Find the gradient of the curve at the point where t = −2

[4 marks] ?

Find a Cartesian equation of the curve.

Give your answer in the form y = f(x)

[2 marks] ?

The equation x^3 − 3x + 1 = 0 has three real roots.


Calculate f(-2) and f(-1) and decide whether there is a root, x, such that −2 < x < −1

f(-2) =

f(-1) =

[2 marks] ?

Taking x_1 = −2 as the first approximation to one of the roots, use the Newton-Raphson method to find x_2, the second approximation.

[3 marks] ?

Will this method work when the first approximation is x_1 = −1?

[1 mark] ?

Express y = 3cos\theta + 3sin\theta in the form y = R cos(\theta − \alpha) where R > 0 and -\frac{\pi}{2}\le\alpha\le\frac{\pi}{2}

R =

𝛼 = [6 marks] ?

Hence or otherwise find the least value and greatest value of 4 + (3cos\theta + 3sin\theta)^2

least value =

greatest value = [3 marks] ?

A curve C, has equation y = x^2 − 4x + k , where k is a constant.

It crosses the x-axis at the points (2 + \sqrt{5}, 0) and (2 − \sqrt{5}, 0)

Find the value of k.

k = [2 marks] ?

Using 2n + 1 and 2n + 3 (where n is a positve integer) as consecutive odd numbers, show that


can be expressed in the form

a^2 − b^2

where a and b are positive integers.

Give a and b in terms of n.

a =

b = [3 marks] ?

A curve has equation y=2x\cos3x+\left(3x^2−4\right)\sin3x

Find \frac{dy}{dx}, giving your answer in the form (mx^2 + n)\cos3x, where m and n are integers.

m =

n = [4 marks] ?

Show that the x-coordinates of the points of inflection of the curve satisfy the equation

\cot ax = \frac{bx^2 − c}{6x}

Find a, b and c where a, b, and c are integers.

a =

b =

c = [4 marks] ?

Three consecutive terms in an arithmetic sequence are 3e^{-p}, 5, 3e^p

Find the possible values of p. Give your answers in an exact form.

p = , [6 marks] ?

A single force of magnitude 4 newtons acts on a particle of mass 50 grams.

Find the magnitude of the acceleration of the particle.

Circle your answer.

12.5 ms−2
0.08 ms−2
0.0125 ms−2
80 ms−2
[1 mark] ?

A uniform rod, AB, has length 3 metres and mass 24 kg.

A particle of mass M kg is attached to the rod at A.

The rod is balanced in equilibrium on a support at C, which is 0.8 metres from A.

Find the value of M.

M = [2 marks] ?

A particle moves on a straight line with a constant acceleration, a ms−2.

The initial velocity of the particle is U ms−1.

After T seconds the particle has velocity V ms−1.

This information is shown on the velocity-time graph.

The displacement, S metres, of the particle from its initial position at time T seconds is given by the formula



By considering the gradient of the graph, or otherwise, write down a formula for a in terms of U, V and T.

a = [1 mark] ?

Find a formula for V^2 in terms of

U, a and S

V^2 = [3 marks] ?

The three forces F1 , F2 and F3 are acting on a particle.

F1 = (25i + 12j)N

F2 = (–7i + 5j)N

F3 = (15i – 28j)N

The unit vectors i and j are horizontal and vertical respectively.

The resultant of these three forces is F newtons.


Find the magnitude of F, giving your answer to three significant figures.

N [2 marks] ?

Find the acute angle that F makes with the horizontal, giving your answer to the nearest 0.1°

° [2 marks] ?

The fourth force, F4, is applied to the particle so that the four forces are in equilibrium.

Find F4, giving your answer in terms of i and j.

F4 = [1 mark] ?

The graph below models the velocity of a small train as it moves on a straight track for 20 seconds.

The front of the train is at the point A when t = 0

The mass of the train is 800kg.


Find the total distance travelled in the 20 seconds.

m [3 marks] ?

Find the distance of the front of the train from the point A at the end of the 20 seconds.

m [1 mark] ?

Find the maximum magnitude of the resultant force acting on the train.

N [2 marks] ?

At time t = 0, a parachutist jumps out of an airplane that is travelling horizontally.

The velocity, v ms-1 , of the parachutist at time t seconds is given by:


The unit vectors i and j are horizontal and vertical respectively.

Assume that the parachutist is at the origin when t = 0

Model the parachutist as a particle.


Find an expression for the position vector of the parachutist at time t.

r = i + j [4 marks] ?

The parachutist opens her parachute when she has travelled 100 metres horizontally.

Find the vertical displacement of the parachutist from the origin when she opens her parachute.

Give your answer to 1 decimal place.

m [4 marks] ?

Deduce the value of g used in the formulation of this model.

ms-2 [3 marks] ?

In this question use g = 9.8 ms−2.

The diagram shows a box, of mass 8.0 kg, being pulled by a string so that the box moves at a constant speed along a rough horizontal wooden board.

The string is at an angle of 40° to the horizontal.

The tension in the string is 50 newtons.

The coefficient of friction between the box and the board is μ.

Model the box as a particle.


Calculate μ giving your answer correct to 2 decimal places.

μ = [4 marks] ?

One end of the board is lifted up so that the board is now inclined at an angle of 5° to the horizontal.

The box is pulled up the inclined board.

The string remains at an angle of 40° to the board.

The tension in the string is increased so that the box accelerates up the board at 3 ms-2

Find the tension in the string as the box accelerates up the slope at 3 ms-2 giving your answer to 2 significant figures.

[7 marks] ?

In this question use g = 9.81 ms−2.

A ball is projected from the origin. After 2.5 seconds, the ball lands at the point with position vector (40i − 10j) metres.

The unit vectors i and j are horizontal and vertical respectively.

Assume that there are no resistance forces acting on the ball.


Find the speed of the ball when it is at a height of 3 metres above its initial position giving your answer to 3 significant figures.

ms-1 [6 marks] ?

State the speed of the ball when it is at its maximum height giving your answer to 2 significant figures.

ms-1 [1 mark] ?
Test check
Total: - of marks