AQA A Level Maths Paper 2 June 2022 question paper 7357/2

A circle has centre (4, 5) and radius 6

Find the equation of the circle.

Tick () one box.

State the value of

Circle your answer.

The function f is concave and is represented by one of the graphs below.

Identify the graph which represents f.

Tick () one box.

The diagram shows a triangle ABC.

AB is the shortest side. The lengths of AC and BC are 6.1 cm and 8.7 cm respectively.

The size of angle ABC is 38

Find the size of the largest angle.

Give your answer to the nearest degree.

The binomial expansion of (2 5x)⁴ is given by

4 2 3 4

Find the value of A and the value of B.

Show that

4 4 3

where C and D are constants to be found.

Hence, or otherwise, find

Asif notices that 24² 576 and 2 4 6 gives the last digit of 576

He checks two more examples:

Last digit 9 Last digit 1

Asif concludes that he can find the last digit of any square number greater than 100 by adding the digits of the number being squared.

Give a counter example to show that Asif’s conclusion is not correct.

Claire tells Asif that he should look only at the last digit of the number being squared.

Last digit 9 Last digit 6

Using Claire’s method determine the last digit of 23456789²

Given Claire’s method is correct, use proof by exhaustion to show that no square number has a last digit of 8

The curve y 15 x² and the isosceles triangle OPQ are shown on the diagram

Vertices P and Q lie on the curve such that Q lies vertically above some point (q, 0)

The line PQ is parallel to the x-axis.

Show that the area, A, of the triangle OPQ is given by

where c is a constant to be found.

Find the exact maximum area of triangle OPQ.

Fully justify your answer.

one direction.

Beth thinks the stretch should be in the y-direction.

Paul thinks the stretch should be in the x-direction.

State, giving reasons for your answer, whether Beth is correct, Paul is correct, both are correct or neither is correct.

Given that

show that

where A is an integer and p is a rational number.

A gardener has a greenhouse containing 900 tomato plants.

The gardener notices that some of the tomato plants are damaged by insects.

Initially there are 25 damaged tomato plants.

The number of tomato plants damaged by insects is increasing by 32% each day.

The total number of plants damaged by insects, x, is modelled by

where A and B are constants and t is the number of days after the gardener first noticed the damaged plants.

Use this model to find the total number of plants damaged by insects 5 days after the gardener noticed the damaged plants.

Explain why this model is not realistic in the long term.

A refined model assumes the rate of increase of the number of plants damaged by insects is given by

Show that

where A and B are positive integers to be found.

Hence, find t in terms of x.

Hence, find the number of days it takes from when the damage is first noticed until half of the plants are damaged by the insects.

Section B

all questions in the spaces provided.

A moon vehicle has a mass of 212 kg and a length of 3 metres.

On the moon the vehicle has a weight of 345 N

Calculate a value for acceleration due to gravity on the moon.

Circle your answer.

A car is travelling along a straight horizontal road with initial velocity u m s¹

Express a in terms of u.

Circle your answer.

In this question use = 98 m s²

A ball is projected from a point on horizontal ground with an initial velocity of 7 m s¹ at an angle above the horizontal.

The ball reaches a maximum vertical height of h metres above the ground.

Show that

Hence, given that 0 60, find the maximum value of h.

Nisha claims that the larger the size of the ball, the greater the maximum vertical height will be.

State whether Nisha is correct, giving a reason for your answer.

A £2 coin has a diameter of 28 mm and a mass of 12 grams.

A uniform rod AB of length 160 mm and a fixed load of mass m grams are used to check that a £2 coin has the correct mass.

The rod rests with its midpoint on a support.

A £2 coin is placed face down on the rod with part of its curved edge directly above A.

The fixed load is hung by a light inextensible string from a point directly below the other end of the rod at B, as shown in the diagram.

Given that the rod is horizontal and rests in equilibrium, find m.

State an assumption you have made about the £2 coin to answer part (a).

A car is moving in a straight line along a horizontal road.

Over the period 0 t 15 the car has a total displacement of 7 metres.

Initially the car has velocity 0 m s¹

Find the next time when the velocity of the car is 0 m s¹

Two particles, P and Q, move in the same horizontal plane.

Particle P is initially at rest at the point with position vector (4i 5j) metres and moves with constant acceleration (3i 4j) m s²

Particle Q moves in a straight line, passing through the points with position vectors

P and Q are moving along parallel paths.

Show that c 13

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

Hence, prove that the paths of P and Q are not collinear.

A particle is moving such that its position vector, r metres, at time t seconds, is given by

Fully justify your answer.

An object, O, of mass m kilograms is hanging from a ceiling by two light, inelastic strings of different lengths.

The shorter string, of length 0.8 metres, is fixed to the ceiling at A.

The longer string, of length 1.2 metres, is fixed to the ceiling at B.

This object hangs 0.6 metres directly below the ceiling as shown in the diagram.

Show that the tension in the shorter string is over 30% more than the tension in the longer string.

The tension in the longer string is known to be 2 g newtons.

Find the value of m.

In this question use 9.8 m s²

A rough wooden ramp is 10 metres long and is inclined at an angle of 25 above the horizontal. The bottom of the ramp is at the point O.

A crate of mass 20 kg is at rest at the point A on the ramp.

The crate is pulled up the ramp using a rope attached to the crate.

Once in motion, the rope remains taut and parallel to the line of greatest slope of the

The tension in the rope is 230 N

The crate accelerates up the ramp at 1.2 m s²

Find the coefficient of friction between the crate and the ramp.

The crate takes 3.8 seconds to reach the top of the ramp.

Find the distance OA.

Other than air resistance, state one assumption you have made about the crate in answering part (b)(i).