AQA A Level Maths Paper 2 November 2021 question paper 7357/2

Four possible sketches of y ax² bx c are shown below.

Given b² 4ac 0 and a, b and c are non-zero constants, which sketch is the only one that could possibly be correct?

Tick () one box.

A curve has equation y f (x)

The curve has a point of inflection at x 7

Identify which one of the statements below must be true.

Circle your answer.

A sequence is defined by

Find ⁹⁵ u_n n1

Circle your answer.

Figure 1 shows the graph of y 2x

On Figure 1 add a sketch of the graph of

Find the coordinates of the points of intersection of the two graphs.

Fully justify your answer.

Express

in the form

where A and B are integers.

Show that the solution of the equation

can be written as

Fully justify your answer.

A circle has equation

State the coordinates of the centre of the circle.

Find the radius of the circle in terms of p.

The circle intersects the coordinate axes at exactly three points.

Find the two possible values of p.

Kai is proving that n³ n is a multiple of 3 for all positive integer values of n.

Kai begins a proof by exhaustion.

non-negative integer
which is a multiple of 3

which is a multiple of 3

Step 5 Therefore n³ n is a multiple of 3 for all positive integer values of n

Explain the two mistakes that Kai has made after Step 3.

Correct Kai’s argument from Step 4 onwards.

A robotic arm which is attached to a flat surface at the origin O, is used to draw a graphic design.

The arm is made from two rods OP and PQ, each of length d, which are joined at P.

A pen is attached to the arm at Q.

The coordinates of the pen are controlled by adjusting the angle OPQ and the angle between OP and the x-axis.

For this particular design the pen is made to move so that the two angles are always equal to each other with 0 as shown in Figure 2.

Show that the x-coordinate of the pen can be modelled by the equation

Hence, show that

It can be shown that

State the greatest possible value of x and the corresponding value of cos

Figure 3 below shows the arm when the x-coordinate is at its greatest possible value.

Find, in terms of d, the exact distance OQ.

The function h is defined by

where h has its maximum possible domain.

Find the domain of h.

Give your answer using set notation.

Alice correctly calculates

She then argues that since there is a change of sign there must be a value of x in the interval 1 x 4 that gives h(x) 0

Explain the error in Alice’s argument.

A particle’s displacement, r metres, with respect to time, t seconds, is defined by the equation

Circle your answer.

A particle has a speed of 6 m s¹ in a direction relative to unit vectors i and j as shown in the diagram below.

Find the correct expression for v₂

Circle your answer.

A vehicle, of total mass 1200 kg, is travelling along a straight, horizontal road at a

This vehicle begins to accelerate at a constant rate.

After 40 metres it reaches a speed of 17 m s¹

Find the resultant force acting on the vehicle during the period of acceleration.

t seconds as shown by the following graph.

Noosha says that, in the period 12 36, the scooter travels approximately 130 metres.

Determine if Noosha is correct, showing clearly any calculations you have used.

A cyclist is towing a trailer behind her bicycle.

She is riding along a straight, horizontal path at a constant speed.

A tension of T newtons acts on the connecting rod between the bicycle and the trailer.

The cyclist is causing a constant driving force of 40 N to be applied whilst pedalling forwards on her bicycle.

The constant resistance force acting on the trailer is 12 N

State the value of T giving a clear reason for your answer.

State one assumption you have made in reaching your answer to part (a).

Find the external resistance force acting on the cyclist and her bicycle.

A straight uniform rod, AB, has length 6 m and mass 0.2 kg

A particle of weight w newtons is fixed at A.

A second particle of weight 3 w newtons is fixed at B.

The rod is suspended by a string from a point x metres from B.

The rod rests in equilibrium with AB horizontal and the string hanging vertically as

Show that

A ball is released from a great height so that it falls vertically downwards towards the surface of the Earth.

Using a simple model, Andy predicts that the velocity of the ball, exactly 2 seconds

Show how Andy has obtained his prediction.

t seconds after being released from rest is

Find an expression for v in terms of t.

Comment on the value of v for the two models as t becomes large.

Two particles, P and Q, are projected at the same time from a fixed point X, on the ground, so that they travel in the same vertical plane.

Both particles land back on the ground at exactly the same point, Y.

Resistance forces to motion may be ignored.

Show that

P takes a total of 0.4 seconds to travel from X to Y.

Find the time taken by Q to travel from X to Y.

State one modelling assumption you have chosen to make in this question.

Two skaters, Jo and Amba, are separately skating across a smooth, horizontal surface of ice.

Both are moving in the same direction, so that their paths are straight and are parallel to each other.

Jo is moving with constant velocity (28i 96j) m s¹

At time t 0 seconds Amba is at position (2i 7j) metres and is moving with a

constant.

Verify that k 08

Find the position vector of Amba when t 4

At both t 0 and t 4 there is a distance of 5 metres between Jo and Amba’s positions.

Determine the shortest distance between their two parallel lines of motion.

Fully justify your answer.