OCR A Level Maths A Pure Mathematics and Mechanics November 2021 question paper H240/03

Show in a sketch the region of the x - y plane within which all three of the following inequalities hold.

y ⩾ x², x + y ⩽ 2, x ⩾ 0.

You should indicate the region for which the inequalities hold by labelling the region R.

The diagram shows triangle ABC in which angle A is 60° and the lengths of AB and AC are (4 + h) cm and (4 - h) cm respectively.

Show that the length of BC is p cm where

p² = 16 + 3h².

Hence show that, when h is small, p ≈ 4 + λ h² + μ h⁴, where λ and μ are rational numbers

whose values are to be determined.

An arithmetic progression has first term 2 and common difference d, where d ≠ 0. The first, third and thirteenth terms of this progression are also the first, second and third terms, respectively, of a geometric progression.

By determining d, show that the arithmetic progression is an increasing sequence.

Hence explain why the equation x|x - 1 | = k has exactly one real root for any negative value

of k.

Determine the real root of the equation x|x - 1 | =- 6.

Express v in the form R cos(t - α), where R > 0 and 0 < α < . Give the value of α correct

to 4 significant figures.

Hence find the two smallest positive values of t for which P is moving, in either direction,

with a speed of 3 m s^-1.

The equation 6 arcsin(2x - 1) - x² = 0 has exactly one real root.

Show by calculation that the root lies between 0.5 and 0.6.

In order to find the root, the iterative formula

x_n+1 = p + q sin(rx_n²),

with initial value x₀ = 0.5, is to be used.

Determine the values of the constants p, q and r.

Hence find the root correct to 4 significant figures. Show the result of each step of the iteration

process.

Express this property in the form of a differential equation.

It is given that C passes through the point (4, 3) and that x > 3 and y > 2 at all points on C.

Determine the equation of C giving your answer in the form y = f (x).

The curve C may be obtained by a transformation of part of the curve y = x³.

Describe fully this transformation.

The diagram shows the curve M with equation y = xe^-2x.

Show that M has a point of inflection at the point P where x = 1.

The line L passes through the origin O and the point P. The shaded region R is enclosed by the curve M and the line L.

Show that the area of R is given by

where a and b are integers to be determined.

There are three checkpoints, A, B and C, in that order, on a straight horizontal road. A car travels along the road, in the direction from A to C, with constant acceleration. The car takes 20 s to travel from B to C. The speed of the car at B is 14 m s^-1 and the speed of the car at C is 18 m s^-1.

Find the acceleration of the car.

It is given that the distance between A and B is 330 m.

Determine the speed of the car at A.

A block D of weight 50 N lies at rest in equilibrium on a fixed rough horizontal surface. A force of magnitude 15 N is applied to D at an angle θ to the horizontal (see diagram).

Complete the diagram in the Printed Answer Booklet showing all the forces acting on D.

It is given that D remains at rest and the coefficient of friction between D and the surface is 0.2.

Show that

15 cos θ - 3 sin θ ⩽ 10.

A golfer hits a ball from a point A with a speed of 25 m s^-1 at an angle of 15° above the horizontal. While the ball is in the air, it is modelled as a particle moving under the influence of gravity. Take the acceleration due to gravity to be 10 m s^-2.

The ball first lands at a point B which is 4 m below the level of A (see diagram).

Determine the time taken for the ball to travel from A to B.

Determine the horizontal distance of B from A.

Determine the direction of motion of the ball 1.5 seconds after the golfer hits the ball.

The horizontal distance from A to B is found to be greater than the answer to part (b).

State one factor that could account for this difference.

A beam, AB, has length 4 m and mass 20 kg. The beam is suspended horizontally by two vertical ropes. One rope is attached to the beam at C, where AC = 0.5 m. The other rope is attached to the beam at D, where DB = 0.7 m (see diagram).

The beam is modelled as a non-uniform rod and the ropes as light inextensible strings.

It is given that the tension in the rope at C is three times the tension in the rope at D.

Determine the distance of the centre of mass of the beam from A.

A particle of mass m kg is now placed on the beam at a point where the magnitude of the moment of the particle’s weight about C is 3.5 mg N m. The beam remains horizontal and in equilibrium.

Determine the largest possible value of m.

In this question the unit vectors i and j are in the directions east and north respectively.

At time t seconds, where t ⩾ 0, a particle P of mass 2 kg is moving on a smooth horizontal surface under the action of a constant horizontal force (- 8 i - 54 j)N and a variable horizontal force (4ti + 6(2t - 1)²j)N.

Determine the value of t when the forces acting on P are in equilibrium.

It is given that P is at rest when t = 0.

Determine the speed of P at the instant when P is moving due north.

Determine the distance between the positions of P when t = 0 and t = 3.

One end of a light inextensible string is attached to a particle A of mass 2 kg. The other end of the string is attached to a second particle B of mass 3 kg. Particle A is in contact with a smooth plane inclined at 30° to the horizontal and particle B is in contact with a rough horizontal plane.

A second light inextensible string is attached to B. The other end of this second string is attached to a third particle C of mass 4 kg. Particle C is in contact with a smooth plane Π inclined at an angle of 60° to the horizontal.

Both strings are taut and pass over small smooth pulleys that are at the tops of the inclined planes. The parts of the strings from A to the pulley, and from C to the pulley, are parallel to lines of greatest slope of the corresponding planes (see diagram).

The coefficient of friction between B and the horizontal plane is μ. The system is released from rest and in the subsequent motion C moves down Π with acceleration a m s^-2.

By considering an equation involving μ, a and g show that a < (2 - 1).

Given that a = , determine the magnitude of the contact force between B and the horizontal

plane. Give your answer correct to 3 significant figures.