OCR A Level Maths A Pure Mathematics and Mechanics June 2024 question paper H240/03

Simplify each of the following.

In this question you must show detailed reasoning.

The diagram shows a sector AOB of a circle with centre O and radius (3 x + 1)cm. The angle AOB is 2 radians. The area of sector AOB is less than (44x - 7)cm².

Find the set of possible values of x. Give your answer in set notation.

Expand (3 - 2x)^-2 in ascending powers of x up to and including the term in x².

State the set of values of x for which this expansion is valid.

When is expanded in ascending powers of x, the coefficient of x is zero.

Determine the value of the constant a.

Show that the equation 2 cot²x - 9 cosec x - 3 = 0 can be expressed in the form

In this question you must show detailed reasoning.

Hence solve, for 0 < θ < π,

Give your answers correct to 3 decimal places.

The small angle approximation for sin2 θ is used to find an approximation for the smallest positive solution of the equation 2 cot²2θ - 9 cosec 2θ - 3 = 0.

Show that this approximate solution is accurate to 2 decimal places.

The diagram shows the curve with equation y = (x³ - 2x²)ln x. The curve has a point of inflection at the point M.

Show that the x-coordinate of M satisfies the equation

Use an iterative formula, based on the equation in part (a)(i), to determine the x-coordinate of M correct to 2 decimal places. Use an initial value of 1.1 and show the

result of each step of the iterative process.

Determine the exact area of the shaded region, giving your answer in the form p ln q - r,

where p and r are positive rational numbers and q is a positive integer.

The curve C is defined, for 0 ⩽ t < 2 π, by the parametric equations

x = 4 k + k sin t, y = 2 + 4 cos t,

where k is a constant.

Find a cartesian equation for C. You do not need to simplify your answer.

You are given that C is a circle.

Determine the radius of C.

Find the possible coordinates for the centre of C.

The diagram shows the curve 5x - 2xy + 2y² - k = 0, where k is a positive integer.

At the points P and Q on the curve, the tangents to the curve are parallel to the y-axis.

Given that the difference in the y-coordinates of P and Q is 3, determine the x-coordinates of

P and Q.

A particle P is moving with constant acceleration (-5i + 2j)m s^-2. At time t = 0 seconds, P is at the origin and has velocity (i + 3j)m s^-1.

Find, in terms of i and j, the displacement of P at time t = 2 seconds.

Determine the speed of P at time t = 2 seconds.

Two horizontal forces of magnitudes 17 N and 12 N act at a point O along bearings of 050° and 340° respectively (see diagram).

Determine the magnitude and bearing of the resultant force.

A third horizontal force F is now applied at O. The three forces are in equilibrium.

State the magnitude of F and give the bearing along which it acts.

A block of mass m kg is on smooth horizontal ground with one end of a light inextensible rope attached to its upper surface. The other end of the rope is attached to an object of mass 5 kg. The rope passes over a small smooth pulley, and the object hangs vertically below the pulley. The part of the rope between the block and the pulley makes an angle of 50° with the horizontal. A force of magnitude X N acts on the block at an angle of 20° above the horizontal in the vertical plane containing the rope (see diagram).

You are given that the block is in equilibrium.

Determine the value of X.

You are also given that the magnitude of the contact force exerted by the ground on the block is 147 N.

Determine the value of m.

By taking moments, show that the magnitude of the normal contact force between the lamina

and the wall is 1.24 N, correct to 3 significant figures.

Determine the coefficient of friction between the lamina and the ground.

A particle P moves in a straight line. The velocity v ms^-1 of P at time t seconds is given by

where k is a positive constant.

Sketch, on the axes in the Printed Answer Booklet, the velocity-time graph for P for values of

t from 0 to 9.

State the value of t in the interval 0 ⩽ t ⩽ 9 when the acceleration of P is zero.

In this question you must show detailed reasoning.

You are given that the total distance travelled by P in the interval 0 ⩽ t ⩽ 9 is 84 m.

Find the value of k.

The points A and B are the lower and upper ends, respectively, of a line of greatest slope on a plane inclined at an angle θ to the horizontal, where sin θ = 0.6 and AB = 1.375 m (see diagram).

A particle P is projected up the plane with speed 6 ms^-1 from A towards B.

The plane at A is fixed to the ground which is horizontal.

The surface of the plane is rough and the coefficient of friction between P and the plane is 0.25.

Show that the speed of P at B is 3.8 ms^-1.

The particle leaves the slope at B and moves freely under gravity.

The particle first lands at a point C on the horizontal ground. The time taken for P to travel from A to C is T seconds.

Determine the value of T.