OCR A Level Maths A Pure Mathematics and Mechanics June 2025 question paper H240/03

Express 3x² - 12x + 17 in the form a(x - b)² + c where a, b and c are constants.

State the coordinates of the minimum point of the curve y = 3x² - 12x + 17.

State the equation of the normal to the curve y = 3x² - 12x + 17 at its minimum point.

The curve y = is translated by 4 units in the positive x-direction.

Write down the equation of the curve after it has been translated.

The curve y = 5x² is stretched parallel to the y-axis with scale factor 4.

The point on the curve y = 5x² with x-coordinate 3 is transformed to the point P.

Write down the coordinates of P.

Sketch the graph of y = | 2 x - 5 | on the grid provided in the Printed Answer Booklet.

Solve the inequality | 2 x - 5 | < 1.

In this question you must show detailed reasoning.

Hence find all the possible integer values N that satisfy the inequality

Functions f and g are defined for all real values of x by

You are given that the equation f^-1g(x) = 4 - 2kx has real distinct roots.

Show that k satisfies the inequality 2k² - k - 6 > 0.

In this question you must show detailed reasoning.

Hence find the set of values of k. Give your answer in set notation.

Use the substitution u = cos x to show that

Fig. 1 shows part of the curve y = The shaded region is bounded by the curve, the x-axis, and the line x = a.

You are given that the area of the shaded region is a square units.

Show that the value of a satisfies the equation a - cos^-1(1 - 2e^-a) = 0.

Show by calculation that the value of a lies between 1.1 and 1.2.

Use the iterative formula

a_n+1 = cos^-1(1 - 2e^-a_n),

with starting value a₁ = 1.2, to find the value of a correct to 2 decimal places. Show the

result of each step of the iterative process.

Fig. 2 shows the curve y = cos^-1(1 - 2e^-x).

Explain why the gradient of the curve y = cos^-1(1 - 2e^-x) at the point where x = a, where a

is the value found in part (d), lies in the interval (- 1, 0).

The compound angle formulae for sin (A + B) and sin (A - B) are

sin (A + B) = sin A cos B + cos A sin B and sin (A - B) = sin A cos B - cos A sin B.

By letting C = A + B and D = A - B, show that

Hence write down a similar inequality interval for the expression

Using parts (a) and (c)(ii), show from first principles that the derivative of sin x is cos x,

where x is measured in radians.

A particle P of mass 3 kg is moving on a smooth horizontal surface under the action of two constant horizontal forces (5i + 3j)N and (ai + 3bj)N. The acceleration of P is (2i - 3j) m s^-2.

Find the value of a and the value of b.

At time t = 0 seconds the velocity of P is u m s^-1 and at time t = 5 seconds the velocity of P is (7i - 6j) m s^-1.

Find, in terms of i and j, an expression for u.

Two particles A and B of masses 0.6 kg and 0.8 kg respectively, are attached to the ends of a light inextensible string. Particle A is held at rest at a fixed point and B hangs vertically below A.

Particle A is now released. As the particles fall the air resistance acting on A has a constant magnitude of R N and the air resistance acting on B has a constant magnitude of 0.3 N. The string remains taut throughout the subsequent motion (see diagram).

The downward acceleration of each of the particles is 9.2 m s^-2 and the tension in the string is T N.

By applying Newton’s second law to B, find the value of T.

Hence, or otherwise, find the value of R.

In this question you must show detailed reasoning.

A particle moves in a straight line. Its velocity t seconds after leaving a fixed point on the line is v m s^-1 where v = 2 + 3t + 0.6t².

Find the distance travelled by the particle from time t = 0 until the instant when its acceleration is

15 m s^-2.

The diagram shows a non-uniform rod AB of mass 4 kg and length 2 m. The end A of the rod rests against a rough vertical wall. The rod is held in a horizontal position, perpendicular to the wall, by a light inextensible string attached to the rod at B. The other end of the string is attached to the wall at a point vertically above A. The string is inclined at an angle of θ° to the horizontal, where

The rod rests in equilibrium in a vertical plane perpendicular to the wall. The rod’s weight acts at the point G on AB.

You are given that the magnitude of the moment of the rod’s weight about A is 47.04 N m.

Show that the distance AG is 1.2 m.

Find the tension in the string.

Determine the magnitude of the contact force exerted on the rod by the wall at A.

In this question you should take the acceleration due to gravity to be 10 m s^-2.

The unit vectors i and j are in a vertical plane with i horizontal and j vertically upwards.

A small ball P is projected from a point O on horizontal ground into the air. When P is in the air, it is to be modelled as a particle moving under the influence of gravity only.

At time t = 2 seconds, P has velocity (2i - 8j) m s^-1. The position vector of a point on the trajectory of P is (x i + y j) m relative to O.

Show that y = 6x - 1.25x².

Determine the maximum height of P above the ground during its motion.

Determine the following as P passes through the point on its trajectory where x = 2.5.

• the speed of P

• the direction of motion of P

In reality P does not move only under the influence of gravity but is also subject to air resistance.

Explain how this would affect your answer to part (b).

A rectangular block B of mass 10 kg lies at rest in limiting equilibrium on a rough plane Π inclined at 30° to the horizontal. A horizontal force of magnitude T N, acting above a line of greatest slope, is applied to B (see Fig. 1).

The coefficient of friction between B and the plane is 0.8.

Show that the value of T is 14.9, correct to 3 significant figures.

For the remainder of the question, you should you use this value of T.

Block B is now cut at an angle of 30° to the horizontal into two smaller blocks. The upper block has a mass of 4 kg, and the lower block has a mass of 6 kg. The two blocks are held at rest with the lower block on Π. The horizontal force of magnitude T N is now applied to the lower block (see Fig. 2).

The two blocks are released from rest and in the subsequent motion the upper block starts to move with acceleration 3.5 m s^-2.

Determine the coefficient of friction between the two blocks.

Show that the lower block does not move.