OCR A Level Maths A Pure Mathematics and Mechanics June 2019 question paper H240/03

The diagram shows triangle ABC, with AC = 13.5cm, BC = 8.3cm and angle ABC = 32°.

Find angle CAB.

A circle with centre C has equation x² + y² - 6x + 4y + 4 = 0.

Find

the coordinates of C,

the radius of the circle.

Determine the set of values of k for which the line y = kx - 3 does not intersect or touch the

circle.

In this question you must show detailed reasoning.

Solve the inequality | x - 2 | ⩽ | 2 x - 6 |.

Give full details of a sequence of two transformations needed to transform the graph of

y = | x - 2 | to the graph of y = | 2 x - 6 |.

The diagram shows the part of the curve y = 3x sin 2x for which 0 ⩽ x ⩽ .

The maximum point on the curve is denoted by P.

Show that the x-coordinate of P satisfies the equation tan 2 x + 2 x = 0.

Use the Newton-Raphson method, with a suitable initial value, to find the x-coordinate of P,

giving your answer correct to 4 decimal places. Show the result of each iteration.

The trapezium rule, with four strips of equal width, is used to find an approximation to

Explain briefly why it is not easy to tell from the diagram alone whether the trapezium rule with four strips of equal width gives an under-or over-estimate for the area of the

region in this case.

In this question you must show detailed reasoning.

Hence solve, for 0 < θ < 2π, 3(cot θ + cosec θ)² = 2 sec θ.

Find the exact area of the shaded region, giving your answer in the form p + q ln r, where p and q

are positive integers and r is a positive rational number.

A cyclist starting from rest accelerates uniformly at 1.5ms^-2 for 4 s and then travels at constant speed.

Sketch a velocity-time graph to represent the first 10 seconds of the cyclist’s motion.

Calculate the distance travelled by the cyclist in the first 10 seconds.

A particle P projected from a point O on horizontal ground hits the ground after 2.4 seconds.

The horizontal component of the initial velocity of P is ms^-1.

Find, in terms of d, the horizontal distance of P from O when it hits the ground.

Find the vertical component of the initial velocity of P.

P just clears a vertical wall which is situated at a horizontal distance d m from O.

Find the height of the wall.

The speed of P as it passes over the wall is 16ms^-1.

Find the value of d correct to 3 significant figures.

The diagram shows a small block B, of mass 0.2 kg, and a particle P, of mass 0.5 kg, which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane.

The block can move on the horizontal surface, which is rough. The particle can move on the inclined plane, which is smooth and which makes an angle of θ with the horizontal where tan θ =

The system is released from rest. In the first 0.4 seconds of the motion P moves 0.3 m down the plane and B does not reach the pulley.

Find the tension in the string during the first 0.4 seconds of the motion.

Calculate the coefficient of friction between B and the horizontal surface.

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

A particle R of mass 2 kg is moving on a smooth horizontal surface under the action of a single horizontal force F N. At time t seconds, the velocity v ms^-1 of R, relative to a fixed origin O, is given by v = (pt² - 3t)i + (8t + q)j, where p and q are constants and p < 0.

Given that when t = 0.5 the magnitude of F is 20, find the value of p.

When t = 0, R is at the point with position vector (2 i - 3 j) m.

Find, in terms of q, an expression for the displacement vector of R at time t.

When t = 1, R is at a point on the line L, where L passes through O and the point with position vector 2 i - 8 j.

Find the value of q.

The diagram shows a ladder AB, of length 2 a and mass m, resting in equilibrium on a vertical wall of height h. The ladder is inclined at an angle of 30° to the horizontal. The end A is in contact with horizontal ground. An object of mass 2 m is placed on the ladder at a point C where AC = d.

The ladder is modelled as uniform, the ground is modelled as being rough, and the vertical wall is modelled as being smooth.

Hence find, in terms of a, the greatest possible value of d.

State one improvement that could be made to the model.