OCR A Level Maths A Pure Mathematics and Mechanics June 2018 question paper H240/03

A circle with centre C has equation x² +y² +8x-2y-7 = 0.

Find

the coordinates of C,

the radius of the circle.

Solve the equation | 2 x - 1 | = | x + 3 |.

In this question you must show detailed reasoning.

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

Write down expressions for

fg (x),

gf (x).

Hence find the values of x for which fg (x) - gf (x) = 24.

Use the trapezium rule, with two strips of equal width, to show that

Use the substitution x = u² to find the exact value of

Using your answers to parts (i) and (ii), show that

where k is a rational number to be determined.

Find the acceleration vector of the particle.

Find the position vector of the particle after 10 seconds.

the magnitude of the reaction of the support on the plank at D,

the value of x.

A stone block, which is modelled as a particle, is now placed at the end of the plank at B and the plank is on the point of tilting about D.

Find the weight of the stone block.

Explain the limitation of modelling

the stone block as a particle,

the plank as a rigid rod.

Three forces, of magnitudes 4 N, 6 N and P N, act at a point in the directions shown in the diagram.

The forces are in equilibrium.

Show that θ = 41.4°, correct to 3 significant figures.

Hence find the value of P.

The force of magnitude 4 N is now removed and the force of magnitude 6 N is replaced by a force of magnitude 3 N acting in the same direction.

Find

the magnitude of the resultant of the two remaining forces,

the direction of the resultant of the two remaining forces.

The velocity v m s^-1 of a car at time t s, during the first 20 s of its journey, is given by v = kt + 0.03t², where k is a constant. When t = 20 the acceleration of the car is 1.3 m s^-2. For t > 20 the car continues its journey with constant acceleration 1.3 m s^-2 until its speed reaches 25 m s^-1.

Find the value of k.

Find the total distance the car has travelled when its speed reaches 25 m s^-1.

One end of a light inextensible string is attached to a particle A of mass m kg. The other end of the string is attached to a second particle B of mass λ m kg, where λ is a constant. Particle A is in contact with a rough plane inclined at 30° to the horizontal. The string is taut and passes over a small smooth pulley P at the top of the plane. The part of the string from A to P is parallel to a line of greatest slope of the plane. The particle B hangs freely below P (see diagram).

The coefficient of friction between A and the plane is μ.

It is given that A is on the point of moving down the plane.

Find the exact value of μ when λ =