Practice Paper 3 (Pure & Mechanics) (OCR A Level Maths: Pure)

In a triangle ABC, cm,  cm and cm, angle °.

Show that .

Given that , find the area of the triangle.

The second and fifth terms of a geometric series are 13.44 and 5.67 respectively.  The series has first term  and common ratio .

By first determining the values of  and , calculate the sum to infinity of the series.

Calculate the difference between the sum to infinity of the series and the sum of the first 20 terms of the series. Give your answer accurate to 2 decimal places.

By sketching the graphs of  and  on the same diagram show that there are two real solutions to the equation .

Show that cos sin can be written in the form  cos, where  and  is an acute angle measured in radians.

Hence, or otherwise, solve the equation cos sin ,for .
Give your answers to three significant figures.

Write down the minimum value of cos sin and the smallest positive value of for which it occurs.  Give your value of   to three significant figures.

Show that the equation   can be rewritten as

Starting with , use the iterative formula

to find a root of the equation , correct to two decimal places.

A bar of soap in the shape of a cuboid is placed in a bowl of warm water and its volume is recorded at regular intervals.  The water is maintained at a constant temperature.

Before being placed in the water the soap measures 3 cm by 6 cm by 10 cm.

Two minutes later the bar of soap measures 2.85 cm by 5.7 cm by 9.5 cm.

The rate of decrease in volume of the bar of soap is modelled as being directly proportional to its volume.

Defining any variables you use, find and solve a differential equation linking the volume of the bar of soap and time.

What happens to the volume of the bar of soap for large values of t?
Briefly explain why this could be considered a criticism of the model.

Use integration by parts to find, in terms of e, the exact value of

In one minute, a particle travels a distance of 1932 m. At this point, its velocity is .  Assuming it is constant, find the acceleration of the particle.

A particle of mass 0.9 kg is at rest on a rough horizontal plane.  A force of magnitude P N is acting on the particle at an angle of 40° to the horizontal.

Given that the coefficient of friction between the plane and the particle is 0.3, and that the particle is on the point of moving to the right under the influence of the force, find the value of P.

A golfer strikes a ball from ground level with velocity

Find the distance the golf ball will travel before first hitting the ground.

Show that by reducing the angle of the strike above the horizontal by  the golfer can achieve approximately 7 m more distance before the ball lands.

Give a reason why the golfer may not want to achieve a longer distance with their shot.

is a non-uniform rod of mass 12 kg and length 4 m.  is held horizontally in equilibrium by a support placed at point  and a vertical wire attached to point such that  and  as shown in the diagram below:

A weight of mass 15 kg is attached to the rod at point  and the rod is at the point of tilting about point .  The weight is then removed.

Find the ratio of the reaction force at  to the tension in the wire at when there are no external weights attached to the rod.  Give your answer in the form  where  and  are integers with no common factors other than 1.

The 15 kg weight is then attached to the rod between points  and .

Find the greatest distance to the left of point  that the weight can be attached without the rod beginning to tilt.

At time t seconds, a particle P has acceleration a m s⁻², where

                                                    .

Initially P starts at the origin O and moves with velocity .

Find the distance between the origin and the position P of when .

Find the value of t at the instant when P is moving in the direction of .

Two particles A and B, of masses 2.7 kg and 2.2 kg respectively, are connected by means of a light inextensible string.  Particle A is held motionless on a rough fixed plane inclined at 25° to the horizontal.  The string passes over a smooth light pulley fixed at the top of the plane so that B is hanging vertically downwards as shown in the diagram below:

The string between A and the pulley lies along a line of greatest slope of the plane, and B hangs freely from the pulley.  The coefficient of friction between particle A and the plane is μ.

The system is released from rest with the string taut.  Given that particle B descends 1.82 m in the first 3 seconds after it is released, find the value of μ.

A particle of mass m kg is released from rest on a rough plane inclined at θ° to the horizontal, where .  The coefficient of friction between the particle and the plane is .

Given that the particle remains motionless after it is released, show that .