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

Show that can be written in the form , where and are constants to be found.

Hence write down the centre and radius of the circle with equation

The line meets the circle with equation .

Solve the equation 

Describe a sequence of transformations that map the graph of  onto the graph of 

Given  ,  where ,  show that

Solve the inequality

and hence determine the set of values for which the graph of   is convex.

The function is defined as

      , where is in radians.

Find .

Use the Newton-Raphson method with to find a root, α, of the equation , correct to four decimal places.

The graph of has a local maximum point at . Briefly explain why the Newton-Raphson method would fail if the exact value of β was used for .

The diagram below shows part of the graph with equation .

The trapezium rule is to be used to estimate the shaded area of the graph which is given by the integral

Prove the identity

Express

as partial fractions.

Hence, or otherwise, find

The diagram below shows the velocity-time graph for a train travelling between two stations, starting at station P and finishing at station Q. The graph indicates velocity in kilometres per hour and time in minutes.

A diver jumps from the edge of a diving board that is 10 m above the surface of the water.  The diver leaves the board at an angle of 80° above the horizontal with a speed of .  The diver is then modelled as a projectile until they splash into the swimming pool below.

Find the time of the dive, giving your answer in seconds to three significant figures.

Find the maximum height above the water achieved by the diver, giving your answer to the nearest tenth of a metre.

Two particles  and , of masses 4 kg and 3 kg respectively, are connected by means of a light inextensible string.  Particle  is held motionless on a rough fixed plane inclined at  to the horizontal.  The string passes over a smooth light pulley fixed at the top of the plane so that  is hanging vertically downwards as shown in the diagram below:

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

The system is released from rest with the string taut.

Calculate the acceleration of the two objects and the tension in the string as  descends.

After descending for 3.2 seconds, particle  strikes the ground and immediately comes to rest.  Particle  continues to move up the slope until the forces of gravity and friction cause it to come momentarily to rest.

Find the total distance travelled by particle  between the time that the system is first released from rest and the time that particle  comes momentarily to rest again after  has struck the ground.

A stone is thrown from the edge of a deep cave such that it will fall into the cave and has velocity

at time t seconds after it is thrown. 

Find the position of the stone at the time it is about to fall into the cave (rather than being in the air above the cave).

Find the maximum height above the cave the stone reaches.

The deepest known cave in the world has a depth of 2212 m. (The Veryovkina Cave in Georgia.) The model above suggests the stone would take around  to reach this depth. Consider the average speed and the acceleration of the stone to highlight a problem with this model.

An ice skater moves across a straight section of a frozen river such that their position, at time t seconds relative to an origin is given by

(i)        Find the distance the skater is from the origin after 25 seconds.

(ii)       As they skate forwards, the skater slowly crosses the width of the river.
            It takes 225 seconds for the skater to cross the river.
            How wide is the river?

Show that the magnitude of acceleration of the skater at time t seconds is given by .

In the following diagram AB is a ladder of length 2a and mass m_l.  End A of the ladder is resting against a rough vertical wall, while end B rests on rough horizontal ground so that the ladder makes an angle of θ with the ground as shown below:

A person with mass m_p is standing on the ladder a distance d from end B. The ladder may be modelled as a uniform rod lying in a vertical plane which is perpendicular to the wall, and the person may be modelled as a particle. The coefficient of friction between the wall and the ladder is μ_A, and the coefficient of friction between the ground and the ladder is μ_B. It may be assumed that .

Given that the ladder is at rest in limiting equilibrium, show that

where R_B is the normal reaction force exerted by the ground on the ladder at point B and where g is the constant of acceleration due to gravity.

Hence find an equivalent expression for R_A, the normal reaction force exerted by the wall on the ladder at point A when the ladder is at rest in limiting equilibrium.