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

A curve has equation .

Describe the transformation of the curve given by the equations below:

The functions  and  are defined as follows

Write down the range of .

Solve the equation 

Determine the number of points of inflection on the curve with equation

and determine their coordinates.

The function is defined as

Use the sign change rule to show there is a root to the equation in the interval .

Find .

Use the Newton-Raphson method with to find the root in the interval correct to three decimal places.

The curve has parametric equations

Show that at the point (0 , 6),and find the value of    at this point.

The tangent at the point (0 , 6) is parallel to the normal at the point P.
Find the exact coordinates of point P

The graph defined by the parametric equations

is shown below.

The point where has coordinates (1 , 1).

The point where has coordinates (8 , 7).

Find the values of and .

Verify that the point A(1 , 1) lies on the curve with equation

The tangent at point A intercepts the -axis at point B and the y-axis at point C.
Find the area of the triangle OBC.

A train leaves station O from rest with constant acceleration
80 seconds later it passes through (but does not stop at) station A at which point its acceleration changes to . 180 seconds later the train passes through station B.

Find the displacement of the train from station O when it passes through station A.

Find the velocity of the train as it passes through station A.

Find the displacement of the train between stations A and B.

In a cheese-rolling competition, a cylindrical block of cheese is rolled down a hill and its acceleration, , is modelled by the equation.

where t is the time in seconds. The block of cheese reaches the bottom of the hill after 20 seconds.

Find the velocity of the block of cheese when it reaches the bottom of the hill.

Show that the distance down the hill, as travelled by the block of cheese, is 330 m to two significant figures.

Two particles A and B, of identical mass, are connected by means of a light inextensible string. Particle A is held motionless on a rough fixed plane inclined at 30° to the horizontal, and that plane is connected at its top to another rough fixed plane inclined at 70° to the horizontal.  The string passes over a smooth light pulley fixed at the top of the two planes so that B is hanging downwards in contact with the second plane.   This situation is shown in the diagram below:

The parts of the string between A and the pulley and between B and the pulley each lie along a line of greatest slope of the respective planes. The coefficient of friction between the particles and the planes is 0.15 in both cases.

The system is released from rest with the string taut, and with particle B a vertical distance of from the ground.  Particle B descends along the slope until it reaches the ground, at which point it immediately comes to rest.  Particle A 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 A between the time that the system is first released from rest and the time that particle A comes momentarily to rest again after B has reached the ground.

In the following diagram  is a ladder of length 10 m and mass 34 kg.  End  of the ladder is resting against a smooth vertical wall, while end  rests on rough horizontal ground so that the ladder makes an angle of  with the ground as shown below:

A housepainter with a mass of 75 kg has decided to climb up the ladder without taking any additional precautions to prevent the bottom of the ladder from slipping.  The ladder may be modelled as a uniform rod lying in a vertical plane perpendicular to the wall, and the housepainter may be modelled as a particle.  The coefficient of friction between the ground and the ladder is 0.4.

Luckily, the housepainter’s partner convinces him not to climb up the ladder without providing some additional support at the bottom to prevent slipping.  If the housepainter had continued with his original plan, however, how far above the ground would he have been when the ladder began to slip?

A particle is projected from a point on a horizontal plane with initial velocity  at an angle of above the horizontal.  The particle moves freely under gravity.  is the constant of acceleration due to gravity.

Show that the time of flight of the particle, T seconds, is given by

Show that the range of the particle, R m, on the horizontal plane is given by

For a particle modelled as a projectile with initial velocity at an angle of α^° above the horizontal, show that the equation of the trajectory of the particle is given by

The flight of a particle projected with an initial velocity of at an angle α above the horizontal is modelled as a projectile moving under gravity only. The particle is projected from the point  (x₀, y₀) with the upward direction being taken as positive, and with the coordinates being expressed in metres.  is the constant of acceleration due to gravity.

Write down expressions for

For a particular projectile, ,  and the particle is projected from the point  Find an expression for the trajectory of the particle, giving your answer in the form

where the constants a, b and c are expressed in terms of g.