Practice Paper 1 (Pure) (OCR A Level Maths: Pure)

A company selling books models the number of books sold per year,, using the formula

where  is the price per book in pounds sterling.

Find the number of books the company can expect to sell if they are priced at £18 each?

Find the number of books the company can expect to sell if they are priced at £16 each and work out the total income the company will receive if they sell all books at this price.

What do your answers to part (a) and (b) suggest about the relationship between the price of a book, the number sold and the total income received?

The leakage rate of water from a pipe,  (litres per second), is directly proportional to the square root of the flow rate,  (meters per second), which is the speed of the water flowing through the pipe.
It was observed that the leaking rate was  when the flow rate was .

Write down an equation connecting and .

Find the flow rate when the leakage rate is .

An alternative model for the leakage rate is .
Apart from when there is no leak find a flow rate and a leakage rate for when both models predict the same result.

Use the factor theorem to show that  is a factor of .

Factorise  completely.

Write down all the real roots of the equation .

The point  lies on the curve with equation . 

State the coordinates of the image of point  on the curves with the following equations:

The diagram below shows a sketch of the graph with equation

On the sketch, mark the approximate locations of the following ...

Also highlight sections of the curve where the graph is convex.

The functions and   are given as follows

Expand , in ascending powers of  up to and including the term in .

Expand , in ascending powers of  up to and including the term in .

Find the expansion of   in ascending powers of , up to and including the term in .

Find the values of  for which your expansion in part (c) is valid.

The function f(x) is defined as

Show that

Use the Newton-Raphson method with to find a root of the equation correct to five significant figures.

Write down the exact value of a root to the equation .

The functionis defined as

Work out the range of .

If the domain of  is changed to , what is the range of ?

Show that the equation  tan² sec   can be written as

sec² sec

Hence, or otherwise, solve the equation

tan²  sec 

Give your answers to three significant figures.

The graph of the curve C shown below is defined by the parametric equations

Write down the value of    at the point (0 , 6).

Write down the value of  at the points (-3 , 3) and (3 , 3).

Find an expression for   in terms of .

The diagram below shows two right-angled triangles.
Angles  and  have been labelled.

Given that   , find the exact values of sin ,cos  and tan .

Using the identities

sin(A + B) sin A cos B + sin B cos A and

cos 2A 1 –  2 sin² A

show that sin 3A  3 sin A – 4 sin³ A

The diagram below shows a sketch of the curves with equations

    and      

Show that the two curves intersect at the points (3, 4), (4, 3) and (5, 0).

By using the substitution  , show that

where c is the constant of integration.

Using calculus, and your results from parts (a) and (b), show that the total shaded area enclosed by the two curves in the diagram is equal to

A tree disease is spreading throughout a large forested area.

The rate of increase in the number of infected trees is modelled by the differential equation

where N is the number of infected trees, t is the time in days since the disease was first identified and k is a positive constant.

Solve the differential equation above, and show that the general solution can be written in the form

where A is a positive constant.

Initially two trees were identified as diseased.
A fortnight later, 4 trees were infected.
Using this information, find the values of the constants A and k.

By considering the solution to the differential equation along with the values of A and k found in part (b), suggest a range of values of  t  for which the model might be considered reliable.