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

Given , find .

Circle your answer.

The diagram below shows a sector of a circle.

The radius of the circle is 7 cm and  radians.

Find the area of the sector.

Circle your answer.

The line l passes through the points (p, 2p) and (3p, 9p).

Find an equation for the line l. 

l intercepts the y-axis at (0, 3).

Find the value of p.

The sum of the first  terms of an arithmetic series is

Show that  .

Hence show that the sum of the first  terms is  .

Given that , find the value of .

The function  is defined as

Show that  can be written in the form

Explain why the inverse of  does not exist and suggest an adaption to its domain so the inverse does exist.

The domain of  is changed to .

Find an expression for  and state its domain and range.

The diagram below shows part of the graph where

Find  and , giving your answers to three significant figures.

One of the solutions to the equation  is , correct to three significant figures.

Hence, use the sign change rule to confirm that this is a solution
(to three significant figures) to the equation .

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 geometric series is given by 

Write down the common ratio, , of the series.

Given that the series is convergent, and that , calculate the value of .

Prove by contradiction that  is an irrational number.  You may use without proof the fact that if is a multiple of 11, then n is a multiple of 11.

In a computer animation, the side length, s mm, of a square is increasing at a constant rate of 2 millimetres per second.

Write down the value of , where t is time and measured in seconds.

Write down a formula for the area, A mm², of the square and hence find  .

Use the chain rule to find an expression for  in terms of s and hence find the rate at which the area is increasing when  s=10.

Given that

Show that

Hence prove that .

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.

Show that if  , then 

Hence find the gradient of the tangent to the curve  at the point with coordinates 

The diagram below shows part of the graph with equation

Use the trapezium rule with 5 strips to find an estimate for the shaded area, giving your answer to three significant figures.

Use a suitable substitution to find the following

A soft ball is thrown upwards from the top of a building.
The height,  m of the ball above the ground after  seconds is modelled by the function

What is the significance of the constant 15 in the function?

At what time is the ball at the same height as when it was thrown?

Find the time at which the ball is at its maximum height and what this maximum height is.

How long does it take for the ball to first hit the ground?

Given that the ball first hits the ground at a distance 20 m from the base of the building find the shortest distance between this point and where the ball was thrown from.

Differentiate with respect to x, simplifying your answers as far as possible:

Integrate

Use calculus to find the value of