Practice Paper 3 (Pure 3) (CIE A Level Maths: Pure 1)

Find, in ascending powers of , the binomial expansion of

up to and including the term in .

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 

Express

as partial fractions.

Hence, or otherwise, find

The diagram below shows the graphs of  and .

Show on the diagram, using the value of indicated, how an iterative process will lead to a sequence of estimates that converge to the x-coordinate of the point P.
Mark the estimates and on your diagram.

By finding a suitable iterative formula, use to estimate a root to the equation correct to two significant figures.

Confirm that your answer to part (b) is correct to two significant figures.

Show that

where k is a constant, and c is the constant of integration.

Use calculus to find the exact value of

writing your answer in the form  a ln b , where a and b are rational numbers to be found.

A large container of water is leaking at a rate directly proportional to the square of the volume of water in the container.

The coordinates of three points are  and  .

Find  and .

By considering the scalar product , or otherwise, calculate the angle between  and  .  Give your answer in degrees, accurate to 1 decimal place.

Given that  and :

Work out   and 

Hence or otherwise work out , giving your answer in the form  where  and  are real numbers.

On an Argand diagram, sketch the loci (i.e., sets of points) for which each of the following equations is true:

Shade the region of your diagram that satisfies both of the following inequalities:

The rare Leaping Unicorn jumps in such a way that the length of a jump is always the same distance.  However, the maximum height a Leaping Unicorn reaches during a jump reduces gradually over time as the unicorn tires.

The way in which Leaping Unicorns jump can be modelled by the function

where  is the horizontal distance covered and  is the height, both measured in metres.
 and  are both positive constants.

(i)      Write down the length of a Leaping Unicorn jump.

(ii)     Briefly describe how changing the value of the constant  would affect the

         model.

During its first jump, a Leaping Unicorn reaches a maximum height of 1.288 metres after covering 1.471 metres over the ground.
Find the values of and .

What is the total distance of ground covered by a Leaping Unicorn when it is at the maximum height of its third jump?