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

Show that  can be written as  where  to three significant figures.

Hence solve the equation

giving your answers correct to 3 significant figures.

Differentiate with respect to . 

Use calculus and the substitution  to find the exact value of

Solve the equation

tate your answers as multiples of .

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.

A tree disease is spreading throughout a large forested area.

When the disease was first discovered, three trees were infected.

Ten days later ten trees were infected.

The differential equation

where k is a positive constant, is used to model the number of infected trees, N, at a time t days after the disease was first discovered.

Find the particular solution to the differential equation.

Scientists believe the majority of the forest can be saved from infection if action is taken before 30 trees are infected.

Measured from the time when the disease was first discovered, how many days does the model predict the scientists have to take action in order to save the majority of the forest from infection?

Use integration by parts to find

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.

Determine whether the following pairs of lines intersect, are parallel, or are skew.

If the lines intersect, find the coordinates of the point of intersection.

and 

Find the perpendicular distance of the point  to the line

, where  is a real constant.

Given that  is a root of the equation  , show that  is a factor of .

Hence find the value of  and solve completely the equation  .

On an Argand diagram, shade the region which satisfies both of the following inequalities: