Practice Paper Pure 4 (Edexcel International A Level Maths: Pure 1)

Given that , find the general solution to the differential equation

Find the general solution to the differential equation

giving your answer in the form .

Write    in the form   , where and  are integers to be found.

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 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 .

A curve C has parametric equations

Find a Cartesian equation for the curve C in the form .

Use calculus and the substitution  to find the exact value of

Find the coordinates of the point on the line  that is closest to the point    and hence determine the minimum distance from point  to the line.

Use integration by parts to find, in terms of e, the exact value of

The diagram below shows the graph of the curve with equation 

Find the -coordinates of the points where  the graph of intercepts the -axis.

The shaded region, , is to be rotated around the  -axis.
Find the volume of the shape generated.

A large weather balloon is being inflated at a rate that is inversely proportional to the square of its volume.

Defining variables for the volume of the balloon (m³) and time (seconds) write down a differential equation to describe the relationship between volume and time as the weather balloon is inflated.

Given that initially the balloon may be considered to have a volume of zero, and that after 400 seconds of inflating its volume is 600 m³, find the particular solution to your differential equation.

Although it can be inflated further, the balloon is considered ready for release when its volume reaches 1250 m^3.  If the balloon needs to be ready for a midday release, what is the latest time that it can start being inflated?

Prove by contradiction that if  is odd, then  must be odd.