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

The functions  and  are given as follows

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

Find the values for  for which the expansion is valid.

The diagram below shows part of the curve defined by the equation  where is a positive constant.  The shaded region  is bounded by the curve, the  -axis, and the lines  and  

Given that the volume of the solid formed when the region   is rotated  about the -axis is  cubic units, find the value of .

A sketch of the graph with parametric equations

is shown below.

Find the coordinates of all points where the graph intersects the coordinate axes.

Find the coordinates of the points where the graph intersects the line with equation .

Find an expression for  in terms of for the parametric equations

The graph of  y  against  x  passes through the point P (1 , 1).

Use the substitution  to show that

where c is the constant of integration.

The curve C is described by the equation

Show that the normal to C at the point where is parallel to the normal to C at the point where

Find the distance between the -axis intercepts of these two normals.

Find the integral

Find an expression for y given that

Integrate

Determine whether each of the following pairs of lines intersect, are parallel, or are skew.  If the lines intersect, find the coordinates of the point of intersection.

   and    

     and    .

A bar of soap in the shape of a cuboid is placed in a bowl of warm water and its volume is recorded at regular intervals.  The water is maintained at a constant temperature.

Before being placed in the water the soap measures 3 cm by 6 cm by 10 cm.

Two minutes later the bar of soap measures 2.85 cm by 5.7 cm by 9.5 cm.

The rate of decrease in volume of the bar of soap is modelled as being directly proportional to its volume.

Defining any variables you use, find and solve a differential equation linking the volume of the bar of soap and time.

What happens to the volume of the bar of soap for large values of t?
Briefly explain why this could be considered a criticism of the model.