Practice Paper Pure 4 (Edexcel International A Level Maths: Pure 3)

Starting with the equation of a semicircle of radius ,  (where ), use calculus to prove that the general formula for the volume, , of a sphere of radius  is

.

The curve C has equation ,

Line L has equation .

Show that the two points where C intersects L have equal gradients.

What else can you deduce about the two points where C and L intercept?

Express

as partial fractions.

Hence, or otherwise, find

Verify that the point A(1 , 1) lies on the curve with equation

The tangent at point A intercepts the -axis at point B and the -axis at point C.
Find the area of the triangle OBC.

The diagram below shows a sketch of the curve defined by the parametric equations

The four tangents from part (a) create a rectangle around the curve as shown below.

Find the percentage of the area of the rectangle enclosed by the curve

(the shaded area on the diagram).

In the enchanted kingdom of Vectoria, a magical flying unicorn takes off from the wizard’s palace at the point known as O and travels 30 km on a bearing of 300°.  Chased by an evil dragon, it then travels an unknown distance of  k km due north before reaching the enchanted grove at point P.  Given that the position vector of P relative to O is , and that the straight-line distance between the grove and the palace is known to be , find the exact values of x and y.

A tree disease is spreading throughout a large forested area.

The rate of increase in the number of infected trees is modelled by the differential equation

where N is the number of infected trees, t is the time in days since the disease was first identified and k is a positive constant.

Solve the differential equation above, and show that the general solution can be written in the form

where A is a positive constant.

Initially two trees were identified as diseased.
A fortnight later, 4 trees were infected.
Using this information, find the values of the constants A and k.

By considering the solution to the differential equation along with the values of A and k found in part (b), suggest a range of values of  t  for which the model might be considered reliable.

In triangle ABC, point F lies on AB and point G lies on BC.  

F divides AB in the ratio m:n.  

The line segment FG is parallel to AC.

Explain why for some constant , where .

Given that   and ,   show that

Using your result from (b), prove that G divides BC in the ratio n:m.

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