Practice Paper Pure 2 (Edexcel International A Level Maths: Pure 4)

On the same axes, sketch the graphs of and .

Label any points of intersection with the coordinate axes.
Write down the equations of any asymptotes.

Write down an equation for the graph that is a reflection of in the -axis.

Solve .

Prove that the square of an even number is a multiple of 4.

Prove that the (positive) difference between an integer and its cube is the product of three consecutive integers.

Find the coefficient of the term in in the expansion of 

Find the first three terms, in ascending powers of , in the expansion of 

Use your answer to part (a) to estimate .

The diagram below shows part of the graph with equation  .

Use the trapezium rule with 5 strips to find an estimate for the shaded area, giving your answer to three significant figures.

Given

find the value of the constant p, where .

Triangle  has vertices (-8, 1), (12, 16) and (12, 1).  A circle with equation touches Triangle  at the three points and , as shown in the diagram below:

Write down the coordinates of points R and Q.

Find the coordinates of point P.

Show that

Find the remainder when  is divided by .

Given that  is a factor of , factorise  completely.

Solve the equation 3 sin  in the interval , giving your answers to 3 significant figures.

Solve the equation cos² sin  for , giving your answers to 1 decimal place where appropriate.

The diagram below shows the part of the curve with equation for which . The marked point P lies on the curve. O is the origin.

Show that

Find the minimum distance from O to the curve, using calculus to prove that your answer is indeed a minimum.