Practice Paper 1 (Pure 1) (CIE AS Maths: Pure 1)

Find the equation of the curve passing through the point (-2, 3) and given by

In a computer animation, the radius of a circle increases at a constant rate of 1 millimetre per second. Find the rate, per second, at which the area of the circle is increasing at the time when the radius is 8 millimetres. Give your answer as a multiple of .

The diagram shows the graph of , where ,  

Write down the maximum value of  when .

Write down the value of  for which this maximum occurs.

Find, in terms of , the combination of transformations that would map the graph of  onto the graph of ,  .

In the expansion of  the coefficient of the  term is 216.

In the expansion of  the coefficient of the  term is 4860.

Find the possible values of  and .

The curve C has equation .  The point P lies on C.

Find an expression for .

Show that an equation of the normal to C at point P is .

Show that .

Hence, or otherwise, solve the equation tan³ tan² tan  for  , giving your answers to 1 decimal place where appropriate.

The th term of a geometric progression is given by .

Calculate, giving your answers as exact values

The sum to infinity of the progression starting with the seventh term.

The sum to infinity of the progression whose th term is given by , where  is defined as above.

A circle has equation

The lines l₁ and l₂ are both tangents to the circle, and they intersect at the point (5, 0).

Find the equations of l₁ and l₂, giving your answers in the form .

The circle sector  is shown in the diagram below.

The angle at the centre is  radians , and the radii  and  are each equal to cm

Additionally, is parallel to , so that   and .

In the case when cm, show that the area of the shaded shape is given by sin .

.

Show that for small values of , the area of is approximately .

The functions f(x) and g(x) are defined as follows

Find 

(i)  fg(x)

(ii)  gf(x)

Write down and state its domain and range.

The graphs of f(x) and  are drawn on the same axes.
Describe the transformation that would map one graph onto the other.

Find the coordinates of the point where the graphs of  y = f(x) and  meet. 

A curve has the equation .

Find expressions forand .

Determine the coordinates of the local minimum of the curve.

The diagram below shows the graphs of  

Find the area of the shaded region, .