Practice Paper 1 (Pure 1) (CIE A Level Maths: Pure 1)

Find the first three terms in the expansion of .

Given that  is small such that  and higher powers of  can be ignored show that 

A ball is dropped and bounces such that the height of each bounce is 80% of the height of the previous bounce.

The first bounce of the ball reaches a height of 1.60 m.

Find the height the ball bounces on its 8th bounce.

Find the total vertical distance travelled by the ball from when it first hits the ground until it hits the ground at the end of its twentieth bounce.

Give one reason why this model may be unrealistic.

The point (3, —12) lies on the curve with equation .

The graph is stretched so that the point  is mapped to the point (3, —4). Write down the equation of the transformed function in the form , where , and  are constants to be found.

The graph is stretched so that the point is mapped to the point . Write down the equation of the transformed function in the form , where  is a constant to be found.

The curve C has equation .  The line l has equation .

Find any points of intersection between C and l.

Sketch the graphs of C and l, showing clearly any points of intersection with the coordinate axes for both graphs, the minimum point of C and any points of intersection found between C and l.

The functions and are defined as follows

Write down the range of .

Find

Solve the equation 

The graph below shows the curve with equation , in the interval  . One value of  has been labelled .

Use the graph, along with the symmetry properties of the sine function, to verify that 

                              .

Solve the equation for .

The diagram below shows the sector of a circle .

Show that the area of the shaded segment is given by  cm².

Find, in terms of , the percentage of the sector that the segment occupies.

A curve is described by the equation , where

Find and .

P is the stationary point on the curve.

Find the coordinates of P and determine its nature.

A circle has equation

The lines and  are both tangents to the circle, and they intersect at the point (0, 14).

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

The line with equation    cuts the curve with equation  at the points P and Q, as shown.

Find the x- and y-coordinates of the points P and Q.

Find the exact area of the region labelled R, giving your answer in the form  , where a and b are integers to be found.

A mathematical model for a bowl is obtained by rotating through  about the -axis the part of the curve   which is between  and   and then adding a flat bottom.   is a constant such that .

Find the capacity of the bowl in terms of .

In the case where the boundaries   and  are used in place of  and ,  the capacity of the bowl is increased by cubic units.
Find the value of .