Practice Paper 1 (Pure 1) (CIE AS Maths: Pure 2)

The equation  has two distinct real roots.

 is a negative constant.

Find the possible values of .

In the case sketch the graph of , labelling all points where the graph crosses the coordinate axes.

Show that the equation  can be written in the form , where ,  and  are integers to be found.

Hence, or otherwise, solve the equation for 

The sum of the first three terms in a geometric series is 8.75.
The sum of the first six terms in the same series is 13.23.

Find the common ratio, , of the series.

On the same set of axes, sketch the graphs of  in the interval . Label the coordinates of points of intersection with the coordinate axes and of maximum and minimum points where appropriate.

Find the solution to the equation within the interval . Hence, determine the coordinates of the corresponding point of intersection between the two graphs in part (a).

The diagram below shows the sector of a circle .

Given that the area of triangle  = 5.64 cm², find the area of the shaded segment.
Give your answer correct to 3 significant figures.

Find the perimeter of the sector , giving your answer correct to 3 significant figures.

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  and .

Find the coordinates of point .

Write the quadratic function  in the form  where and  are integers to be found.

Write down the minimum point on the graph of 

Sketch the graph of, clearly labelling the minimum point and any point where the graph intersects the coordinate axes.

The point  lies on the curve with equation 

The graph is translated so that the point  is mapped to the point .

Write down the equation of the transformed function.

The graph is translated 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.

A function, , has second derivative given by

Given that , and  , find .

A curve has the equation 

The point  is the stationary point of the curve.

Find the coordinates of  and determine its nature.

Use calculus to find the value of

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  

The curve  has equation . The point  lies on .

Find an equation of the tangent to  at .