Practice Paper Pure 1 (Edexcel International A Level Maths: Pure 4)

Write the quadratic function  in the form  where a, b and c 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.

Solve the inequality 

The circle sector  is shown in the diagram below.

The angle at the centre is radians , and the line segments  and  have lengths of cm and cm respectively.

Additionally,  is parallel to , so that  and .

Show that the area of the sector is   cm².

Show that the area of the triangle is cm².

Given that the area of the shaded shape is cm².  find the value of .

Solve the simultaneous equations

The diagram below shows the graph of . The graph intersects the coordinate axes at the two marked points and .  The graph has two asymptotes as shown, with equations     and  .

In separate diagrams, sketch the curves with equation

On each diagram, give the coordinates of the images of points  and  under the given transformation, as well as stating the equations of the transformed asymptotes.

The graph of  has an asymptote with equation .  Find the value of .

The graph below shows part of the curve with equation ,  where the angle is measured in radians and k is a constant.

A student states that there are an infinite number of possible values for k. Is the student correct?  You must explain your answer fully.

Another student claims that the curve could also be the graph of the equation . Find a value for k to show that the student is correct.

The line has equation and crosses the -axis at point

The line  is perpendicular to  and crosses the x-axis at (9, 0).

 crosses the -axis at point .

Find the area of the triangle , where  is the origin.

The line  passes through the points  and .

Find an equation for the line . 

intercepts the y-axis at (0, 3).

Find the value of .

On the same diagram, sketch the graphs of  and .

Use your graph to determine the number of solutions to the equation

.

The equation  has two distinct real roots, .
k is a negative constant and .

Sketch the graph of , labelling the points where the graph crosses the coordinate axes.

Find the possible values of k.

A curve has equation .  

A is the point on the curve with x coordinate 0, and B is the point on the curve with x coordinate 6.  

C is the point of intersection of the tangents to the curve at A and B. 

Find the coordinates of point C.

Calculate the area of triangle ABC.

Use calculus to work out the following integrals: