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

Factorise completely .

Solve the equation .

Solve .

Solve the equation .

A triangle ABC has sides , and angle .

The area of the triangle is   cm².

Show that  satisfies the equation .

Hence, or otherwise, find the perimeter of the triangle. Giving your answer to 3 significant figures.

Given angle ABC is obtuse, find the ratio of the angles of the triangle, to the nearest degree.

Find the equation of the curve passing through the point (4, -8) and given by

A stone is thrown vertically upwards from the top of a cliff.  The path of the stone is modelled by the quadratic function , , where h is the height, in meters, of the stone above the sea and t is the time in seconds since the stone was thrown.

Write down the height of the cliff from which the stone was thrown.

Find the maximum height the stone reaches above the sea.

How long does it take for the stone to hit the sea?

How long does the stone stay above it’s starting height for?

A firework is launched inside a large shed with a sloping roof.  In relation to the horizontal distance from the point it was launched, the height of the firework, m, can be modelled by the quadratic equation

The sloping roof of the shed can be modelled with the equation

Determine whether, according to the model, the firework will hit the roof of the shed before escaping out the open end of the shed on the right of the diagram.

The curve C has equation .  The point P(2,  2) lies on C.

Find an equation of the tangent to C at P.

Show that   can be written as .

Show that , where a is a rational number to be found.

Show that    can be written as , where and a and b are integers to be found.

The point  lies on the line , crosses the -axis at the point R.

Another line, , is perpendicular to  at the point and crosses the -axis at the point (-1, 0).

Find the area of the triangle .

The diagram shows a prism with cross-section in the shape of a sector of a circle. 

The radius of the sector is cm, and the angle at the centre is radians.

The height of the prism is cm.

Given that the volume of the prism is cm³, find the possible value(s) of .

On the same diagram, sketch the graphs of  and .

Write down the number of solutions to the equation

.

The graph below shows two curves with equations  and , in the interval , where  and are integers.

Using the graph above, find the values of and and label the points of intersection each graph has with the coordinate axes.

Within the stated interval, the curves intersect at the two points and as shown in the diagram. The coordinates of point  are (9.90°, 0.34), accurate to 2 decimal places. By considering the graph, as well as the properties of the sine and cosine functions, state the coordinates of Point , to two decimal places.