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Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
1. | GCSE Higher [788] |
An ornamental lake is in the shape of a semicircle with a diameter of 900 metres.
(a) Calculate the perimeter of the lake to the nearest metre.
(b) The lake has a uniform depth of 1.5 metres. Calculate the volume of the water in the lake in cubic metres to the nearest integer.
(c) Give the volume of the lake in litres. Give your answer correct to three significant figures in standard form.
2. | GCSE Higher [834] |
\(3x^2+9x-12\) can be written in the form \(a(x+b)^2-c\).
Find the values of \(a\), \(b\) and \(c\).
3. | GCSE Higher [802] |
(a) Write \(x^2 - 6x + 5\) in the form \((x-a)^2 + b\).
(b) Write down the coordinates of the turning point of the graph of \(y = x^2 - 6x + 5\).
4. | GCSE Higher [490] |
The diagram below shows two rectangles not drawn to scale.
The perimeter of rectangle EFGH is 41cm and the area of rectangle ABCD is 55cm2.
Find the length of AD.
6. | GCSE Higher [154] |
B, C, D and E are points on the circumference of a circle, centre O.
BD is a diameter of the circle and the diagram is not drawn to scale.
(a) Find the size of angle DBE and give a reason for your answer.
(b) Find the size of angle DBC and give a reason for your answer.
7. | GCSE Higher [726] |
Walter Wall is planning to renovate a remote building by installing new flooring. A diagram showing the area required to be covered by the flooring can be seen below.
ABCDEFG is a regular heptagon with sides of length 5m.
O is the centre of the heptagon and M is the midpoint of AB.
(a) Find angle AOM
(b) Calculate the area of the heptagon in square metres.
Walter may be able to save money by purchasing some second-hand flooring material in the shape of a circle.
(c) Find the area of the circle that passes through the vertices of the heptagon.
8. | GCSE Higher [742] |
A red square has sides of length \(x\) cm. A blue rectangle has sides of length \(3x-1\) cm and \(x-2\) cm. The diagrams below are not drawn to scale.
The area of the rectangle is \(132\) cm2 greater than the area of the square.
The difference between the perimeters of the two shapes is \(p\) cm.
Find the value of \(p\).
You must show all your working.
9. | GCSE Higher [284] |
Find the length of the sides of a square from the following clues:
Give your answer in centimetres to three significant figures.
10. | GCSE Higher [623] |
What is the equation of a circle, centered at the origin, that has a tangent passing through the points (-30, 0) and (0, -15)?
Show all of your working.
11. | GCSE Higher [252] |
The diagram shows a circle with equation \(x^2+y^2=13\).
A tangent to the circle touches the circle at the point J. The x-coordinate of J is 2. The tangent intersects the x-axis at K. Find the coordinates of the point K.
12. | GCSE Higher [270] |
(a) Find the coordinates of the point at which the curve \(y = k^x\) intersects the y-axis.
The equation of circle \(A\) is \(x^2+y^2=25\). This circle is translated by the vector \( \begin{pmatrix} 0 \\ 4 \\ \end{pmatrix} \) to give circle \(B\).
(b) Draw a sketch of circle \(B\) clearly labelling the points of intersection with the y-axis.
13. | GCSE Higher [293] |
A circle with equation \(x^2 + y^2 = 6 \) meets a one of its tangents at point \(S\).
The coordinates of \(S\) are:
$$\left( \frac{\sqrt 5}{3} , \frac{7}{3} \right)$$Find the equation of the tangent to the circle at the point \(S\).
14. | GCSE Higher [295] |
Five identical circles fit exactly inside a rectangle as shown in the diagram.
Find the area of the rectangle in terms of \(r\), the radius of the circles.
15. | GCSE Higher [172] |
Two identical small yellow circles are drawn inside one large circle, as shown in the diagram. The centres of the small circles lie on the diameter of the large circle. The part of the large circle that is outside both small circles is painted red.
Find the fraction of the large circle that is painted red.
16. | GCSE Higher [253] |
(a) The circumference of a circle is \(16 \pi \) cm and its centre is at the origin. Find the equation of the circle.
(b) The line \(12x+ ay = b\) is a tangent at the point (6, 5) to a different circle with centre at the origin. Find the values of a and b.
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The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.
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