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Exam-Style Questions.

Problems adapted from questions set for previous Mathematics exams.


GCSE Higher

Multiply out and simplify:



GCSE Higher

(a) Simplify:

$$ p^6 \times p^3$$

(b) Simplify:

$$ \dfrac{a^5b^7}{a^4b} $$

(c) Solve:

$$ \dfrac{2w}{9} \gt 10$$


GCSE Higher

Find the highest common factor of the following two expressions:

$$ 8x^5y^3 $$ $$ 6x^2y $$


GCSE Higher

Simplify then find the square root of this expression:

$$\frac{y}{(1-y)^2} - \frac{y}{1-y}$$


GCSE Higher

(a) The expression \( (x+1)(2x-3)(3x+4) \) can be written in the form \(ax^3 + bx^2 + cx + d \) where \(a, b, c\) and \(d\) are integers. Find the values of \(a, b, c\) and \(d\).

(b) Solve the following inequality:

$$(x-2)^2 \lt \frac{16}{49}$$


GCSE Higher

(a) Simplify \( \left(\dfrac{3a}{a^3 - 3}\right)^0 \)


(b) Simplify \( \dfrac{9(2b-1)}{(2b-1)^2}\)


(c) Simplify \( (2c^3d^4)^5 \)


GCSE Higher
$$ 9^{\frac{1}{3}} \times 3^n = 27^{\frac{4}{5}} $$

Work out the exact value of \(n\).


GCSE Higher

(a) Express the following as a single fraction in its simplest form.

$$ \dfrac{6}{x-3} - \dfrac{2}{x-1} $$

(b) Expand and simplify the following:

$$ (x+ 2)(3x-5)(5x+1) $$


GCSE Higher

(a) Simplify \( 6(3a-2)-2(a-3) \)

(b) Simplify \( \frac{3x}{4} - \frac{2x}{3} \)

(c) Rearrange the formula to make m the subject:

$$ w = \frac{m+n}{3-m} + 7 $$

(d) Solve \( x^8 = 5600 \)

(e) Solve \( |x+5| = 9 \)

(f) Solve by factorising:

$$ 9z^2 - 27z + 20 = 0 $$


GCSE Higher

(a) The \(n\)th term of a sequence is \(2^n+2^{n+1}\)

Work out the 8th term of the sequence.

(b) The \(n\)th term of a different sequence is \(9(3^n + 3^{n+1})\)

Expand and express this expression as the sum of two powers of three.


GCSE Higher

(a) Simplify the following expression.

$$ \frac{x^2 - 4}{3x^2 + 13x + 14}$$

(b) Make b the subject of the following formula.

$$ a = \frac{7(3b-c)}{b}$$


GCSE Higher

(a) Without using a calculator, show that \(\sqrt{28}=2\sqrt7\)

(b) The point \(X\) is shown on the unit grid below. The point \(Y\) is \(\sqrt17\) units from \(X\) and lies on the intersection of two grid lines. Mark one possible position for \(Y\).

Unit Grid


GCSE Higher

If a, b and c are positive integers use the following statements to find the values of a, b and c.

$$ (ab^c)^3 = 27b^{21} $$ $$ b= 9a $$


GCSE Higher

Factorise the following expression



GCSE Higher

Show that:

$$3 - (4x^2+11x+7) \div (x^2+4x+3) $$

simplifies to \( \frac{a-x}{x+b}\) where \(a\) and \(b\) are integers.


GCSE Higher

The expression below can be written as a single fraction in the form \( \dfrac{a-bx}{x^2-25} \) where \(a\) and \(b\) are integers.

$$ \frac{x-4}{x-5} - 2 + \frac{x+1}{x+5}$$

Work out the value of \(a\) and the value of \(b\).


GCSE Higher

\(y = a \times b^{x – 2}\) where \(a\) and \(b\) are numbers.

\(y = 5\) when \(x = 2\)

\(y = 0.005\) when \(x = 5\)

Work out the value of \(y\) when \(x = 4\)

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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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