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GCSE Maths Higher — Sample Paper 1

A complete GCSE Higher (grades 4–9) paper you can work on paper or on screen. Each question has its mark scheme underneath, collapsed until you want it, so you mark yourself the way an examiner would.

90 minutes
Non-calculator
40 marks · Edexcel style

Question 1
(2 marks)
Write $0.000\,45$ in standard form.
Mark scheme
M1
For $4.5$ with a negative power of 10
A1
cao $4.5 \times 10^{-4}$
Answer:
$4.5 \times 10^{-4}$
Question 2
(3 marks)
Expand and simplify $(x + 5)(x - 3)$ .
Mark scheme
M1
For 3 of 4 terms correct from expansion
M1
For collecting $5x - 3x$
A1
cao $x^{2} + 2x - 15$
Answer:
$x^{2} + 2x - 15$
Question 3
(3 marks)
Simplify fully $\dfrac{6x^{3}y^{2}}{9xy^{5}}$ .
Mark scheme
M1
For $\frac{6}{9} = \frac{2}{3}$
M1
For $x^{3-1} = x^{2}$ or $y^{2-5} = y^{-3}$
A1
cao $\frac{2x^{2}}{3y^{3}}$
Answer:
$\dfrac{2x^{2}}{3y^{3}}$
Question 4
(4 marks)
Solve the simultaneous equations

$3x + 2y = 16$

$x - 2y = 0$
Mark scheme
M1
For adding the equations to eliminate $y$
A1
$x = 4$
M1
For substituting their $x$ into either equation
A1
$y = 2$
Answer:
$x = 4$ , $y = 2$
Question 5
(3 marks)
A shop reduces the price of a coat by $20\%$ to £68. Work out the original price.
Mark scheme
M1
For recognising £68 represents $80\%$
M1
For $68 \div 0.8$ (or $68 \div 80 \times 100$ )
A1
cao £85
Answer:
£85
Question 6
(4 marks)
Rationalise the denominator and simplify $\dfrac{6}{\sqrt{3}} + \sqrt{12}$ .
Mark scheme
M1
For $\frac{6}{\sqrt3} \times \frac{\sqrt3}{\sqrt3}$
A1
For $2\sqrt3$
M1
For $\sqrt{12} = 2\sqrt3$
A1
cao $4\sqrt3$
Answer:
$4\sqrt{3}$
Question 7
(4 marks)
The $n$ th term of a sequence is $n^{2} + 3n$ .

(a) Work out the 5th term.

(b) Show that 40 is a term of the sequence.
Mark scheme
M1
(a) For $5^{2} + 3 \times 5$
A1
(a) cao 40
M1
(b) For solving $n^{2} + 3n = 40$
A1
(b) For $n = 5$ with a conclusion
Answer:
(a) 40 (b) $n = 5$ , so 40 is the 5th term
Question 8
(5 marks)
A right-angled triangle has hypotenuse $13\,\text{cm}$ and one shorter side $5\,\text{cm}$ .

(a) Work out the third side.

(b) Hence work out the area of the triangle.
Mark scheme
M1
(a) For $13^{2} - 5^{2}$
A1
(a) cao $12$ (cm)
M1
(b) For $\frac12 \times 5 \times$ their 12
A1
(b) cao $30$
B1
(b) For correct units $\text{cm}^{2}$
Answer:
(a) $12\,\text{cm}$ (b) $30\,\text{cm}^{2}$
Question 9
(4 marks)
The probability that it rains on any given day is $0.3$ , independently of other days.

Work out the probability that it rains on exactly one of two consecutive days.
Mark scheme
M1
For $0.3 \times 0.7$
M1
For recognising two cases
M1
For $0.3 \times 0.7 + 0.7 \times 0.3$
A1
cao $0.42$
Answer:
$0.42$
Question 10
(4 marks)
Prove that the sum of any three consecutive integers is always a multiple of 3.
Mark scheme
M1
For defining consecutive integers as $n$ , $n+1$ , $n+2$
M1
For summing to $3n + 3$
A1
For factorising to $3(n + 1)$
A1
For a conclusion referring to a multiple of 3
Answer:
$n + (n+1) + (n+2) = 3n + 3 = 3(n+1)$ , a multiple of 3
Question 11
(4 marks)
$y$ is inversely proportional to $x^{2}$ . When $x = 3$ , $y = 4$ .

Work out $y$ when $x = 6$ .
Mark scheme
M1
For $y = \frac{k}{x^{2}}$
M1
For $k = 4 \times 9 = 36$
M1
For $\frac{36}{36}$
A1
cao $y = 1$
Answer:
$y = 1$

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