Sample paper · GCSE Higher · grades 4–9

GCSE Higher sample Paper 1 (non-calculator)

The real Paper 1 is 1.5 hour 30 minutes and 80 marks, non-calculator, and all three papers carry equal weight. This sample is 20 original questions in the same content proportions as the Higher qualification — number 15%, algebra 30%, ratio, proportion and rates of change 20%, geometry and measures 20%, probability 7.5%, statistics 7.5% — with no calculator-only items. AO1 / AO2 / AO3 at this tier: 40% / 30% / 30%.

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GCSE Higher sample Paper 1 (non-calculator)

Maths · GCSE Higher · Paper 1 · Non-calculator · 20 questions

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1.The two shorter sides of a right-angled triangle are √12 cm and √24 cm. Work out the exact length of the hypotenuse.
(a)2√3 + 2√6
(b)12√2
(c)36
(d)6
2.A pattern is made from tiles. Pattern 1 uses 4 tiles, pattern 2 uses 7 tiles, pattern 3 uses 10 tiles and pattern 4 uses 13 tiles, with each pattern using 3 more tiles than the one before. Work out an expression, in terms of n, for the number of tiles used in pattern n.
(a)3n − 2
(b)3n + 1
(c)3n + 4
(d)n + 3
3.y is inversely proportional to x, so y = k ÷ x. When x = 5, the value of y is 8. Work out the value of k.
(a)3
(b)1.6
(c)13
(d)40
4.Triangle ABC has AB = (2x + 3) cm, BC = 11 cm and angle B = 90°. Triangle DEF has DE = 13 cm, EF = 11 cm and angle E = 90°. Given that the two triangles are congruent by SAS, work out the value of x.
(a)4
(b)10
(c)8
(d)5
5.At a garden centre, 3/5 of the plants for sale are perennials. 1/4 of the perennials are in flower. Work out the probability that a plant picked at random from the garden centre is a perennial and is in flower.
(a)4/9
(b)17/20
(c)3/20
(d)1/4
6.A garden centre's sales, in thousands of pounds, at the end of each quarter last year were: quarter 1 — 18, quarter 2 — 34, quarter 3 — 30, quarter 4 — 22. Work out the increase in sales from quarter 1 to the quarter with the highest sales.
(a)£16,000
(b)£12,000
(c)£34,000
(d)£16
7.Write 12 as a product of its prime factors.
(a)2² × 3
(b)2 × 3²
(c)2 × 3
(d)2 × 6
8.A sequence has the position-to-term rule n² + 1, where n is the position number. Work out how much bigger the 5th term is than the 4th term.
(a)9
(b)7
(c)1
(d)11
9.A cordial drink is made by mixing cordial concentrate and water in the ratio 1 : 6. Freya says that a 700 ml jug of this drink contains 100 ml of concentrate. Is she correct? Give a reason for your answer.
(a)No — half of 700 ml (350 ml) should be concentrate
(b)No — 600 ml is concentrate, 100 ml is water
(c)Yes — 700 ÷ 7 = 100 ml for the 1 part of concentrate
(d)No — divide 700 by 6, not 7: about 117 ml
10.Line 1 passes through (0, 1) and (2, 5). Line 2 passes through (3, 2) and (5, 6). Work out the gradient of each line. Then write down whether the two lines are parallel.
(a)Parallel, since both gradients are 2
(b)Not parallel, since the gradients are 2 and 4
(c)Parallel, since both lines pass through positive coordinates
(d)Not parallel, since the two lines cross the y-axis at different points
11.At a coffee shop, each customer buys tea, coffee, water or juice, and never more than one of these. The probability that a customer buys tea is 0.36 and the probability that they buy juice is 0.04. The probability that a customer buys coffee is three times the probability that they buy water. Work out the probability that a customer buys water.
(a)0.15
(b)0.30
(c)0.60
(d)0.45
12.Work out how many factors 100 has.
(a)8
(b)4
(c)9
(d)10
13.Expand and simplify 4(2x − 1) − 3(x + 2) − 5x
(a)−10
(b)5x − 10
(c)2
(d)−7
14.The height of a ball, in metres, above the ground is plotted against time, in seconds, after it is thrown. The tangent to the graph at t = 1.5 seconds has gradient 2. Which of these four statements about the ball at t = 1.5 seconds is correct?
(a)Height is 2 m at t = 1.5 s
(b)Speed is 2 m/s² at t = 1.5 s
(c)Height rising at 2 m/s at t = 1.5 s
(d)Ball travelled 2 m from t = 1 to t = 2 s
15.A pop-up canopy has two sloping supports that meet at the top. Each support makes an angle of 30° with the ground and is 6 m long. Using the exact value of cos 30°, work out the total width of the canopy's base.
(a)3√3 m
(b)6√3 m
(c)12 m
(d)6 m
16.The nth term of a sequence is n² − 2n + 5. Work out the 10th term.
(a)68
(b)85
(c)80
(d)125
17.Noah runs 2 km in 10 minutes. Work out how long he takes to run 5 km at the same speed.
(a)50 minutes
(b)20 minutes
(c)15 minutes
(d)25 minutes
18.Triangle ABC is similar to triangle PQR. AB corresponds to PQ, and BC corresponds to QR. AB = 5 cm, PQ = 15 cm and BC = 7 cm. Work out the length of QR.
(a)35 cm
(b)7 cm
(c)2.33 cm
(d)21 cm
19.The mean of four numbers is 12.5. Three of the numbers are 8, 15 and 19. Form an equation using n for the fourth number, and solve it to find n.
(a)−4.5
(b)50
(c)29.5
(d)8
20.Solve the simultaneous equations x + y = 1 and 2x + y = 5.
(a)x = 4, y = −3
(b)x = −2, y = 3
(c)x = 6, y = −5
(d)x = 4, y = 5

How the 20 questions are shared out

Number — 3 questions (15% of the qualification)
Algebra — 6 questions (30% of the qualification)
Ratio, proportion and rates of change — 4 questions (20% of the qualification)
Geometry and measures — 4 questions (20% of the qualification)
Probability — 2 questions (7.5% of the qualification)
Statistics — 1 question (7.5% of the qualification)

Where an area has fewer printable questions than its share, the shortfall is filled from the other areas. These are original questions, not past papers.