Printable · GCSE Higher · ages 14-16

Algebra worksheet — GCSE Higher

Fifteen questions across the algebra statements at Higher tier. Choose the non-calculator filter to rehearse Paper 1, which counts for a third of the marks.

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Algebra worksheet — GCSE Higher

Maths · GCSE Higher · Algebra · 15 questions

MathsUKwww.geekhero.co.uk

1.A rectangular vegetable plot has an area of 30 m² and its length is 4 m greater than its width, x metres. This gives x² + 4x − 30 = 0, which can be solved using the iterative formula xₙ₊₁ = √(30 − 4xₙ). The starting value is x₀ = 3, so x₁ is the value after the formula has been used once. Work out x₃, and use it to find an estimate for the length of the plot, giving your answer correct to 1 decimal place.
(a)7.9 m
(b)3.9 m
(c)7.6 m
(d)15.8 m
2.The graph of y = sin x is reflected in the x-axis to give the graph of y = g(x). Work out the value of g(90), where x is measured in degrees.
y = sin(x)
(a)−1
(b)1
(c)−0.5
(d)0
3.By completing the square, find the turning point of the curve y = x² − 4x + 9.
y = x² − 4x + 9
(a)x = 4, y = −7
(b)x = 2, y = −5
(c)x = 2, y = 5
(d)x = −2, y = 5
4.The graph of y = x² − 5x + 6 crosses the x-axis at two points. By factorising, work out the x-coordinates of these two points.
y = x² − 5x + 6
(a)x = −1 and x = −6
(b)x = 2 and x = −3
(c)x = 2 and x = 3
(d)x = −2 and x = −3
5.The equation x² − 5x − 2 = 0 can be solved using the iterative formula xₙ₊₁ = √(5xₙ + 2). The starting value is x₀ = 2, so x₁ is the value after the formula has been used once. Work out x₃ correct to 3 decimal places.
(a)4.897
(b)4.396
(c)5.146
(d)3.464
6.The solution set of a quadratic inequality is {x : x ≤ −3} ∪ {x : x ≥ 5}. Which of these inequalities has this solution set?
(a)x² − 8x + 15 ≥ 0
(b)x² + 2x − 15 ≥ 0
(c)x² − 2x − 15 ≥ 0
(d)x² − 2x − 15 ≤ 0
7.Which expression is equivalent to 7x − 3(2x − 6)?
(a)13x − 18
(b)x − 18
(c)x + 18
(d)x + 6
8.A circular pond has equation x² + y² = 20, with lengths in metres from the centre of the garden. A straight path runs along the line y = 2x, entering the pond and leaving it again. Work out the coordinates of the two points where the path meets the edge of the pond.
y = 2x
(a)(2, 4)
(b)(2, 4) and (−2, −4)
(c)(2, 1) and (−2, −1)
(d)(2, −4) and (−2, 4)
9.A cinema sells adult tickets for £10 and child tickets for £6. A family group buys 9 tickets in total, spending £74. Work out how many adult tickets were bought.
(a)7.4
(b)1.25
(c)5
(d)4
10.The equation x² − 4x + k = 0 has two different real solutions. Work out the range of values of k.
(a)k < 16
(b)k ≤ 4
(c)k > 4
(d)k < 4
11.Write down the expression that means the same as m² × m × 3.
(a)m⁶
(b)m³
(c)3m²
(d)3m³
12.A geometric sequence begins 162, 54, 18, 6, ... Work out the next term in the sequence.
(a)2
(b)3
(c)−6
(d)18
13.A circle has centre (0, 0) and equation x² + y² = 25. Work out the x-coordinates of the two points where the circle crosses the line y = 3.
(a)x = 3 and x = −3
(b)x = 4
(c)x = 5 and x = −5
(d)x = 4 and x = −4
14.A circle has centre (0, 0) and passes through the point (5, 12). Work out the equation of the circle.
(a)x² + y² = 119
(b)x² + y² = 169
(c)x² + y² = 17
(d)x² + y² = 13
15.Work out the value of n² − n when n = −3
(a)6
(b)−3
(c)−6
(d)12

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