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 · Calculator · 15 questions

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1.The graph of y = 60 × (0.5)ˣ is sketched for x ≥ 0. Which statement correctly describes what happens to the curve as x increases?
(a)It gets closer to zero but never actually reaches it.
(b)It decreases steadily and reaches zero when x = 60.
(c)It levels off at y = 0.5 as x increases.
(d)It decreases at first, then becomes negative for large x.
2.A gas company charges a standing charge plus a rate per unit used. The total cost, y in pounds, for using x units is given by y = 0.15x + 20. Work out the cost of using 200 units.
y = 0.15x + 20
(a)£50
(b)£30
(c)£203
(d)£320
3.The tangent to a curve at the point (5, 2) is the line y = mx + c. This tangent crosses the y-axis at (0, −8). Work out the gradient, m, of the tangent.
(a)0.5
(b)−1.2
(c)−2
(d)2
4.Solve the inequality 6.3x + 1.8 ≤ 40. Hence write down the greatest number with exactly one decimal place that satisfies the inequality.
(a)6.6
(b)6.1
(c)6.3
(d)6.0
5.f(x) = x³ − 3x − 20, and the equation f(x) = 0 has exactly one solution. Work out the pair of consecutive integers between which that solution lies.
y = x
(a)2 and 3
(b)3 and 4
(c)4 and 5
(d)6 and 7
6.Work out the value of 5m² + 3m when m = −2.4
(a)36
(b)29.4
(c)21.6
(d)−36
7.A company's weekly profit, P thousand pounds, when it makes x thousand items is modelled by P = x² − 8x + 12, for x ≥ 0. Work out the values of x for which the company makes a loss.
(a)x < 2 or x > 6
(b)2 < x < 6
(c)−6 < x < −2
(d)2 ≤ x ≤ 6
8.A rectangular photo has length (x + 3) cm and width x cm. Its area is 40 cm². Work out the value of x.
(a)x = −8
(b)x = 5
(c)x = 5 or x = −8
(d)x = 4
9.A cyclist's speed rises from rest to a maximum — quickly at first, then more and more slowly — so her speed-time graph is a curve that is concave down (its gradient decreases as time goes on). A trapezium estimate is made for the distance travelled during this phase, using two points on the curve. Is the trapezium estimate an overestimate or an underestimate of the true distance, and why?
(a)Underestimate: the chord lies below the curve
(b)Overestimate: the trapezium rule always overestimates
(c)Exact: a trapezium and a curve enclose equal areas
(d)Overestimate: the chord lies above the curve
10.The trapezium rule is used to estimate the area under a curve between x = 2 and x = 6. The curve is concave up (it curves upwards, like the inside of a bowl) throughout this interval. Which statement about the estimate is correct?
(a)Overestimates — trapezium edges lie above the curve.
(b)Underestimates — trapezium edges lie below the curve.
(c)Exact — straight lines model any curve with no error.
(d)Overestimates, because more strips always increase it.
11.An allotment is in the shape of a rectangle. Its length is 5 m more than its width, x metres, and its area is 20 m². This gives x² + 5x − 20 = 0, which can be solved using the iterative formula xₙ₊₁ = 20 ÷ (xₙ + 5). Taking x₀ = 2, so that x₁ is the value found after the formula has been used once, work out x₂ correct to 2 decimal places.
(a)2.55
(b)−1.71
(c)2.00
(d)2.86
12.The iterative formula xₙ₊₁ = 12 ÷ (xₙ + 2) is used repeatedly, starting from x₀ = 1. Work out the value that xₙ approaches, correct to 2 decimal places.
(a)3.00
(b)6.00
(c)2.61
(d)4.00
13.A number, n, is tripled and then 4.5 is added. The result is 22.5. Form an equation and solve it to find n.
(a)12
(b)6
(c)7.5
(d)9
14.Solve 5x² − 15x = 0.
(a)x = 0 or x = 15
(b)x = 3
(c)x = 0 or x = 3
(d)x = 5 or x = 3
15.Four sequences are shown below. Sequence P: 4, 8, 16, 32, ... Sequence Q: 4, 8, 12, 16, ... Sequence R: 4, 7, 12, 19, ... Sequence S: 4, 9, 16, 25, ... Work out which of the sequences P, Q, R and S is geometric.
(a)R
(b)Q
(c)P
(d)S

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