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.A phone plan costs a fixed £15 per month plus 8p per minute of calls. In one month, Jamal's bill was £23.80. Form an equation using m for the number of minutes, and solve it to find how many minutes Jamal used that month.
(a)485
(b)1.1
(c)297.5
(d)110
2.A water tank is a cuboid with a square base of side x metres and height (x + 1) metres. Its volume is 10 m³. This gives x³ + x² − 10 = 0, which can be solved using the iterative formula xₙ₊₁ = ∛(10 − xₙ²). Taking x₀ = 2, so that x₁ is the value found after the formula has been used once, work out x₃ correct to 3 decimal places.
(a)1.861
(b)1.885
(c)2.535
(d)1.817
3.A model rocket's height is given by h = −5t² + 20t, where h is in metres and t is in seconds. A student says the graph of h against t is n-shaped. Which statement gives the correct verdict on the SHAPE and the reason that settles it from the equation?
(a)Yes — the t² coefficient is negative, giving an n-shape.
(b)No — it is U-shaped, since height starts by increasing.
(c)No — the shape comes from the story, not the coefficient.
(d)Yes, but only because the rocket eventually lands.
4.A circle has centre (0, 0) and passes through the point (12, 35). Work out the equation of the circle.
(a)x² + y² = 1225
(b)x² + y² = 1369
(c)x² + y² = 144
(d)x² + y² = 2209
5.A quadratic graph y = ax² + bx + c has its turning point on the y-axis. Which statement about its roots must be true?
(a)The two roots must both be positive.
(b)The graph has no roots at all.
(c)The graph must touch the x-axis at exactly one point.
(d)If the graph has two real roots, they are equal and opposite in value, so they sum to zero.
6.Mia buys 3 identical notebooks and a pen costing £1.50. She pays a total of £9.90. Form an equation using n for the cost of one notebook, and solve it to find n.
(a)£2.48
(b)£3.30
(c)£2.80
(d)£3.80
7.A distance-time graph is a straight line from (0, 0) to (4, 100), where time is in hours and distance is in kilometres. Work out the gradient of the line.
(a)25
(b)400
(c)0.04
(d)100
8.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.
9.Two students each use the trapezium rule to estimate the area under the same curve between x = 0 and x = 8. Student A uses 4 strips of width 2. Student B uses 8 strips of width 1. Which estimate is more likely to be closer to the true area, and why?
(a)Student A's — fewer strips means fewer rounding errors.
(b)Student B's — narrower strips fit the curve more closely.
(c)They are always the same; the trapezium rule is exact.
(d)Student A's — wider strips give a better average height.
10.A cup of tea cools after being made. Its temperature-time graph is a curve. Three tangents are drawn on the graph: at t = 2 minutes the tangent has gradient −8; at t = 10 minutes the tangent has gradient −3; at t = 20 minutes the tangent has gradient −0.5. Which of these statements about the tea's cooling is correct?
(a)Fastest at t = 20 min — largest gradient.
(b)Cools at the same rate throughout.
(c)Fastest at t = 2 min — steepest gradient.
(d)Fastest at t = 10 min — the middle reading.
11.Solve the inequality 2(3x − 1) ≥ 4x + 8.
(a)x ≥ 4.5
(b)x ≥ 5
(c)x ≥ 1
(d)x ≥ 20
12.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.
13.Two numbers have a sum of 45. The larger number is 9 more than the smaller number. Form an equation using n for the smaller number, and solve it to find the larger number.
(a)27
(b)22.5
(c)18
(d)36
14.Work out the value of 4p² − 3 when p = 2.5
(a)97
(b)7
(c)22
(d)1
15.Six years ago Harry was three times as old as his brother Leo. Harry is now 24 years old. Work out Leo's age now.
(a)12
(b)8
(c)36
(d)6

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