Algebra worksheet — GCSE Higher

1.The simultaneous equations 2x + 3y = 12 and x − y = 1 are given. Work out the value of y.

2.A circle has centre (0, 0) and equation x² + y² = 50. Work out the radius of the circle, giving your answer as a surd in its simplest form.

3.The table shows the velocity, in m/s, of a train at various times, in seconds: 0 at t = 0, 20 at t = 4, 35 at t = 10, 27 at t = 14. Assuming the velocity changes at a constant rate between each pair of readings, estimate the distance the train travels between t = 0 and t = 12, using the trapezium rule.

4.Solve the inequality x² ≥ 16, giving your answer using set notation.

5.The 2nd term of a geometric sequence is 6 and the 4th term is 24. Given that all the terms of the sequence are positive, work out the common ratio.

6.A van's value depreciates by 18% each year. After 2 years it is worth £5,379.20. Work out its value when it was new, correct to the nearest pound.

7.There are 50 adults on a coach to the Lake District. Every adult is either a man or a woman. There are 10 more men than women. Work out the number of men.

8.A garden path runs along the line 4x + y = 12. A drainage pipe must be laid perpendicular to the path, passing through the point (3, 1) where a sprinkler sits. Work out the equation of the pipe's line, giving your answer in the form y = mx + c.

9.Write down the first four terms of the sequence with nth term 6n − 5.

10.A gardener charges a fixed fee of £15 plus £8 for each hour worked. The total charge, £T, for a job lasting h hours is given by T = 15 + 8h. Work out the charge for a job lasting 4 hours, and identify what kind of statement T = 15 + 8h is.

11.A car accelerates uniformly from rest at 2.5 m/s² until it reaches a speed of 20 m/s, then travels at this constant speed for a further 30 seconds. Work out the total distance travelled.

12.By completing the square, find the turning point of the curve y = 3x² + 12x + 7.

13.Solve 7x − 5 = 3x + 11

14.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.

15.A tank already contains some water and is then filled at a constant rate. After 2 minutes it contains 50 litres in total; after 7 minutes it contains 150 litres in total. Work out the rate at which water is added, in litres per minute.