Algebra worksheet — GCSE Higher

1.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.

2.A hiker's distance-time graph shows the following: he walks 5 km in the first 1 hour at a steady speed, rests for 1 hour, then walks a further 9 km in the next 2 hours, also at a steady speed. Work out his average speed for the whole journey, in km/h.

3.The solution set of a quadratic inequality is {x : x ≤ −3} ∪ {x : x ≥ 5}. Which of these inequalities has this solution set?

4.Work out the smallest positive value of x, in degrees, for which y = tan x is undefined.

5.Solve 2x² = 18.

6.The first five terms of a quadratic sequence are 6, 11, 18, 27, 38. Work out an expression, in terms of n, for the nth term.

7.A taxi fare in pounds is given by F = 2.50 + 1.20m, where m is the number of miles travelled. Sam takes a taxi for a journey of 6 miles. Work out the fare.

8.Which expression is equivalent to 3(x + 4) − 2(x − 1)?

9.For 0 ≤ x ≤ 360, the equation sin x = 0.5 has two solutions. What are they?

10.A circle has centre (0, 0) and equation x² + y² = 50. Work out the length of the diameter of the circle, correct to 1 decimal place.

11.The first five terms of a sequence are 7, 9, 13, 19, 27. By finding the second difference, work out the coefficient of n² in the nth term.

12.Points A(8, 15) and B(−8, 15) both lie on the circle x² + y² = 289, which has centre (0, 0). Work out the length of the chord AB.

13.The mass of a chemical sample decays by 15% every hour. It starts at 500 g. Work out the mass remaining after 3 hours, correct to 1 decimal place.

14.The first four terms of a sequence are 4, 9, 14, 19. Work out an expression, in terms of n, for the nth term.

15.The first five terms of a quadratic sequence are 5, 8, 13, 20, 29. Work out the next term in the sequence.