Algebra worksheet — GCSE Higher

1.Expand and simplify 4(2x − 1) − 3(x + 2) − 5x

2.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.

3.Points A(−1, 2) and B(5, 8) are the endpoints of a line segment. Work out the equation of the perpendicular bisector of AB.

4.A circle has centre (0, 0) and equation x² + y² = 64. Write down the radius of the circle.

5.Solve x² + 3x − 10 = 0.

6.A car's speed, in m/s, during a journey is: 0 at t = 0 s, 8 at t = 4 s, and 8 (constant) from t = 4 s to t = 10 s, increasing at a constant rate between t = 0 and t = 4. Estimate the total distance the car travels, using the areas of a triangle and a rectangle.

7.A tram sets off from a stop. Its velocity-time graph rises in a straight line from 0 m/s to 12 m/s over the first 8 seconds, then stays constant at 12 m/s for a further 10 seconds. Work out the total distance the tram travels in these 18 seconds, using the area under the graph.

8.The numbers x and y satisfy x + y = 10 and xy = 21. Work out the pair of values.

9.Which expression is equivalent to 7x − 3(2x − 6)?

10.A taxi firm charges a fixed fee of £3.50 plus £2.20 per mile. Work out the total cost of a journey of 6 miles.

11.A table shows y = x² − 6x + 5 at these points (x, y): (0, 5), (1, 0), (2, −3), (3, −4), (4, −3), (5, 0), (6, 5). Using the symmetry shown, write down the x-coordinate of the turning point.

12.The nth term of a sequence is n² + 2n − 4. Work out the 7th term.

13.f(x) = (x + 1)/2. Find f⁻¹(x).

14.The first five terms of a quadratic sequence are 2, 5, 10, 17, 26. Work out the next term in the sequence.

15.Work out the gradient of a line that is perpendicular to the line with equation 2x + 3y = 6.