Algebra worksheet — GCSE Higher

1.A circle has centre (0, 0) and passes through the point (12, 35). Work out the equation of the circle.

2.The point (3, 4) lies on the circle x² + y² = 25, which has centre (0, 0). Work out the gradient of the tangent to the circle at (3, 4).

3.A phone company works out a monthly bill, £B, using the formula B = 18 + 0.05t, where £18 is the fixed monthly charge and t is the number of extra text messages sent beyond the free allowance, each charged at 5p. Farida's bill for one month is £24.50. Work out the number of extra text messages, t, she sent.

4.A candle is 30 cm tall when lit and burns at a constant rate. After burning for 5 minutes, its height is 20 cm. The height h cm after t minutes is given by h = 30 − mt. Work out the value of m.

5.The line y = 3x + 6 meets the x-axis at one point. Work out the coordinates of that point.

6.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.

7.Which of these statements is an identity?

8.Work out the value of 2x² when x = −3

9.Work out the value of n² − n when n = −3

10.A car's journey is represented by a velocity-time graph. The total distance travelled, found from the area under the graph, is 350 metres, and the total time for the journey is 25 seconds. Work out the car's average speed for the whole journey, in m/s.

11.A runner's journey is shown on a distance-time graph: she runs 8 km in the first hour, rests for 30 minutes with no distance gained, then runs a further 4 km in the next 30 minutes. Work out her average speed for the whole journey, in kilometres per hour.

12.A sequence has the position-to-term rule n² + 1, where n is the position number. Work out how much bigger the 5th term is than the 4th term.

13.The nth term of a sequence is n² + 3. Work out the first term of the sequence that is greater than 50.

14.f(x) = 3x − 2. Find f⁻¹(x).

15.f(x) = 2x − 1. Work out ff(x).