Algebra worksheet — GCSE Higher

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

2.f(x) = 2x + 1. Work out the value of x for which f⁻¹(x) = 5.

3.To solve 6x − 4 = 2x + 20, Yusuf's first step is to subtract 2x from both sides. Work out what equation this gives.

4.A region R satisfies y ≤ x + 1, x + y ≤ 5 and y ≥ 0. Which of these correctly describes R in words?

5.A candidate solves the simultaneous equations y = x − 2 and y = x² − 4x + 2 by substitution. They write: "x − 2 = x² − 4x + 2, so x² − 3x + 4 = 0." Which of these is a correct comment on the candidate's working?

6.Factorise x² − 64.

7.Matchsticks are laid out as a row of squares, with each new square sharing a side with the square before it. The first square uses 4 matchsticks and every extra square uses 3 more. Work out how many matchsticks a row of 4 squares uses.

8.f(x) = 2x − 1. Work out ff(x).

9.A school trip costs a £15 deposit plus £9 per student for the coach. The total cost for a class is £186. Work out how many students went on the trip.

10.A straight line has equation y = 4 − 3x. Work out the gradient of the line.

11.f(x) = x² − 1 and g(x) = 3x. Work out fg(4).

12.The nth term of a sequence is 2n² + 1. Work out the 4th term of the sequence.

13.The first four terms of a sequence are 6, 13, 20, 27. Work out an expression, in terms of n, for the nth term.

14.Aisha saves money each week. In week 1 she has saved £15 in total, in week 2 she has saved £23 in total, in week 3 she has saved £31 in total and in week 4 she has saved £39 in total, with the total increasing by the same amount each week. Work out an expression, in terms of n, for the total amount she has saved, in pounds, after week n.

15.A rectangular sheep pen has perimeter P, length l and width w, connected by the formula P = 2l + 2w. A farmer has 60 m of fencing for the perimeter and sets the width at 8 m. Work out the length of the pen.