Algebra worksheet — GCSE Higher

1.A candidate solves the inequality x² > 9 and writes: "x² > 9, so x > 3." Which of these is a correct comment on the candidate's work?

2.A proof sets out to show that the sum of the squares of two consecutive odd numbers, written as 2n + 1 and 2n + 3, is always 2 more than a multiple of 8. Four attempts to expand (2n + 1)² + (2n + 3)² and reach a conclusion are shown below. Which attempt correctly proves this claim?

3.A plumber charges a call-out fee of £30 plus £25 for each hour worked. The total bill for a job was £130. Work out how many hours the plumber worked.

4.Work out the period of the graph of y = cos x, in degrees.

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

6.Find the common ratio of the geometric sequence 5, 5√2, 10, 10√2, ... Give your answer in surd form.

7.The first four terms of a sequence are 11, 18, 25, 32. Ravi thinks the nth term is 7n. Work out the correct expression for the nth term.

8.A bath is filled with water; the graph of volume (litres) against time (minutes) is a curve, since the flow rate changes. The tangent to the curve at t = 10 minutes passes through the points (8, 130) and (12, 190), and the volume in the bath at t = 10 minutes is 160 litres. Use the gradient of this tangent to estimate the volume in the bath 5 minutes after t = 10 minutes.

9.The tangent to a curve at the point where x = 4 passes through the points (2, 5) and (6, 21). Use these two points to estimate the gradient of the curve at x = 4.

10.Solve x² − 2x − 24 = 0.

11.Which of these equations gives a graph with two separate curved branches — one where x and y are both positive, and one where x and y are both negative?

12.To rearrange the formula y = 3x − 8 to make x the subject, Priya writes: 'Add 8 to both sides to get y + 8 = 3x, then divide both sides by 3 to get x = y + 8 ÷ 3.' Work out the correct expression for x.

13.A geometric sequence begins 162, 54, 18, 6, ... Work out the next term in the sequence.

14.Point P has coordinates (2, −5). P is reflected in the x-axis to point Q. Write down the coordinates of Q.

15.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.