Algebra worksheet — GCSE Higher

1.A student is proving that (n + 1)² − n² is always an odd number. Which of these correctly completes the first line of algebra?

2.f(x) = x + 2 and g(x) = x². Work out the value of x for which fg(x) = gf(x).

3.Solve 5(x + 3) = 40

4.The nth term of a geometric sequence is given by aₙ = 3 × 2ⁿ⁻¹. Work out the position of the first term that is greater than 1,000.

5.A ferry's distance-time graph is a straight line from (0 minutes, 0 km) to (25 minutes, 15 km). Work out the ferry's speed in kilometres per hour.

6.Work out the coordinates of the point where the graph of y = 2x − 6 crosses the x-axis.

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

8.Ben writes n + 4 ≤ 9. Which statement describes what Ben has written? Give a reason for your answer.

9.The graph of W = 840 ÷ L shows how the width, W metres, of a rectangular field of area 840 m² depends on its length, L metres. Use the relationship to work out the width when the length is 35 m, and hence the perimeter of the field.

10.Look at the statement 2x + 5 = 17. Considering whether it is an expression, an equation, a formula, or an identity, which classification is correct?

11.Factorise fully 5x + 5y − 5

12.A cup of tea cools after being made. Its temperature-time graph is a curve. Three tangents are drawn on the graph: at t = 2 minutes the tangent has gradient −8; at t = 10 minutes the tangent has gradient −3; at t = 20 minutes the tangent has gradient −0.5. Which of these statements about the tea's cooling is correct?

13.A ribbon of length L metres is cut into 5 equal pieces, and then 3 metres is removed from one piece. Write down an expression, in metres, for the length of that piece after 3 metres is removed.

14.The first five terms of a quadratic sequence are −1, 4, 13, 26, 43. Work out an expression, in terms of n, for the nth term.

15.The point (9, 40) lies on the circle x² + y² = 1681, which has centre (0, 0). Work out the equation of the tangent to the circle at (9, 40), giving your answer in the form ax + by = c.