Algebra worksheet — GCSE Higher

1.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?

2.Work out the value of 3x² − 4x when x = −2.

3.A walker's distance-time graph is described as follows: she walks 3 km in the first 15 minutes at a steady speed, rests for 10 minutes, then walks a further 3 km in the next 20 minutes, also at a steady speed. Work out her speed, in km/h, during the first 15 minutes.

4.A Fibonacci-type sequence is defined by aₙ = aₙ₋₁ + aₙ₋₂ for n ≥ 3. Given that a₁ = 5 and a₃ = 17, work out a₂.

5.A geometric sequence has first term 4 and common ratio √2. Work out the sum of the first three terms of the sequence, giving your answer in the form a + b√2.

6.The graph of y = f(x) has a maximum turning point at (−1, 6). Write down the coordinates of the maximum turning point of the graph of y = f(x − 3).

7.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.

8.Line L has equation y = 2x + 3. Work out the equation of the line perpendicular to L that passes through the point (4, 1), giving your answer in the form x + 2y = c.

9.A company's profit is £500 in its first year. Each year after that, the profit is £300 more than the year before. Work out the profit in the company's 6th year.

10.A water tank is a cuboid with a square base of side x metres and height (x + 1) metres. Its volume is 10 m³. This gives x³ + x² − 10 = 0, which can be solved using the iterative formula xₙ₊₁ = ∛(10 − xₙ²). Taking x₀ = 2, so that x₁ is the value found after the formula has been used once, work out x₃ correct to 3 decimal places.

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

12.The nth term of a sequence is 3n² + 2n − 1. Work out the 10th term.

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

14.Write down the expression that means the same as 'subtract 5 from n, then divide the result by 2'.

15.The function f(x) = x² for all real values of x has no inverse function, but g(x) = x² for x ≥ 0 does have one. Which statement correctly explains this?