Algebra worksheet — GCSE Higher

1.The table shows the speed, in m/s, of a cyclist at times, in seconds, 1 second apart: 0 at t = 0, 7 at t = 1, 12 at t = 2, 15 at t = 3, 16 at t = 4, 14 at t = 5, 9 at t = 6. Using the trapezium rule with strip width 1 second, estimate the distance the cyclist travels between t = 2 and t = 5 only.

2.The equation x² − x − 6 = 0 has roots x = 3 and x = −2. It can be rearranged as xₙ₊₁ = xₙ² − 6. This formula is used with starting value x₀ = 2.9, close to the root x = 3. Work out what happens to the sequence of values as n increases.

3.A line has equation y = −4x + 1. Work out the equation of the line perpendicular to it that passes through the point (0, 3).

4.A company's weekly profit, P thousand pounds, when it makes x thousand items is modelled by P = x² − 8x + 12, for x ≥ 0. Work out the values of x for which the company makes a loss.

5.A gardener charges a fixed fee of £15 plus £8 for each hour worked. The total charge, £T, for a job lasting h hours is given by T = 15 + 8h. Work out the charge for a job lasting 4 hours, and identify what kind of statement T = 15 + 8h is.

6.The first four terms of a sequence are 4, 9, 14, 19. Work out an expression, in terms of n, for the nth term.

7.Write down the expression that means the same as m² × m × 3.

8.A point has coordinates (x, y). In which two quadrants is the product x × y positive?

9.The 2nd term of a geometric sequence is 6 and the 4th term is 24. Given that all the terms of the sequence are positive, work out the common ratio.

10.Solve the inequality 2(3x − 1) ≥ 4x + 8.

11.The equation 3(2x − 1) = 4x + 9 is rearranged by expanding the brackets. Which of these is the correctly expanded equation?

12.The graph of y = f(x) has a minimum turning point at (3, −5), crosses the x-axis at x = 1, and crosses the y-axis at (0, −2). Exactly one of these statements about the graph of y = −f(x + 2) is true. Which statement is true?

13.Work out the value of (x − 4)/2 + 3 when x = 10

14.A gym charges a reduced price for the first month of membership and a standard price for every month after that. The total cost is £5 for 1 month, £12 for 2 months, £19 for 3 months and £26 for 4 months. Work out an expression, in terms of n, for the total cost, in pounds, of n months of membership.

15.Simplify fully: (x + 2)/(3x) × 6x²/(x² − 4)