Algebra worksheet — GCSE Higher

1.The equation x³ − 3x − 4 = 0 has a root near x = 2. Four students each try a different iterative formula, all starting from x₀ = 2: xₙ₊₁ = ∛(3xₙ + 4); xₙ₊₁ = (xₙ³ − 4) ÷ 3; xₙ₊₁ = 4 ÷ (xₙ² − 3); xₙ₊₁ = xₙ³ − 2xₙ − 4. Only one of these formulas keeps producing values that settle near the root when it is repeated. Work out x₁, correct to 3 decimal places, for the formula that does this.

2.Work out the value of m² + 3m when m = −2

3.How many turning points does the graph of y = x² − 7x + 10 have?

4.The nth term of a sequence is n² + 4n. Work out the term number, n, for which the term equals 45.

5.A car's fuel tank contains 50 litres when it starts a journey. After travelling for 2 hours, it contains 38 litres, and fuel is used at a constant rate. Work out the rate at which fuel is used, in litres per hour.

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

7.The mean of four numbers is 12.5. Three of the numbers are 8, 15 and 19. Form an equation using n for the fourth number, and solve it to find n.

8.A regular hexagon has sides of length (x + 2) cm. Write down an expression, in terms of x, for the perimeter of the hexagon.

9.The table shows some values of f(x): when x = 0, f(x) = 5; when x = 1, f(x) = 8; when x = 2, f(x) = 4; when x = 3, f(x) = 1. Work out the value of f(x − 1) when x = 2.

10.Three vertices of a rectangle are (−4, −1), (2, −1) and (2, 3). The sides of the rectangle are parallel to the axes. Write down the coordinates of the fourth vertex.

11.The formula y = x² + 1 gives y in terms of x. Work out the value of y when x = −2.

12.A straight line has equation 4y = 12x + 20. Work out the gradient of the line.

13.Two numbers have a sum of 50. The larger number is twice the smaller number. Work out the smaller number.

14.A tree is 2 metres tall. Each year its height increases by 10% of its height at the start of that year. Work out the height of the tree after 3 years, giving your answer to 1 decimal place.

15.The formula connecting distance (d), speed (s) and time (t) is d = st. Make s the subject of the formula.