Algebra worksheet — GCSE Higher

1.A circle has centre (0, 0) and equation x² + y² = 3721. The point (11, 60) lies on the circle. One of these is the gradient of the tangent to the circle at (11, 60). Work out which one.

2.A geometric sequence has first term 3 and common ratio 2. Work out the 5th term of the sequence.

3.Solve the inequality 5x − 3 > 2x + 9.

4.Two numbers have a sum of 24. Their difference is 6. Work out the two numbers.

5.Write down the coefficient of xy in the expression 4x²y − 7xy + 2.

6.A pattern is made from tiles. Pattern 1 uses 4 tiles, pattern 2 uses 7 tiles, pattern 3 uses 10 tiles and pattern 4 uses 13 tiles, with each pattern using 3 more tiles than the one before. Work out an expression, in terms of n, for the number of tiles used in pattern n.

7.A curve has equation y = x² − 9. Which statement about its graph is correct?

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

9.The nth term of a sequence is n² − 2n + 5. Work out the 10th term.

10.A colony of bacteria doubles in number every hour. At 9am there are 5 bacteria in the colony. Work out how many bacteria there will be at 12 noon.

11.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.

12.A rectangular garden has length (x + 7) m and width (x − 7) m. Work out an expression for the area of the garden, giving your answer in its simplest form.

13.Factorise x² − 3x − 10.

14.Solve the inequality 3x + 6 ≤ 0.

15.A circle has equation x² + y² = 25. Does the point (3, 4) lie on this circle?