Algebra worksheet — GCSE Higher

1.The graph of y = 2x² + 6x − 1 is translated by the vector (−2, 0). Work out the equation of the image, giving your answer in the form y = 2x² + bx + c.

2.The point (20, 21) lies on the circle x² + y² = 841, which has centre O(0, 0). The tangent to the circle at (20, 21) crosses the x-axis at P and the y-axis at Q. Work out the area of triangle OPQ, correct to 1 decimal place.

3.Solve the simultaneous equations 3x + y = 14 and x + y = 6. Work out the value of x.

4.A cafe sells coffees at £c each and pastries at £p each. Three coffees and two pastries cost £9.60. Two coffees and two pastries cost £7.60. Work out the price of one coffee.

5.(2x + 3)(x + a) ≡ 2x² + 11x + 12 is an identity. Work out the value of a.

6.The first five terms of a sequence are 7, 9, 13, 19, 27. By finding the second difference, work out the coefficient of n² in the nth term.

7.For 0 ≤ x ≤ 360, the equation sin x = 0.5 has two solutions. What are they?

8.A geometric sequence begins 162, 54, 18, 6, ... Work out the next term in the sequence.

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

10.The equation x² = 5x − 3 is to be solved using iteration. Work out which of these iterative formulas comes from a correct rearrangement of the equation.

11.A market trader builds a display of oranges in a rectangular block. The block is n oranges wide and 4 oranges longer than it is wide. The number of oranges needed is 5 when n = 1, 12 when n = 2, 21 when n = 3, 32 when n = 4, and 45 when n = 5. Work out an expression, in terms of n, for the number of oranges needed for a display of width n.

12.A geometric sequence has first term 5, and each term after that is 3 times the term before it. Work out the first term of the sequence that is greater than 1000.

13.The equation x² − 3x − 7 = 0 can be solved using the iterative formula xₙ₊₁ = √(3xₙ + 7). The starting value is x₀ = 4, so x₁ is the value after the formula has been used once. Work out x₃ correct to 3 decimal places.

14.f(x) = 2x + 1. Work out the value of x for which f⁻¹(x) = 5.

15.A rule turns each input x into an output y. The inputs are x = 0, 1, 2, 3 and the outputs are y = 4, 7, 10, 13. Work out the output when x = 5.