Algebra worksheet — GCSE Higher

1.Which expression is equivalent to 3(x + 4) − 2(x − 1)?

2.An equation has exactly one value of x that makes it true, but an identity is true for every value of x. Which of these best explains why 3x + 5 = 20 is an equation rather than an identity?

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

4.A closed cylinder has radius r cm and height (r + 5) cm. Its volume is 300 cm³, giving the equation πr²(r + 5) = 300, which can be solved using the iterative formula rₙ₊₁ = √(300 ÷ (π(rₙ + 5))). Taking r₀ = 3, work out r₃ correct to 2 decimal places.

5.A circular pond has equation x² + y² = 20, with lengths in metres from the centre of the garden. A straight path runs along the line y = 2x, entering the pond and leaving it again. Work out the coordinates of the two points where the path meets the edge of the pond.

6.The simultaneous equations kx + 2y = 4 and 3x + y = 5 have no solution. Work out the value of k.

7.Solve 7x − 5 = 3x + 11

8.f(x) = (x + 1)/2. Find f⁻¹(x).

9.Simplify 5a/6 − a/3

10.Solve the inequality 3x − 1 ≤ 11.

11.The diagram shows the graph of a function. Which equation could it represent?

12.Work out the equation of the straight line that passes through (−2, 3) and (4, −9).

13.Harry will spend at most £150 on a party. The cake costs £60 and each helium balloon costs £3. Solve an inequality to find all the possible numbers of balloons, x, that he can buy.

14.A student attempts to prove that the sum of any three consecutive integers is a multiple of 3. Line 1: Let the three consecutive integers be n, n + 1 and n + 2. Line 2: Their sum is n + (n + 1) + (n + 2) = 3n + 2. Line 3: 3n + 2 leaves a remainder of 2 when divided by 3, so it is not a multiple of 3. Line 4: So the sum of three consecutive integers is not always a multiple of 3. Which line contains the FIRST error?

15.A candle is 30 cm tall and burns down at a steady rate of 1 cm per hour. Write down the function for the height of the candle y, in centimetres, after x hours.