Algebra worksheet — GCSE Higher

1.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?

2.Work out the value of 3a − 2b when a = 5 and b = 4

3.A student attempts to prove that the product of two consecutive integers is always even: (i) Let the two consecutive integers be n and n + 1. (ii) Since n(n + 1) is even, one of n and n + 1 must be an even number. (iii) Therefore, n(n + 1) is even. At which statement does the proof first assume the very fact it is trying to prove?

4.f(x) = x + 3 and g(x) = 2x. Work out fg(x).

5.A car park charges by the hour. Parking for 1 hour costs £5, 2 hours costs £9, 3 hours costs £13 and 4 hours costs £17, with the cost increasing by the same amount for each extra hour. Work out an expression, in terms of n, for the cost, in pounds, of parking for n hours.

6.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.

7.A bath is filled with water; the graph of volume (litres) against time (minutes) is a curve, since the flow rate changes. The tangent to the curve at t = 10 minutes passes through the points (8, 130) and (12, 190), and the volume in the bath at t = 10 minutes is 160 litres. Use the gradient of this tangent to estimate the volume in the bath 5 minutes after t = 10 minutes.

8.A line segment has one endpoint at (−6, 2) and its midpoint at (−1, 5). Work out the coordinates of the other endpoint.

9.A ball's height, h metres, t seconds after being thrown follows h = (t − 1)(9 − t). Given that the ball is at ground level at t = 1 and t = 9, work out at what time t the ball reaches its maximum height, using symmetry.

10.Solve 4x − (2x − 6) = 18

11.Which of these values of x is a solution of x² + 2x − 15 = 0?

12.A circle has centre (0, 0) and passes through the point (5, 12). Work out the equation of the circle.

13.Solve x² + 7x = 0.

14.A company's profit was £2000 in its first year. Each following year, the profit increases by £800. Work out the first year in which the profit is more than £7000.

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