Algebra worksheet — GCSE Higher

1.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?

2.One solution of the equation x² − (k + 1)x + k = 0 is x = 3. Work out the value of k.

3.Point P has coordinates (2, −5). P is reflected in the x-axis to point Q. Write down the coordinates of Q.

4.x = 5. Work out the value of 3x² − 4.

5.A car's journey is represented by a velocity-time graph. The total distance travelled, found from the area under the graph, is 350 metres, and the total time for the journey is 25 seconds. Work out the car's average speed for the whole journey, in m/s.

6.Work out the gradient of the straight line with equation y = (2x/5) − 3.

7.Work out the equation of the straight line through the points (−2, 1) and (0, 7).

8.A rule turns each input into an output. An input of 0 gives an output of −1, an input of 1 gives an output of 1, and an input of 2 gives an output of 3. Work out the rule, writing the input as x and the output as y.

9.Point P has coordinates (5, 2). Point P is rotated 180° about the origin to point Q. Write down the coordinates of Q.

10.A sequence begins at 50, and each term after that is found by subtracting 8 from the term before it. Priya says the 8th term of the sequence is negative. Is Priya correct?

11.Solve the inequality 2(x − 1) ≤ 8.

12.p = 4. Work out the value of 2p³.

13.A rectangle has width x cm. Its length is 3 cm more than its width. The perimeter of the rectangle is 38 cm. Form an equation and solve it to find x.

14.Make x the subject of the formula y = 5x + 3.

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