Algebra worksheet — GCSE Higher

1.A car's velocity–time graph is a straight line from (0 s, 4 m/s) rising to (6 s, V m/s), followed by a straight line falling from (6 s, V m/s) to (9 s, 0 m/s). The gradient of the second line is −8 m/s². Work out the total distance travelled between t = 0 and t = 9 seconds.

2.The first four terms of a sequence are 9, 14, 19, 24. Work out an expression, in terms of n, for the nth term.

3.A company charges for hiring chairs for a day. Hiring 1 chair costs £12, hiring 2 chairs costs £19, hiring 3 chairs costs £26, and hiring 4 chairs costs £33. Work out an expression, in terms of n, for the cost in pounds of hiring n chairs for a day.

4.The formula y = x² + 1 gives y in terms of x. Work out the value of y when x = −2.

5.f(x) = 2x − 1. Work out ff(x).

6.Describe a sequence of two transformations that maps the graph of y = x² onto the graph of y = −(x − 5)².

7.The height, h metres, of a ball t seconds after a stopwatch is started follows h = (t − 1)(5 − t) for 1 ≤ t ≤ 5, where h = 0 means the ball is at ground level. Work out the two times at which the ball is at ground level.

8.Three vertices of a rectangle are (−4, −1), (2, −1) and (2, 3). The sides of the rectangle are parallel to the axes. Write down the coordinates of the fourth vertex.

9.A circle has centre (0, 0) and equation x² + y² = 100. A straight line is drawn from the origin through the point P(8, 9). Work out whether P lies inside, on, or outside the circle.

10.A geometric sequence has first term 3 and common ratio √3. Work out the position of the first term of the sequence that exceeds 30.

11.Which expression means 'y squared, multiplied by 3'?

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

13.Solve the inequality 4x + 1 > 2x + 9.

14.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.

15.A rectangle has area 24x + 18. Ffion factorises the area as 6(4x + 3). Work out the value of the area when x = 2, and decide whether 6 and (4x + 3) are correctly identified as factors of 24x + 18.