Algebra worksheet — GCSE Higher

1.A pattern is made from tiles. Pattern 1 uses 4 tiles, pattern 2 uses 7 tiles, pattern 3 uses 10 tiles and pattern 4 uses 13 tiles, with each pattern using 3 more tiles than the one before. Work out an expression, in terms of n, for the number of tiles used in pattern n.

2.f(x) = (x + 1)/2. Find f⁻¹(x).

3.Line L has equation y = 2x + 3. Work out the equation of the line perpendicular to L that passes through the point (4, 1), giving your answer in the form x + 2y = c.

4.The function f(x) = x² for all real values of x has no inverse function, but g(x) = x² for x ≥ 0 does have one. Which statement correctly explains this?

5.Points A(8, 15) and B(−8, 15) both lie on the circle x² + y² = 289, which has centre (0, 0). Work out the length of the chord AB.

6.Which of these equations is equivalent to 5x − 2 = 3x + 8?

7.The line y = 3x + 6 meets the x-axis at one point. Work out the coordinates of that point.

8.A company's weekly profit, P thousand pounds, when it makes x thousand items is modelled by P = x² − 8x + 12, for x ≥ 0. Work out the values of x for which the company makes a loss.

9.The graph of y = 60 × (0.5)ˣ is sketched for x ≥ 0. Which statement correctly describes what happens to the curve as x increases?

10.Solve 2x² = 18.

11.Which expression means 'triple c, then add double d'?

12.The graph of y = x² − 5x + 6 crosses the x-axis at two points. By factorising, work out the x-coordinates of these two points.

13.Expand and simplify 2(a + 6) − 2(a − 7)

14.A rule multiplies the input by a fixed number and then adds a fixed number. An input of 1 gives an output of 5, and an input of 3 gives an output of 11. Work out the rule, writing the input as x and the output as y.

15.A walker's distance-time graph is described as follows: she walks 3 km in the first 15 minutes at a steady speed, rests for 10 minutes, then walks a further 3 km in the next 20 minutes, also at a steady speed. Work out her speed, in km/h, during the first 15 minutes.