Algebra worksheet — GCSE Higher

1.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.

2.The point (3, 4) lies on the circle x² + y² = 25, which has centre (0, 0). Work out the gradient of the tangent to the circle at (3, 4).

3.A company's weekly profit, in £, is modelled by y = f(x), where x is the number of weeks since launch. The graph of y = f(x) has a maximum at (10, 45000). A rival company uses the same marketing strategy but starts trading 6 weeks later and has fixed costs £8000 higher every week, so its profit is modelled by y = f(x − 6) − 8000. In which week does the rival's maximum weekly profit occur, and what is it?

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

5.The graph of y = f(x) has a maximum turning point at (5, 8). Which of these correctly gives the corresponding turning point on the graph of y = −f(x + 1), and its type?

6.Simplify (x² − 9) ÷ (x² + 5x + 6).

7.Which of these is a formula, rather than an expression, an equation, or an identity?

8.The discriminant of a quadratic equation is greater than zero. Which statement about the solutions of that equation is correct?

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

10.A circle has equation x² + y² = 25. Does the point (3, 4) lie on this circle?

11.The graph of y = x² + 2x − 15 crosses the x-axis at two points. By factorising, work out the x-coordinates of these two points.

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

14.f(x) = x³ − 3x − 5. Given that f(2.2) = −0.952 and f(2.3) = 0.267, work out what this shows about the equation x³ − 3x − 5 = 0.

15.Which of these statements is an equation with exactly one solution?