Algebra worksheet — GCSE Higher

1.A distance-time graph is a straight line from (0, 0) to (4, 100), where time is in hours and distance is in kilometres. Work out the gradient of the line.

2.Write down the first four terms of the sequence with nth term 6n − 5.

3.The equation x² = 5x − 3 is to be solved using iteration. Work out which of these iterative formulas comes from a correct rearrangement of the equation.

4.Yusuf has £40 saved and adds £6 each week. His sister Aisha has £10 saved and adds £11 each week. After how many weeks will they have saved the same amount?

5.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?

6.Which expression is equivalent to 6x − (2x − 5)?

7.A ramp rises 3 m over a horizontal distance of 12 m. Work out the gradient of the ramp.

8.f(x) = 2x² + 1 and g(x) = x − 1. Work out fg(x), giving your answer in expanded form.

9.An exponential graph y = A × rˣ passes through the points (0, 8) and (2, 18). Work out the value of r, correct to 2 decimal places.

10.A line has equation y = −4x + 1. Work out the equation of the line perpendicular to it that passes through the point (0, 3).

11.On Monday, a runner covers 15 km in 2.5 hours. On Tuesday, she covers 12 km in 1.5 hours. Using the formula speed = distance ÷ time, work out on which day she ran faster, and by how much.

12.A circle has centre (0, 0) and equation x² + y² = 50. Work out the length of the diameter of the circle, correct to 1 decimal place.

13.The simultaneous equations 2x + 3y = 12 and x − y = 1 are given. Work out the value of y.

14.Find the common ratio of the geometric sequence 5, 5√2, 10, 10√2, ... Give your answer in surd form.

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