Algebra worksheet — GCSE Higher

1.Two students each use the trapezium rule to estimate the area under the same curve between x = 0 and x = 8. Student A uses 4 strips of width 2. Student B uses 8 strips of width 1. Which estimate is more likely to be closer to the true area, and why?

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

3.Which expression is equivalent to 0.5(4x + 6) − x?

4.Tom buys 2 notebooks and 6 pencils on Monday, priced at p pence each for a notebook and q pence each for a pencil. On Tuesday he buys 3 more notebooks and 1 more pencil. Write a simplified expression, in pence, for the total amount Tom has spent on notebooks and pencils.

5.A garden path runs along the line 4x + y = 12. A drainage pipe must be laid perpendicular to the path, passing through the point (3, 1) where a sprinkler sits. Work out the equation of the pipe's line, giving your answer in the form y = mx + c.

6.A circle has centre O(0, 0) and equation x² + y² = 169. The point Q has coordinates (10, 11). Work out which of these gives the correct position of Q together with correct working.

7.The first four terms of a sequence are 11, 18, 25, 32. Ravi thinks the nth term is 7n. Work out the correct expression for the nth term.

8.A delivery van's distance-time graph shows the following: it travels 9 km in the first 15 minutes at a steady speed, stops for 45 minutes to unload, then travels a further 21 km in the next 30 minutes, also at a steady speed. Work out the average speed, in km/h, for the whole journey.

9.A number machine multiplies its input by 2 and then subtracts 5. Work out the output when the input is 6.

10.Solve 2x² − 7x + 3 = 0.

11.The graph of y = f(x) has x-intercepts at x = −1 and x = 4. Which statement correctly describes the x-intercepts of y = f(2 − x)?

12.Write down the coordinates of the point where the line y = 2x + 4 crosses the y-axis.

13.A geometric sequence has first term 4 and common ratio √2. Work out the sum of the first three terms of the sequence, giving your answer in the form a + b√2.

14.A sequence is defined by aₙ = aₙ₋₁ + aₙ₋₂ for n ≥ 3. Given that a₃ = 11 and a₅ = 29, work out a₁.

15.A straight line has gradient −2 and passes through the point (3, 1). Work out the equation of the line.