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.Harry is 14 years old and Mia is 8 years old. Will Harry ever be exactly twice as old as Mia? Give a reason for your answer.

3.A rectangle has width x cm. Its length is 3 cm more than its width. The perimeter of the rectangle is 38 cm. Form an equation and solve it to find x.

4.Ethan is 10 years old and his father is 40 years old. Work out how many years ago Ethan's father was seven times as old as Ethan.

5.The curve y = x² − 3x. Use the chord between x = 1 and x = 4 to estimate the gradient of the curve at x = 2.5.

6.A colony of bacteria doubles in number every hour. At 9am there are 5 bacteria in the colony. Work out how many bacteria there will be at 12 noon.

7.The equation x³ − 2x − 7 = 0 has exactly one solution. It can be found using the iterative formula xₙ₊₁ = ∛(2xₙ + 7), with starting value x₀ = 2, so that x₁ is the value after the formula has been used once. Work out the solution correct to 2 decimal places, iterating until two consecutive values round to the same 2 decimal places.

8.Explain why the equation 4x + 3 = 4x + 10 has no solution for x.

9.Three consecutive integers add up to 72. Using n for the smallest integer, form an equation and solve it to find the smallest of the three integers.

10.The equation x³ − 5x − 3 = 0 can be rearranged to give an iterative formula of the form xₙ₊₁ = ∛(…). Work out which one of these is a correct rearrangement.

11.Two numbers have a sum of 45. The larger number is 9 more than the smaller number. Form an equation using n for the smaller number, and solve it to find the larger number.

12.The point (20, 21) lies on the circle x² + y² = 841, which has centre O(0, 0). The tangent to the circle at (20, 21) crosses the x-axis at P and the y-axis at Q. Work out the area of triangle OPQ, correct to 1 decimal place.

13.The iterative formula xₙ₊₁ = √(2xₙ + 15) is used repeatedly, starting from x₀ = 1. Work out the value that xₙ approaches, correct to 2 decimal places.

14.A phone plan costs a fixed £15 per month plus 8p per minute of calls. In one month, Jamal's bill was £23.80. Form an equation using m for the number of minutes, and solve it to find how many minutes Jamal used that month.

15.A cup of tea cools after being made. Its temperature-time graph is a curve. Three tangents are drawn on the graph: at t = 2 minutes the tangent has gradient −8; at t = 10 minutes the tangent has gradient −3; at t = 20 minutes the tangent has gradient −0.5. Which of these statements about the tea's cooling is correct?