Algebra worksheet — GCSE Higher

1.Solve the inequality 3x − 1 ≤ 11.

2.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.

3.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?

4.Work out the value of 3x² − 4x when x = −2.

5.The graph of y = 20/x is drawn for x > 0. As the value of x increases, what happens to the value of y?

6.The area under a speed-time graph between t = 0 and t = 6 seconds is estimated using three strips of equal width, using the speeds, in m/s, at t = 0, 2, 4 and 6: 0, 5, 9 and 12. Using the trapezium rule with these three trapezia, estimate the distance travelled.

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

8.A taxi fare, in pounds, for a journey of m miles is given by the formula F = 3 + 2.5m. A journey costs £15.50. Work out the distance travelled, m, by first rearranging the formula to make m the subject, then substituting F = 15.50.

9.The nth term of a sequence is 4n − 3. Work out the 7th term of the sequence.

10.Factorise 3x² + 10x − 8.

11.The tangent to a curve at the point where x = 4 passes through the points (2, 5) and (6, 21). Use these two points to estimate the gradient of the curve at x = 4.

12.A table shows y = x² − 6x + 5 at these points (x, y): (0, 5), (1, 0), (2, −3), (3, −4), (4, −3), (5, 0), (6, 5). Using the symmetry shown, write down the x-coordinate of the turning point.

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

14.Work out the coordinates of the midpoint of the line segment joining (−3, 5) and (7, 9).

15.Solve x² − 8x + 3 = 0 by completing the square. Give your answers in surd form.