Algebra worksheet — GCSE Higher

1.A candle is 30 cm tall when lit and burns at a constant rate. After burning for 5 minutes, its height is 20 cm. The height h cm after t minutes is given by h = 30 − mt. Work out the value of m.

2.Expand and simplify (x − 3)²

3.The graph of y = f(x) crosses the x-axis at x = −3 and x = 7, and crosses the y-axis at (0, 21). A second graph crosses the x-axis at x = −7 and x = 3, and crosses the y-axis at the same point, (0, 21). The second graph is y = g(x). Which of these could be the equation of g(x)?

4.A point has coordinates (x, y), where x + y = 0 and x is not 0. In which two quadrants could this point lie?

5.A point has coordinates (x, y). In which two quadrants is the product x × y positive?

6.A rectangle has length (2x + 5) cm and width (x − 2) cm. Work out an expression, in terms of x, for the perimeter of the rectangle. Give your answer in its simplest form.

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

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

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

10.A rectangular garden has length (x + 7) m and width (x − 7) m. Work out an expression for the area of the garden, giving your answer in its simplest form.

11.A straight line passes through the points (1, 3) and (2, 5). Work out the equation of the line.

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

13.Factorise fully 5x + 5y − 5

14.On a distance-time graph, a horizontal line segment shows a period when the graph's gradient is 0. What does this tell you about the journey during that time?

15.A cyclist's speed rises from rest to a maximum — quickly at first, then more and more slowly — so her speed-time graph is a curve that is concave down (its gradient decreases as time goes on). A trapezium estimate is made for the distance travelled during this phase, using two points on the curve. Is the trapezium estimate an overestimate or an underestimate of the true distance, and why?