Algebra worksheet — GCSE Higher

1.Solve 2x² = 18.

2.A sequence has the position-to-term rule n² − 3, where n is the position number. Work out the difference between the 6th term and the 5th term.

3.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.

4.f(x) = 2x − 1. Work out ff(x).

5.A car's speed, in m/s, during a journey is: 0 at t = 0 s, 8 at t = 4 s, and 8 (constant) from t = 4 s to t = 10 s, increasing at a constant rate between t = 0 and t = 4. Estimate the total distance the car travels, using the areas of a triangle and a rectangle.

6.The graph of y = sin x is reflected in the x-axis to give the graph of y = g(x). Work out the value of g(90), where x is measured in degrees.

7.The table shows the rate at which water flows from a tank, in litres per minute, over a 6-minute period: 12 at t = 0 min, 12 at t = 2 min, 3 at t = 5 min, and 0 at t = 6 min. Between t = 0 and t = 2 the rate is constant. Between t = 2 and t = 5, and between t = 5 and t = 6, the rate decreases at a constant rate in each section. Estimate the total volume of water that has flowed out, using the areas of a rectangle and two trapeziums.

8.The curve y = x² − 6 and the line y = 2x − 3 intersect at two points. Which pair of points is correct?

9.A gym charges a reduced price for the first month of membership and a standard price for every month after that. The total cost is £5 for 1 month, £12 for 2 months, £19 for 3 months and £26 for 4 months. Work out an expression, in terms of n, for the total cost, in pounds, of n months of membership.

10.How many turning points does the graph of y = x² − 7x + 10 have?

11.A machine bought for £2,400 loses 12% of its value every year. After how many complete years does its value first fall below £1,500?

12.Work out the smallest positive value of x, in degrees, for which y = tan x is undefined.

13.A square patio of side length n slabs is surrounded by a single border of square paving slabs of the same size. For a patio with side length n, the total number of slabs used for the patio and its border together is 9 when n = 1, 16 when n = 2, 25 when n = 3, and 36 when n = 4. Work out an expression, in terms of n, for the total number of slabs.

14.Kai writes the expression 8 − 3x + 5x² and says it has three terms. Ryan says it only has two terms because a number on its own is not a term. Work out the correct number of terms in the expression.

15.A zip-line runs from a platform 25 m high to a landing point 5 m high, over a horizontal distance of 40 m. Work out the gradient of the zip-line.