Algebra worksheet — GCSE Higher

1.A tap fills a tank at a varying rate. A graph shows the rate of flow, in litres per minute, against time, in minutes. What does the area under this graph represent?

2.Work out the maximum value of y = sin x, and the smallest positive value of x, in degrees, at which it occurs.

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

4.The point (−4, 3) lies on the circle x² + y² = 25, which has centre (0, 0). Work out the equation of the tangent to the circle at (−4, 3), giving your answer in the form y = mx + c.

5.An equation has exactly one value of x that makes it true, but an identity is true for every value of x. Which of these best explains why 3x + 5 = 20 is an equation rather than an identity?

6.The simultaneous equations 2x + 3y = 12 and x − y = 1 are given. Work out the value of y.

7.A car park charges a £4 fixed fee plus £3 for each hour. Kofi has exactly £25 to spend on parking. Using the inequality 4 + 3h ≤ 25, work out the greatest number of whole hours, h, he can park for.

8.Factorise fully 5x + 5y − 5

9.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.

10.Write down the expression that means 5 more than half of n.

11.A square has vertices at (−2, 3), (−2, −3) and (4, −3). Write down the coordinates of the fourth vertex.

12.f(x) = x + 2 and g(x) = x². Work out the value of x for which fg(x) = gf(x).

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

14.A circle has centre (0, 0) and passes through the point (5, 12). Work out the equation of the circle.

15.A cyclist's distance-time graph shows the following: she travels 12 km in the first 30 minutes at a steady speed, then rests for 30 minutes, then travels a further 8 km in the next 15 minutes. Work out her average speed, in km/h, for the whole journey.