Algebra worksheet — GCSE Higher

1.f(x) = x³ − 3x − 20, and the equation f(x) = 0 has exactly one solution. Work out the pair of consecutive integers between which that solution lies.

2.A number machine multiplies its input by 2 and then subtracts 5. Work out the output when the input is 6.

3.Write y = x² + 12x + 40 in the form (x + a)² + b, and hence write down the minimum value of y.

4.Expand and simplify (x + 1)(x + 2)(x + 3).

5.A circle has centre (0, 0) and equation x² + y² = 100. A straight line is drawn from the origin through the point P(8, 9). Work out whether P lies inside, on, or outside the circle.

6.The equation 3(2x − 1) = 4x + 9 is rearranged by expanding the brackets. Which of these is the correctly expanded equation?

7.The first four terms of a sequence are 6, 13, 20, 27. Work out an expression, in terms of n, for the nth term.

8.The graph of y = sin x is transformed onto the graph of y = sin x + 1. Which statement correctly describes the transformation and the new range of the graph?

9.The equation 7x = x² + 3 can be solved using the iterative formula xₙ₊₁ = (xₙ² + 3) ÷ 7. Taking x₀ = 0.4, x₁ = 0.4514 correct to 4 decimal places. Using the full unrounded value of x₁, work out x₂ correct to 3 decimal places.

10.The area of a circle is given by the formula A = πr², where r is the radius. Rearrange the formula to make r the subject.

11.A ribbon of length L metres is cut into 5 equal pieces, and then 3 metres is removed from one piece. Write down an expression, in metres, for the length of that piece after 3 metres is removed.

12.A geometric sequence has first term 3 and common ratio √2. Work out the 6th term of the sequence, giving your answer in the form a√2.

13.A company's weekly profit, in £, is modelled by y = f(x), where x is the number of weeks since launch. The graph of y = f(x) has a maximum at (10, 45000). A rival company uses the same marketing strategy but starts trading 6 weeks later and has fixed costs £8000 higher every week, so its profit is modelled by y = f(x − 6) − 8000. In which week does the rival's maximum weekly profit occur, and what is it?

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

15.An allotment is in the shape of a rectangle. Its length is 5 m more than its width, x metres, and its area is 20 m². This gives x² + 5x − 20 = 0, which can be solved using the iterative formula xₙ₊₁ = 20 ÷ (xₙ + 5). Taking x₀ = 2, so that x₁ is the value found after the formula has been used once, work out x₂ correct to 2 decimal places.