GCSE Maths Formulae, functions & proof questions by topic

Substitution practice.

1. (a)Given T = 4p − 3q, find T when p = 5 and q = 2.[2]
2. (b)Given A = 6w², find A when w = 3.[2]

Using a scientific formula.

1. (a)The distance fallen from rest is d = 4.9t². Find d when t = 3.[2]
2. (b)A temperature conversion is F = 1.8C + 32. Find F when C = 25.[2]

Simple rearranging.

1. (a)Make t the subject of d = 8t.[1]
2. (b)Make h the subject of C = 12h + 5.[2]
3. (c)Make b the subject of a = b + 9.[1]

Negative substitution.

1. (a)Given y = x² + 2x, find y when x = −3.[2]
2. (b)Given s = ½(u + v)t, find s when u = −4, v = 6 and t = 3.[2]

A taxi firm charges by the mile.

1. (a)The fare in pounds is C = 3 + 1.8m, where m is the distance in miles. Work out the fare for a 7-mile journey.[2]
2. (b)Another journey cost £24.60. Work out its distance in miles.[2]

Formulas are rearranged.

1. (a)In y = 5x − 7, write x in terms of y.[2]
2. (b)Make q the subject of p = 3(q − 4).[2]

Changing the subject with roots and powers.

1. (a)Make r the subject of V = ⅓πr²h.[3]
2. (b)Make x the subject of y = √(2x + 1).[2]

A letter appearing twice.

1. (a)Make x the subject of 3x + 5 = y − 2x.[3]
2. (b)Make p the subject of a(p − 3) = b.[1]

Using a rearranged formula in context.

1. (a)The formula for the area of a trapezium is A = ½(a + b)h. Make b the subject.[3]
2. (b)Use your answer to find b when A = 40, a = 6 and h = 5.[1]

Substitution with a swap.

1. (a)Given f = ma, find f when m = 2.5 and a = −4.[1]
2. (b)Make m the subject of f = ma.[1]
3. (c)Use your subject to find m when f = −18 and a = −4.[2]

A stone is dropped from a tower.

1. (a)The distance fallen in metres is s = 4.9t², where t is the time in seconds. Work out s when t = 3.[2]
2. (b)The stone falls 19.6 m. Show that t = 2.[2]

Two functions are combined.

1. (a)f(x) = 4x + 3 and g(x) = x². Work out fg(2).[2]
2. (b)Find f⁻¹(x).[2]
3. (c)Use your answer to part (b) to find f⁻¹(11).[1]

Functions and inverses. (Higher)

1. (a)f(x) = 4x − 7. Find f⁻¹(x).[3]
2. (b)Show that f(3) = 5 and check your inverse with that value.[2]
3. (c)Solve f(x) = 21.[1]

Composite functions. (Higher)

1. (a)f(x) = 2x + 3 and g(x) = x². Find fg(4) and gf(4).[3]
2. (b)Find and simplify fg(x).[2]
3. (c)Solve fg(x) = 75.[1]

Algebraic proof. (Higher)

1. (a)Prove that the sum of three consecutive even numbers is a multiple of 6.[3]
2. (b)Prove that the product of an even number and any integer is even.[3]

Iteration. (Higher)

1. (a)Show that the equation x³ + 4x = 30 can be rearranged into x = ∛(30 − 4x).[2]
2. (b)Use the iteration xₙ₊₁ = ∛(30 − 4xₙ) with x₀ = 2 to find x₁, x₂, x₃ and x₄, giving each to 4 decimal places.[3]
3. (c)Write down a root of the original equation to 3 decimal places, and check it by substitution.[2]

Squares of consecutive even numbers. (Higher)

1. (a)Two consecutive even numbers are 2n and 2n + 2. Show that the difference of their squares is 8n + 4.[3]
2. (b)Prove that 8n + 4 is a multiple of 4 for every integer n.[1]
3. (c)Explain why 8n + 4 is never a multiple of 8.[2]

An iterative method finds a root. (Higher)

1. (a)Show that x³ − 6x + 2 = 0 can be rearranged as x = ∛(6x − 2).[1]
2. (b)Using x₀ = 2, find x₁ and x₂ to 3 decimal places.[3]
3. (c)Explain how to check that the root lies between 2.26 and 2.27.[2]