Identities and algebraic proof - Edexcel GCSE Maths (Non-calculator) Exam Questions

Practise 36 exam-style questions on Identities and algebraic proof for Edexcel GCSE Maths (Non-calculator). Write your answer as you would in the exam and get it marked in seconds against the mark scheme, point by point, with an AI tutor on hand if you get stuck.

36 questions
1–5 marks each
Instant marking
No sign-up needed

20 questions, marked as you go

Sample questions with model answers

Here are 4 of the questions in this topic, each with a model answer, an explanation of why it earns the marks and the mark scheme an examiner would use.

Question 1

2 marks

Here is a number puzzle.
Think of a number.
Add 3.
Multiply the result by 2.
Subtract 6.
Divide by the number you first thought of.

Use algebra to show that the answer is always 2 (for any starting number except 0).

Model answer

Start with \(n\).
Add 3: \(n + 3\)
Multiply by 2: \(2(n + 3) = 2n + 6\)
Subtract 6: \(2n\)
Divide by \(n\): \(\dfrac{2n}{n} = 2\)
So the answer is always 2.

Explanation

Use a letter for the starting number and follow each instruction with algebra. Trying a few numbers can suggest the result but never proves it; the algebra shows it works for every number. The starting number cannot be 0 because you cannot divide by 0.

Mark scheme

1 mark
M1 for \(2(n + 3) - 6\) simplified to \(2n\)
oe with any letter
1 mark
C1 for \(\dfrac{2n}{n} = 2\) with a correct conclusion
numerical examples only score 0

Question 2

3 marks

Prove that the sum of any three consecutive even numbers is a multiple of 6.

Model answer

Let the numbers be \(2n\), \(2n + 2\) and \(2n + 4\), where \(n\) is an integer.
Sum \(= 2n + 2n + 2 + 2n + 4 = 6n + 6 = 6(n + 1)\)
\(n + 1\) is an integer, so \(6(n + 1)\) is a multiple of 6.

Explanation

Any even number can be written \(2n\), and the next even numbers are 2 and 4 more. Factorise the sum to show the factor 6, then state the conclusion. Writing \(n\), \(n + 2\), \(n + 4\) is not enough, because \(n\) might be odd.

Mark scheme

1 mark
P1 for representing three consecutive even numbers, e.g. \(2n\), \(2n + 2\), \(2n + 4\)
1 mark
P1 for the sum \(6n + 6\)
1 mark
C1 for \(6(n + 1)\) with a conclusion that this is a multiple of 6
using n, n + 2, n + 4 does not guarantee even numbers: max P0 P1 C0

Question 3

3 marks

\(n\) is a positive integer.

Prove that \(n^2 + 3n\) is always even.

Model answer

\(n^2 + 3n = n^2 + n + 2n = n(n + 1) + 2n\)
\(n\) and \(n + 1\) are consecutive integers, so one of them is even and \(n(n + 1)\) is even.
\(2n\) is even.
Even + even = even, so \(n^2 + 3n\) is always even.

Explanation

One neat approach is to split off \(2n\), leaving \(n(n + 1)\), the product of two consecutive integers, which must include an even number. Another is to factorise to \(n(n + 3)\) and look at the cases \(n\) even and \(n\) odd: in either case one factor is even.

Mark scheme

1 mark
P1 for a useful rearrangement, e.g. \(n^2 + 3n = n(n + 1) + 2n\) or \(n(n + 3)\)
1 mark
P1 for explaining that \(n(n + 1)\) is the product of consecutive integers so is even, or for considering n odd and n even separately
e.g. if n is odd, n + 3 is even; if n is even, n is even
1 mark
C1 for a complete proof with a conclusion

Question 4

4 marks

A rectangle has length \((x + 3)\) cm and width \((x - 1)\) cm, where \(x > 1\)
A square has the same perimeter as the rectangle.

Prove that the area of the square is always 4 cm² more than the area of the rectangle.

Model answer

Perimeter of rectangle \(= 2(x + 3) + 2(x - 1) = 4x + 4\)
Side of square \(= \dfrac{4x + 4}{4} = x + 1\)
Area of square \(= (x + 1)^2 = x^2 + 2x + 1\)
Area of rectangle \(= (x + 3)(x - 1) = x^2 + 2x - 3\)
Difference \(= (x^2 + 2x + 1) - (x^2 + 2x - 3) = 4\)
The difference is always 4 cm², whatever the value of \(x\).

Explanation

Use the equal perimeters to find the side of the square in terms of \(x\). Then expand both areas and subtract; every term in \(x\) cancels, leaving a constant. This is why a proof needs algebra - it shows the result for every possible rectangle at once.

Mark scheme

1 mark
P1 for the perimeter of the rectangle, \(4x + 4\)
oe, e.g. \(2(x + 3) + 2(x - 1)\)
1 mark
P1 for the side of the square, \(x + 1\), and its area \((x + 1)^2 = x^2 + 2x + 1\)
1 mark
P1 for the area of the rectangle, \((x + 3)(x - 1) = x^2 + 2x - 3\)
1 mark
C1 for \((x^2 + 2x + 1) - (x^2 + 2x - 3) = 4\) with a conclusion

Ready to try the rest? Every answer you write is marked against the mark scheme, so you can see exactly where the marks are.

AI Tutor Hints first, answers after you try

Hi! Stuck? Ask me for a hint and I’ll help you work it out. Once you’ve checked your answer I can explain exactly where the marks are.