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Algebraic Proof

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AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen and paper handy, to work out your answers.

Question 12 marks

k is an integer.

A student is proving that (5k+2)(k+3)-(5k-1)(k+2) is always a multiple of 8

Which expression does (5k+2)(k+3)-(5k-1)(k+2) simplify to?

Hint

Expand each product separately, then put the second expansion in a bracket before you subtract.

Worked solution
  1. (5k+2)(k+3)=5k^2+15k+2k+6=5k^2+17k+6
  2. (5k-1)(k+2)=5k^2+10k-k-2=5k^2+9k-2
  3. Subtract, keeping the bracket: 5k^2+17k+6-(5k^2+9k-2)
  4. =5k^2+17k+6-5k^2-9k+2
  5. =8k+8
  6. =8(k+1), which is a multiple of 8
  7. Answer: 8k+8

Question 21 mark

n is an integer.

Which expression is always odd?

Hint

Try factorising the terms involving n and think about whether the product is odd or even when n is odd and when n is even.

Worked solution
  1. n^2+3n=n(n+3)
  2. If n is even, n(n+3) is even
  3. If n is odd, n+3 is even, so n(n+3) is even
  4. So n^2+3n is always even, and n^2+3n+5 is always odd
  5. The others are even when n is odd (e.g. n=1 gives 6, 8 and 8)
  6. Answer: n^2+3n+5

Question 3Challenge5 marks

n is an integer.

(a)

Expand and simplify (2n+7)^2-(2n-3)^2 Give your answer fully factorised.

3 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(b)

When n is odd, (2n+7)^2-(2n-3)^2 is always a multiple of m.

Work out the largest possible value of m.

2 marks

Hint

Bracket the second expansion before subtracting so every sign changes, then think about what n+1 is when n is odd.

Worked solution

Part (a)

  1. (2n+7)^2=4n^2+28n+49
  2. (2n-3)^2=4n^2-12n+9
  3. Subtract, keeping the bracket: 4n^2+28n+49-(4n^2-12n+9)
  4. =40n+40
  5. =40(n+1)
  6. (Or use the difference of two squares: (2n+7-2n+3)(2n+7+2n-3)=10(4n+4)=40(n+1))

Part (b)

  1. If n is odd, n+1 is even
  2. So n+1=2p for some integer p
  3. 40(n+1)=40\times 2p=80p, a multiple of 80
  4. n=1 gives 80 and n=3 gives 160, so no larger number always works
  5. m=80

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