Higher January 2023 Paper 2R Q19
19 Prove algebraically that, for any three consecutive even numbers,
the sum of the squares of the smallest even number and the largest even number is 8 more than twice the square of the middle even number.
(3)
| Scheme | Marks |
|---|---|
| eg \(2n,\; 2n + 2,\; 2n + 4\) or \(2n - 2,\; 2n,\; 2n + 2\) etc | M1 |
| eg \((2n)^2 + (2n + 4)^2\;(= 4n^2 + 4n^2 + 16n + 16 = 8n^2 + 16n + 16)\) or \(2(2n + 2)^2\;(= 2(4n^2 + 8n + 4) = 8n^2 + 16n + 8)\) or \(2(2n + 2)^2 + 8\;(= 2(4n^2 + 8n + 4) + 8 = 8n^2 + 16n + 16)\) | M1 |
| eg \((2n)^2 + (2n + 4)^2 = 8n^2 + 16n + 16\) and \(2(2n + 2)^2 + 8 = 8n^2 + 16n + 16\) or \((2n)^2 + (2n + 4)^2 = 8n^2 + 16n + 16\) and \(2(2n + 2)^2 = 8n^2 + 16n + 8\) and \(8n^2 + 16n + 16 - (8n^2 + 16n + 8) = 8\) or \((2n)^2 + (2n + 4)^2 = 8n^2 + 16n + 16\) and \(8n^2 + 16n + 16 = 8n^2 + 16n + 8 + 8 = 2(2n + 2)^2 + 8\) or \(2(2n + 2)^2 + 8 = 8n^2 + 16n + 16\) and \(8n^2 + 16n + 16 = 4n^2 + 4n^2 + 16n + 16 = (2n)^2 + (2n + 4)^2\) Working required Answer: Correctly shown | A1 |
| (3) | |
| (3 marks) |
Notes
M1: for 3 consecutive even numbers in algebraic form (any letter can be used)
M1: for the sum of the squares of the largest and smallest even numbers and adding or the square of the middle even number multiplied by 2
(no need to expand or simplify for this mark)
A1: dep on M2 for use of algebra to show correct conclusion
(SCB1 for eg \((p + 4)^2 + p^2\) or \(2(p + 2)^2\) or \(2(p + 2)^2 + 8\))
(SCB2 for use of eg \((p + 4)^2 + p^2 = 2p^2 + 8p + 16\) and \(2(p + 2)^2 + 8 = 2p^2 + 8p + 16\)
If the student shows this and also says “it is true for all numbers, so it must be true for even numbers” oe or defines \(p\), \(p + 2\), \(p + 4\) as even numbers, then this would gain M2A1