Higher June 2021 Paper 1 Q17
17 Using algebra, prove that, given any 3 consecutive even numbers, the difference between the square of the largest number and the square of the smallest number is always 8 times the middle number.
(3)
| Scheme | Marks |
|---|---|
| eg \(2n, 2n + 2, 2n + 4\) or \(2n - 2, 2n, 2n + 2\) etc | M1 |
| eg \((2n + 4)^2 - (2n)^2\) \((= 4n^2 + 8n + 8n + 16 - 4n^2\ (= 16n + 16))\) or \((2n + 2)^2 - (2n - 2)^2\) \((= 4n^2 + 4n + 4n + 4 - (4n^2 - 4n - 4n + 4)\ (= 16n))\) | M1 |
eg \(8(2n + 2) = 16n + 16\) or eg \(16n + 16 = 8(2n + 2)\) or eg \(16n = 8(2n)\) or eg \(8n + 8n = 8(n + n)\) or eg \(\dfrac{16n + 16}{2n + 2} = 8\) Answer: Correctly shown | A1 |
| (3) | |
| (3 marks) |
Notes
M1: 3 consecutive even numbers in algebraic form (any letter can be used)
M1: for squaring the largest and smallest even numbers and subtracting
(no need to expand or simplify for this mark)
A1: dep on M2 for use of algebra to show correct conclusion
(SCB1 for eg \((p + 4)^2 - p^2\))
(SCB2 for use of eg \((p + 4)^2 - p^2 = 8p + 16 = 8(p + 2)\)
If the student shows this and also says “it is true for all numbers, so it must be true for even numbers” oe then this would gain M2A1
| Scheme | Marks |
|---|---|
eg \(a\), \(b\), \(c\) are consecutive even numbers where \(a \lt b \lt c\) and one of \(b = \dfrac{a + c}{2}\) or \(a + c = 2b\) or \(c - a = 4\) oe | M1 |
eg \(a\), \(b\), \(c\) are consecutive even numbers where \(a \lt b \lt c\) and all of \(b = \dfrac{a + c}{2}\) and \(a + c = 2b\) and \(c - a = 4\) oe | M1 |
| Now \(c^2 - a^2 = (c - a)(c + a) = 4 \times 2b = 8b\) Answer: Correctly shown | A1 |
Notes
M1: 3 numbers defined as consecutive even numbers with one correct equation, writing one term in terms of one or more of the others or \(c - a = 4\)
M1: 3 numbers defined as consecutive even numbers with three correct equations that involve all letters in some place
A1: dep on M2 for use of algebra to show correct conclusion