Higher June 2023 Paper 1 Q13
13
Show that when the input increases by 2 the output increases by \(2a\). [2 marks]
Kai says that \(\dfrac{\text{f}(6)}{\text{f}(2)}\) is equal to \(\text{f}(3)\) because \(\dfrac{6}{2} = 3\)
Is he correct?
Show working to support your answer. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \((y =)\ ax + b\) and \((y =)\ ax + 2a + b\) | B2 | any letter for \(x\) other than \(a\) or \(b\) or \(y\) B1 \((y =)\ ax + b\) or \((y =)\ a(x + 2) + b\) or \((y =)\ ax + 2a + b\) or substitution of two values for \(x\) with a difference of 2 and correct working to show that the output increases by \(2a\) eg substituting \(x = 3\) and \(x = 5\) to get \(3a + b\) and \(5a + b\) |
Additional guidance
| Allow \(xa\) for \(ax\) throughout | |
| Do not allow \(a \times x + b\) for \(ax + b\) unless recovered | |
| Allow, eg \((x + 2) \times a + b\) for \(a(x + 2) + b\) | |
| Do not allow missing brackets unless recovered eg do not allow \(x + 2 \times a\) for \(a(x + 2)\) | |
| Do not accept written answers without the necessary algebra eg The input has increased by 2 and will then be multiplied by \(a\), so the output will increase by \(2a\) | B0 |
| Ignore further non-contradictory work if B2 awarded |
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 – using \(k\) | ||
| \(\dfrac{\text{f}(6)}{\text{f}(2)} \left(= \dfrac{36k}{4k}\right) = 9\) or \(\text{f}(3) = 9k\) | M1 | condone eg \(k36\) |
| \(\dfrac{\text{f}(6)}{\text{f}(2)} = 9\) and \(\text{f}(3) = 9k\) and No | A1 | condone \(k9\) |
| Alternative method 2 – substituting a value for \(k\) | ||
| Identifies a value of \(k\) other than 1 and correctly evaluates \(\dfrac{\text{f}(6)}{\text{f}(2)}\) or \(\text{f}(3)\) | M1 | eg \(k = 2\) and \(\dfrac{\text{f}(6)}{\text{f}(2)} = 9\) or \(\text{f}(3) = 18\) |
| Identifies a value of \(k\) other than 1 and correctly evaluates \(\dfrac{\text{f}(6)}{\text{f}(2)}\) and \(\text{f}(3)\) and No | A1 | eg \(k = 2\) and \(\dfrac{\text{f}(6)}{\text{f}(2)} = 9\) and \(\text{f}(3) = 18\) and No |
Additional guidance
\(9k\) from \(\dfrac{\text{f}(6)}{\text{f}(2)}\) is M0, but M1 can be awarded if accompanied by \(\text{f}(3) = 9k\)
Do not allow 9 from \(\dfrac{36}{4}\) (unless \(\dfrac{36}{4}\) is from \(\dfrac{36k}{4k}\))
Do not allow 9 from \(\dfrac{36k^2}{4k^2}\)
Students may correctly state that \(\dfrac{\text{f}(6)}{\text{f}(2)}\) and \(\text{f}(3)\) are (only) equal when \(k = 1\)
This may replace ‘No’ in their answer, but does not score without \(9k\) and 9
Do not allow unprocessed values, eg \(6^2\), \(2^2\) or \(3^2\)