Higher June 2024 Paper 4 Q15
15
(a) Solve by factorisation.
\(3x^2 + 10x - 8 = 0\) [3]
(b) Write \(x^2 + 8x + 11\) in the form \((x + a)^2 - b\). [3]
(c)
(i) Write down the coordinates of the turning point of the graph \(y = (x - 3)^2 + 8\). [2]
(ii) Describe the single transformation which maps the graph of \(y = x^2\) onto the graph of \(y = (x - 3)^2 + 8\). [2]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \((3x - 2)(x + 4)\) oe | M2 | M1 for \((3x + a)(x + b)\) with \(a + 3b = 10\) or \(ab = -8\) or \(3x(x + 4) - 2(x + 4)\) or \(x(3x - 2) + 4(3x - 2)\) | For M2 and M1 condone the omission of the final bracket. \(\frac{(3x - 2)(3x + 12)}{3}\) followed by \(3x - 2\) [= 0] and \(x + 4\) [= 0] scores M2. If no products of factors shown then \(3x - 2 = 0\) and \(x + 4 = 0\) scores M1. After \(3x(x + 4) - 2(x + 4)\) or \(x(3x - 2) + 4(3x - 2)\) followed by correct answers award M2B1 BOD |
| – 4 and \(\frac{2}{3}\) | B1FT | correct or FT their linear factors | Condone \(0.\dot{6}\) or 0.666… or 0.67 for \(\frac{2}{3}\) |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \((x + 4)^2 - 5\) | 3 | B1 for \((x + 4)^2\) B2FT their \((x + a)^2\) for their – 5 If 0 scored SC2 for ‘correct’ answer with missing power or power written in the wrong place | e.g. \((x + 11)^2 - 110\) scores B2FT e.g. SC2 for \((x + 4) - 5\) or \((x + 4^2) - 5\) |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) 3 8 | 2 | B1 for either 3 or 8 in the correct place | |
| (ii) Translation \(\begin{pmatrix} 3 \\ 8 \end{pmatrix}\) | 2 | B1 for each correct or FT the vector from their answer from (c)(i) | If more than one transformation score 0 |