Higher June 2022 Paper 1 Q24
24 Express each of \(a\), \(b\) and \(c\) in terms of \(q\) so that
\[q + 12x - qx^2\]can be written as \(a - b(x - c)^2\)
(4)
| Scheme | Marks |
|---|---|
| \(-q\left(x^2 - \dfrac{12}{q}x\right) + q\) or \(-q\left(x^2 - \dfrac{12}{q}x - \dfrac{q}{q}\right)\) oe | M1 |
| \(-q\left[\left(x - \dfrac{12}{2q}\right)^2 \ldots..\right]\) oe or \(-q\left[\left(x - \dfrac{6}{q}\right)^2 \ldots..\right]\) oe | M1 |
E.g. \(-q\left(x - \dfrac{6}{q}\right)^2 + \dfrac{36}{q} + q\) oe or \(-q\left(x - \dfrac{12}{2q}\right)^2 + \dfrac{144q}{4q^2} + q\) oe | M1 |
\(a = \dfrac{36}{q} + q\) \(b = q\) \(c = \dfrac{6}{q}\) | A1 |
| (4) | |
| (4 marks) |
Notes
M1: for a correct factorisation of the expression or \(b = q\) (must be stated)
M1: for starting the correct process to complete the square
M1: for a complete process of completing the square. (Does not need to be simplified)
A1: oe \(a\) and \(c\) must come from a correct process of completing the square. (Does not need to be simplified)
| Scheme | Marks |
|---|---|
| \(a - bx^2 + 2bcx - bc^2\) oe or \(-bx^2 + 2bcx - bc^2 + a\) oe or \(b = q\) | M1 |
| \(2bc = 12\) or \(a - bc^2 = q\) oe | M1 |
\(c = \dfrac{12}{2q}\) or \(a = q\left(\dfrac{12}{2q}\right)^2 + q\) or \(c = \dfrac{6}{q}\) or \(a = q\left(\dfrac{6}{q}\right)^2 + q\) | M1 |
\(a = \dfrac{36}{q} + q\) \(b = q\) \(c = \dfrac{6}{q}\) | A1 |
Notes
M1: for correctly multiplying out \(a - b(x - c)^2\)
M1: for correctly equating coefficients
M1: for correctly finding \(a\) or \(c\) (Does not need to be simplified)
A1: oe (Does not need to be simplified)