Higher November 2021 Paper 1 Q17
17
(a) Factorise \(6y^2 - y - 5\) (2)
(b) Make \(f\) the subject of \(w = \dfrac{2f + 3}{8 - f}\) (3)
(c) Express \(4x^2 - 8x + 7\) in the form \(a(x + b)^2 + c\) where \(a\), \(b\) and \(c\) are integers. (3)
| Scheme | Marks |
|---|---|
| \(6y(y - 1) + 5(y - 1)\) or \(y(6y + 5) - 1(6y + 5)\) | M1 |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: \((6y + 5)(y - 1)\) | A1 |
| (2) |
Notes
M1: for \((6y \pm 5)(y \pm 1)\) or \((6y \pm 1)(y \pm 5)\)
or \((ay + 5)(by - 1)\) where \(ab = 6\) or \(5b - a = -1\)
or \((6y + p)(y + q)\) where \(pq = -5\) or \(6q + p = -1\)
Condone use of a different letter to \(y\)
A1: oe
| Scheme | Marks |
|---|---|
| \(8w - fw = 2f + 3\) oe | M1 |
| \(8w - 3 = 2f + fw\) oe or \(-2f - fw = 3 - 8w\) oe | M1 |
Correct answer scores full marks (unless from obvious incorrect working) Answer: \(f = \dfrac{8w - 3}{2 + w}\) | A1 |
| (3) |
Notes
M1: for multiplying by denominator and expanding in a correct equation
M1: for gathering terms in \(f\) on one side and other terms the other side in a correct equation
ft their equation dep on 2 terms in \(f\) and two other terms
A1: oe accept \(f = \dfrac{3 - 8w}{-2 - w}\) oe
| Scheme | Marks |
|---|---|
| \(4(x^2 - 2x) + 7\) or \(4\left(x^2 - 2x + \dfrac{7}{4}\right)\) oe | M1 |
| \(4\left[(x - 1)^2 - 1^2\right] + 7\) oe or \(4\left[(x - 1)^2 - 1^2 + \dfrac{7}{4}\right]\) oe | M1 |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: \(4(x - 1)^2 + 3\) | A1 |
| (3) | |
| (8 marks) |
Notes
M1: for a complete method
A1: allow \(a = 4\), \(b = -1\) and \(c = 3\)
| Scheme | Marks |
|---|---|
| \(ax^2 + 2bax + b^2a + c\) | M1 |
| \(2ba = -8\) and \(b^2a + c = 7\) | M1 |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: \(4(x - 1)^2 + 3\) | A1 |
Notes
M1: for correctly expanding \(a(x + b)^2 + c\) to give \(ax^2 + 2bax + b^2a + c\)
M1: for a complete method (equating coefficients)
A1: allow \(a = 4\), \(b = -1\) and \(c = 3\)