Higher January 2021 Paper 1R Q16
16
(a) Expand and simplify \((x + 4)(x - 2)(x + 1)\) (3)
(b) Express \(x^2 - 10x + 40\) in the form \((x + a)^2 + b\), where \(a\) and \(b\) are integers. (2)
| Scheme | Marks |
|---|---|
| E.g. \(x^2 + 4x - 2x - 8\;(= x^2 + 2x - 8)\) or \(x^2 - 2x + x - 2\;(= x^2 - x - 2)\) or \(x^2 + 4x + x + 4\;(= x^2 + 5x + 4)\) | M1 |
| E.g. \(x^3 + 2x^2 - 8x + x^2 + 2x - 8\) or \(x^3 + 4x^2 - 2x^2 - 8x + x^2 + 4x - 2x - 8\) or \(x^3 - x^2 - 2x + 4x^2 - 4x - 8\) or \(x^3 - 2x^2 + x^2 - 2x + 4x^2 - 8x + 4x - 8\) or \(x^3 + 5x^2 + 4x - 2x^2 - 10x - 8\) or \(x^3 + 4x^2 + x^2 + 4x - 2x^2 - 8x - 2x - 8\) | M1 |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: \(x^3 + 3x^2 - 6x - 8\) | A1 |
| (3) |
Notes
M1: for multiplying out two brackets correctly with no more than one error
M1: for at least 3 terms correct out of a maximum of 6 terms
or
for at least 4 terms correct out of a maximum of 8 terms
| Scheme | Marks |
|---|---|
E.g. \((x - 5)^2 - 5^2\;(+ 40)\) or \((x - 5)^2 - 25\;(+ 40)\) \(\left(x^2 + 2ax + a^2\;(+b^2)\right)\;\; 2a = -10\) or \(a = -5\) | M1 |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: \((x - 5)^2 + 15\) | A1 |
| (2) | |
| (5 marks) |
Notes
M1: for a correct first step or
for equating coefficients
A1: accept \(a = -5\), \(b = 15\)
SC B1 for \((-x + 5)^2 + 15\) or \((5 - x)^2 + 15\)