Higher January 2022 Paper 1R Q12
12
(a) Expand and simplify \(n(n - 4)(3n + 5)\) (2)
(b) Express \[\frac{3}{x} + \frac{x + 2}{2x} + \frac{1}{4}\] as a single fraction in its simplest form. (3)
| Scheme | Marks |
|---|---|
| \(n(3n^2 + 5n - 12n - 20)\) or \(n(3n^2 - 7n - 20)\) or \((3n^2 + 5n)(n - 4)\) or \((n^2 - 4n)(3n + 5)\) or \(3n^3 + 5n^2 - 12n^2 - 20n\) | M1 |
| \(3n^3 - 7n^2 - 20n\) | A1 |
| (2) |
Notes
M1: for a correct partial expansion (may be unsimplified) (allow one error in the expansion of \((n - 4)(3n + 5)\) e.g.
for any 3 correct terms
or
for 4 out of 4 correct terms ignoring signs or
for \(3n^2 - 7n\ldots\)
or
for \(\ldots -7n - 20\))
A1: oe e.g. if correct answer seen allow further factorisation to \(n(3n^2 - 7n - 20)\)
| Scheme | Marks |
|---|---|
\(\dfrac{12}{4x} + \dfrac{2(x + 2)}{4x} + \dfrac{x}{4x}\) oe or \(\dfrac{12 + 2(x + 2) + x}{4x}\) oe \(\dfrac{3(8x)}{8x^2} + \dfrac{4x(x + 2)}{8x^2} + \dfrac{2x^2}{8x^2}\) oe or \(\dfrac{3(8x) + 4x(x + 2) + 2x^2}{8x^2}\) oe | M1 |
\(\dfrac{12 + 2x + 4 + x}{4x}\) oe or \(\dfrac{24x + 4x^2 + 8x + 2x^2}{8x^2}\) oe or \(\dfrac{6x^2 + 32x}{8x^2}\) oe or \(\dfrac{3x^2 + 16x}{4x^2}\) oe or \(\dfrac{6x + 32}{8x}\) oe | M1 |
| \(\dfrac{3x + 16}{4x}\) | A1 |
| (3) | |
| (5 marks) |
Notes
M1: for three correct fractions with a common denominator or a single correct fraction
M1: for a correct single fraction with brackets expanded
A1: oe \(\dfrac{16 + 3x}{4x}\)