Higher November 2021 Paper 2 Q12
12
(a) Simplify \(\left(64p^9q^{12}\right)^{\frac{2}{3}}\) (2)
(b) Write as a single fraction \(\dfrac{2}{3x} + \dfrac{4}{5x} - \dfrac{9}{10x}\)
Give your answer in its simplest form. (2)
Give your answer in its simplest form. (2)
(c) Expand and simplify \(4x(x - 5)(2x + 3)\)
Show your working clearly. (3)
Show your working clearly. (3)
| Scheme | Marks |
|---|---|
| \(16p^6q^8\) | B2 |
| (2) |
Notes
B2: for all three correct terms
(B1 for 2 correct terms in a product of 3 terms or for \((4p^3q^4)^2\) or \(\left(4096p^{18}q^{24}\right)^{\frac{1}{3}}\))
| Scheme | Marks |
|---|---|
| eg \(\dfrac{2 \times 10}{3x \times 10} + \dfrac{4 \times 6}{5x \times 6} - \dfrac{9 \times 3}{10x \times 3}\ \left(= \dfrac{20}{30x} + \dfrac{24}{30x} - \dfrac{27}{30x}\right)\) | M1 |
Correct answer scores full marks (unless from obvious incorrect working) Answer: \(\dfrac{17}{30x}\) | A1 |
| (2) |
Notes
M1: for a common denominator for all 3 terms with at least 2 correct equivalent fractions (no need for signs)
[NB: fraction can be done in 2 parts]
A1: or \(\dfrac{17}{30}x^{-1}\)
| Scheme | Marks |
|---|---|
| eg \(4x(x - 5) = 4x^2 - 20x\) or \(4x(2x + 3) = 8x^2 + 12x\) or \((x - 5)(2x + 3) = 2x^2 + 3x - 10x - 15\) \(= 2x^2 - 7x - 15\) | M1 |
| eg \((4x^2 - 20x)(2x + 3) = 8x^3 + 12x^2 - 40x^2 - 60x\) or \((8x^2 + 12x)(x - 5) = 8x^3 + 12x^2 - 40x^2 - 60x\) or \(4x(2x^2 + 3x - 10x - 15) = 8x^3 + 12x^2 - 40x^2 - 60x\) or \(4x(2x^2 - 7x - 15) = 8x^3 - 28x^2 - 60x\) | M1ft |
| Working required Answer: \(8x^3 - 28x^2 - 60x\) | A1 |
| (3) | |
| (7 marks) |
Notes
M1: allow one error in the expansion of
\(4x(x - 5)\) or
\(4x(2x + 3)\) or
\((x - 5)(2x + 3)\)
M1ft: but dep on previous M1 for correctly expanding – allow one extra error or one omission.
A1: dep on M1
May be factorised if \(8x^3 - 28x^2 - 60x\) seen