Higher November 2021 Unit 1 Q14
14.
(a) Evaluate \(4^{-\frac{3}{2}}\). [2]
(b) Evaluate \(\dfrac{1}{3} + 0{\cdot}0\dot{2}\).
Express your answer as a fraction. [3]
Express your answer as a fraction. [3]
| Scheme | Marks | Comments |
|---|---|---|
| \(\dfrac{1}{8}\) | B2 | B1 for \(8^{-1}\) or \(\dfrac{1}{2^3}\) or \(\left(\tfrac{1}{2}\right)^3\) or \(\dfrac{1}{\sqrt{64}}\) or \(\sqrt{\dfrac{1}{64}}\) or \(\dfrac{1}{64^{\frac{1}{2}}}\) or \(\left(\dfrac{1}{64}\right)^{\frac{1}{2}}\) |
| Scheme | Marks | Comments |
|---|---|---|
| \(x = 0{\cdot}02222.....\) \(10x = 0{\cdot}2222.....\) with an attempt to subtract | M1 | Or \(10x\) and \(100x\), or equivalent. Or an alternative method. |
| \(\left(\tfrac{1}{3}\,+\right) \tfrac{2}{90}\) OR \(\left(\tfrac{1}{3}\,+\right) \tfrac{1}{45}\) | A1 | Sight of \(\tfrac{0\cdot2}{9}\) gains M1 only. |
| \(x = \tfrac{32}{90}\ \left(= \tfrac{16}{45}\right)\) | B1 | FT ‘their \(\tfrac{2}{90}\)’ provided equivalent difficulty. Mark final answer. Do not ignore incorrect cancelling. |
Alternative method 1
| Scheme | Marks | Comments |
|---|---|---|
| \(x = \left(\tfrac{1}{3}\,+\right) \tfrac{0\cdot2}{9}\) | B1 | |
| \(= \tfrac{3\cdot2}{9}\) | M1 | Mark final answer |
| \(= \tfrac{32}{90}\ \left(= \tfrac{16}{45}\right)\) | A1 |
Alternative method 2
| Scheme | Marks | Comments |
|---|---|---|
| \(x = 0{\cdot}35555.\ldots..\) | B1 | |
| \(10x = 3{\cdot}5555.....\) with an attempt to subtract | M1 | Or \(10x\) and \(100x\), or equivalent. ‘FT ‘their 0.35555…’ provided equivalent difficulty’. |
| \(x = \tfrac{32}{90}\ \left(= \tfrac{16}{45}\right)\) | A1 | Sight of \(\tfrac{3\cdot2}{9}\) gains B1 M1 only Mark final answer |
Alternative method 3
| Scheme | Marks | Comments |
|---|---|---|
| \(x = 0{\cdot}35555.\ldots..\ (= 0{\cdot}3 + 0{\cdot}05555)\) | B1 | |
| \(= \tfrac{3}{10} + \tfrac{0\cdot5}{9}\) or equivalent | M1 | |
| \(= \tfrac{32}{90}\ \left(= \tfrac{16}{45}\right)\) | A1 | Mark final answer |