June 2023 Paper 2 Q13
13
(a) Circle the expression equivalent to \(\quad 3x^{-7}\) [1 mark]
- \(-\dfrac{3}{x^7}\)
- \(-\dfrac{1}{3x^7}\)
- \(\dfrac{1}{3x^7}\)
- \(\dfrac{3}{x^7}\)
(b) Simplify fully \(\quad \dfrac{12w^8}{\left(4w^3\right)^2}\) [2 marks]
(c) \(\sqrt{y} \times \sqrt[3]{y} = \sqrt[c]{y^d} \quad\) where \(c\) and \(d\) are positive integers.
Work out the least possible values of \(c\) and \(d\). [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{3}{x^7}\) | B1 |
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{3w^2}{4}\) or \(0.75w^2\) | B2 | B1 unsimplified equivalent fraction with brackets processed eg \(\dfrac{12w^8}{16w^6}\) or \(\dfrac{12w^2}{16}\) or \(\dfrac{6w^2}{8}\) or \(\dfrac{3w^5}{4w^3}\) SC1 \(\dfrac{3w^3}{4}\) or \(0.75w^3\) or \(3w^2\) |
Additional guidance
| B1 may be awarded for correct work with no answer or incorrect answer, even if this is seen amongst multiple attempts | |
| \(\dfrac{3}{4}w^2\) is equivalent to \(\dfrac{3w^2}{4}\) etc |
| Answer | Mark | Comments |
|---|---|---|
| \(c = 6\) and \(d = 5\) | B3 | B2 \(y^{\frac{5k}{6k}}\) or \(\sqrt[6k]{y^{5k}}\) or \(\left(y^{5k}\right)^{\frac{1}{6k}}\) or \(\left(y^{\frac{1}{6k}}\right)^{5k}\) where \(k\) is an integer \(\geqslant 1\) or \(\left(\dfrac{1}{2} + \dfrac{1}{3} =\right) \dfrac{5}{6}\) oe fractions or \(\left(\dfrac{c}{2} + \dfrac{c}{3} =\right) \dfrac{5c}{6}\) oe fractions or \(5c = 6d\) oe equation B1 \(y^{\frac{1}{2}}\) or \(y^{\frac{1}{3}}\) or \(y^{\frac{c}{2}}\) or \(y^{\frac{c}{3}}\) or \(\left(y^c\right)^{\frac{1}{2}}\) or \(\left(y^c\right)^{\frac{1}{3}}\) or \(y^{\frac{d}{c}}\) or \(\left(y^{\frac{1}{c}}\right)^d\) or \(\left(y^d\right)^{\frac{1}{c}}\) or \(\dfrac{1}{2} + \dfrac{1}{3}\) oe sum of fractions or \(\dfrac{c}{2} + \dfrac{c}{3}\) oe sum of fractions SC2 \(c = 6k\) and \(d = 5k\) where \(k\) is an integer \(\geqslant 2\) |
Additional guidance
| Up to B2 may be awarded for correct work with no answer or incorrect answer, even if this is seen amongst multiple attempts | |
| \(\dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}\) | B2 |
| \(\dfrac{3}{6} + \dfrac{2}{6}\) | B1 |
| \(\dfrac{10}{12}\) or \(y^{\frac{10}{12}}\) | B2 |
| \(c = 18\) and \(d = 15\) | SC2 |
| \(\dfrac{1}{2}\) may be 0.5 | |
| Only trialling numerical values for \(y\) is B0 or B3 |