Higher November 2019 Paper 1 Q16
16. It is known that \(y\) varies inversely as the cube root of \(x\) and that \(y = 2\) when \(x = 27\).
(a) Find a formula for \(y\) in terms of \(x\). [3]
(b) Using your answer to part (a), find
(i) \(y\) when \(x = 1000\), [1]
(ii) \(x\) when \(y = 3\). [2]
| Scheme | Marks | Comments |
|---|---|---|
| \(y \propto \dfrac{1}{\sqrt[3]{x}}\) OR \(y = \dfrac{k}{\sqrt[3]{x}}\) oe | B1 | Allow \(y \propto \dfrac{k}{\sqrt[3]{x}}\); |
| \(2 = \dfrac{k}{\sqrt[3]{27}}\) OR \(2 = \dfrac{k}{3}\) OR \(k = 6\) | M1 | M1 implies B1 FT (for possible B0 M1 A0) for use of \(y \propto x^{\frac{1}{3}}\) or \(y \propto \dfrac{1}{x^n}\) with \(n \gt 0\) and \(n \ne \dfrac{1}{3}\) |
| \(y = \dfrac{6}{\sqrt[3]{x}}\) | A1 | CAO; may be seen (explicitly) in part (b) |
| Scheme | Marks | Comments |
|---|---|---|
| (i) \((y =)\ 0.6\) oe | B1 | FT ‘their derived \(k\)’ or use of \(y \propto x^{\frac{1}{3}}\) or \(y \propto \dfrac{1}{x^n}\) with \(n \gt 0\) and \(n \ne \dfrac{1}{3}\) |
| (ii) \(\sqrt[3]{x} = \dfrac{6}{3}\) oe | M1 | FT ‘their derived \(k\)’ or use of \(y \propto x^{\frac{1}{3}}\) or \(y \propto \dfrac{1}{x^n}\) with \(n \gt 0\) and \(n \ne \dfrac{1}{3}\) |
| \((x =)\ 8\) | A1 | FT |
| (6) |