June 2023 Paper 2 Q11
11 In this question you must show detailed reasoning.
The variables \(x\) and \(y\) are such that \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) is directly proportional to the square root of \(x\).
When \(x = 4\), \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 3\).
(a) Find \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \(x\). [3]
When \(x = 4\), \(y = 10\).
(b) Find \(y\) in terms of \(x\). [3]
| Scheme | Marks | AO |
|---|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = k\sqrt{x}\) | B1 | 2.1 |
| \(3 = k \times \sqrt{4}\) | M1 | 1.1 |
| \(k = \frac{3}{2}\) or \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \frac{3}{2}\sqrt{x}\) isw | A1 | 2.2a |
| [3] |
Notes
B1: may be implied by final answer
A1: if B0M0 allow SC1 for \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{k}{\sqrt{x}}\) and \(k = 6\) as final answer
or \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{6}{\sqrt{x}}\) as final answer
| Scheme | Marks | AO |
|---|---|---|
| their \(k \times \dfrac{x^{\frac{3}{2}}}{\frac{3}{2}}\) oe | B1 | 3.1a |
| \(10 = \left(\sqrt{4}\right)^3 + c\) | M1 | 1.1 |
| \(c = 2\) or \(y = x^{\frac{3}{2}} + 2\) or \(y = x\sqrt{x} + 2\) isw | A1 | 1.1 |
| [3] |
Notes
B1: FT their \(k\)
M1: FT their integration, one term in \(x\) with index 1.5
A1: must see ‘\(y =\)’ at some point
if B0M0 allow SC2 for \(y = 12x^{\frac{1}{2}} - 14\) or \(y = 12\sqrt{x} - 14\)
or \(y = 12x^{\frac{1}{2}} + c\) and \(c = -14\)