October 2020 Paper 1 Q10
10 In this question you must show detailed reasoning.
Fig. 10 shows the curve given parametrically by the equations \(x = \dfrac{1}{t^2},\ y = \dfrac{1}{t^3} - \dfrac{1}{t}\), for \(t > 0\).
| Scheme | Marks | AO |
|---|---|---|
| DR \(\dfrac{\mathrm{d}x}{\mathrm{d}t} = -2t^{-3}\) and \(\dfrac{\mathrm{d}y}{\mathrm{d}t} = -3t^{-4} + t^{-2}\) | M1 | 2.1 |
| So \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{-3t^{-4} + t^{-2}}{-2t^{-3}}\) | M1 | 2.1 |
| Multiply top and bottom by \(t^4\) \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{-3+t^2}{-2t} = \dfrac{3-t^2}{2t}\) | A1 | 2.1 |
| [3] |
Notes
M1: Attempt to differentiate both equations
M1: Combining derivatives for \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\)
Note that \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \left(-\dfrac{3}{t^4} + \dfrac{1}{t^2}\right) \times \left(-\dfrac{t^3}{2}\right)\)
(corrected from the printed mark scheme: the first bracket is printed as \(\left(-\frac{3}{t^4} - \frac{1}{t^2}\right)\))
A1: AG Correct derivative in required form.
| Scheme | Marks | AO |
|---|---|---|
| DR tangent parallel when \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = -\dfrac{1}{4}\) | B1 | 3.1a |
| \(\dfrac{3-t^2}{2t} = -\dfrac{1}{4}\) \(4t^2 - 2t - 12 = 0\) roots \(2, \left[-\dfrac{3}{2}\right]\) [but since \(t > 0\ \ t = 2\)] | M1 | 1.1a |
| When \(t = 2,\quad x = \dfrac{1}{4},\quad y = \dfrac{1}{8} - \dfrac{1}{2} = -\dfrac{3}{8}\) So the coordinates are \(\left(\dfrac{1}{4}, -\dfrac{3}{8}\right)\) | A1 | 1.1 |
| [3] |
Notes
B1: Establishing gradient \(-\dfrac{1}{4}\)
\(y = -\dfrac{1}{4}x + \dfrac{1}{4}\) not sufficient on its own
M1: Forming and solving quadratic equation.
A1: Using the value of \(t\) for both coordinates
Ignore any point based on \(t = -\dfrac{3}{2}\)
| Scheme | Marks | AO |
|---|---|---|
| DR Rearrange \(t = x^{-\frac{1}{2}}\) | B1 | 3.1a |
| Substitute \(y = \left(x^{-\frac{1}{2}}\right)^{-3} - \left(x^{-\frac{1}{2}}\right)^{-1} = x^{\frac{3}{2}} - x^{\frac{1}{2}}\) | M1 | 1.1 |
| \(= x^{\frac{1}{2}}(x-1) = (x-1)\sqrt{x}\) | A1 | 1.1 |
| [3] |
Notes
B1: or equivalent eg \(\dfrac{1}{t} = \sqrt{x}\)
\(y = \dfrac{1}{\left(\frac{1}{\sqrt{x}}\right)^3} - \dfrac{1}{\left(\frac{1}{\sqrt{x}}\right)}\)
M1: Attempt to eliminate \(t\)
A1: factorised form. Allow surd or index form
Do not allow for \(= \pm(x-1)\sqrt{x}\)