October 2021 Paper 3 Q7
7 A curve \(C\) in the \(x\)-\(y\) plane has the property that the gradient of the tangent at the point \(P(x, y)\) is three times the gradient of the line joining the point \((3, 2)\) to \(P\).
It is given that \(C\) passes through the point \((4, 3)\) and that \(x \gt 3\) and \(y \gt 2\) at all points on \(C\).
The curve \(C\) may be obtained by a transformation of part of the curve \(y = x^3\).
| Scheme | Marks | AO |
|---|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 3\left(\dfrac{y - 2}{x - 3}\right)\) | M1 A1 | 2.5 1.1 |
| [2] |
Notes
M1: Correct left-hand side (including equals sign) and right-hand side must be of the form \(\dfrac{\mathrm{f}(y)}{\mathrm{g}(x)}\) or \(\dfrac{\mathrm{g}(x)}{\mathrm{f}(y)}\)
A1: cao (oe)
| Scheme | Marks | AO |
|---|---|---|
| \(\displaystyle\int \frac{1}{y - 2}\,\mathrm{d}y = 3\int \frac{1}{x - 3}\,\mathrm{d}x\) | M1* | 1.1a |
| \(\ln(y - 2) = 3\ln(x - 3)\,(+c)\) | A1ft | 1.1 |
| \((4, 3) \Rightarrow c = 0,\ y - 2 = (x - 3)^3\) | M1dep* | 1.1 |
| \(y = (x - 3)^3 + 2\) | A1 | 2.2a |
| [4] |
Notes
M1*: Separation of variables – dependent on the M mark in (a)
With an indication of integration
A1ft: Follow through their differential eq. from (a)
Condone no constant
M1dep*: Attempt to find \(c\) and eliminate logs
A1: oe (but must be of the form \(y = \mathrm{f}(x)\))
| Scheme | Marks | AO |
|---|---|---|
| Translation | B1 | 1.1 |
| \(\begin{pmatrix}3\\2\end{pmatrix}\) | B1ft | 1.1 |
| [2] |
Notes
B1: B0 if another type of transformation stated or if shift/move etc. used
B1ft: Follow through their \(y = (x - p)^3 + q\) - B0 if this transformation is given as a stretch/rotation/reflection/enlargement etc. (but condone no transformation stated or shift/move etc.) – need not be given as a vector
\(p, q \neq 0\)