C3 January 2006 Q3
3. The point \(P\) lies on the curve with equation \(y = \ln\left(\dfrac{1}{3}x\right)\). The \(x\)-coordinate of \(P\) is 3.
Find an equation of the normal to the curve at the point \(P\) in the form \(y = ax + b\), where \(a\) and \(b\) are constants. (5)
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{1}{x}\) accept \(\dfrac{3}{3x}\) | M1 A1 |
| At \(x = 3\), \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{1}{3} \Rightarrow m' = -3\) Use of \(mm' = -1\) | M1 |
| \(y - \ln 1 = -3(x - 3)\) | M1 |
| \(y = -3x + 9\) Accept \(y = 9 - 3x\) | A1 |
| (5) | |
| (5 marks) |
Notes
\(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{1}{3x}\) leading to \(y = -9x + 27\) is a maximum of M1 A0 M1 M1 A0 = 3/5