C3 January 2011 Q2
2.
(a) Express\[\frac{4x - 1}{2(x - 1)} - \frac{3}{2(x - 1)(2x - 1)}\]as a single fraction in its simplest form. (4)
Given that
\[\mathrm{f}(x) = \frac{4x - 1}{2(x - 1)} - \frac{3}{2(x - 1)(2x - 1)} - 2, \qquad x \gt 1,\](b) show that\[\mathrm{f}(x) = \frac{3}{2x - 1}\] (2)
(c) Hence differentiate \(\mathrm{f}(x)\) and find \(\mathrm{f}'(2)\). (3)
| Scheme | Marks |
|---|---|
| \(\dfrac{4x - 1}{2(x - 1)} - \dfrac{3}{2(x - 1)(2x - 1)}\) | |
| \(= \dfrac{(4x - 1)(2x - 1) - 3}{2(x - 1)(2x - 1)}\) | M1 |
| \(= \dfrac{8x^2 - 6x - 2}{\{2(x - 1)(2x - 1)\}}\) | A1 aef |
| \(= \dfrac{2(x - 1)(4x + 1)}{\{2(x - 1)(2x - 1)\}}\) | M1 |
| \(= \dfrac{4x + 1}{2x - 1}\) | A1 |
| (4) |
Notes
M1: An attempt to form a single fraction
A1 aef: Simplifies to give a correct quadratic numerator over a correct quadratic denominator
M1: An attempt to factorise a 3 term quadratic numerator
| Scheme | Marks |
|---|---|
| \(\mathrm{f}(x) = \dfrac{4x - 1}{2(x - 1)} - \dfrac{3}{2(x - 1)(2x - 1)} - 2,\quad x \gt 1\) | |
| \(\mathrm{f}(x) = \dfrac{(4x + 1)}{(2x - 1)} - 2\) | |
| \(= \dfrac{(4x + 1) - 2(2x - 1)}{(2x - 1)}\) | M1 |
| \(= \dfrac{4x + 1 - 4x + 2}{(2x - 1)}\) | |
| \(= \dfrac{3}{(2x - 1)}\) | A1 * |
| (2) |
Notes
M1: An attempt to form a single fraction
A1 *: Correct result
| Scheme | Marks |
|---|---|
| \(\mathrm{f}(x) = \dfrac{3}{(2x - 1)} = 3(2x - 1)^{-1}\) | |
| \(\mathrm{f}'(x) = 3(-1)(2x - 1)^{-2}(2)\) | M1 A1 aef |
| \(\mathrm{f}'(2) = \dfrac{-6}{9} = -\dfrac{2}{3}\) | A1 |
| (3) | |
| (9 marks) |
Notes
M1: \(\pm k(2x - 1)^{-2}\)
A1: Either \(\dfrac{-6}{9}\) or \(-\dfrac{2}{3}\)