2. (a) Express
$$\frac { 4 x - 1 } { 2 ( x - 1 ) } - \frac { 3 } { 2 ( x - 1 ) ( 2 x - 1 ) }$$
as a single fraction in its simplest form.
Given that
$$f ( x ) = \frac { 4 x - 1 } { 2 ( x - 1 ) } - \frac { 3 } { 2 ( x - 1 ) ( 2 x - 1 ) } - 2 , \quad x > 1$$
(b) show that
$$f ( x ) = \frac { 3 } { 2 x - 1 }$$
(c) Hence differentiate \(\mathrm { f } ( x )\) and find \(\mathrm { f } ^ { \prime } ( 2 )\).
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Question 2:
Part (a)
Answer Marks
Guidance
Answer/Working Marks
Guidance
\(\frac{(4x-1)(2x-1)-3}{2(x-1)(2x-1)}\) M1
An attempt to form a single fraction
\(= \frac{8x^2 - 6x - 2}{\{2(x-1)(2x-1)\}}\) A1 aef
Correct quadratic numerator over correct quadratic denominator
\(= \frac{2(x-1)(4x+1)}{\{2(x-1)(2x-1)\}}\) M1
An attempt to factorise a 3-term quadratic numerator
\(= \frac{4x+1}{2x-1}\) A1
(4 marks total)
Part (b)
Answer Marks
Guidance
Answer/Working Marks
Guidance
\(f(x) = \frac{4x+1}{2x-1} - 2\) \(= \frac{(4x+1) - 2(2x-1)}{(2x-1)}\) M1
An attempt to form a single fraction
\(= \frac{3}{(2x-1)}\) A1 *
Correct result (2 marks total)
Part (c)
Answer Marks
Guidance
Answer/Working Marks
Guidance
\(f(x) = 3(2x-1)^{-1}\) \(f'(x) = 3(-1)(2x-1)^{-2}(2)\) M1
\(\pm k(2x-1)^{-2}\)
A1 aef \(f'(2) = \frac{-6}{9} = -\frac{2}{3}\) A1
Either \(\frac{-6}{9}\) or \(-\frac{2}{3}\) (3 marks total) [9]
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## Question 2:
**Part (a)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{(4x-1)(2x-1)-3}{2(x-1)(2x-1)}$ | M1 | An attempt to form a single fraction |
| $= \frac{8x^2 - 6x - 2}{\{2(x-1)(2x-1)\}}$ | A1 aef | Correct quadratic numerator over correct quadratic denominator |
| $= \frac{2(x-1)(4x+1)}{\{2(x-1)(2x-1)\}}$ | M1 | An attempt to factorise a 3-term quadratic numerator |
| $= \frac{4x+1}{2x-1}$ | A1 | (4 marks total) |
**Part (b)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(x) = \frac{4x+1}{2x-1} - 2$ | | |
| $= \frac{(4x+1) - 2(2x-1)}{(2x-1)}$ | M1 | An attempt to form a single fraction |
| $= \frac{3}{(2x-1)}$ | A1 * | Correct result (2 marks total) |
**Part (c)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(x) = 3(2x-1)^{-1}$ | | |
| $f'(x) = 3(-1)(2x-1)^{-2}(2)$ | M1 | $\pm k(2x-1)^{-2}$ |
| | A1 aef | |
| $f'(2) = \frac{-6}{9} = -\frac{2}{3}$ | A1 | Either $\frac{-6}{9}$ or $-\frac{2}{3}$ (3 marks total) **[9]** |
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2. (a) Express
$$\frac { 4 x - 1 } { 2 ( x - 1 ) } - \frac { 3 } { 2 ( x - 1 ) ( 2 x - 1 ) }$$
as a single fraction in its simplest form.
Given that
$$f ( x ) = \frac { 4 x - 1 } { 2 ( x - 1 ) } - \frac { 3 } { 2 ( x - 1 ) ( 2 x - 1 ) } - 2 , \quad x > 1$$
(b) show that
$$f ( x ) = \frac { 3 } { 2 x - 1 }$$
(c) Hence differentiate $\mathrm { f } ( x )$ and find $\mathrm { f } ^ { \prime } ( 2 )$.\\
\hfill \mbox{\textit{Edexcel C3 2011 Q2 [9]}}