| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2020 |
| Session | October |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Determine increasing/decreasing intervals |
| Difficulty | Standard +0.8 This question requires quotient rule differentiation, sign analysis of a rational function, and solving logarithmic inequalities. Part (b) demands proving a derivative is always positive (requiring careful algebraic manipulation), and part (c) involves analyzing when a rational function with logarithms is positive. The combination of proof, inequality solving with logs, and multi-step reasoning elevates this above standard differentiation exercises. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.06d Natural logarithm: ln(x) function and properties1.06f Laws of logarithms: addition, subtraction, power rules1.07l Derivative of ln(x): and related functions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(k = e^2\) or \(x \neq e^2\) | B1 | Condone \(k =\) awrt 7.39 or \(x \neq\) awrt 7.39 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(g'(x) = \frac{(\ln x-2)\times\frac{3}{x}-(3\ln x-7)\times\frac{1}{x}}{(\ln x-2)^2} = \frac{1}{x(\ln x-2)^2}\) | M1 | Attempts quotient rule with \(\ln x \to \frac{1}{x}\) |
| \(\frac{1}{x(\ln x-2)^2}\) | A1 | Numerator simplified to \(\frac{1}{x}\) |
| As \(x>0\) and \((\ln x-2)^2>0\), so \(g'(x)>0\) | A1cso | Must state both \(x>0\) AND \((\ln x-2)^2\) is squared |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts to solve \(3\ln x - 7 = 0\) or \(\ln x - 2 = 0\) | M1 | May treat as equality or inequality |
| \(0 < a < e^2\), \(a > e^{\frac{7}{3}}\) | A1 | Both conditions required |
## Question 13(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $k = e^2$ or $x \neq e^2$ | B1 | Condone $k =$ awrt 7.39 or $x \neq$ awrt 7.39 |
## Question 13(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $g'(x) = \frac{(\ln x-2)\times\frac{3}{x}-(3\ln x-7)\times\frac{1}{x}}{(\ln x-2)^2} = \frac{1}{x(\ln x-2)^2}$ | M1 | Attempts quotient rule with $\ln x \to \frac{1}{x}$ |
| $\frac{1}{x(\ln x-2)^2}$ | A1 | Numerator simplified to $\frac{1}{x}$ |
| As $x>0$ and $(\ln x-2)^2>0$, so $g'(x)>0$ | A1cso | Must state both $x>0$ AND $(\ln x-2)^2$ is squared |
## Question 13(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts to solve $3\ln x - 7 = 0$ or $\ln x - 2 = 0$ | M1 | May treat as equality or inequality |
| $0 < a < e^2$, $a > e^{\frac{7}{3}}$ | A1 | Both conditions required |
---\begin{enumerate}
\item The function $g$ is defined by
\end{enumerate}
$$g ( x ) = \frac { 3 \ln ( x ) - 7 } { \ln ( x ) - 2 } \quad x > 0 \quad x \neq k$$
where $k$ is a constant.\\
(a) Deduce the value of $k$.\\
(b) Prove that
$$\mathrm { g } ^ { \prime } ( x ) > 0$$
for all values of $x$ in the domain of g .\\
(c) Find the range of values of $a$ for which
$$g ( a ) > 0$$
\hfill \mbox{\textit{Edexcel Paper 2 2020 Q13 [6]}}