| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Show derivative equals given algebraic form |
| Difficulty | Standard +0.8 This question requires applying the quotient rule correctly, then algebraically manipulating to match a given form (part a), followed by analyzing stationary points using the discriminant of a quadratic (part b). The algebraic manipulation and discriminant analysis for the range of k elevate this beyond a routine differentiation exercise, though it remains within standard A-level techniques. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07q Product and quotient rules: differentiation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(x) = \frac{e^{3x}}{4x^2+k} \Rightarrow f'(x) = \frac{(4x^2+k)3e^{3x} - 8xe^{3x}}{(4x^2+k)^2}\) | M1 | Attempts quotient rule to obtain form \(\frac{\alpha(4x^2+k)e^{3x} - \beta xe^{3x}}{(4x^2+k)^2}\), \(\alpha,\beta > 0\); condone bracketing errors if intention clear |
| Correct unsimplified derivative | A1 | If quotient rule formula quoted it must be correct; condone missing brackets if implied by subsequent work |
| \(f'(x) = \frac{(12x^2 - 8x + 3k)e^{3x}}{(4x^2+k)^2}\) | A1 | Must obtain \(f'(x) = (12x^2-8x+3k)g(x)\) where \(g(x) = \frac{e^{3x}}{(4x^2+k)^2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| If \(y=f(x)\) has at least one stationary point then \(12x^2 - 8x + 3k = 0\) has at least one root | B1 | B0M1A1 not possible; no requirement to formally state \(\frac{e^{3x}}{(4x^2+k)^2} > 0\) |
| Applies \(b^2 - 4ac \geq 0\) with \(a=12, b=-8, c=3k\) | M1 | Also allow completing the square approach; condone \(b^2-4ac > 0\) |
| \(0 < k \leqslant \frac{4}{9}\) | A1 | Condone \(k \leqslant \frac{4}{9}\) and condone \(0 \leqslant k \leqslant \frac{4}{9}\); must be in terms of \(k\) not \(x\) |
# Question 12:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(x) = \frac{e^{3x}}{4x^2+k} \Rightarrow f'(x) = \frac{(4x^2+k)3e^{3x} - 8xe^{3x}}{(4x^2+k)^2}$ | M1 | Attempts quotient rule to obtain form $\frac{\alpha(4x^2+k)e^{3x} - \beta xe^{3x}}{(4x^2+k)^2}$, $\alpha,\beta > 0$; condone bracketing errors if intention clear |
| Correct unsimplified derivative | A1 | If quotient rule formula quoted it must be correct; condone missing brackets if implied by subsequent work |
| $f'(x) = \frac{(12x^2 - 8x + 3k)e^{3x}}{(4x^2+k)^2}$ | A1 | Must obtain $f'(x) = (12x^2-8x+3k)g(x)$ where $g(x) = \frac{e^{3x}}{(4x^2+k)^2}$ |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| If $y=f(x)$ has at least one stationary point then $12x^2 - 8x + 3k = 0$ has at least one root | B1 | B0M1A1 not possible; no requirement to formally state $\frac{e^{3x}}{(4x^2+k)^2} > 0$ |
| Applies $b^2 - 4ac \geq 0$ with $a=12, b=-8, c=3k$ | M1 | Also allow completing the square approach; condone $b^2-4ac > 0$ |
| $0 < k \leqslant \frac{4}{9}$ | A1 | Condone $k \leqslant \frac{4}{9}$ and condone $0 \leqslant k \leqslant \frac{4}{9}$; must be in terms of $k$ not $x$ |
---\begin{enumerate}
\item The function f is defined by
\end{enumerate}
$$f ( x ) = \frac { e ^ { 3 x } } { 4 x ^ { 2 } + k }$$
where $k$ is a positive constant.\\
(a) Show that
$$f ^ { \prime } ( x ) = \left( 12 x ^ { 2 } - 8 x + 3 k \right) g ( x )$$
where $\mathrm { g } ( x )$ is a function to be found.
Given that the curve with equation $y = \mathrm { f } ( x )$ has at least one stationary point, (b) find the range of possible values of $k$.
\hfill \mbox{\textit{Edexcel Paper 2 2022 Q12 [6]}}