| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Determine increasing/decreasing intervals |
| Difficulty | Standard +0.3 This is a straightforward application of quotient and product rules with standard follow-up tasks. Part (i)(a) requires quotient rule and factorisation; part (i)(b) is routine sign analysis of the derivative. Part (ii) uses product and chain rules, then sets g'(x)=0 and rearranges—all mechanical steps with no novel insight required. Slightly easier than average due to the structured guidance. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| (i)(a) \(f'(x) = \frac{4(x-3)(2x+5) - (2x+5)^2}{(x-3)^2}\) or \(\frac{(x-3)(8x+20) - (4x^2+20x+25)}{(x-3)^2}\) | M1 A1 | |
| \(= \frac{(2x+5)(2x-17)}{(x-3)^2}\) | M1 A1 | (4) |
| (b) Attempts both critical values or finds one "correct" end \(x < -2.5, x > 8.5\) (accept \(x \leq -2.5, x \geq 8.5\)) | M1 A1 | (2) |
| (ii) Attempts the chain rule on \((\sin 4x)^{\frac{1}{2}} \rightarrow A(\sin 4x)^{-\frac{1}{2}} \times \cos 4x\) | M1 | |
| \(g(x) = x(\sin 4x)^{\frac{1}{2}} \Rightarrow g'(x) = (\sin 4x)^{\frac{1}{2}} + x \times \frac{1}{2}(\sin 4x)^{-\frac{1}{2}} 4\cos 4x\) | M1 A1 | |
| Sets \(g'(x) = 0 \Rightarrow (\sin 4x)^{\frac{1}{2}} + x \times \frac{2\cos 4x}{(\sin 4x)^{\frac{1}{2}}} = 0\) and \(x \times \frac{(\sin 4x)^{\frac{1}{2}}}{\cos 4x}\) oe | M1 | |
| \(\rightarrow \tan 4x + 2x = 0\) | A1 | (5) |
**(i)(a)** $f'(x) = \frac{4(x-3)(2x+5) - (2x+5)^2}{(x-3)^2}$ or $\frac{(x-3)(8x+20) - (4x^2+20x+25)}{(x-3)^2}$ | M1 A1 |
$= \frac{(2x+5)(2x-17)}{(x-3)^2}$ | M1 A1 | (4)
**(b)** Attempts both critical values or finds one "correct" end $x < -2.5, x > 8.5$ (accept $x \leq -2.5, x \geq 8.5$) | M1 A1 | (2)
**(ii)** Attempts the chain rule on $(\sin 4x)^{\frac{1}{2}} \rightarrow A(\sin 4x)^{-\frac{1}{2}} \times \cos 4x$ | M1 |
$g(x) = x(\sin 4x)^{\frac{1}{2}} \Rightarrow g'(x) = (\sin 4x)^{\frac{1}{2}} + x \times \frac{1}{2}(\sin 4x)^{-\frac{1}{2}} 4\cos 4x$ | M1 A1 |
Sets $g'(x) = 0 \Rightarrow (\sin 4x)^{\frac{1}{2}} + x \times \frac{2\cos 4x}{(\sin 4x)^{\frac{1}{2}}} = 0$ and $x \times \frac{(\sin 4x)^{\frac{1}{2}}}{\cos 4x}$ oe | M1 |
$\rightarrow \tan 4x + 2x = 0$ | A1 | (5)
**Total: 11 marks**
---4. (i)
$$f ( x ) = \frac { ( 2 x + 5 ) ^ { 2 } } { x - 3 } \quad x \neq 3$$
\begin{enumerate}[label=(\alph*)]
\item Find $\mathrm { f } ^ { \prime } ( x )$ in the form $\frac { P ( x ) } { Q ( x ) }$ where $P ( x )$ and $Q ( x )$ are fully factorised quadratic expressions.
\item Hence find the range of values of $x$ for which $\mathrm { f } ( x )$ is increasing.\\
(ii)
$$g ( x ) = x \sqrt { \sin 4 x } \quad 0 \leqslant x < \frac { \pi } { 4 }$$
The curve with equation $y = g ( x )$ has a maximum at the point $M$.
Show that the $x$ coordinate of $M$ satisfies the equation
$$\tan 4 x + k x = 0$$
where $k$ is a constant to be found.\\
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P3 2020 Q4 [11]}}