| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | October |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find stationary points - mixed transcendental products |
| Difficulty | Standard +0.3 This question requires product rule with exponential and trigonometric functions, then solving g'(x)=0, plus implicit differentiation with logarithms. All techniques are standard P3/C3 material with straightforward application—slightly easier than average due to clear structure and routine methods, though the mixed transcendental functions add minor complexity. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.07s Parametric and implicit differentiation |
| VIXV SIHIANI III IM IONOO | VIAV SIHI NI JYHAM ION OO | VI4V SIHI NI JLIYM ION OO |
(i)(a) $g'(x) = 3e^{3x}\sec 2x + 2e^{3x}\sec 2x \tan 2x$ M1 A1
(2 marks)
(b) $g'(x) = 0 \Rightarrow e^{3x}\sec 2x(3 + 2\tan 2x) = 0$ M1
$\tan 2x = -1.5 \Rightarrow x = -0.491$ d\begin{enumerate}
\item (i) The curve $C$ has equation $y = \mathrm { g } ( x )$ where
\end{enumerate}
$$g ( x ) = e ^ { 3 x } \sec 2 x \quad - \frac { \pi } { 4 } < x < \frac { \pi } { 4 }$$
(a) Find $\mathrm { g } ^ { \prime } ( x )$\\
(b) Hence find the $x$ coordinate of the stationary point of $C$.\\
(ii) A different curve has equation
$$x = \ln ( \sin y ) \quad 0 < y < \frac { \pi } { 2 }$$
Show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \mathrm { e } ^ { x } } { \mathrm { f } ( x ) }$$
where $\mathrm { f } ( x )$ is a function of $\mathrm { e } ^ { x }$ that should be found.\\
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VIXV SIHIANI III IM IONOO & VIAV SIHI NI JYHAM ION OO & VI4V SIHI NI JLIYM ION OO \\
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\hfill \mbox{\textit{Edexcel P3 2020 Q8 [9]}}