| Exam Board | Edexcel |
|---|---|
| Module | F3 (Further Pure Mathematics 3) |
| Year | 2023 |
| Session | January |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find gradient at point |
| Difficulty | Standard +0.8 This Further Maths question requires applying the product rule to a function involving arcsin, knowing the derivative of arcsin(2x) requires the chain rule, then substituting x=1/4 which involves evaluating arcsin(1/2)=π/6 and simplifying surds. While systematic, it combines multiple techniques (product rule, chain rule, inverse trig derivatives, exact value manipulation) beyond standard A-level, making it moderately challenging for Further Maths. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{dy}{dx} = 3\arcsin 2x + 3x \cdot \frac{1}{\sqrt{1-(2x)^2}} \times 2\) | M1 | Obtains \(p\arcsin qx + \frac{rx}{\sqrt{1-(sx)^2}}\) or \(p\arcsin qx + \frac{rx}{\sqrt{1-tx^2}}\), where \(p, q, r, s, t > 0\) |
| \(= 3\arcsin 2x + \frac{6x}{\sqrt{1-4x^2}}\) | A1 | Correct derivative. Allow unsimplified and isw. Allow \(\sin^{-1}\) and condone "arsin" but "arsinh" is M0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x = \frac{1}{4} \Rightarrow \frac{dy}{dx} = \frac{\pi}{2} + \sqrt{3}\) | B1dep | \(\frac{\pi}{2} + \sqrt{3}\) only but allow \(\frac{1}{2}\pi\) or \(0.5\pi\). Terms as a sum in either order. Allow \(a = \frac{1}{2}\), \(b = \sqrt{3}\). Isw following a correct answer. This is a "Hence" question so mark can only be awarded following full marks in part (a) |
# Question 1:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{dy}{dx} = 3\arcsin 2x + 3x \cdot \frac{1}{\sqrt{1-(2x)^2}} \times 2$ | M1 | Obtains $p\arcsin qx + \frac{rx}{\sqrt{1-(sx)^2}}$ or $p\arcsin qx + \frac{rx}{\sqrt{1-tx^2}}$, where $p, q, r, s, t > 0$ |
| $= 3\arcsin 2x + \frac{6x}{\sqrt{1-4x^2}}$ | A1 | Correct derivative. Allow unsimplified and isw. Allow $\sin^{-1}$ and condone "arsin" but "arsinh" is M0 |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x = \frac{1}{4} \Rightarrow \frac{dy}{dx} = \frac{\pi}{2} + \sqrt{3}$ | B1dep | $\frac{\pi}{2} + \sqrt{3}$ only but allow $\frac{1}{2}\pi$ or $0.5\pi$. Terms as a sum in either order. Allow $a = \frac{1}{2}$, $b = \sqrt{3}$. Isw following a correct answer. This is a "Hence" question so mark can only be awarded following full marks in part (a) |
---\begin{enumerate}
\item Given that
\end{enumerate}
$$y = 3 x \arcsin 2 x \quad 0 \leqslant x \leqslant \frac { 1 } { 2 }$$
(a) determine an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$\\
(b) Hence determine the exact value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ when $x = \frac { 1 } { 4 }$, giving your answer in the form $a \pi + b$ where $a$ and $b$ are fully simplified constants to be found.
\hfill \mbox{\textit{Edexcel F3 2023 Q1 [3]}}