| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2010 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Second derivative calculation |
| Difficulty | Standard +0.3 This is a structured two-part differentiation question with clear guidance. Part (a) is routine quotient rule application with a standard trigonometric identity (p=4). Part (b) requires product rule on the result from (a) and substitution, but the form of the answer is given. While it involves second derivatives and some algebraic manipulation, the scaffolding makes it slightly easier than average for C3. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{dy}{dx} = \frac{\cos 4x \cdot 4\cos 4x - \sin 4x \cdot (-4\sin 4x)}{\cos^2 4x}\) | M1 | \(\frac{\pm A\cos^2 4x \pm B\sin^2 4x}{\cos^2 4x}\) |
| \(= \frac{4\cos^2 4x + 4\sin^2 4x}{\cos^2 4x}\) or better | A1 | Both terms correct |
| \(= 4(1 + \tan^2 4x)\) | A1 (CSO) | All correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{d^2y}{dx^2} = 4 \times 2\tan 4x \times \ldots\) | M1 | \(A\tan 4x \times f(4x)\) |
| \(4\sec^2 4x\) | m1 | \(f(4x) = B\sec^2 4x\) |
| \(= 32\tan 4x \sec^2 4x\) | A1F | ft \(8\times\) their \(p\) from part (a) |
| \(= 32\tan 4x(1 + \tan^2 4x)\) | m1 | Previous two method marks must have been earned |
| \(= 32y(1 + y^2)\) | A1 | CSO |
## Question 7(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dy}{dx} = \frac{\cos 4x \cdot 4\cos 4x - \sin 4x \cdot (-4\sin 4x)}{\cos^2 4x}$ | M1 | $\frac{\pm A\cos^2 4x \pm B\sin^2 4x}{\cos^2 4x}$ |
| $= \frac{4\cos^2 4x + 4\sin^2 4x}{\cos^2 4x}$ or better | A1 | Both terms correct |
| $= 4(1 + \tan^2 4x)$ | A1 (CSO) | All correct |
## Question 7(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{d^2y}{dx^2} = 4 \times 2\tan 4x \times \ldots$ | M1 | $A\tan 4x \times f(4x)$ |
| $4\sec^2 4x$ | m1 | $f(4x) = B\sec^2 4x$ |
| $= 32\tan 4x \sec^2 4x$ | A1F | ft $8\times$ their $p$ from part (a) |
| $= 32\tan 4x(1 + \tan^2 4x)$ | m1 | Previous two method marks must have been earned |
| $= 32y(1 + y^2)$ | A1 | CSO |
**Alternative solutions also shown with equivalent mark schemes (M1)(m1)(A1F)(m1)(A1)**
---7 It is given that $y = \tan 4 x$.
\begin{enumerate}[label=(\alph*)]
\item By writing $\tan 4 x$ as $\frac { \sin 4 x } { \cos 4 x }$, use the quotient rule to show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = p \left( 1 + \tan ^ { 2 } 4 x \right)$, where $p$ is a number to be determined.
\item Show that $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = q y \left( 1 + y ^ { 2 } \right)$, where $q$ is a number to be determined.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2010 Q7 [8]}}