| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find derivative of product |
| Difficulty | Standard +0.3 Part (i) is a straightforward application of the product rule with a standard trigonometric function. Part (ii) requires integration by parts, which is slightly more challenging but still a routine C3 technique. The question is slightly above average difficulty due to the two-part nature and the need to apply two different calculus techniques, but both are standard textbook exercises with no novel insight required. |
| Spec | 1.07q Product and quotient rules: differentiation1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Rightarrow \frac{dy}{dx} = -2x\sin 2x + \cos 2x\) | M1, B1, A1 [3] | Product rule; \(\frac{d}{dx}(\cos 2x) = -2\sin 2x\); oe cao |
| Answer | Marks | Guidance |
|---|---|---|
| \(= \frac{1}{2}x\sin 2x + \frac{1}{4}\cos 2x + c\) | M1, A1, A1ft, A1 [4] | Parts with \(u = x\), \(v = \frac{1}{2}\sin 2x\); \(+\frac{1}{4}\cos 2x\); cao — must have \(+c\) |
## Question 3:
**(i)** $y = x\cos 2x$
$\Rightarrow \frac{dy}{dx} = -2x\sin 2x + \cos 2x$ | M1, B1, A1 [3] | Product rule; $\frac{d}{dx}(\cos 2x) = -2\sin 2x$; oe cao
**(ii)** $\int x\cos 2x \, dx = \int x\frac{d}{dx}\left(\frac{1}{2}\sin 2x\right)dx$
$= \frac{1}{2}x\sin 2x - \int\frac{1}{2}\sin 2x \, dx$
$= \frac{1}{2}x\sin 2x + \frac{1}{4}\cos 2x + c$ | M1, A1, A1ft, A1 [4] | Parts with $u = x$, $v = \frac{1}{2}\sin 2x$; $+\frac{1}{4}\cos 2x$; cao — must have $+c$
---3 (i) Differentiate $x \cos 2 x$ with respect to $x$.\\
(ii) Integrate $x \cos 2 x$ with respect to $x$.
\hfill \mbox{\textit{OCR MEI C3 Q3 [7]}}