| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 20 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find derivative of product |
| Difficulty | Standard +0.3 This is a multi-part question testing product rule differentiation and related concepts. While it has many parts, each individual step is straightforward: (i) solving cos 2x = 0, (ii) checking f(-x) = -f(x), (iii) applying product rule, (iv) algebraic manipulation of the derivative, (v) substitution, and (vi) integration by parts. These are all standard C3 techniques with no novel problem-solving required. The length and multiple parts add some complexity, but each component is routine, making it slightly easier than average overall. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.07q Product and quotient rules: differentiation1.08i Integration by parts |
1 Fig. 8 shows part of the curve $y = x \cos 2 x$, together with a point P at which the curve crosses the $x$-axis.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{aee8da6a-7d5c-442f-9729-55d81d9a606f-1_427_968_432_584}
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\caption{Fig. 8}
\end{center}
\end{figure}
(i) Find the exact coordinates of P .\\
(ii) Show algebraically that $x \cos 2 x$ is an odd function, and interpret this result graphically.\\
(iii) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.\\
(iv) Show that turning points occur on the curve for values of $x$ which satisfy the equation $x \tan 2 x = \frac { 1 } { 2 }$.\\
(v) Find the gradient of the curve at the origin.
Show that the second derivative of $x \cos 2 x$ is zero when $x = 0$.\\
(vi) Evaluate $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x \cos 2 x \mathrm {~d} x$, giving your answer in terms of $\pi$. Interpret this result graphically.
\hfill \mbox{\textit{OCR MEI C3 Q1 [20]}}