OCR C4 2016 June — Question 3 5 marks

Exam BoardOCR
ModuleC4 (Core Mathematics 4)
Year2016
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicImplicit equations and differentiation
TypeShow dy/dx equals given expression
DifficultyStandard +0.3 This is a straightforward implicit differentiation question requiring the product rule, chain rule, and basic differentiation of standard functions. While it involves multiple terms and algebraic rearrangement to isolate dy/dx, it follows a standard procedure with no conceptual surprises—slightly easier than average for C4 level.
Spec1.07s Parametric and implicit differentiation
3 Given that \(y \sin 2 x + \frac { 1 } { x } + y ^ { 2 } = 5\), find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
Question 3:
AnswerMarks Guidance
AnswerMarks Guidance
\(2y\frac{dy}{dx}\)B1 from differentiation of \(y^2\)
\(\sin 2x\frac{dy}{dx} + 2y\cos 2x\)M1 correct use of Product Rule; allow sign error or one incorrect coefficient
\(\sin 2x\frac{dy}{dx} + 2y\cos 2x - \frac{1}{x^2} + 2y\frac{dy}{dx} = 0\)A1
\((\sin 2x + 2y)\frac{dy}{dx} = \frac{1}{x^2} - 2y\cos 2x\)M1 collection of like terms on separate sides; must be two terms in \(\frac{dy}{dx}\)
\(\frac{dy}{dx} = \frac{1 - 2x^2 y\cos 2x}{(\sin 2x + 2y)x^2}\)A1 e.g. \(\frac{dy}{dx} = \frac{x^{-2} - 2y\cos 2x}{(\sin 2x + 2y)}\); A0 for e.g. \(y = \ldots\) 
## Question 3:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $2y\frac{dy}{dx}$ | B1 | from differentiation of $y^2$ |
| $\sin 2x\frac{dy}{dx} + 2y\cos 2x$ | M1 | correct use of Product Rule; allow sign error or one incorrect coefficient |
| $\sin 2x\frac{dy}{dx} + 2y\cos 2x - \frac{1}{x^2} + 2y\frac{dy}{dx} = 0$ | A1 | |
| $(\sin 2x + 2y)\frac{dy}{dx} = \frac{1}{x^2} - 2y\cos 2x$ | M1 | collection of like terms on separate sides; must be two terms in $\frac{dy}{dx}$ |
| $\frac{dy}{dx} = \frac{1 - 2x^2 y\cos 2x}{(\sin 2x + 2y)x^2}$ | A1 | e.g. $\frac{dy}{dx} = \frac{x^{-2} - 2y\cos 2x}{(\sin 2x + 2y)}$; A0 for e.g. $y = \ldots$ |

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3 Given that $y \sin 2 x + \frac { 1 } { x } + y ^ { 2 } = 5$, find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.

\hfill \mbox{\textit{OCR C4 2016 Q3 [5]}}
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Q1 3 Q2 5 Q3 5 Q4 5 Q5 6 Q6 6 Q7 6 Q8 9 Q9 15 Q10 12