OCR MEI C3 — Question 7 5 marks

Exam BoardOCR MEI
ModuleC3 (Core Mathematics 3)
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProduct & Quotient Rules
TypeShow derivative equals given algebraic form
DifficultyModerate -0.3 This is a straightforward product rule application with a chain rule component for the square root term. The algebra simplification to reach the given form requires combining fractions and factoring, but follows a standard pattern. Slightly easier than average as it's a 'show that' question with a clear target and uses routine C3 techniques.
Spec1.07q Product and quotient rules: differentiation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates
7 Given that \(y = x ^ { 2 } \sqrt { 1 + 4 x }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x ( 5 x + 1 ) } { \sqrt { 1 + 4 x } }\).
Question 7:
AnswerMarks Guidance
AnswerMarks Guidance
\(y = x^2(1+4x)^{1/2}\)
\(\dfrac{dy}{dx} = x^2 \cdot \dfrac{1}{2}(1+4x)^{-1/2} \cdot 4 + 2x(1+4x)^{1/2}\)M1 Product rule with \(u=x^2\), \(v=\sqrt{1+4x}\); consistent with their derivatives; condone wrong index in \(v\) used for M1 only
B1\(\frac{1}{2}(\ldots)^{-1/2}\) soi
A1Correct expression
\(= 2x(1+4x)^{-1/2}(x+1+4x)\)M1 Factorising or combining fractions; (need not factor out the \(2x\)); must have evidence of \(x+1+4x\) oe
\(= \dfrac{2x(5x+1)}{\sqrt{1+4x}}\) *A1 NB AG; or \(2x(5x+1)(1+4x)^{-1/2}\) or \(2x(5x+1)/(1+4x)^{1/2}\)
[5] 
## Question 7:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = x^2(1+4x)^{1/2}$ | | |
| $\dfrac{dy}{dx} = x^2 \cdot \dfrac{1}{2}(1+4x)^{-1/2} \cdot 4 + 2x(1+4x)^{1/2}$ | M1 | Product rule with $u=x^2$, $v=\sqrt{1+4x}$; consistent with their derivatives; condone wrong index in $v$ used for M1 only |
| | B1 | $\frac{1}{2}(\ldots)^{-1/2}$ soi |
| | A1 | Correct expression |
| $= 2x(1+4x)^{-1/2}(x+1+4x)$ | M1 | Factorising or combining fractions; (need not factor out the $2x$); must have evidence of $x+1+4x$ oe |
| $= \dfrac{2x(5x+1)}{\sqrt{1+4x}}$ * | A1 | **NB AG**; or $2x(5x+1)(1+4x)^{-1/2}$ or $2x(5x+1)/(1+4x)^{1/2}$ |
| **[5]** | | |
7 Given that $y = x ^ { 2 } \sqrt { 1 + 4 x }$, show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x ( 5 x + 1 ) } { \sqrt { 1 + 4 x } }$.

\hfill \mbox{\textit{OCR MEI C3  Q7 [5]}}
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Q1 6 Q2 5 Q3 6 Q4 17 Q5 3 Q6 4 Q7 5