OCR C3 2011 June — Question 5 8 marks

Exam BoardOCR
ModuleC3 (Core Mathematics 3)
Year2011
SessionJune
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProduct & Quotient Rules
TypeSecond derivative calculation
DifficultyStandard +0.3 This is a straightforward two-step differentiation problem requiring the product rule twice, followed by substitution. While it involves ln differentiation and some algebraic manipulation, it's a standard C3 exercise with no conceptual challenges—slightly easier than the typical multi-part question but requires careful execution.
Spec1.07d Second derivatives: d^2y/dx^2 notation1.07q Product and quotient rules: differentiation
5 The equation of a curve is \(y = x ^ { 2 } \ln ( 4 x - 3 )\). Find the exact value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point on the curve for which \(x = 2\).
AnswerMarks Guidance
Attempt use of product rule*M1 to produce \(k_1 y \ln(4x - 3) + \frac{k_2 x^2}{4x - 3}\) form
Obtain \(2x \ln(4x - 3)\)A1
Obtain \(\cdots + \frac{4x^2}{4x - 3}\)A1 or equiv
Attempt second use of product rule*M1
Attempt use of quotient (or product) rule*M1 allow numerator the wrong way round
Obtain \(2\ln(4x - 3) + \frac{8x}{4x - 3} + \frac{8x(4x - 3) - 16x^2}{(4x - 3)^2}\)A1 or equiv
Substitute 2 into attempt at second derivM1 dep *M *M *M
Obtain \(2\ln 5 + \frac{25}{8}\)A1 8 marks: or exact equiv consisting of two terms 
Attempt use of product rule | *M1 | to produce $k_1 y \ln(4x - 3) + \frac{k_2 x^2}{4x - 3}$ form

Obtain $2x \ln(4x - 3)$ | A1 |

Obtain $\cdots + \frac{4x^2}{4x - 3}$ | A1 | or equiv

Attempt second use of product rule | *M1 |

Attempt use of quotient (or product) rule | *M1 | allow numerator the wrong way round

Obtain $2\ln(4x - 3) + \frac{8x}{4x - 3} + \frac{8x(4x - 3) - 16x^2}{(4x - 3)^2}$ | A1 | or equiv

Substitute 2 into attempt at second deriv | M1 | dep *M *M *M

Obtain $2\ln 5 + \frac{25}{8}$ | A1 | 8 marks: or exact equiv consisting of two terms

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5 The equation of a curve is $y = x ^ { 2 } \ln ( 4 x - 3 )$. Find the exact value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at the point on the curve for which $x = 2$.

\hfill \mbox{\textit{OCR C3 2011 Q5 [8]}}
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Q1 5 Q2 4 Q3 8 Q4 8 Q5 8 Q6 9 Q7 8 Q8 10 Q9 12