CAIE P1 2013 November — Question 2 5 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2013
SessionNovember
Marks5
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Mark schemeDownload PDF ↗
TopicStandard Integrals and Reverse Chain Rule
TypeFind curve equation from derivative (reverse chain rule / composite functions)
DifficultyModerate -0.3 This is a straightforward integration question requiring recognition of standard forms (x^n and (ax+b)^n) and application of a boundary condition. While it involves two terms and the reverse chain rule for √(x+6), these are routine P1 techniques with no conceptual challenges, making it slightly easier than average.
Spec1.08b Integrate x^n: where n != -1 and sums1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)
2 A curve has equation \(y = f ( x )\). It is given that \(f ^ { \prime } ( x ) = \frac { 1 } { \sqrt { } ( x + 6 ) } + \frac { 6 } { x ^ { 2 } }\) and that \(f ( 3 ) = 1\). Find \(f ( x )\).
AnswerMarks Guidance
Attempt integrationM1
\(f(x) = 2(x+6)\frac{1}{2} - \frac{6}{x} (+c)\)A1A1 Accept unsimplified terms
\(2(3) - \frac{6}{3} + c = 1\)M1 Sub \(x=3, y=1\). \(c\) must be present
\(c = -3\)A1 [5] 
Attempt integration | M1 |
$f(x) = 2(x+6)\frac{1}{2} - \frac{6}{x} (+c)$ | A1A1 | Accept unsimplified terms
$2(3) - \frac{6}{3} + c = 1$ | M1 | Sub $x=3, y=1$. $c$ must be present
$c = -3$ | A1 | [5]
2 A curve has equation $y = f ( x )$. It is given that $f ^ { \prime } ( x ) = \frac { 1 } { \sqrt { } ( x + 6 ) } + \frac { 6 } { x ^ { 2 } }$ and that $f ( 3 ) = 1$. Find $f ( x )$.

\hfill \mbox{\textit{CAIE P1 2013 Q2 [5]}}
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