CAIE P1 2020 Specimen — Question 4 5 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2020
SessionSpecimen
Marks5
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Mark schemeDownload PDF ↗
TopicIntegration by Substitution
TypeFinding curve equation from derivative
DifficultyModerate -0.3 This is a straightforward integration question requiring standard techniques: reverse chain rule for the first term (or substitution u = x+6) and power rule for the second term, followed by using the boundary condition to find the constant. While it requires careful execution, it's a routine application of basic integration methods with no novel problem-solving required, making it slightly easier than average.
Spec1.07l Derivative of ln(x): and related functions1.08a Fundamental theorem of calculus: integration as reverse of differentiation
4 A curve has equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 1 } { \sqrt { x + 6 } } + \frac { 6 } { x ^ { 2 } }\) and that \(\mathrm { f } ( 3 ) = 1\). Find \(\mathrm { f } ( x )\).
Question 4:
AnswerMarks Guidance
AnswerMarks Guidance
Attempt integration1 M1
\(f(x) = 2(x+6)^{\frac{1}{2}} - \frac{6}{x}(+c)\)2 A1A1 Accept unsimplified terms, A1 for each term
\(2(3) - \frac{6}{3} + c = 1\)1 M1 Substitute \(x=3\), \(y=1\), \(c\) must be present
\([c = -3]\)
\(f(x) = 2(x+6)^{\frac{1}{2}} - \frac{6}{x} - 3\)1 A1
Total5 
## Question 4:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt integration | 1 M1 | |
| $f(x) = 2(x+6)^{\frac{1}{2}} - \frac{6}{x}(+c)$ | 2 A1A1 | Accept unsimplified terms, A1 for each term |
| $2(3) - \frac{6}{3} + c = 1$ | 1 M1 | Substitute $x=3$, $y=1$, $c$ must be present |
| $[c = -3]$ | | |
| $f(x) = 2(x+6)^{\frac{1}{2}} - \frac{6}{x} - 3$ | 1 A1 | |
| **Total** | **5** | |
4 A curve has equation $y = \mathrm { f } ( x )$. It is given that $\mathrm { f } ^ { \prime } ( x ) = \frac { 1 } { \sqrt { x + 6 } } + \frac { 6 } { x ^ { 2 } }$ and that $\mathrm { f } ( 3 ) = 1$. Find $\mathrm { f } ( x )$.\\

\hfill \mbox{\textit{CAIE P1 2020 Q4 [5]}}
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