CAIE P1 2018 June — Question 4 6 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2018
SessionJune
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration by Substitution
TypeFinding curve equation from derivative
DifficultyModerate -0.3 This is a straightforward integration problem requiring substitution u = 3x - 1, followed by using the given point to find the constant of integration, then evaluating at x = 0. While it requires multiple steps, the substitution is standard and the process is routine for A-level students who have learned this technique.
Spec1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits
4 A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(A ( 3,1 )\) and crosses the \(y\)-axis at \(B\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = ( 3 x - 1 ) ^ { - \frac { 1 } { 3 } }\). Find the \(y\)-coordinate of \(B\).
Question 4:
AnswerMarks Guidance
\(f(x) = \left[\frac{(3x-1)^{\frac{2}{3}}}{\frac{2}{3}}\right] \div 3 \quad (+c)\)B1B1
\(1 = \frac{8^{\frac{2}{3}}}{2} + c\)M1 Sub \(y=1, x=3\). Dep. on attempt to integrate and \(c\) present
\(c = -1 \to y = \frac{1}{2}(3x-1)^{\frac{2}{3}} - 1\) SOIA1
When \(x = 0\), \(y = \frac{1}{2}(-1)^{\frac{2}{3}} - 1 = -\frac{1}{2}\)DM1A1 Dep. on previous M1
Total: 6 
## Question 4:

| $f(x) = \left[\frac{(3x-1)^{\frac{2}{3}}}{\frac{2}{3}}\right] \div 3 \quad (+c)$ | B1B1 | |
|---|---|---|
| $1 = \frac{8^{\frac{2}{3}}}{2} + c$ | M1 | Sub $y=1, x=3$. Dep. on attempt to integrate and $c$ present |
| $c = -1 \to y = \frac{1}{2}(3x-1)^{\frac{2}{3}} - 1$ SOI | A1 | |
| When $x = 0$, $y = \frac{1}{2}(-1)^{\frac{2}{3}} - 1 = -\frac{1}{2}$ | DM1A1 | Dep. on previous M1 |
| **Total: 6** | | |

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4 A curve with equation $y = \mathrm { f } ( x )$ passes through the point $A ( 3,1 )$ and crosses the $y$-axis at $B$. It is given that $\mathrm { f } ^ { \prime } ( x ) = ( 3 x - 1 ) ^ { - \frac { 1 } { 3 } }$. Find the $y$-coordinate of $B$.\\

\hfill \mbox{\textit{CAIE P1 2018 Q4 [6]}}
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