CAIE P1 2018 June — Question 3 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=2x+1, followed by using a boundary condition to find the constant, then solving y=0. The substitution is standard and the algebraic steps are routine, making it slightly easier than average but still requiring proper technique.
Spec1.08b Integrate x^n: where n != -1 and sums
3 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 12 } { ( 2 x + 1 ) ^ { 2 } }\). The point \(( 1,1 )\) lies on the curve. Find the coordinates of the point at which the curve intersects the \(x\)-axis.
Question 3:
AnswerMarks Guidance
\(\frac{dy}{dx} = \frac{12}{(2x+1)^2} \rightarrow y = \frac{-12}{2x+1} \div 2 \ (+c)\)B1 B1 Correct without "\(\div 2\)". For "\(\div 2\)". Ignore \(c\)
Uses \((1,1) \rightarrow c = 3 \ (\rightarrow y = \frac{-6}{2x+1} + 3)\)M1 A1 Finding \(c\) following integration. CAO
Sets \(y\) to \(0\) and attempts to solve for \(x \rightarrow x = \frac{1}{2} \rightarrow ((\frac{1}{2}, 0))\)DM1 A1 Sets \(y\) to \(0\). \(x = \frac{1}{2}\) is sufficient for A1 
## Question 3:

| $\frac{dy}{dx} = \frac{12}{(2x+1)^2} \rightarrow y = \frac{-12}{2x+1} \div 2 \ (+c)$ | B1 B1 | Correct without "$\div 2$". For "$\div 2$". Ignore $c$ |
|---|---|---|
| Uses $(1,1) \rightarrow c = 3 \ (\rightarrow y = \frac{-6}{2x+1} + 3)$ | M1 A1 | Finding $c$ following integration. CAO |
| Sets $y$ to $0$ and attempts to solve for $x \rightarrow x = \frac{1}{2} \rightarrow ((\frac{1}{2}, 0))$ | DM1 A1 | Sets $y$ to $0$. $x = \frac{1}{2}$ is sufficient for A1 |

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3 A curve is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 12 } { ( 2 x + 1 ) ^ { 2 } }$. The point $( 1,1 )$ lies on the curve. Find the coordinates of the point at which the curve intersects the $x$-axis.\\

\hfill \mbox{\textit{CAIE P1 2018 Q3 [6]}}
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