CAIE P1 2024 March — Question 3 5 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2024
SessionMarch
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration by Substitution
TypeFinding curve equation from derivative
DifficultyModerate -0.8 This is a straightforward integration problem requiring substitution of u=4x+5, followed by applying a boundary condition to find the constant, then evaluating at x=5. The substitution is obvious, the integration is routine (power rule), and it's a standard single-method question worth modest marks. Easier than average A-level questions.
Spec1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits
3 A curve is such that \(\frac { \mathrm { dy } } { \mathrm { dx } } = 3 ( 4 \mathrm { x } + 5 ) ^ { \frac { 1 } { 2 } }\). It is given that the points \(( 1,9 )\) and \(( 5 , a )\) lie on the curve. Find the value of \(a\).
Question 3:
AnswerMarks Guidance
AnswerMarks Guidance
Integrate to obtain form \(k(4x+5)^{\frac{3}{2}}\)\*M1
Obtain correct \(\frac{1}{2}(4x+5)^{\frac{3}{2}}\)A1 Or unsimplified equivalent. Condone missing \(+c\) so far.
Substitute \(x=1\), \(y=9\) to form an equation in \(c\)DM1
Obtain or imply \([y=]\frac{1}{2}(4x+5)^{\frac{3}{2}} - \frac{9}{2}\)A1 May be implied by \([a=]\frac{1}{2}(4(1)+5)^{\frac{3}{2}} - \frac{9}{2}\)
Substitute \(x=5\) to obtain \(a=58\)A1
Total: 5 
## Question 3:

| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrate to obtain form $k(4x+5)^{\frac{3}{2}}$ | \*M1 | |
| Obtain correct $\frac{1}{2}(4x+5)^{\frac{3}{2}}$ | A1 | Or unsimplified equivalent. Condone missing $+c$ so far. |
| Substitute $x=1$, $y=9$ to form an equation in $c$ | DM1 | |
| Obtain or imply $[y=]\frac{1}{2}(4x+5)^{\frac{3}{2}} - \frac{9}{2}$ | A1 | May be implied by $[a=]\frac{1}{2}(4(1)+5)^{\frac{3}{2}} - \frac{9}{2}$ |
| Substitute $x=5$ to obtain $a=58$ | A1 | |
| **Total: 5** | | |

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3 A curve is such that $\frac { \mathrm { dy } } { \mathrm { dx } } = 3 ( 4 \mathrm { x } + 5 ) ^ { \frac { 1 } { 2 } }$. It is given that the points $( 1,9 )$ and $( 5 , a )$ lie on the curve. Find the value of $a$.\\

\hfill \mbox{\textit{CAIE P1 2024 Q3 [5]}}
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