CAIE P1 2022 March — Question 1 4 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2022
SessionMarch
Marks4
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Mark schemeDownload PDF ↗
TopicStandard Integrals and Reverse Chain Rule
TypeFind curve equation from derivative (straightforward integration + point)
DifficultyModerate -0.8 This is a straightforward integration question requiring only the power rule for two terms, followed by substituting a point to find the constant. The powers (−1/3 and 1/3) are simple fractional indices with no algebraic manipulation needed before integrating. This is easier than average A-level content—a routine P1/C1 exercise testing basic integration mechanics.
Spec1.08a Fundamental theorem of calculus: integration as reverse of differentiation1.08b Integrate x^n: where n != -1 and sums
1 A curve with equation \(y = \mathrm { f } ( x )\) is such that \(\mathrm { f } ^ { \prime } ( x ) = 2 x ^ { - \frac { 1 } { 3 } } - x ^ { \frac { 1 } { 3 } }\). It is given that \(\mathrm { f } ( 8 ) = 5\).
Find \(\mathrm { f } ( x )\).
Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
\(\left[f(x)=\right] \frac{2x^{\frac{2}{3}}}{\frac{2}{3}} - \frac{x^{\frac{4}{3}}}{\frac{4}{3}}\) \([+c]\)B1 B1 \(\frac{2}{3}\) and \(\frac{4}{3}\) may be seen as sums of 1 and a fraction.
\(5 = 12 - 12 + c\)M1 Substituting \((8,5)\) into an integral.
\(\left[f(x)=\right] 3x^{\frac{2}{3}} - \frac{3}{4}x^{\frac{4}{3}} + 5\)A1 Fractions in the denominators scores A0.
4 
**Question 1:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left[f(x)=\right] \frac{2x^{\frac{2}{3}}}{\frac{2}{3}} - \frac{x^{\frac{4}{3}}}{\frac{4}{3}}$ $[+c]$ | B1 B1 | $\frac{2}{3}$ and $\frac{4}{3}$ may be seen as sums of 1 and a fraction. |
| $5 = 12 - 12 + c$ | M1 | Substituting $(8,5)$ into an integral. |
| $\left[f(x)=\right] 3x^{\frac{2}{3}} - \frac{3}{4}x^{\frac{4}{3}} + 5$ | A1 | Fractions in the denominators scores A0. |
| | **4** | |

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1 A curve with equation $y = \mathrm { f } ( x )$ is such that $\mathrm { f } ^ { \prime } ( x ) = 2 x ^ { - \frac { 1 } { 3 } } - x ^ { \frac { 1 } { 3 } }$. It is given that $\mathrm { f } ( 8 ) = 5$.\\
Find $\mathrm { f } ( x )$.\\

\hfill \mbox{\textit{CAIE P1 2022 Q1 [4]}}
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