CAIE P1 2021 June — Question 1 4 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2021
SessionJune
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 basic power rule application and finding a constant using given coordinates. It involves routine manipulation of negative and positive powers with no conceptual challenges or problem-solving insight needed—simpler than the average A-level question.
Spec1.07i Differentiate x^n: for rational n and sums1.08a Fundamental theorem of calculus: integration as reverse of differentiation1.08b Integrate x^n: where n != -1 and sums
1 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { x ^ { 4 } } + 32 x ^ { 3 }\). It is given that the curve passes through the point \(\left( \frac { 1 } { 2 } , 4 \right)\). Find the equation of the curve.
Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
\([y =] -\frac{1}{x^3} + 8x^4 \ [+c]\)B1 B1 OE. Accept unsimplified.
\(4 = -8 + \frac{1}{2} + c\)M1 Substituting \(\left(\frac{1}{2}, 4\right)\) into an integrated expression
\(y = -\frac{1}{x^3} + 8x^4 + \frac{23}{2}\)A1 OE. Accept \(-x^{-3}\); must be 8; \(y =\) must be seen in working.
4 
**Question 1:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $[y =] -\frac{1}{x^3} + 8x^4 \ [+c]$ | B1 B1 | OE. Accept unsimplified. |
| $4 = -8 + \frac{1}{2} + c$ | M1 | Substituting $\left(\frac{1}{2}, 4\right)$ into an integrated expression |
| $y = -\frac{1}{x^3} + 8x^4 + \frac{23}{2}$ | A1 | OE. Accept $-x^{-3}$; must be 8; $y =$ must be seen in working. |
| | **4** | |

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1 The equation of a curve is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { x ^ { 4 } } + 32 x ^ { 3 }$. It is given that the curve passes through the point $\left( \frac { 1 } { 2 } , 4 \right)$.

Find the equation of the curve.\\

\hfill \mbox{\textit{CAIE P1 2021 Q1 [4]}}
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