CAIE P2 2021 November — Question 1 4 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2021
SessionNovember
Marks4
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Mark schemeDownload PDF ↗
TopicStandard Integrals and Reverse Chain Rule
TypeDefinite integral with exponentials
DifficultyModerate -0.8 This is a straightforward application of standard exponential integration rules with simple coefficients. Students need only recall that ∫e^(ax)dx = (1/a)e^(ax) and substitute limits—no problem-solving or insight required, making it easier than average but not trivial since it involves negative exponents and careful arithmetic with exact values.
Spec1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits
1 Find the exact value of \(\int _ { - 1 } ^ { 2 } \left( 4 \mathrm { e } ^ { 2 x } - 2 \mathrm { e } ^ { - x } \right) \mathrm { d } x\).
Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
Integrate to obtain \(2e^{2x}\)B1
Integrate to obtain \(2e^{-x}\)B1
Apply limits correctly to integral of the form \(k_1e^{2x} + k_2e^{-x}\)M1 \(k_1 \neq 4\). Condone one error.
Obtain \(2e^4 - 2e\)A1 or exact equivalent. 
## Question 1:

| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrate to obtain $2e^{2x}$ | **B1** | |
| Integrate to obtain $2e^{-x}$ | **B1** | |
| Apply limits correctly to integral of the form $k_1e^{2x} + k_2e^{-x}$ | **M1** | $k_1 \neq 4$. Condone one error. |
| Obtain $2e^4 - 2e$ | **A1** | or exact equivalent. |

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1 Find the exact value of $\int _ { - 1 } ^ { 2 } \left( 4 \mathrm { e } ^ { 2 x } - 2 \mathrm { e } ^ { - x } \right) \mathrm { d } x$.\\

\hfill \mbox{\textit{CAIE P2 2021 Q1 [4]}}
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Q1 4 Q2 6 Q3 5 Q4 8 Q5 8 Q6 10 Q7 9