CAIE P1 2009 November — Question 1 4 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2009
SessionNovember
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 problem requiring only basic standard integrals (x^{-1/2} and x) followed by using a boundary condition to find the constant. It's simpler than average A-level questions as it involves direct application of power rule integration with no algebraic manipulation or problem-solving insight needed.
Spec1.04a Binomial expansion: (a+b)^n for positive integer n
1 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { \sqrt { x } } - x\). Given that the curve passes through the point (4,6), find the equation of the curve.
\(\frac{dy}{dx} = \frac{3}{\sqrt{x}} - x\)
AnswerMarks Guidance
\((y) = 6\sqrt{x} - \frac{x^2}{2} (+c)\)B1, B1 B1 for each term
\((4, 6)\) fits \(6 = 12 - 8 + c \rightarrow c = 2\)M1, A1 Uses \((4, 6)\) in an integration with \(+c\) co
Total: [4]
$\frac{dy}{dx} = \frac{3}{\sqrt{x}} - x$

$(y) = 6\sqrt{x} - \frac{x^2}{2} (+c)$ | B1, B1 | B1 for each term

$(4, 6)$ fits $6 = 12 - 8 + c \rightarrow c = 2$ | M1, A1 | Uses $(4, 6)$ in an integration with $+c$ co

**Total: [4]**

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1 The equation of a curve is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { \sqrt { x } } - x$. Given that the curve passes through the point (4,6), find the equation of the curve.

\hfill \mbox{\textit{CAIE P1 2009 Q1 [4]}}
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