CAIE P1 2023 November — Question 1 4 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2023
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 question requiring only basic power rule application and using a boundary condition to find the constant. The integration is routine (x^1 and x^(-1/2) are standard), and substituting a point to find C is a standard textbook exercise with no problem-solving insight required.
Spec1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits
1 A curve is such that its gradient at a point \(( x , y )\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x - 3 x ^ { - \frac { 1 } { 2 } }\). It is given that the curve passes through the point \(( 4,1 )\). Find the equation of the curve.
Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
\([y] = \left\{\dfrac{x^2}{2}\right\} \left\{\dfrac{-3x^{\frac{1}{2}}}{\frac{1}{2}}\right\} [+c]\)B1 B1 Any unsimplified correct form, ISW for extra terms, allow lists.
\(1 = 8 - 12 + c\)M1 Substitute (into an integrated expression) \(x = 4\), \(y = 1\). \(c\) must be present. Expect \(c = 5\).
\(y = \dfrac{1}{2}x^2 - 6x^{\frac{1}{2}} + 5\), allow \(f(x) =\)A1 Condone \(c = 5\) as the final line so long as '\(y =\)' present.
4 
**Question 1:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $[y] = \left\{\dfrac{x^2}{2}\right\} \left\{\dfrac{-3x^{\frac{1}{2}}}{\frac{1}{2}}\right\} [+c]$ | **B1 B1** | Any unsimplified correct form, ISW for extra terms, allow lists. |
| $1 = 8 - 12 + c$ | **M1** | Substitute (into an integrated expression) $x = 4$, $y = 1$. $c$ must be present. Expect $c = 5$. |
| $y = \dfrac{1}{2}x^2 - 6x^{\frac{1}{2}} + 5$, allow $f(x) =$ | **A1** | Condone $c = 5$ as the final line so long as '$y =$' present. |
| | **4** | |

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1 A curve is such that its gradient at a point $( x , y )$ is given by $\frac { \mathrm { d } y } { \mathrm {~d} x } = x - 3 x ^ { - \frac { 1 } { 2 } }$. It is given that the curve passes through the point $( 4,1 )$.

Find the equation of the curve.\\

\hfill \mbox{\textit{CAIE P1 2023 Q1 [4]}}
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