CAIE P1 2022 November — Question 3 5 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2022
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicChain Rule
TypeFind constant using stationary point
DifficultyModerate -0.8 This is a straightforward application of differentiation to find a stationary point. Students differentiate using power rule, set derivative to zero, substitute x=9 to find a, then calculate y. All steps are routine with no conceptual challenges beyond basic calculus mechanics.
Spec1.02a Indices: laws of indices for rational exponents1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives
3 A curve has equation \(y = a x ^ { \frac { 1 } { 2 } } - 2 x\), where \(x > 0\) and \(a\) is a constant. The curve has a stationary point at the point \(P\), which has \(x\)-coordinate 9 . Find the \(y\)-coordinate of \(P\).
Question 3:
AnswerMarks Guidance
AnswerMarks Guidance
\(\frac{dy}{dx} = \frac{1}{2}ax^{-\frac{1}{2}}-2\)B2, 1, 0
\(0 = \frac{1}{2}a(9)^{-\frac{1}{2}}-2 \Rightarrow \frac{a}{6}-2=0 \Rightarrow a=[12]\)M1 Substitute \(x=9\) and \(\frac{dy}{dx}=0\) into *their* derivative and solve a linear equation for \(a\).
\([a=]12\)A1
\([y = \text{their } a \times (9)^{\frac{1}{2}}-18=]18\)A1 FT FT on *their* \(a\).
5 
## Question 3:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = \frac{1}{2}ax^{-\frac{1}{2}}-2$ | **B2, 1, 0** | |
| $0 = \frac{1}{2}a(9)^{-\frac{1}{2}}-2 \Rightarrow \frac{a}{6}-2=0 \Rightarrow a=[12]$ | **M1** | Substitute $x=9$ and $\frac{dy}{dx}=0$ into *their* derivative and solve a linear equation for $a$. |
| $[a=]12$ | **A1** | |
| $[y = \text{their } a \times (9)^{\frac{1}{2}}-18=]18$ | **A1 FT** | FT on *their* $a$. |
| | **5** | |

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3 A curve has equation $y = a x ^ { \frac { 1 } { 2 } } - 2 x$, where $x > 0$ and $a$ is a constant. The curve has a stationary point at the point $P$, which has $x$-coordinate 9 .

Find the $y$-coordinate of $P$.\\

\hfill \mbox{\textit{CAIE P1 2022 Q3 [5]}}
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