AQA AS Paper 2 2022 June — Question 3 5 marks

Exam BoardAQA
ModuleAS Paper 2 (AS Paper 2)
Year2022
SessionJune
Marks5
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Mark schemeDownload PDF ↗
TopicChain Rule
TypeSecond derivative and nature determination
DifficultyModerate -0.8 This is a straightforward differentiation question requiring two applications of the power rule (rewriting √x as x^(1/2)) and substitution of x=4. It's simpler than average A-level questions as it involves only routine calculus with no problem-solving, though the algebraic manipulation and working with the constant k adds minor complexity beyond pure recall.
Spec1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums
3 A curve has equation \(y = k \sqrt { x }\) where \(k\) is a constant. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(( 4,2 k )\) on the curve, giving your answer as an expression in terms of \(k\).
Question 3:
AnswerMarks Guidance
AnswerMark Guidance
\(y = k\sqrt{x} = kx^{\frac{1}{2}}\)B1 (AO1.2) Uses fractional power to represent square root; PI by \(\frac{dy}{dx}\) involving \(x^{-\frac{1}{2}}\)
\(\frac{dy}{dx} = \frac{k}{2}x^{-\frac{1}{2}}\)M1 (AO1.1a) Differentiates to obtain an expression in \(x^{-\frac{1}{2}}\) or \(\frac{1}{\sqrt{x}}\)
\(\frac{d^2y}{dx^2} = -\frac{k}{4}x^{-\frac{3}{2}}\)A1 (AO1.1b) Obtains fully correct expression for \(\frac{dy}{dx}\)
At \((4, 2k)\): \(\frac{d^2y}{dx^2} = -\frac{k}{32}\)M1 (AO1.1a) Differentiates to obtain expression in \(x^{-\frac{3}{2}}\) or \(\frac{1}{x\sqrt{x}}\) OE
\(-\frac{k}{32}\)A1 (AO1.1b) Obtains \(-\frac{k}{32}\) seen anywhere following a correct \(\frac{d^2y}{dx^2}\) 
**Question 3:**

| Answer | Mark | Guidance |
|--------|------|----------|
| $y = k\sqrt{x} = kx^{\frac{1}{2}}$ | B1 (AO1.2) | Uses fractional power to represent square root; PI by $\frac{dy}{dx}$ involving $x^{-\frac{1}{2}}$ |
| $\frac{dy}{dx} = \frac{k}{2}x^{-\frac{1}{2}}$ | M1 (AO1.1a) | Differentiates to obtain an expression in $x^{-\frac{1}{2}}$ or $\frac{1}{\sqrt{x}}$ |
| $\frac{d^2y}{dx^2} = -\frac{k}{4}x^{-\frac{3}{2}}$ | A1 (AO1.1b) | Obtains fully correct expression for $\frac{dy}{dx}$ |
| At $(4, 2k)$: $\frac{d^2y}{dx^2} = -\frac{k}{32}$ | M1 (AO1.1a) | Differentiates to obtain expression in $x^{-\frac{3}{2}}$ or $\frac{1}{x\sqrt{x}}$ OE |
| $-\frac{k}{32}$ | A1 (AO1.1b) | Obtains $-\frac{k}{32}$ seen anywhere following a correct $\frac{d^2y}{dx^2}$ |
3 A curve has equation $y = k \sqrt { x }$ where $k$ is a constant.

Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at the point $( 4,2 k )$ on the curve, giving your answer as an expression in terms of $k$.\\

\hfill \mbox{\textit{AQA AS Paper 2 2022 Q3 [5]}}
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