| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2008 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Differentiate after index conversion |
| Difficulty | Moderate -0.8 This is a straightforward C2 question testing basic index law recall (converting a surd to fractional power) and routine differentiation using the power rule. Part (a) is trivial recall, parts (b)(i) and (b)(ii) are standard textbook exercises requiring no problem-solving insight—just mechanical application of learned techniques. |
| Spec | 1.02a Indices: laws of indices for rational exponents1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\sqrt{x^3} = x^{\frac{3}{2}}\) | B1 (1 mark) | OE; accept '\(k = 1.5\)' |
| (b)(i) \(\frac{dy}{dx} = 2x - \frac{3}{2}x^{-\frac{1}{2}}\) | M1, B1, A1F (3 marks) | At least one index reduced by 1 and no term of the form \(\sqrt{ax^2}\); For \(2x\); For \(-1.5x^{0.5}\) Ft on ans (a) non-integer \(k\) |
| (b)(ii) When \(x = 4\), \(y = 8\) | B1, M1, A1F (3 marks) | Attempt to find \(\frac{dy}{dx}\) when \(x = 4\); Ft on one earlier error provided non-integer powers in (a) and (b)(i) |
| Tangent: \(y - 8 = 5(x - 4)\); \(y = 5x - 12\) | m1, A1 | \(y - y(4) = y'(4)[x - 4]\) OE; CSO; must be \(y = 5x - 12\) |
**(a)** $\sqrt{x^3} = x^{\frac{3}{2}}$ | B1 (1 mark) | OE; accept '$k = 1.5$'
**(b)(i)** $\frac{dy}{dx} = 2x - \frac{3}{2}x^{-\frac{1}{2}}$ | M1, B1, A1F (3 marks) | At least one index reduced by 1 and no term of the form $\sqrt{ax^2}$; For $2x$; For $-1.5x^{0.5}$ Ft on ans (a) non-integer $k$
**(b)(ii)** When $x = 4$, $y = 8$ | B1, M1, A1F (3 marks) | Attempt to find $\frac{dy}{dx}$ when $x = 4$; Ft on one earlier error provided non-integer powers in (a) and (b)(i)
Tangent: $y - 8 = 5(x - 4)$; $y = 5x - 12$ | m1, A1 | $y - y(4) = y'(4)[x - 4]$ OE; CSO; must be $y = 5x - 12$
**Total for Q1: 9 marks**
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\begin{enumerate}[label=(\alph*)]
\item Write $\sqrt { x ^ { 3 } }$ in the form $x ^ { k }$, where $k$ is a fraction.\\
(1 mark)
\item A curve, defined for $x \geqslant 0$, has equation
$$y = x ^ { 2 } - \sqrt { x ^ { 3 } }$$
\begin{enumerate}[label=(\roman*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
\item Find the equation of the tangent to the curve at the point where $x = 4$, giving your answer in the form $y = m x + c$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C2 2008 Q1 [9]}}