| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Equation of normal line |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question requiring basic algebraic manipulation (simplifying indices), routine differentiation using power rule, and finding a normal gradient. All techniques are standard C2 level with no problem-solving insight needed—easier than average A-level questions. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| 7(a) | ........ = \(x^2 - x^{-3}\) | M1, A1 |
| 7(b)(i) | \(f'(x) = \frac{5x^4 + 3x^{-4}}{}\) | B1 ft, B1 ft |
| 7(b)(ii) | \(f'(x) = \{5x^4 + \frac{3}{x^4}\} > 0\) ⟹ f is increasing {function} | M1, A1 |
| 7(c) | At (1,0), f'(1) = 5 + 3 = 8 ⟹ grad. of normal = \(-\frac{1}{8}\) | M1, m1, A1 ft |
7(a) | ........ = $x^2 - x^{-3}$ | M1, A1 | One power correct; Accept p = 5, q = −3 |
7(b)(i) | $f'(x) = \frac{5x^4 + 3x^{-4}}{}$ | B1 ft, B1 ft | ft on $px^{p-1}$; ft on $-qx^{q-1}$ provided q < 0 |
7(b)(ii) | $f'(x) = \{5x^4 + \frac{3}{x^4}\} > 0$ ⟹ f is increasing {function} | M1, A1 | M1 Considers sign of f'(x); a statement "f'(x) > 0" OE " with 'f increasing'; A1 needs f'(x) of the form $ax^c + \frac{b}{x^d}$, where a and b both ≥ 0 and no incorrect statements based on f'(x) at different values of x |
7(c) | At (1,0), f'(1) = 5 + 3 = 8 ⟹ grad. of normal = $-\frac{1}{8}$ | M1, m1, A1 ft | Attempts to find f'(1); Use of $m \times m' = -1$ PI; ft on wrong f'(x) |7 A curve is defined, for $x > 0$, by the equation $y = \mathrm { f } ( x )$, where
$$\mathrm { f } ( x ) = \frac { x ^ { 8 } - 1 } { x ^ { 3 } }$$
\begin{enumerate}[label=(\alph*)]
\item Express $\frac { x ^ { 8 } - 1 } { x ^ { 3 } }$ in the form $x ^ { p } - x ^ { q }$, where $p$ and $q$ are integers.
\item \begin{enumerate}[label=(\roman*)]
\item Hence differentiate $\mathrm { f } ( x )$ to find $\mathrm { f } ^ { \prime } ( x )$.
\item Hence show that f is an increasing function.
\end{enumerate}\item Find the gradient of the normal to the curve at the point $( 1,0 )$.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2005 Q7 [9]}}