| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find derivative of simple polynomial (integer powers) |
| Difficulty | Easy -1.3 This is a straightforward differentiation exercise testing basic rules (simplification, power rule, negative/fractional powers). All three parts are routine C1 content requiring only direct application of standard techniques with no problem-solving or conceptual insight needed. Easier than average A-level questions. |
| Spec | 1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| M1 | Obtain \(kx^5\) | If individual terms are differentiated then M0A0B0 |
| A1 | Correct expression for \(y\) \((9x^5)\) | |
| B1 ft | Follow through from their single \(kx^5\), \(n \neq 0\). Must be simplified. | \(3x^2 + x^4\) is not a misread M0A0B0 |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| B1 | \(\sqrt[3]{x} = x^{\frac{1}{3}}\) | |
| B1 | \(kx^{-\frac{2}{3}}\) | SC \(\sqrt[3]{x} = x^{-\frac{1}{3}}\) differentiated to \(-\frac{1}{3}x^{-\frac{4}{3}}\) B1 |
| B1 | \(\frac{1}{3}x^{-\frac{2}{3}}\). Allow 0.3 (not finite) | |
| [3] |
| Answer | Marks |
|---|---|
| M1 | \(kx^{-4}\) seen |
| A1 | |
| [2] |
### (i)
$y = 9x^5$, $\frac{dy}{dx} = 45x^4$
| M1 | Obtain $kx^5$ | If individual terms are differentiated then M0A0B0 |
| A1 | Correct expression for $y$ $(9x^5)$ | |
| B1 ft | Follow through from their single $kx^5$, $n \neq 0$. Must be simplified. | $3x^2 + x^4$ is not a misread M0A0B0 |
| [3] | | |
### (ii)
$y = x^{\frac{1}{3}}$, $\frac{dy}{dx} = \frac{1}{3}x^{-\frac{2}{3}}$
| B1 | $\sqrt[3]{x} = x^{\frac{1}{3}}$ | |
| B1 | $kx^{-\frac{2}{3}}$ | SC $\sqrt[3]{x} = x^{-\frac{1}{3}}$ differentiated to $-\frac{1}{3}x^{-\frac{4}{3}}$ B1 |
| B1 | $\frac{1}{3}x^{-\frac{2}{3}}$. Allow 0.3 (not finite) | |
| [3] | | |
### (iii)
$y = \frac{1}{2}x^{-3}$, $\frac{dy}{dx} = -\frac{3}{2}x^{-4}$
| M1 | $kx^{-4}$ seen | |
| A1 | | |
| [2] | | |Find $\frac{dy}{dx}$ in each of the following cases:
\begin{enumerate}[label=(\roman*)]
\item $y = \frac{(3x)^2 \times x^4}{x}$, [3]
\item $y = ^3\sqrt{x}$, [3]
\item $y = \frac{1}{2x^3}$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR C1 2013 Q7 [8]}}