| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find derivative after algebraic simplification (fractional/mixed powers) |
| Difficulty | Easy -1.2 This is a straightforward C1 differentiation question requiring only direct application of basic power rule. Part (a) involves rewriting 3/x as 3x^(-1) and differentiating, while part (b) requires differentiating x^(5/2) and substituting x=4. Both are routine textbook exercises with no problem-solving or conceptual challenge beyond recalling standard differentiation rules. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07l Derivative of ln(x): and related functions |
| Answer | Marks | Guidance |
|---|---|---|
| \(f(x) = x + 3x^{-1}\) | M1 | Attempt to differentiate |
| \(f'(x) = 1 - 3x^{-2}\) | A1 | First term correct |
| A1 | \(x^{-2}\) soi www | |
| A1 (4) | Fully correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dy}{dx} = \frac{5}{2}x^{\frac{3}{2}}\) | M1 | Use of differentiation to find gradient |
| B1 | \(\frac{5}{2}x^c\) | |
| B1 | \(kx^{\frac{3}{2}}\) | |
| When \(x = 4\), \(\frac{dy}{dx} = \frac{5}{2}\sqrt{4^3}\) | M1 | \(\sqrt{4^3}\) soi |
| \(= 20\) | A1 (5) | SR If 0 scored for first 3 marks, award B1 if \(\sqrt{4^n}\) correctly evaluated |
# Question 7:
**(a)**
$f(x) = x + 3x^{-1}$ | M1 | Attempt to differentiate
$f'(x) = 1 - 3x^{-2}$ | A1 | First term correct
| A1 | $x^{-2}$ soi **www**
| A1 (4) | Fully correct answer
**(b)**
$\frac{dy}{dx} = \frac{5}{2}x^{\frac{3}{2}}$ | M1 | Use of differentiation to find gradient
| B1 | $\frac{5}{2}x^c$
| B1 | $kx^{\frac{3}{2}}$
When $x = 4$, $\frac{dy}{dx} = \frac{5}{2}\sqrt{4^3}$ | M1 | $\sqrt{4^3}$ soi
$= 20$ | A1 (5) | **SR** If 0 scored for first 3 marks, award B1 if $\sqrt{4^n}$ correctly evaluated
---7
\begin{enumerate}[label=(\alph*)]
\item Given that $f ( x ) = x + \frac { 3 } { x }$, find $f ^ { \prime } ( x )$.
\item Find the gradient of the curve $\mathrm { y } = \mathrm { x } ^ { \frac { 5 } { 2 } }$ at the point where $\mathrm { x } = 4$.
\end{enumerate}
\hfill \mbox{\textit{OCR C1 2007 Q7 [9]}}