| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find tangent at given point (polynomial/algebraic) |
| Difficulty | Moderate -0.3 This is a straightforward C1 differentiation question requiring standard techniques: differentiating powers of x, finding stationary points by setting dy/dx = 0, and finding a tangent equation. All parts are routine textbook exercises with no problem-solving insight required, though the fractional power and 12 total marks make it slightly more substantial than the most basic questions. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks |
|---|---|
| \(\frac{dy}{dx} = 3x^{\frac{1}{2}} - 4x^{-\frac{1}{2}}\) | M1 A2 |
| Answer | Marks |
|---|---|
| \(3x^{\frac{1}{2}} - 4x^{-\frac{1}{2}} = 0\) | M1 |
| \(x^{-\frac{1}{2}}(3x - 4) = 0\) | M1 |
| \(x = \frac{4}{3}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = 2 \therefore y = 2(2\sqrt{2}) - 8(\sqrt{2}) = -4\sqrt{2}\) | M1 A1 | |
| \(\text{grad} = 3\sqrt{2} - \frac{4}{\sqrt{2}} = 3\sqrt{2} - 2\sqrt{2} = \sqrt{2}\) | M1 A1 | |
| \(\therefore y + 4\sqrt{2} = \sqrt{2}(x-2)\) | M1 | |
| \(y = \sqrt{2}x - 6\sqrt{2}\) | A1 | (12) |
| Answer | Marks |
|---|---|
| Total | (72) |
## (i)
$\frac{dy}{dx} = 3x^{\frac{1}{2}} - 4x^{-\frac{1}{2}}$ | M1 A2 |
## (ii)
$3x^{\frac{1}{2}} - 4x^{-\frac{1}{2}} = 0$ | M1 |
$x^{-\frac{1}{2}}(3x - 4) = 0$ | M1 |
$x = \frac{4}{3}$ | A1 |
## (iii)
$x = 2 \therefore y = 2(2\sqrt{2}) - 8(\sqrt{2}) = -4\sqrt{2}$ | M1 A1 |
$\text{grad} = 3\sqrt{2} - \frac{4}{\sqrt{2}} = 3\sqrt{2} - 2\sqrt{2} = \sqrt{2}$ | M1 A1 |
$\therefore y + 4\sqrt{2} = \sqrt{2}(x-2)$ | M1 |
$y = \sqrt{2}x - 6\sqrt{2}$ | A1 | (12) |
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**Total** | (72) |The curve with equation $y = 2x^3 - 8x^{\frac{1}{3}}$ has a minimum at the point $A$.
\begin{enumerate}[label=(\roman*)]
\item Find $\frac{dy}{dx}$. [3]
\item Find the $x$-coordinate of $A$. [3]
\end{enumerate}
The point $B$ on the curve has $x$-coordinate 2.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Find an equation for the tangent to the curve at $B$ in the form $y = mx + c$. [6]
\end{enumerate}
\hfill \mbox{\textit{OCR C1 Q9 [12]}}