| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Normal meets curve/axis — further geometry |
| Difficulty | Standard +0.3 This is a standard C1 differentiation question requiring the quotient/power rule, finding a normal equation, and solving a simultaneous equation. While part (iii) involves solving a cubic, it factors easily. The techniques are routine and well-practiced, making it slightly easier than average for a multi-part calculus question. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
| Answer | Marks |
|---|---|
| \(\frac{dy}{dx} = 1 - 3x^{-2}\) | M1 A1 |
| grad \(= 1 - 3(1)^{-2} = 1 - 3 = -2\) | A1 |
| Answer | Marks |
|---|---|
| \(x = 1\) ∴ \(y = 4\) | M1 |
| grad \(= \frac{-1}{-2} = \frac{1}{2}\) | M1 |
| ∴ \(y - 4 = \frac{1}{2}(x - 1)\) | M1 |
| \(y = \frac{1}{2}x + \frac{7}{2}\) | A1 |
| Answer | Marks |
|---|---|
| \(x + 3 = \frac{1}{2}x + \frac{7}{2}\) | M1 |
| \(2x^2 + 6 = x^2 + 7x\) | M1 |
| \(x^2 - 7x + 6 = 0\) | M1 |
| \((x - 1)(x - 6) = 0\) | M1 |
| \(x = 1\) (at \(P\)), \(6\) | A1 |
| ∴ \((6, 6\frac{1}{2})\) | A1 |
| (10) |
## (i)
$\frac{dy}{dx} = 1 - 3x^{-2}$ | M1 A1
grad $= 1 - 3(1)^{-2} = 1 - 3 = -2$ | A1
## (ii)
$x = 1$ ∴ $y = 4$ | M1
grad $= \frac{-1}{-2} = \frac{1}{2}$ | M1
∴ $y - 4 = \frac{1}{2}(x - 1)$ | M1
$y = \frac{1}{2}x + \frac{7}{2}$ | A1
## (iii)
$x + 3 = \frac{1}{2}x + \frac{7}{2}$ | M1
$2x^2 + 6 = x^2 + 7x$ | M1
$x^2 - 7x + 6 = 0$ | M1
$(x - 1)(x - 6) = 0$ | M1
$x = 1$ (at $P$), $6$ | A1
∴ $(6, 6\frac{1}{2})$ | A1
| (10)
---A curve has the equation $y = x + \frac{3}{x}$, $x \neq 0$.
The point $P$ on the curve has $x$-coordinate $1$.
\begin{enumerate}[label=(\roman*)]
\item Show that the gradient of the curve at $P$ is $-2$. [3]
\item Find an equation for the normal to the curve at $P$, giving your answer in the form $y = mx + c$. [3]
\item Find the coordinates of the point where the normal to the curve at $P$ intersects the curve again. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR C1 Q9 [10]}}