| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 11 |
| 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 basic derivative of x^{-1}, gradient calculation, and finding a normal line. Part (c) adds mild algebraic challenge by requiring substitution and solving a quadratic, but all techniques are routine for this level. |
| Spec | 1.07a Derivative as gradient: of tangent to curve1.07m Tangents and normals: gradient and equations |
| Answer | Marks |
|---|---|
| (a) \(\frac{dy}{dx} = 1 - 3x^{-2}\) | M1 A1 |
| \(\text{grad} = 1 - 3(1)^2 = 1 - 3 = -2\) | A1 |
| (b) \(x = 1 \therefore y = 4\) | |
| \(\text{grad} = \frac{-1}{-2} = \frac{1}{2}\) | M1 A1 |
| \(\therefore y - 4 = \frac{1}{2}(x - 1)\) | M1 |
| \(y = \frac{1}{2}x + \frac{7}{2}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x^2 - 7x + 6 = 0\), \((x-1)(x-6) = 0\) | M1 | |
| M1 | ||
| \(x = 1\) (at P), \(6\) | A1 | |
| \(\therefore (6, 6\frac{1}{2})\) | A1 | (11 marks) |
(a) $\frac{dy}{dx} = 1 - 3x^{-2}$ | M1 A1 |
$\text{grad} = 1 - 3(1)^2 = 1 - 3 = -2$ | A1 |
(b) $x = 1 \therefore y = 4$ | |
$\text{grad} = \frac{-1}{-2} = \frac{1}{2}$ | M1 A1 |
$\therefore y - 4 = \frac{1}{2}(x - 1)$ | M1 |
$y = \frac{1}{2}x + \frac{7}{2}$ | A1 |
(c) $x + \frac{3}{x} = \frac{1}{2}x + \frac{7}{2}$
$2x^2 + 6 = x^2 + 7x$
$x^2 - 7x + 6 = 0$, $(x-1)(x-6) = 0$ | M1 |
| M1 |
$x = 1$ (at P), $6$ | A1 |
$\therefore (6, 6\frac{1}{2})$ | A1 | (11 marks)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=(\alph*)]
\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 $y = mx + c$. [4]
\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{Edexcel C1 Q10 [11]}}