| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Tangent parallel to given line |
| Difficulty | Moderate -0.8 This is a straightforward C1 differentiation question requiring standard techniques: differentiate a simple rational function, find gradient at a point, verify tangent equation, and find another point with the same gradient. All steps are routine with no problem-solving insight needed, making it easier than average but not trivial due to the multi-part structure and 10 total marks. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\frac{dy}{dx} = \frac{1}{2} + x^{-2}\) | M1 A1 | |
| \(\text{grad} = \frac{1}{2} + 2^{-3} = \frac{3}{4}\) | M1 A1 | |
| (b) \(x = 2 \therefore y = \frac{7}{2}\) | B1 | |
| \(y - \frac{7}{2} = \frac{3}{4}(x - 2)\) | M1 | |
| \(4y - 14 = 3x - 6\) | ||
| \(3x - 4y + 8 = 0\) | A1 | |
| (c) at \(B\), grad \(= \frac{3}{4}\) | ||
| \(\therefore \frac{1}{2} + x^{-2} = \frac{3}{4}\) | M1 | |
| \(x^2 = 4, \quad x = 2\) (at A), \(-2\) | A1 | |
| \(\therefore B(-2, \frac{2}{3})\) | A1 | (10) |
**(a)** $\frac{dy}{dx} = \frac{1}{2} + x^{-2}$ | M1 A1 |
$\text{grad} = \frac{1}{2} + 2^{-3} = \frac{3}{4}$ | M1 A1 |
**(b)** $x = 2 \therefore y = \frac{7}{2}$ | B1 |
$y - \frac{7}{2} = \frac{3}{4}(x - 2)$ | M1 |
$4y - 14 = 3x - 6$ |
$3x - 4y + 8 = 0$ | A1 |
**(c)** at $B$, grad $= \frac{3}{4}$ |
$\therefore \frac{1}{2} + x^{-2} = \frac{3}{4}$ | M1 |
$x^2 = 4, \quad x = 2$ (at A), $-2$ | A1 |
$\therefore B(-2, \frac{2}{3})$ | A1 | (10)A curve has the equation $y = \frac{x}{2} + 3 - \frac{1}{x}$, $x \neq 0$.
The point $A$ on the curve has $x$-coordinate 2.
\begin{enumerate}[label=(\alph*)]
\item Find the gradient of the curve at $A$. [4]
\item Show that the tangent to the curve at $A$ has equation
$$3x - 4y + 8 = 0.$$ [3]
\end{enumerate}
The tangent to the curve at the point $B$ is parallel to the tangent at $A$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the coordinates of $B$. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q7 [10]}}