| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find normal line equation at given point |
| Difficulty | Standard +0.3 This is a standard C1 differentiation question requiring finding a tangent equation, then using the parallel condition to find where a normal has the same gradient. All steps are routine: differentiate, find gradient at a point, use y=mx+c, and solve a linear equation. Slightly above average difficulty due to the two-part structure and the conceptual link between parallel lines, but still a textbook exercise with no novel insight required. |
| Spec | 1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(A(0, 2)\) | B1 | |
| \(\frac{dy}{dx} = 3 - 2x\) | M1 A1 | |
| grad \(= 3\) | M1 | |
| \(\therefore y = 3x + 2\) | A1 | |
| (b) grad of \(m = 3\) | ||
| grad of curve at \(B = \frac{-1}{3} = -\frac{1}{3}\) | M1 A1 | |
| at \(B\): \(3 - 2x = -\frac{1}{3}\) | ||
| \(x = \frac{5}{3}\) | M1 A1 | |
| \(y = 2 + 3(\frac{5}{3}) - (\frac{5}{3})^2 = 4\frac{2}{9}\) \(\therefore B(1\frac{2}{3}, 4\frac{2}{9})\) | M1 A1 | (11) |
**(a)** $A(0, 2)$ | B1 |
$\frac{dy}{dx} = 3 - 2x$ | M1 A1 |
grad $= 3$ | M1 |
$\therefore y = 3x + 2$ | A1 |
**(b)** grad of $m = 3$ | |
grad of curve at $B = \frac{-1}{3} = -\frac{1}{3}$ | M1 A1 |
at $B$: $3 - 2x = -\frac{1}{3}$ | |
$x = \frac{5}{3}$ | M1 A1 |
$y = 2 + 3(\frac{5}{3}) - (\frac{5}{3})^2 = 4\frac{2}{9}$ $\therefore B(1\frac{2}{3}, 4\frac{2}{9})$ | M1 A1 | (11)
**Total: (75)**\includegraphics{figure_1}
Figure 1 shows the curve with equation $y = 2 + 3x - x^2$ and the straight lines $l$ and $m$.
The line $l$ is the tangent to the curve at the point $A$ where the curve crosses the $y$-axis.
\begin{enumerate}[label=(\alph*)]
\item Find an equation for $l$. [5]
\end{enumerate}
The line $m$ is the normal to the curve at the point $B$.
Given that $l$ and $m$ are parallel,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the coordinates of $B$. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q10 [11]}}