| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Normal meets curve/axis — further geometry |
| Difficulty | Standard +0.3 This is a standard C3 calculus question requiring differentiation of ln x and x^(-1), finding stationary points, and working with normals. Part (a) involves routine differentiation and solving f'(x)=0. Parts (b)-(c) are straightforward substitution and algebraic manipulation. Part (d) requires setting up an equation for intersection and using sign-change method for bounds—all standard techniques with no novel insight required. Slightly easier than average due to clear structure and guided steps. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.07l Derivative of ln(x): and related functions1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives1.09b Sign change methods: understand failure cases |
The curve $C$ has equation $y = f(x)$, where
$$f(x) = 3 \ln x + \frac{1}{x}, \quad x > 0.$$
The point $P$ is a stationary point on $C$.
\begin{enumerate}[label=(\alph*)]
\item Calculate the $x$-coordinate of $P$. [4]
\item Show that the $y$-coordinate of $P$ may be expressed in the form $k - k \ln k$, where $k$ is a constant to be found. [2]
\end{enumerate}
The point $Q$ on $C$ has $x$-coordinate $1$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find an equation for the normal to $C$ at $Q$. [4]
\end{enumerate}
The normal to $C$ at $Q$ meets $C$ again at the point $R$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Show that the $x$-coordinate of $R$
\begin{enumerate}[label=(\roman*)]
\item satisfies the equation $6 \ln x + x + \frac{2}{x} - 3 = 0$,
\item lies between $0.13$ and $0.14$. [4]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q32 [14]}}