| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Tangent meets curve/axis — further geometry |
| Difficulty | Moderate -0.8 This is a straightforward C1 differentiation question requiring routine application of basic rules (quotient rule or simplification first), then using point-gradient form for the tangent. All steps are standard textbook procedures with no problem-solving insight needed, making it easier than average but not trivial due to the algebraic manipulation required. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\frac{5-x}{x} = \frac{5}{x} - 1\) (= \(5x^{-1} - 1\)) | M1 | |
| \(\frac{dy}{dx} = 8x - 5x^{-2}\) | M1 A1 A1 | |
| When \(x = 1\): \(\frac{dy}{dx} = 3\) | A1 | (5 marks) |
| (b) At \(P\), \(y = 8\) | B1 | |
| Equation of tangent: \(y - 8 = 3(x - 1)\) (\(y = 3x + 5\)) (or equiv.) | M1 A1ft | (3 marks) |
| (c) Where \(y = 0\), \(x = -\frac{5}{3}\) (= \(k\)) (or exact equiv.) | M1 A1 | (2 marks) |
**(a)** $\frac{5-x}{x} = \frac{5}{x} - 1$ (= $5x^{-1} - 1$) | M1 |
$\frac{dy}{dx} = 8x - 5x^{-2}$ | M1 A1 A1 |
When $x = 1$: $\frac{dy}{dx} = 3$ | A1 | (5 marks)
**(b)** At $P$, $y = 8$ | B1 |
Equation of tangent: $y - 8 = 3(x - 1)$ ($y = 3x + 5$) (or equiv.) | M1 A1ft | (3 marks)
**(c)** Where $y = 0$, $x = -\frac{5}{3}$ (= $k$) (or exact equiv.) | M1 A1 | (2 marks)
**Total: 10 marks**
**Notes:**
(a) First M1 can also be scored by an attempt to use the quotient or product rule to differentiate $\frac{5-x}{x}$.
(b) The B mark may be earned in part (a).
---7. The curve $C$ has equation $y = 4 x ^ { 2 } + \frac { 5 - x } { x } , x \neq 0$. The point $P$ on $C$ has $x$-coordinate 1 .
\begin{enumerate}[label=(\alph*)]
\item Show that the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at $P$ is 3 .
\item Find an equation of the tangent to $C$ at $P$.
This tangent meets the $x$-axis at the point $( k , 0 )$.
\item Find the value of $k$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2005 Q7 [10]}}