| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Tangent meets curve/axis — further geometry |
| Difficulty | Moderate -0.3 This is a straightforward tangent question requiring finding an intersection point, computing a derivative using the chain rule, finding the tangent equation, and calculating a distance. All steps are routine A-level techniques with no novel insight required. The 5 marks for part (i) reflect multiple standard steps rather than conceptual difficulty. Slightly easier than average due to the algebraic simplicity and clear structure. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.07l Derivative of ln(x): and related functions1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{dy}{dx} = \frac{-10}{(2x+1)^2} \times 2\); At 4, \(y = 0, \to x = 2\); \(m\) at \(x = 2\) is \(-\frac{4}{5}\); Eqn of tangent is \(y = -\frac{4}{5}(x-2) \to 5y + 4x = 8\) | B1, B1, B1, M1, A1 | Without the "×2"; For x = 2; For x = 2; Must be using differential as m; co – answer given |
| (ii) \(C(0, 1.6)\); \(d = \sqrt{(1.6^2 + 2^2)} = 2.56\) | M1, A1 | Correct method – needs ✓; co |
$y = \frac{10}{2x+1} - 2$
**(i)** $\frac{dy}{dx} = \frac{-10}{(2x+1)^2} \times 2$; At 4, $y = 0, \to x = 2$; $m$ at $x = 2$ is $-\frac{4}{5}$; Eqn of tangent is $y = -\frac{4}{5}(x-2) \to 5y + 4x = 8$ | B1, B1, B1, M1, A1 | Without the "×2"; For x = 2; For x = 2; Must be using differential as m; co – answer given
**(ii)** $C(0, 1.6)$; $d = \sqrt{(1.6^2 + 2^2)} = 2.56$ | M1, A1 | Correct method – needs ✓; coThe curve $y = \frac{10}{2x + 1} - 2$ intersects the $x$-axis at $A$. The tangent to the curve at $A$ intersects the $y$-axis at $C$.
\begin{enumerate}[label=(\roman*)]
\item Show that the equation of $AC$ is $5y + 4x = 8$. [5]
\item Find the distance $AC$. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2012 Q7 [7]}}