| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find gradient at a point - direct evaluation |
| Difficulty | Moderate -0.3 Part (a) requires chain rule application to differentiate tan²(x/2), then direct substitution - a standard A-level technique. Part (b) involves integrating tan²(x) using the identity tan²(x) = sec²(x) - 1, plus a routine sin(x) integral. Both parts are textbook exercises requiring familiar techniques without novel problem-solving, making this slightly easier than average. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Differentiate to obtain form \(k\tan\frac{1}{2}x\sec^2\frac{1}{2}x\) | M1 | OE. May use identities before differentiation |
| Obtain correct \(\tan\frac{1}{2}x\sec^2\frac{1}{2}x\) | A1 | OE. Allow unsimplified |
| Substitute \(\frac{2}{3}\pi\) to obtain \(4\sqrt{3}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Express integrand as \(\sec^2\frac{1}{2}x - 1 + \sin x\) | B1 | |
| Integrate to obtain \(k_1\tan\frac{1}{2}x - x + k_2\cos x\) | M1 | Where \(k_1k_2 \neq 0\) |
| Obtain correct \(2\tan\frac{1}{2}x - x - \cos x\) | A1 | |
| Apply limits correctly to obtain \(3 - \frac{1}{2}\pi\) or exact equivalent | A1 |
## Question 3(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Differentiate to obtain form $k\tan\frac{1}{2}x\sec^2\frac{1}{2}x$ | M1 | OE. May use identities before differentiation |
| Obtain correct $\tan\frac{1}{2}x\sec^2\frac{1}{2}x$ | A1 | OE. Allow unsimplified |
| Substitute $\frac{2}{3}\pi$ to obtain $4\sqrt{3}$ | A1 | |
---
## Question 3(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Express integrand as $\sec^2\frac{1}{2}x - 1 + \sin x$ | B1 | |
| Integrate to obtain $k_1\tan\frac{1}{2}x - x + k_2\cos x$ | M1 | Where $k_1k_2 \neq 0$ |
| Obtain correct $2\tan\frac{1}{2}x - x - \cos x$ | A1 | |
| Apply limits correctly to obtain $3 - \frac{1}{2}\pi$ or exact equivalent | A1 | |
---3 The function f is defined by $\mathrm { f } ( x ) = \tan ^ { 2 } \left( \frac { 1 } { 2 } x \right)$ for $0 \leqslant x < \pi$.
\begin{enumerate}[label=(\alph*)]
\item Find the exact value of $\mathrm { f } ^ { \prime } \left( \frac { 2 } { 3 } \pi \right)$.\\
\includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-05_2726_33_97_22}
\item Find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } ( \mathrm { f } ( x ) + \sin x ) \mathrm { d } x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q3 [7]}}