| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Separable variables |
| Difficulty | Standard +0.3 This is a straightforward separable differential equation with a helpful hint provided in part (a). Students must recognize the derivative result to simplify the right-hand side, separate variables, and integrate using standard techniques. The integration is routine (involving tan²θ and the given derivative), and applying the initial condition is mechanical. Slightly above average difficulty due to the trigonometric manipulation required, but the structure is standard for P3 level. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show sufficient working to justify the given statement | B1 | e.g. see \(2\cot\theta \times -\csc^2\theta\) in the working or express in terms of \(\sin\theta\) and \(\cos\theta\) and use quotient rule to obtain the given result. Solution must have \(\theta\) present throughout and must reach the given answer. |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Separate variables correctly and check for relevant working in (a) | B1 | \(\int x\,dx = \int\dfrac{\tan^2\theta}{\sin^2\theta} - \dfrac{2\cot\theta}{\sin^2\theta}\,d\theta\). Condone incorrect notation e.g. missing \(dx\). Need either the integral sign or the \(dx\), \(d\theta\). |
| Obtain term \(\dfrac{1}{2}x^2\) | B1 | |
| Obtain terms \(\tan\theta + \cot^2\theta\) | B1+B1 | Alternative: \(\int\dfrac{2\cot\theta}{\sin^2\theta}\,d\theta = \int\dfrac{2\cos\theta}{\sin^3\theta}\,d\theta = -\dfrac{1}{\sin^2\theta}(+C)\) |
| Form an equation for the constant of integration, or use limits \(x=2\), \(\theta=\frac{1}{4}\pi\), in a solution with at least two correctly obtained terms of the form \(ax^2\), \(b\tan\theta\) and \(c\cot^2\theta\), where \(abc\neq 0\) | M1 | Need to have 3 terms. Constant of correct form. |
| State correct solution in any form, e.g. \(\dfrac{1}{2}x^2 = \tan\theta + \cot^2\theta\) | A1 | or \(\frac{1}{2}x^2 = \tan\theta + \csc^2\theta - 1\). If everything else is correct, allow a correct final answer to imply this A1. |
| Substitute \(\theta = \frac{1}{6}\pi\) and obtain answer \(x = 2.67\) | A1 | \(2.6748\ldots\) \(\sqrt{\dfrac{18+2\sqrt{3}}{3}}\). If see a correctly rounded value ISW. |
| 7 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show sufficient working to justify the given statement | B1 | e.g. see $2\cot\theta \times -\csc^2\theta$ in the working or express in terms of $\sin\theta$ and $\cos\theta$ and use quotient rule to obtain the given result. Solution must have $\theta$ present throughout and must reach the given answer. |
| | **1** | |
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Separate variables correctly and check for relevant working in (a) | B1 | $\int x\,dx = \int\dfrac{\tan^2\theta}{\sin^2\theta} - \dfrac{2\cot\theta}{\sin^2\theta}\,d\theta$. Condone incorrect notation e.g. missing $dx$. Need either the integral sign or the $dx$, $d\theta$. |
| Obtain term $\dfrac{1}{2}x^2$ | B1 | |
| Obtain terms $\tan\theta + \cot^2\theta$ | B1+B1 | Alternative: $\int\dfrac{2\cot\theta}{\sin^2\theta}\,d\theta = \int\dfrac{2\cos\theta}{\sin^3\theta}\,d\theta = -\dfrac{1}{\sin^2\theta}(+C)$ |
| Form an equation for the constant of integration, or use limits $x=2$, $\theta=\frac{1}{4}\pi$, in a solution with at least two correctly obtained terms of the form $ax^2$, $b\tan\theta$ and $c\cot^2\theta$, where $abc\neq 0$ | M1 | Need to have 3 terms. Constant of correct form. |
| State correct solution in any form, e.g. $\dfrac{1}{2}x^2 = \tan\theta + \cot^2\theta$ | A1 | or $\frac{1}{2}x^2 = \tan\theta + \csc^2\theta - 1$. If everything else is correct, allow a correct final answer to imply this A1. |
| Substitute $\theta = \frac{1}{6}\pi$ and obtain answer $x = 2.67$ | A1 | $2.6748\ldots$ $\sqrt{\dfrac{18+2\sqrt{3}}{3}}$. If see a correctly rounded value ISW. |
| | **7** | |7 The variables $x$ and $\theta$ satisfy the differential equation
$$x \sin ^ { 2 } \theta \frac { \mathrm {~d} x } { \mathrm {~d} \theta } = \tan ^ { 2 } \theta - 2 \cot \theta$$
for $0 < \theta < \frac { 1 } { 2 } \pi$ and $x > 0$. It is given that $x = 2$ when $\theta = \frac { 1 } { 4 } \pi$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\frac { \mathrm { d } } { \mathrm { d } \theta } \left( \cot ^ { 2 } \theta \right) = - \frac { 2 \cot \theta } { \sin ^ { 2 } \theta }$.\\
(You may assume without proof that the derivative of $\cot \theta$ with respect to $\theta$ is $- \operatorname { cosec } ^ { 2 } \theta$.)
\item Solve the differential equation and find the value of $x$ when $\theta = \frac { 1 } { 6 } \pi$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q7 [8]}}