| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | March |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Separable variables - standard (polynomial/exponential x-side) |
| Difficulty | Standard +0.3 This is a standard separable differential equation requiring routine techniques: separate variables, integrate using a standard substitution (recognizing 1-cos x = 2sin²(x/2)), apply initial conditions, and simplify. The sketch is straightforward once the solution is found. Slightly above average due to the algebraic manipulation needed, but well within typical P3 expectations. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks |
|---|---|
| 4 | Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored (isw). |
| Answer | Marks | Guidance |
|---|---|---|
| 4(a) | Separate variables correctly and attempt integration of at least one side | M1 |
| Obtain term lny | A1 | |
| Obtain term of the form ±ln(1−cosx) | M1 | |
| Obtain term ln ( 1−cosx ) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| terms of the form alny and bln(1−cosx) | M1 | |
| Obtain final answer y=2(1−cosx) | A1 | OE |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 4(b) | Show a correct graph for 0< x<2π with the maximum at x = π | B1 FT |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 4:
4 | Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored (isw).
--- 4(a) ---
4(a) | Separate variables correctly and attempt integration of at least one side | M1
Obtain term lny | A1
Obtain term of the form ±ln(1−cosx) | M1
Obtain term ln ( 1−cosx ) | A1
x=π
Use , y = 4 to evaluate a constant, or as limits, in a solution containing
terms of the form alny and bln(1−cosx) | M1
Obtain final answer y=2(1−cosx) | A1 | OE
6
Question | Answer | Marks | Guidance
--- 4(b) ---
4(b) | Show a correct graph for 0< x<2π with the maximum at x = π | B1 FT | The FT is for graphs of the form y=a(1−cosx),
where a is positive.
1
Question | Answer | Marks | GuidanceThe variables $x$ and $y$ satisfy the differential equation
$$(1 - \cos x)\frac{dy}{dx} = y \sin x.$$
It is given that $y = 4$ when $x = \pi$.
\begin{enumerate}[label=(\alph*)]
\item Solve the differential equation, obtaining an expression for $y$ in terms of $x$. [6]
\item Sketch the graph of $y$ against $x$ for $0 < x < 2\pi$. [1]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q4 [7]}}