Standard +0.3 This is a straightforward separable variables question requiring standard manipulation (separating variables, integrating both sides, applying initial condition). The substitution u = ½θ makes integration routine, and expressing the answer in terms of cos θ uses a standard double-angle identity. Slightly above average due to the trigonometric manipulation required, but still a standard textbook exercise.
6 The variables \(x\) and \(\theta\) satisfy the differential equation
$$\sin \frac { 1 } { 2 } \theta \frac { d x } { d \theta } = ( x + 2 ) \cos \frac { 1 } { 2 } \theta$$
for \(0 < \theta < \pi\). It is given that \(x = 1\) when \(\theta = \frac { 1 } { 3 } \pi\). Solve the differential equation and obtain an expression for \(x\) in terms of \(\cos \theta\).
Separate variables correctly to obtain \(\int \frac{1}{x+2}dx = \int \cot\frac{1}{2}\theta \, d\theta\)
B1
Or equivalent integrands. Integral signs SOI
Obtain term \(\ln(x+2)\)
B1
Modulus signs not needed
Obtain term of the form \(k\ln\sin\frac{1}{2}\theta\)
M1
Obtain term \(2\ln\sin\frac{1}{2}\theta\)
A1
Use \(x=1\), \(\theta=\frac{1}{3}\pi\) to evaluate a constant, or as limits, in an expression containing \(p\ln(x+2)\) and \(q\ln\left(\sin\frac{1}{2}\theta\right)\)
M1
Reach \(C\) = an expression or a decimal value
Obtain correct solution in any form e.g. \(\ln(x+2) = 2\ln\sin\frac{1}{2}\theta + \ln 12\)
A1
\(\ln 12 = 2.4849\ldots\) Accept constant to at least 3 s.f. Accept with \(\ln 3 - 2\ln\frac{1}{2}\)
Remove logarithms and use correct double angle formula
M1
Need correct algebraic process. \(\left(\frac{x+2}{12} = \frac{1-\cos\theta}{2}\right)\)
Obtain answer \(x = 4 - 6\cos\theta\)
A1
Total
8
## Question 6:
| Answer | Mark | Guidance |
|--------|------|----------|
| Separate variables correctly to obtain $\int \frac{1}{x+2}dx = \int \cot\frac{1}{2}\theta \, d\theta$ | B1 | Or equivalent integrands. Integral signs SOI |
| Obtain term $\ln(x+2)$ | B1 | Modulus signs not needed |
| Obtain term of the form $k\ln\sin\frac{1}{2}\theta$ | M1 | |
| Obtain term $2\ln\sin\frac{1}{2}\theta$ | A1 | |
| Use $x=1$, $\theta=\frac{1}{3}\pi$ to evaluate a constant, or as limits, in an expression containing $p\ln(x+2)$ and $q\ln\left(\sin\frac{1}{2}\theta\right)$ | M1 | Reach $C$ = an expression or a decimal value |
| Obtain correct solution in any form e.g. $\ln(x+2) = 2\ln\sin\frac{1}{2}\theta + \ln 12$ | A1 | $\ln 12 = 2.4849\ldots$ Accept constant to at least 3 s.f. Accept with $\ln 3 - 2\ln\frac{1}{2}$ |
| Remove logarithms and use correct double angle formula | M1 | Need correct algebraic process. $\left(\frac{x+2}{12} = \frac{1-\cos\theta}{2}\right)$ |
| Obtain answer $x = 4 - 6\cos\theta$ | A1 | |
| **Total** | **8** | |
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6 The variables $x$ and $\theta$ satisfy the differential equation
$$\sin \frac { 1 } { 2 } \theta \frac { d x } { d \theta } = ( x + 2 ) \cos \frac { 1 } { 2 } \theta$$
for $0 < \theta < \pi$. It is given that $x = 1$ when $\theta = \frac { 1 } { 3 } \pi$. Solve the differential equation and obtain an expression for $x$ in terms of $\cos \theta$.\\
\hfill \mbox{\textit{CAIE P3 2019 Q6 [8]}}