Moderate -0.3 This is a straightforward separable variables question requiring standard technique: separate variables, integrate both sides (using a standard trigonometric identity for sin²2θ), apply initial conditions, and evaluate. The integration requires knowing sin²2θ = (1-cos4θ)/2, which is a standard A-level identity. Slightly easier than average due to being a routine application of a well-practiced method with no conceptual surprises.
8 The variables \(x\) and \(\theta\) satisfy the differential equation
$$\frac { \mathrm { d } x } { \mathrm {~d} \theta } = ( x + 2 ) \sin ^ { 2 } 2 \theta$$
and it is given that \(x = 0\) when \(\theta = 0\). Solve the differential equation and calculate the value of \(x\) when \(\theta = \frac { 1 } { 4 } \pi\), giving your answer correct to 3 significant figures.
Use cos \(2A\) formula to express \(\sin^2 2\theta\) in the form \(a + b\cos 4\theta\)
M1
Obtain correct form \((1 - \cos 4\theta)/2\), or equivalent
A1
Integrate and obtain term \(\frac{1}{2}\theta - \frac{1}{8}\sin 4\theta\), or equivalent
A1\(\ft\)
Evaluate a constant, or use \(\theta = 0\), \(x = 0\) as limits in a solution containing terms \(c\ln(x+2)\), \(d\sin(4\theta)\), \(e\theta\)
M1
Obtain correct solution in any form, e.g. \(\ln(x+2) = \frac{1}{2}\theta - \frac{1}{8}\sin 4\theta + \ln 2\)
A1
Use correct method for solving an equation of the form \(\ln(x+2) = f\)
M1
Obtain answer \(x = 0.962\)
A1
Total: 9
## Question 8:
| Answer | Mark | Guidance |
|--------|------|----------|
| Separate variables and integrate one side | B1 | |
| Obtain term $\ln(x+2)$ | B1 | |
| Use cos $2A$ formula to express $\sin^2 2\theta$ in the form $a + b\cos 4\theta$ | M1 | |
| Obtain correct form $(1 - \cos 4\theta)/2$, or equivalent | A1 | |
| Integrate and obtain term $\frac{1}{2}\theta - \frac{1}{8}\sin 4\theta$, or equivalent | A1$\ft$ | |
| Evaluate a constant, or use $\theta = 0$, $x = 0$ as limits in a solution containing terms $c\ln(x+2)$, $d\sin(4\theta)$, $e\theta$ | M1 | |
| Obtain correct solution in any form, e.g. $\ln(x+2) = \frac{1}{2}\theta - \frac{1}{8}\sin 4\theta + \ln 2$ | A1 | |
| Use correct method for solving an equation of the form $\ln(x+2) = f$ | M1 | |
| Obtain answer $x = 0.962$ | A1 | |
| **Total: 9** | | |
8 The variables $x$ and $\theta$ satisfy the differential equation
$$\frac { \mathrm { d } x } { \mathrm {~d} \theta } = ( x + 2 ) \sin ^ { 2 } 2 \theta$$
and it is given that $x = 0$ when $\theta = 0$. Solve the differential equation and calculate the value of $x$ when $\theta = \frac { 1 } { 4 } \pi$, giving your answer correct to 3 significant figures.\\
\hfill \mbox{\textit{CAIE P3 Q8 [9]}}