Question
$\int {\left( {\sin \left( {101x} \right).{{\sin }^{99}}x} \right)} dx = \frac{{\sin \left( {100x} \right){{\left( {\sin x} \right)}^\lambda }}}{\mu } + C$ where $C$ is constant of integration then $\frac{\lambda }{\mu }$ is equal to
$\int(\sin 100 x \cos x+\cos 100 x \sin x) \cdot \sin ^{99} x d x$
$ = \int {\underbrace {\sin 100x}_I} \underbrace {\cos x{{\sin }^{99}}x}_{II}dx + \int {\cos } (100x){(\sin x)^{100}}dx$
$\Rightarrow I=\frac{\sin (100 x)(\sin x)^{100}}{100}-\frac{100}{100}$
$\int \cos (100 x)(\sin x)^{100} d x+\int \cos (100 x)(\sin x)^{100} d x$
$\Rightarrow \mathrm{I}=\frac{\sin (100 \mathrm{x})(\sin \mathrm{x})^{100}}{100}+\mathrm{c}$
$\lambda=100, \quad \mu=100 \Rightarrow \frac{\lambda}{\mu}=1$
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| Size | Mean | Variance | |
| Observation $I$ | $10$ | $2$ | $2$ |
| Observation $II$ | $n$ | $3$ | $1$ |
If the variance of the combined set of these two observations is $\frac{17}{9},$ then the value of $n$ is equal to ..... .
$f(x)=\left\{\begin{array}{cc}\left(\frac{8}{7}\right)^{\frac{\tan 8 x}{\tan 7 x}}, & 0 < x < \frac{\pi}{2} \\ a-8, & x=\frac{\pi}{2} \\ (1+\mid \cot x)^{\frac{b}{a}|\tan x|}, & \frac{\pi}{2} < x < \pi\end{array}\right.$
Where $a, b \in Z$. If $f$ is continuous at $x=\frac{\pi}{2}$, then $\mathrm{a}^2+\mathrm{b}^2$ is equal to ..........