$\ln y=\sin ^2 x \cdot \ln \left(\frac{\sqrt{3 e}}{2 \sin x}\right)$
$\frac{1}{y} y^{\prime}=\ln \left(\frac{\sqrt{3 e}}{2 \sin x}\right) 2 \sin x \cos x+\sin ^2 x \frac{2 \sin x}{\sqrt{3 e}} \frac{\sqrt{3 e}}{2}(-\operatorname{cosec} x \cot x)$
$\frac{ dy }{ dx }=0 \Rightarrow \ln \left(\frac{\sqrt{3 e }}{2 \sin x }\right) 2 \sin x \cos x -\sin x \cos x =0$
$\Rightarrow \sin x \cos x\left[2 \ln \left(\frac{\sqrt{3 e }}{2 \sin x }\right)-1\right]=0$
$\Rightarrow \ln \left(\frac{3 e }{4 \sin ^2 x }\right)=1 \Rightarrow \frac{3 e }{4 \sin ^2 x }= e \Rightarrow \sin ^2 x =\frac{3}{4}$
$\Rightarrow \sin x =\frac{\sqrt{3}}{2} \quad\left(\text { as } x \in\left(0, \frac{\pi}{2}\right)\right)$
$\Rightarrow \text { local max value }=\left(\frac{\sqrt{3 e }}{\sqrt{3}}\right)^{3 / 4}= e ^{3 / 8}=\frac{ k }{ e }$
$\Rightarrow k ^8= e ^{11}$
$\Rightarrow\left(\frac{ k }{ e }\right)^8+\frac{ k ^8}{ e ^5}+ k ^8= e ^3+ e ^6+ e ^{11}$
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$1 + \frac{1}{{1 + 2}} + \frac{1}{{1 + 2 + 3}} + .......$ up to $10$ terms, is
$g(x)=\left\{\begin{array}{ccc}0 & \text { if } & x < a, \\ \int_a^x f(t) d t & \text { if } & a \leq x \leq b, \\ \int_a^b f(t) d t & \text { if } & x > b .\end{array}\right.$, Then
$(A)$ $g(x)$ is continuous but not differentiable at a
$(B)$ $g(x)$ is differentiable on $R$
$(C)$ $g(x)$ is continuous but not differentiable at $b$
$(D)$ $g(x)$ is continuous and differentiable at either a or $b$ but not both
| Class: | $10-20$ | $20-30$ | $30-40$ | $40-50$ | $50-60$ |
| Freq: | $\alpha$ | $110$ | $54$ | $30$ | $\beta$ |
If the sum of all frequencies is $584$ and median is $45$ , then $|\alpha-\beta|$ is equal to $.....$