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અહી $S$ એ અંતરાલ $[0,4 \pi]$ માં સમીકરણ $\sin ^{4} \theta+\cos ^{4} \theta-\sin \theta \cos \theta=0$ ઉકેલનો સરવાળો દર્શાવે છે તો $\frac{8 \mathrm{~S}}{\pi}$ ની કિમંત મેળવો.
$87$
$78$
$56$
$65$
Solution
Given equation
$\sin ^{4} \theta+\cos ^{4} \theta-\sin \theta \cos \theta=0$
$\Rightarrow 1-\sin ^{2} \theta \cos ^{2} \theta-\sin \theta \cos \theta=0$
$\Rightarrow 2-(\sin 2 \theta)^{2}-\sin 2 \theta=0$
$\Rightarrow(\sin 2 \theta)^{2}+(\sin 2 \theta)-2=0$
$\Rightarrow(\sin 2 \theta+2)(\sin 2 \theta-1)=0$
$\Rightarrow \sin 2 \theta=1 \text { or } \sin 2 \theta=-2$ $\rightarrow$ not possible
$\Rightarrow \quad 2 \theta=\frac{\pi}{2}, \frac{5 \pi}{2}, \frac{9 \pi}{2}, \frac{13 \pi}{2}$
$\Rightarrow \quad \theta=\frac{\pi}{4}, \frac{5 \pi}{4}, \frac{9 \pi}{4}, \frac{13 \pi}{4}$
$\Rightarrow \frac{8 S}{\pi}=\frac{8 \times 7 \pi}{\pi}=56.00$
