MCQ
If $\sin 2\theta + \sin 2\phi = 1/2$ and $\cos 2\theta + \cos 2\phi = 3/2$, then ${\cos ^2}(\theta - \phi ) = $
- A$3/8$
- ✓$5/8$
- C$3/4$
- D$5/4$
and $\cos 2\,\theta + \cos 2\,\varphi = 3/2$…..$(ii)$
Square અને adding ,
$\therefore \,({\sin ^2}2\theta + {\cos ^2}2\theta ) + ({\sin ^2}2\phi + {\cos ^2}2\phi )$
$ + 2\,[\sin 2\,\theta \,\sin 2\,\phi + \cos 2\,\theta \,\cos 2\,\phi ] = 1/4 + 9/4$
==> $\cos 2\theta \cos 2\,\phi + \sin 2\theta \sin 2\phi = 1/4$
==> $\cos (2\theta - 2\phi ) = 1/4$
==> ${\cos ^2}(\theta - \phi ) = 5/8$.
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$(a)$ reflection about the line $y=x$.
$(b)$ translation through $2$ units along the positive direction of $x$-axis.
$(c)$ rotation through angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction.
If the co-ordinates of the final position of the point $P$ are $\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)$, then the value of $2 a+b$ is equal to: