If $A$ and $B$ are acute angles satisfying $3 \cos ^2 A + 2 \cos ^2 B = 4$ and $\frac{3 \sin A}{\sin B} = \frac{2 \cos B}{\cos A}$,then $A + 2B =$ (in $^{\circ}$)

If $O$ is the circumcentre of the $\Delta ABC$ and $R_1, R_2$ and $R_3$ are the radii of the circumcircles of triangles $OBC, OCA$ and $OAB$ respectively,then the value of $\frac{a}{R_1} + \frac{b}{R_2} + \frac{c}{R_3}$ is equal to:

If $u = \log \tan \left(\frac{\pi}{4} + \frac{\theta}{2}\right)$,then $\cosh u =$

Let $a, b, c$ be three non-zero real numbers such that the equation $\sqrt{3} a \cos x + 2 b \sin x = c$,where $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$,has two distinct real roots $\alpha$ and $\beta$ with $\alpha + \beta = \frac{\pi}{3}$. Then,the value of $\frac{b}{a}$ is:

The value of $\cos ^4 \frac{\pi}{8}+\cos ^4 \frac{2 \pi}{8}+\cos ^4 \frac{3 \pi}{8}+\cos ^4 \frac{4 \pi}{8}+\cos ^4 \frac{5 \pi}{8}+\cos ^4 \frac{6 \pi}{8}+\cos ^4 \frac{7 \pi}{8}+\cos ^4 \frac{8 \pi}{8}$ is equal to:

Let $\alpha = 3 + 4 + 8 + 9 + 13 + 14 + \dots$ up to $40$ terms. If $(\tan \beta)^{1020}$ is a root of the equation $x^2 + x - 2 = 0$,where $\beta \in (0, \frac{\pi}{2})$,then $\sin^2 \beta + 3 \cos^2 \beta$ is equal to: