In any $\triangle ABC$,$b^2 \sin 2C + c^2 \sin 2B$ is equal to

In $\triangle ABC$,if $\sin^2 B = \sin C$ and $3 \cos^2 B = 2 \cos^2 C$,then $\triangle ABC$ is

Let $x, y$ and $z$ be positive real numbers. Suppose $x, y$ and $z$ are lengths of the sides of a triangle opposite to its angles $X, Y$ and $Z$,respectively. If $\tan \frac{X}{2} + \tan \frac{Z}{2} = \frac{2y}{x+y+z}$,then which of the following statements is/are $TRUE$?
$(A) 2Y = X + Z$
$(B) Y = X + Z$
$(C) \tan \frac{X}{2} = \frac{x}{y+z}$
$(D) x^2 + z^2 - y^2 = xz$

With usual notations,in $\Delta ABC$,if $b \cos ^{2} \frac{C}{2}+c \cos ^{2} \frac{B}{2}=\frac{3 a}{2}$,then

$A$ spherical balloon of radius $r$ subtends an angle $\alpha$ at the eye of an observer. If the angle of elevation of the centre of the balloon is $\beta$,then the height of the centre of the balloon is:

With usual notation in a $\Delta ABC$,$\left( \frac{1}{r_1} + \frac{1}{r_2} \right) \left( \frac{1}{r_2} + \frac{1}{r_3} \right) \left( \frac{1}{r_3} + \frac{1}{r_1} \right) = \frac{K R^3}{a^2 b^2 c^2}$ where $K$ has the value equal to