In a triangle $ABC$,if $\cos A + 2 \cos B + \cos C = 2$ and the lengths of the sides opposite to the angles $A$ and $C$ are $3$ and $7$ respectively,then $\cos A - \cos C$ is equal to

In $\triangle ABC$,$\frac{a}{s-a}+\frac{b}{s-b}+\frac{c}{s-c} =$

In a triangle $ABC$ with a fixed base $BC$,the vertex $A$ moves such that $\cos B + \cos C = 4 \sin^2 \frac{A}{2}$. If $a, b,$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A, B,$ and $C$,respectively,then:
$(A) b+c=4a$
$(B) b+c=2a$
$(C) \text{locus of point } A \text{ is an ellipse}$
$(D) \text{locus of point } A \text{ is a pair of straight lines}$

$A$ 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:

The two adjacent sides of a cyclic quadrilateral are $2$ and $5$ and the angle between them is $60^{\circ}$. If the area of the quadrilateral is $4\sqrt{3}$,then the perimeter of the quadrilateral is

The number of solutions to $\sin(\pi \sin^2 \theta) + \sin(\pi \cos^2 \theta) = 2 \cos(\frac{\pi}{2} \cos \theta)$ satisfying $0 \leq \theta \leq 2\pi$ is