Which of the following is true?

In a $\Delta ABC$,a semicircle is inscribed,whose diameter lies on the side $c$. Then the radius of the semicircle is
Where $\Delta$ is the area of the triangle $ABC$.

Number of solutions of $5 \cos^2 \theta - 3 \sin^2 \theta + 6 \sin \theta \cos \theta = 7$ in the interval $[0, 2 \pi]$ is:

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 $\triangle ABC$,with usual notations,if $a \cos B = b \cos A$ and $a \cos C \neq c \cos A$,then the area of $\triangle ABC$ is . . . . . . sq. units.

$A$ tower is situated on a horizontal plane. Two points lie on the line passing through the base of the tower,at distances $a$ and $b$ from the base. The angles of elevation of the top of the tower from these points are $\alpha$ and $90^\circ - \alpha$. If the line segment joining the two points subtends an angle $\theta$ at the top of the tower,find the height of the tower.