In a triangle $A B C$ with usual notations if $\angle A=30^{\circ}$, then the value of $\left(1+\frac{a}{c}+\frac{b}{c}\right)\left(1+\frac{c}{b}-\frac{a}{b}\right)=$

In a triangle PQR with usual notations, $\angle \mathrm{R}=\frac{\pi}{2}$. If $\tan \frac{\mathrm{P}}{2}$ and $\tan \frac{\mathrm{Q}}{2}$ are the roots of the equation $a x^2+b x+c=0(a \neq 0)$, then

If the angles $\mathrm{A}, \mathrm{B}$ and C of a triangle are in A.P. and if $\mathrm{a}, \mathrm{b}$ and c denote the length of the sides opposite to $\mathrm{A}, \mathrm{B}$ and C respectively, then the value of $\frac{a}{b} \sin 2 B+\frac{b}{a} \sin 2 A$ is

In a triangle $A B C$, with usual notations, if $a=5$, $\mathrm{b}=7 \sin \mathrm{~A}=\frac{3}{4}$, then total number of triangles possible are