If in a triangle $ABC$,$a\cos^2\frac{C}{2} + c\cos^2\frac{A}{2} = \frac{3b}{2}$,then its sides will be in

In a triangle $ABC$,with usual notations,the sides $a, b, c$ are the roots of the equation $x^3-11x^2+38x-40=0$. Then,find the value of $\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}$.

For a triangle $ABC$,the value of $\cos 2A + \cos 2B + \cos 2C$ is least. If its inradius is $3$ and incentre is $M$,then which of the following is $NOT$ correct?

In $\triangle PQR$,$\angle R = \frac{\pi}{4}$. If $\tan \left(\frac{P}{3}\right)$ and $\tan \left(\frac{Q}{3}\right)$ are the roots of the equation $ax^2 + bx + c = 0$,then:

In an isosceles triangle $ABC$,$\angle C = \angle A$. If the point of intersection of the bisectors of internal angles $\angle A$ and $\angle C$ divides the median of side $AC$ in the ratio $3 : 1$ (from vertex $B$ to side $AC$),then the value of $\csc \frac{B}{2}$ is equal to

The median $AD$ of a triangle $ABC$ is perpendicular to $AB$. Then the value of $\tan A + 2\tan B$ is: