Suppose that the sides $a, b, c$ of a triangle $ABC$ satisfy $b^2 = ac$. Then the set of all possible values of $\frac{\sin A \cot C + \cos A}{\sin B \cot C + \cos B}$ is

In a triangle $PQR$,$P$ is the largest angle and $\cos P = \frac{1}{3}$. Further,the incircle of the triangle touches the sides $PQ, QR$ and $RP$ at $N, L$ and $M$ respectively,such that the lengths of $PN, QL$ and $RM$ are consecutive even integers. Then the possible length$(s)$ of the side$(s)$ of the triangle is (are):
$(A) 16$
$(B) 18$
$(C) 24$
$(D) 22$

In a triangle $ABC$ with usual notations,if $\frac{\cos A}{a} = \frac{\cos B}{b} = \frac{\cos C}{c}$,then the triangle is equilateral. If the side length is $a = \sqrt{6}$,find the area of the triangle.

In a $\triangle ABC$,$(b^2-c^2) \cot A + (c^2-a^2) \cot B =$

If $2 \cos \theta + \sin \theta = 1$,then the value of $4 \cos \theta + 3 \sin \theta$ is equal to

In a triangle $ABC$,if $A, B$ and $C$ are in arithmetic progression,$rr_3 = r_1 r_2$ and $c = 10$,then $a^2 + b^2 + c^2 =$