For $x \in (0, \pi)$,the equation $\sin x + 2 \sin 2x - \sin 3x = 3$ has

If $p_1, p_2, p_3$ are the altitudes and $a=4, b=5, c=6$ are the sides of a triangle $ABC$,then $\frac{1}{p_1^2} + \frac{1}{p_2^2} + \frac{1}{p_3^2} =$

In a $\triangle ABC$,if $A-B=120^{\circ}$ and $R=8r$,then find the value of $\frac{1+\cos C}{1-\cos C}$.

In $\triangle ABC$,if the sides $a, b, c$ are in geometric progression and the largest angle exceeds the smallest angle by $60^{\circ}$,then $\cos B$ is equal to

The angles $\alpha, \beta, \gamma$ of a triangle satisfy the equations $2 \sin \alpha + 3 \cos \beta = 3 \sqrt{2}$ and $3 \sin \beta + 2 \cos \alpha = 1$. Then,angle $\gamma$ equals (in $^{\circ}$)

Statement $-1$: The number of common solutions of the trigonometric equations $2\sin^2\theta - \cos 2\theta = 0$ and $2\cos^2\theta - 3\sin\theta = 0$ in the interval $[0, 2\pi]$ is two.
Statement $-2$: The number of solutions of the equation $2\cos^2\theta - 3\sin\theta = 0$ in the interval $[0, \pi]$ is two.