In $\triangle ABC$,if $A, B, C$ are in arithmetic progression,then $\frac{c}{a} \sin 2A + \frac{a}{c} \sin 2C =$

In a $\triangle ABC$,$\cos \left(\frac{B+2C+3A}{2}\right) + \cos \left(\frac{A-B}{2}\right)$ is equal to

The number of solutions of $\sin ^2 x + (2 + 2x - x^2) \sin x - 3(x - 1)^2 = 0$,where $-\pi \leq x \leq \pi$,is....................

The number of solutions to $\sin x = \frac{6}{x}$ with $0 \leq x \leq 12 \pi$ is

In $\Delta ABC,$ if $\sin^2 \frac{A}{2}, \sin^2 \frac{B}{2}, \sin^2 \frac{C}{2}$ are in $H.P.,$ then $a, b, c$ will be in

Let $PQRS$ be a quadrilateral in a plane,where $QR = 1$,$\angle PQR = \angle QRS = 70^{\circ}$,$\angle PQS = 15^{\circ}$ and $\angle PRS = 40^{\circ}$. If $\angle RPS = \theta^{\circ}$,$PQ = \alpha$ and $PS = \beta$,then the interval$(s)$ that contain$(s)$ the value of $4 \alpha \beta \sin \theta^{\circ}$ is/are
$(A)$ $(0, \sqrt{2})$
$(B)$ $(1, 2)$
$(C)$ $(\sqrt{2}, 3)$
$(D)$ $(2 \sqrt{2}, 3 \sqrt{2})$