One of the solutions of the equation $8 \sin ^3 \theta-7 \sin \theta+\sqrt{3} \cos \theta=0$ lies in the interval

If both roots of quadratic equation ${x^2} + \left( {\sin \,\theta  + \cos \,\theta } \right)x + \frac{3}{8} = 0$ are positive and distinct then complete set of values of $\theta $ in $\left[ {0,2\pi } \right]$ is 

If $tanA + cotA = 4$, then $tan^4A + cot^4A$ is equal to

The general solution of the equation $sin^{100}x\,-\,cos^{100} x= 1$ is

In a triangle $P Q R, P$ is the largest angle and $\cos P=\frac{1}{3}$. Further the incircle of the triangle touches the sides $P Q, Q R$ and $R P$ at $N, L$ and $M$ respectively, such that the lengths of $P N, Q L$ and $R M$ are consecutive even integers. Then possible length$(s)$ of the side$(s)$ of the triangle is (are)

$(A)$ $16$ $(B)$ $18$ $(C)$ $24$ $(D)$ $22$

If $\sin x=\frac{3}{5}, \cos y=-\frac{12}{13},$ where $x$ and $y$ both lie in second quadrant, find the value of $\sin (x+y)$.