Trigonometry Questions

If the roots of the quadratic equation $x^2+px+q=0$ are $\tan 30^{\circ }$ and $\tan 15^{\circ }$, respectively, then the value of $2+q-p$ is:

The number of values of $x$ in the interval $[0,3\pi ]$ satisfying the equation $2{\sin }^{2}x+5\sin x-3=0$ is:

The value of ${\sum }_{k=1}^{10}\left(\sin \frac{2k\pi }{11}+i\cos \frac{2k\pi }{11}\right)$ is:

A particle has two velocities of equal magnitude $u$ inclined to each other at an angle $\theta$. If one of them is halved, the angle between the other and the original resultant velocity is bisected by the new resultant. Then $\theta$ is:

If $0<x<\pi$ and $\cos x+\sin x=1/2$, then $\tan x$ is: