If $$\omega \,$$ is an imaginary cube root of unity then the value of $$\sin \left\{ {\left( {{\omega ^{10}} + {\omega ^{23}}} \right)\pi - {\pi \over 4}} \right\}$$ is

Suppose Z₁, Z₂, Z₃ are the vertices of an equilateral triangle inscribed in the circle $$\left| Z \right| = 2.$$ If Z₁ = $$1 + i\sqrt 3 $$ then Z2 = ......., Z₃ =..............

Let $$2{\sin ^2}x + 3\sin x - 2 > 0$$ and $${x^2} - x - 2 < 0$$ ($$x$$ is measured in radians). Then $$x$$ lies in the interval

The number of points of intersection of two curves y = 2 sin x and y $$ = 5{x^2} + 2x + 3$$ is