The points with position vectors $$a+b,$$ $$a-b,$$ and $$a+kb$$ are collinear for all real values of $$k.$$

If the complex numbers, $${Z_1},{Z_2}$$ and $${Z_3}$$ represent the vertics of an equilateral triangle such that
$$\left| {{Z_1}} \right| = \left| {{Z_2}} \right| = \left| {{Z_3}} \right|$$ then $${Z_1} + {Z_2} + {Z_3} = 0.$$

Evaluate the following $$\int {{{dx} \over {{x^2}{{\left( {{x^4} + 1} \right)}^{3/4}}}}} $$

There exists a value of $$\theta $$ between 0 and $$2\pi $$ that satisfies the equation $$\,\,{\sin ^4}\theta - 2{\sin ^2}\theta - 1 = 0.$$