If $$p$$ and $$q$$ are chosen randomly from the set $$\left\{ {1,2,3,4,5,6,7,8,9,10} \right\},$$ with replacement, determine the probability that the roots of the equation $${x^2} + px + q = 0$$ are real.

Let $$OA=a,$$ $$OB=10a+2b$$ and $$OC=b$$ where $$O,A$$ and $$C$$ are non-collinear points. Let $$p$$ denote the area of the quadrilateral $$OABC,$$ and let $$q$$ denote the area of the parallelogram with $$OA$$ and $$OC$$ as adjacent sides. If $$p=kq,$$ then $$k=$$.........

If $$A,B$$ and $$C$$ are vectors such that $$\left| B \right| = \left| C \right|.$$ Prove that
$$\left[ {\left( {A + B} \right) \times \left( {A + C} \right)} \right] \times \left( {B \times C} \right)\left( {B + C} \right) = 0\,\,.$$

Determine the value of $$\int_\pi ^\pi {{{2x\left( {1 + \sin x} \right)} \over {1 + {{\cos }^2}x}}} \,dx.$$