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

For each natural number k, let $${C_k}$$ denote the circle with radius k centimetres and centre at the origin. On the circle $${C_k}$$, a-particle moves k centimetres in the counter-clockwise direction. After completing its motion on $${C_k}$$, the particle moves to $${C_{k + 1}}$$ in the radial direction. The motion of the patticle continues in the manner. The particle starts at (1, 0). If the particle crosses the positive direction of the x-axis for the first time on the circle $${C_n}$$ then n = ..............