The system of equations

$$ - kx + 3y - 14z = 25$$

$$ - 15x + 4y - kz = 3$$

$$ - 4x + y + 3z = 4$$

is consistent for all k in the set

$$\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{{\tan }^2}x\left( {{{(2{{\sin }^2}x + 3\sin x + 4)}^{{1 \over 2}}} - {{({{\sin }^2}x + 6\sin x + 2)}^{{1 \over 2}}}} \right)} \right)$$ is equal to

The area of the region enclosed between the parabolas y² = 2x $$-$$ 1 and y² = 4x $$-$$ 3 is

The coefficient of x¹⁰¹ in the expression $${(5 + x)^{500}} + x{(5 + x)^{499}} + {x^2}{(5 + x)^{498}} + \,\,.....\,\, + \,\,{x^{500}}$$, x > 0, is