The oxygen dissolved in water exerts a partial pressure of 20 kPa in the vapour above water. The molar solubility of oxygen in water is __________ $$\times$$ 10^$$-$$5 mol dm^$$-$$3. (Round off to the Nearest Integer).

[Given : Henry's law constant = K_H = 8.0 $$\times$$ 10⁴ kPa for O₂. Density of water with dissolved oxygen = 1.0 kg dm^$$-$$3 ]

In a triangle PQR, the co-ordinates of the points P and Q are ($$-$$2, 4) and (4, $$-$$2) respectively. If the equation of the perpendicular bisector of PR is 2x $$-$$ y + 2 = 0, then the centre of the circumcircle of the $$\Delta$$PQR is :

The value of
$$\mathop {\lim }\limits_{x \to {0^ + }} {{{{\cos }^{ - 1}}(x - {{[x]}^2}).{{\sin }^{ - 1}}(x - {{[x]}^2})} \over {x - {x^3}}}$$, where [ x ] denotes the greatest integer $$ \le $$ x is :

Which of the following statements is correct for the function g($$\alpha$$) for $$\alpha$$ $$\in$$ R such that

$$g(\alpha ) = \int\limits_{{\pi \over 6}}^{{\pi \over 3}} {{{{{\sin }^\alpha }x} \over {{{\cos }^\alpha }x + {{\sin }^\alpha }x}}dx} $$