75.2 g of C₆H₅OH (phenol) is dissolved in a solvent of K_f = 14. If the depression in freezing point is 7 K then find the % of phenol that dimerises.

We have taken a saturated solution of AgBr. K_sp of AgBr is 12 $$\times$$ 10^-14. If 10^-7 mole of AgNO₃ are added to 1 litre of this solution find conductivity (specific conductance) of this solution in terms of 10^-7 S m^-1 units. Given, molar conductance of Ag⁺, Br^- and $$NO_3^-$$ are 6 $$\times$$ 10^-3 Sm² mol^-1, 8 $$\times$$ 10^-3 Sm² mol^-1 and 7 $$\times$$ 10^-3 Sm² mol^-1

A plane which is perpendicular to two planes $$2x - 2y + z = 0$$ and $$x - y + 2z = 4,$$ passes through $$(1, -2, 1).$$ The distance of the plane from the point $$(1, 2, 2)$$ is

Match the following :

Column $$I$$
(A) $$\int\limits_0^{\pi /2} {{{\left( {\sin x} \right)}^{\cos x}}\left( {\cos x\cot x - \log {{\left( {\sin x} \right)}^{\sin x}}} \right)dx} $$
(B) Area bounded by $$ - 4{y^2} = x$$ and $$x - 1 = - 5{y^2}$$
(C) Cosine of the angle of intersection of curves $$y = {3^{x - 1}}\log x$$ and $$y = {x^x} - 1$$ is
(D) Let $${{dy} \over {dx}} = {6 \over {x + y}}$$ where $$y(0)=0$$ then value of $$y$$ when $$x+y=6$$ is

Column $$II$$
(p) $$1$$
(q) $$0$$
(r) $$6\ln 2$$
(s) $${4 \over 3}$$