Radium decomposes at the rate proportional to the amount present at any time. If $$\mathrm{P} \%$$ of amount disappears in one year, then amount of radium left after 2 years is

The minimum value of the objective function $$z=4 x+6 y$$ subject to $$x+2 y \geq 80,3 x+y \geq 75, x, y \geq 0$$ is

The joint equation of pair of lines through the origin and having slopes $$(1+\sqrt{2})$$ and $$\frac{1}{(1+\sqrt{2})}$$ is

If $$4 a b=3 h^2$$, then the ratio of slopes of the lines represented by $$a x^2+2 h x y+b y^2=0$$ is