Three critics review a book. For the three critics the odds in favour of the book are $$2: 5, 3: 4$$ and $$4: 3$$ respectively. The probability that the majority is in favour of the book, is given by

Consider the lines $$\mathrm{L}_1: \frac{x+1}{3}=\frac{y+2}{1}=\frac{\mathrm{z}+1}{2}$$

$$\mathrm{L}_2: \frac{x-2}{1}=\frac{y+2}{2}=\frac{\mathrm{z}-3}{3}$$, then the unit vector perpendicular to both $$\mathrm{L}_1$$ and $$\mathrm{L}_2$$ is

The area bounded by the curves $$y=(x-1)^2, y=(x+1)^2$$ and $$y=\frac{1}{4}$$ is

A sphere and a cube, both of copper have equal volumes and are black. They are allowed to cool at same temperature and in same atmosphere. The ratio of their rate of loss of heat will be