The number of distinct solutions of the equation
$${\log _{{1 \over 2}}}\left| {\sin x} \right| = 2 - {\log _{{1 \over 2}}}\left| {\cos x} \right|$$ in the interval [0, 2$$\pi $$], is ____.

The value of
$${\cos ^3}\left( {{\pi \over 8}} \right)$$$${\cos}\left( {{3\pi \over 8}} \right)$$+$${\sin ^3}\left( {{\pi \over 8}} \right)$$$${\sin}\left( {{3\pi \over 8}} \right)$$
is :

If for all real triplets (a, b, c), ƒ(x) = a + bx + cx²; then $$\int\limits_0^1 {f(x)dx} $$ is equal to :

A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness the melts at a rate of 50 cm³/min. When the thickness of ice is 5 cm, then the rate (in cm/min.) at which of the thickness of ice decreases, is :