If ƒ'(x) = tan^–1(secx + tanx), $$ - {\pi \over 2} < x < {\pi \over 2}$$,
and ƒ(0) = 0, then ƒ(1) is equal to :

The value of
$$\int\limits_0^{2\pi } {{{x{{\sin }^8}x} \over {{{\sin }^8}x + {{\cos }^8}x}}} dx$$ is equal to :

In a box, there are 20 cards, out of which 10 are lebelled as A and the remaining 10 are labelled as B. Cards are drawn at random, one after the other and with replacement, till a second A-card is obtained. The probability that the second A-card appears before the third B-card is :

Let z be complex number such that
$$\left| {{{z - i} \over {z + 2i}}} \right| = 1$$ and |z| = $${5 \over 2}$$.
Then the value of |z + 3i| is :