Let the lines (2 $$-$$ i)z = (2 + i)$$\overline z $$ and (2 $$+$$ i)z + (i $$-$$ 2)$$\overline z $$ $$-$$ 4i = 0, (here i² = $$-$$1) be normal to a circle C. If the line iz + $$\overline z $$ + 1 + i = 0 is tangent to this circle C, then its radius is :

$$\mathop {\lim }\limits_{n \to \infty } {\left( {1 + {{1 + {1 \over 2} + ........ + {1 \over n}} \over {{n^2}}}} \right)^n}$$ is equal to :

Let $$\alpha$$ be the angle between the lines whose direction cosines satisfy the equations l + m $$-$$ n = 0 and l² + m² $$-$$ n² = 0. Then the value of sin⁴$$\alpha$$ + cos⁴$$\alpha$$ is :

The value of the integral
$$\int {{{\sin \theta .\sin 2\theta ({{\sin }^6}\theta + {{\sin }^4}\theta + {{\sin }^2}\theta )\sqrt {2{{\sin }^4}\theta + 3{{\sin }^2}\theta + 6} } \over {1 - \cos 2\theta }}} \,d\theta $$ is :