$${\rm{z }} \ne {\rm{0}}$$ is a complex number

Column I

(A) Re z = 0
(B) Arg $$z = {\pi \over 4}$$

Column II

(p) Re$${z^2}$$ = 0
(q) Im$${z^2}$$ = 0
(r) Re$${z^2}$$ = Im$${z^2}$$

Show that the value of $${{\tan x} \over {\tan 3x}},$$ wherever defined never lies between $${1 \over 3}$$ and 3.

Let $$\alpha \,,\,\beta $$ be the roots of the equation (x - a) (x - b) = c, $$c \ne 0$$. Then the roots of the equation $$(x - \alpha \,)\,(x - \beta ) + c = 0$$ are

The expansion $${\left( {x + {{\left( {{x^3} - 1} \right)}^{{1 \over 2}}}} \right)^5} + {\left( {x - {{\left( {{x^3} - 1} \right)}^{{1 \over 2}}}} \right)^5}$$ is a polynomial of degree