If the area of the region bounded by the curves, $$y = {x^2},y = {1 \over x}$$ and the lines y = 0 and x= t (t >1) is 1 sq. unit, then t is equal to :

Let N denote the set of all natural numbers. Define two binary relations on N as _R = {(x, y) $$ \in $$ N $$ \times $$ N : 2x + y = 10} and R₂ = {(x, y) $$ \in $$ N $$ \times $$ N : x + 2y = 10}. Then :

Let p, q and r be real numbers (p $$ \ne $$ q, r $$ \ne $$ 0), such that the roots of the equation $${1 \over {x + p}} + {1 \over {x + q}} = {1 \over r}$$ are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to :

The least positive integer n for which $${\left( {{{1 + i\sqrt 3 } \over {1 - i\sqrt 3 }}} \right)^n} = 1,$$ is :