The coefficient of x⁻⁵ in the binomial expansion of

$${\left( {{{x + 1} \over {{x^{{2 \over 3}}} - {x^{{1 \over 3}}} + 1}} - {{x - 1} \over {x - {x^{{1 \over 2}}}}}} \right)^{10}},$$ where x $$ \ne $$ 0, 1, is :

The number of ways in which 5 boys and 3 girls can be seated on a round table if a particular boy B₁ and a particular girl G₁ never sit adjacent to each other, is :

For two 3 × 3 matrices A and B, let A + B = 2B^T and 3A + 2B = I₃, where B^T is the transpose of B and I₃ is 3 × 3 identity matrix. Then :

The equation
Im $$\left( {{{iz - 2} \over {z - i}}} \right)$$ + 1 = 0, z $$ \in $$ C, z $$ \ne $$ i
represents a part of a circle having radius equal to :