Let E^C denote the complement of an event E. Let E₁ , E₂ and E₃ be any pairwise independent events with P(E₁) > 0

and P(E₁ $$ \cap $$ E₂ $$ \cap $$ E₃) = 0.

Then P($$E_2^C \cap E_3^C/{E_1}$$) is equal to :

For some $$\theta \in \left( {0,{\pi \over 2}} \right)$$, if the eccentricity of the
hyperbola, x²–y²sec²$$\theta $$ = 10 is $$\sqrt 5 $$ times the
eccentricity of the ellipse, x²sec²$$\theta $$ + y² = 5, then the length of the latus rectum of the ellipse, is :

$$\mathop {\lim }\limits_{x \to 0} {\left( {\tan \left( {{\pi \over 4} + x} \right)} \right)^{{1 \over x}}}$$ is equal to :

The imaginary part of
$${\left( {3 + 2\sqrt { - 54} } \right)^{{1 \over 2}}} - {\left( {3 - 2\sqrt { - 54} } \right)^{{1 \over 2}}}$$ can be :