The probabilities of three events A, B and C are given by
P(A) = 0.6, P(B) = 0.4 and P(C) = 0.5.
If P(A$$ \cup $$B) = 0.8, P(A$$ \cap $$C) = 0.3, P(A$$ \cap $$B$$ \cap $$C) = 0.2, P(B$$ \cap $$C) = $$\beta $$
and P(A$$ \cup $$B$$ \cup $$C) = $$\alpha $$, where 0.85 $$ \le \alpha \le $$ 0.95, then $$\beta $$ lies in the interval :

If the constant term in the binomial expansion of
$${\left( {\sqrt x - {k \over {{x^2}}}} \right)^{10}}$$ is 405, then |k| equals :

Let $$\theta = {\pi \over 5}$$ and $$A = \left[ {\matrix{ {\cos \theta } & {\sin \theta } \cr { - \sin \theta } & {\cos \theta } \cr } } \right]$$.

If B = A + A⁴ , then det (B) :

Assuming the nitrogen molecule is moving with r.m.s. velocity at 400 K, the de-Broglie wavelength of nitrogen molecule is close to :
(Given : nitrogen molecule weight : 4.64 $$ \times $$ 10^–26 kg,
Boltzman constant: 1.38 $$ \times $$ 10^–23 J/K,
Planck constant : 6.63 $$ \times $$ 10^–34 J.s)