Let $A, B$ and $C$ be three events such that the probability that exactly one of $A$ and $B$ occurs is $(1-k)$, the probability that exactly one of $B$ and $C$ occurs is $(1-2 k)$, the probability that exactly one of $C$ and $A$ occurs is $(1-k)$ and the probability of all $A, B$ and $C$ occur simultaneously is $k^{2}$, where $0\,<\,\mathrm{k}\,<\,1$. Then the probability that at least one of $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ occur is:
→Let $p$ and $q$ be the position vectors of $P$ and $Q$ respectively with respect to $O$ and $|p|\, = p,\,\,|q|\,\, = q.$ The points $R$ and $S$ divide $PQ$ internally and externally in the ratio $2 : 3$ respectively. If $\overrightarrow {OR} $ and $\overrightarrow {OS} $ are perpendicular, then
→If $f(x) = \left\{ \begin{array}{l}\,\,\,\,\,\,\,\,\,\frac{{1 - \cos 4x}}{{{x^2}}},\;\;{\rm{when}}\,x < 0\\\,\,\,\,\,\,\,\,\,\,\,a,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm{when}}\,\,x = 0\\\frac{{\sqrt x }}{{\sqrt {(16 + \sqrt x )} - 4}},\,\,{\rm{when}}\,\, x > 0\end{array} \right.$, is continuous at $x = 0$, then the value of $'a'$ will be
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