A) \[\Omega \]
B) \[\Omega \]
C) \[\frac{i}{{{i}_{0}}}=\frac{4}{11}\]
D) 2
Correct Answer: B
Solution :
We have,\[\left| \begin{matrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \\ \end{matrix} \right|=0\]\[\Rightarrow \]\[\left| \begin{matrix} 1 & a & b \\ 0 & c-a & a-b \\ 0 & b-a & c-b \\ \end{matrix} \right|=0\] [applying\[{{R}_{2}}\to {{R}_{2}}-{{R}_{1}}\]and\[{{R}_{3}}\to {{R}_{3}}-{{R}_{1}}\]] \[\Rightarrow \]\[(c-a)(c-b)+{{(a-b)}^{2}}=0\] \[\Rightarrow \]\[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}-ab-bc-ca=0\] \[\Rightarrow \]\[2{{a}^{2}}+2{{b}^{2}}+2{{c}^{2}}-2ab-2bc-2ca=0\] \[\Rightarrow \]\[{{(a-b)}^{2}}+{{(b-c)}^{2}}+{{(c-a)}^{2}}=0\] \[\Rightarrow \]\[a=b=c\] \[\therefore \]\[\Delta ABC\] is an equilateral triangle. \[\Rightarrow \]\[A=B=C=\frac{\pi }{3}\] \[\therefore \]\[{{\sin }^{2}}A+{{\sin }^{2}}B+{{\sin }^{2}}C=3{{\sin }^{2}}\frac{\pi }{3}=\frac{9}{4}\]
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