MCQ
જો $\vec a \,$ અને $\vec b $ એકમ સદિશો હોય કે જેથી સદિશ $\vec a \,\, + \;\,3\vec b \,$ એ $ \,\,7\vec a \,\, - \;\,5\vec b $ ને લંબ હોય , તો $\vec a \,$ અને $ \,\vec b $ વચ્ચેનો ખૂણો મેળવો.
- A$\pi /6$
- B$\pi /4$
- ✓$\pi /3$
- D$\pi /2$
આપણી પાસે $\left( {\vec a \,\, + \;\,3\vec b } \right)\,\, \bot \,\,\left( {2\vec a \,\, - \;\,5\vec b } \right)\,\,\, $
$\Rightarrow \,\,\left( {\vec a \,\, + \;\,3\vec b } \right)\,\,.\,\,\left( {7\vec a \,\, - \;\,5\vec b } \right)\,\, = \,\,0$
$ \Rightarrow \,\,7\,|\vec a {|^2}\,\, + \;\,16\,\left( {\vec a .\,\,\vec b } \right)\, - \,\,\left( {15} \right)\,\,\,\,|\vec b {|^2}\, = \,\,0$
$ \Rightarrow \,\,7\,\, + \;\,16\,\,\cos \,\,\theta \,\, - \,\,15\,\, = \,\,0\,\, $
$\Rightarrow \,\,\cos \,\theta \,\, = \,\,\frac{1}{2}\,\, $
$\Rightarrow \,\,\theta \,\, = \,\,\frac{\pi }{3}$
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$E_1=\{A \in S: \operatorname{det} A=0\} \text { and }$ $E_2=\{A \in S: \text { sum of entries of } A \text { is } 7\}.$ If a matrix is chosen at random from $S$, then the conditional probability $P\left(E_1 \mid E_2\right)$ equals. . . . . . . .