MHTCET 11 May 2024 Shift 2 Paper

Concept:Use the vector triple product expansion and the linear independence of non-coplanar vectors.Explanation:Given:$< br>\overline{a} \times (\overline{b}times \overline{c})=\frac{\overline{b}+\overline{c}}{\sqrt{2}} < br>$Expand using the vector triple product formula:$< br>(\overline{a}\cdot \overline{c})\overline{b}-(\overline{a}\cdot \overline{b})\overline{c}=\frac{\overline{b}+\overline{c}}{\sqrt{2}} < br>$Rearrange all terms to one side:$< br>\left(\overline{a}\cdot \overline{c}-\frac{1}{\sqrt{2}}\right)\overline{b}-\left(\overline{a}\cdot \overline{b}+\frac{1}{\sqrt{2}}\right)\overline{c}=0 < br>$Since $\overline{a},\overline{b},\overline{c}$ are non-coplanar unit vectors, they are linearly independent. Therefore, each coefficient must be zero.So:$< br>\overline{a}\cdot \overline{b}+\frac{1}{\sqrt{2}}=0 < br>$$< br>\overline{a}\cdot \overline{b}=-\frac{1}{\sqrt{2}} < br>$Using the dot product formula:$< br>|\overline{a}||\overline{b}|\cos\theta=-\frac{1}{\sqrt{2}} < br>$Because they are unit vectors, $|\overline{a}|=|\overline{b}|=1$:$< br>\cos\theta=-\frac{1}{\sqrt{2}} < br>$Hence:$< br>\theta=\frac{3\pi}{4} < br>$Answer:$< br>\frac{3\pi}{4} < br>$So, the correct option is A.