A) \[3\sqrt{\frac{3}{2}}<r<5\sqrt{\frac{3}{2}}\]
B) \[0<r\le \sqrt{\frac{3}{2}}\]
C) \[\sqrt{\frac{3}{2}}<r\le 3\sqrt{\frac{3}{2}}\]
D) \[r\ge 5\sqrt{\frac{3}{2}}\]
Correct Answer: D
Solution :
\[\vec{a}\times \vec{b}=\left| \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ 3 & 2 & x \\ 1 & -1 & 1 \\ \end{matrix} \right|\] \[=(2+x)\hat{i}+(x-3)\hat{j}-5k\] \[\left| \vec{a}\times \vec{b} \right|=\sqrt{4+{{x}^{2}}+4x+{{x}^{2}}+9-6x+25}\] \[=\sqrt{2{{x}^{2}}-2x+38}\] \[\Rightarrow \]\[\left| \vec{a}\times \vec{b} \right|\ge \sqrt{\frac{75}{2}}\]\[\Rightarrow \]\[\left| \vec{a}\times \vec{b} \right|\ge 5\sqrt{\frac{3}{2}}\]
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