A) \[\left\{ -\frac{2}{3},2 \right\}\]
B) \[\left\{ \frac{1}{3},2 \right\}\]
C) \[\left\{ \frac{2}{3},0 \right\}\]
D) \[\left\{ 2,7 \right\}\]
Correct Answer: A
Solution :
Since, the vector \[\hat{i}+x\hat{j}+3\hat{k}\] doubled in magnitude, then it becomes\[4\hat{i}+(4x-2)\hat{j}+2\hat{k}\]. \[\therefore \] \[3\left| \hat{i}+x\,\hat{j}+3\hat{k} \right|=\left| 4\,\hat{i}+(4x-2)\,\hat{j}+2\hat{k} \right|\] \[\Rightarrow \] \[2\sqrt{1+{{x}^{2}}+9}=\sqrt{16+{{(4x-2)}^{2}}+4}\] \[\Rightarrow \] \[40+4{{x}^{2}}=20+{{(4x-2)}^{2}}\] \[\Rightarrow \,12{{x}^{2}}-16x-16=0\] \[\Rightarrow \] \[3{{x}^{2}}-4x-4=0\] \[\Rightarrow \] \[(x-2)\,(3x+2)=0\] \[\Rightarrow \] \[x=2,-\frac{2}{3}\]
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