A) \[\left\{ -\frac{2}{3},\,\,0 \right\}\]
B) \[\left\{ \frac{1}{3},\,\,2 \right\}\]
C) \[\left\{ \frac{2}{3},\,\,0 \right\}\]
D) \[\{2,\,\,7\}\]
Correct Answer: A
Solution :
We have,\[2|\widehat{\mathbf{i}}+x\widehat{\mathbf{j}}+3\widehat{\mathbf{k}}|\] \[=|4\widehat{\mathbf{i}}+(4x-2)\widehat{\mathbf{j}}+2\widehat{\mathbf{k}}|\] \[\Rightarrow \] \[2\sqrt{1+{{x}^{2}}+9}=\sqrt{{{4}^{2}}{{(4x-2)}^{2}}+{{2}^{2}}}\] \[\Rightarrow \] \[4({{x}^{2}}+10)=16{{x}^{2}}+4-16x+4\] \[\Rightarrow \] \[12{{x}^{2}}-16x-16=0\] \[\Rightarrow \] \[3{{x}^{2}}-4x-4=0\] \[\Rightarrow \] \[(3x+2)(x-2)=0\] \[\Rightarrow \] \[x=2,\,\,-\frac{2}{3}\]
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