A) \[a<x<1/2\]
B) \[1/2<x<15\]
C) \[x>1/2\]or\[x<0\]
D) None of these
Correct Answer: D
Solution :
The angle between \[\mathbf{a}\] and \[\mathbf{b}\] is obtuse. \[\therefore \] \[\mathbf{a}\cdot \mathbf{b}<0\] \[(2{{x}^{2}}\widehat{\mathbf{i}}+4x\widehat{\mathbf{j}}+\widehat{k})\cdot (7\widehat{\mathbf{i}}-2\widehat{\mathbf{j}}+x\widehat{\mathbf{k}})<0\] \[\Rightarrow \] \[14{{x}^{2}}-8x+x<0\] \[\Rightarrow \] \[7x(2x-1)<0\] \[0<x<1/2\] ... (i) The angle between \[\mathbf{b}\] and \[Z-axis\] is acute and less than\[\pi /6\]. \[\therefore \] \[\frac{\mathbf{b}\cdot \mathbf{k}}{|\mathbf{b}||\widehat{\mathbf{k}}|}>\cos \frac{\pi }{6}\] \[\left[ \because \,\,\theta <\frac{\pi }{6}\Rightarrow \cos \theta >\cos \frac{\pi }{6} \right]\] \[\Rightarrow \] \[\frac{(7\widehat{\mathbf{i}}-2\widehat{\mathbf{j}}+x\mathbf{k})\widehat{\mathbf{k}}}{\sqrt{{{7}^{2}}+{{2}^{2}}+{{x}^{2}}|1|}}>\frac{\sqrt{3}}{2}\] \[\Rightarrow \] \[\frac{x}{\sqrt{{{x}^{2}}+53}}>\frac{\sqrt{3}}{2}\] \[\Rightarrow \] \[2x>\sqrt{3}\sqrt{{{x}^{2}}+53}\] \[\Rightarrow \] \[4{{x}^{2}}>3{{x}^{2}}+159\] \[\Rightarrow \] \[{{x}^{2}}>159\] \[\Rightarrow \] \[x>\sqrt{159}\]or\[x<-\sqrt{159}\] ? (ii) Clearly, Eqs. (i) and (ii) cannot hold together,
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