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Question

The values of $$x$$ for which the angle between the vectors $$\overrightarrow { a } =x\hat { i } -3\hat { j } -\hat { k } $$ and $$\overrightarrow { b } =2x\hat { i } +x\hat { j } -\hat { k } $$ is acute, and the angle between the vectors $$\overrightarrow{b}$$ and the $$y$$-axis of ordinates is obtuse, are


A
1,2
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B
2,3
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C
All x<0
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D
All x>0
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Solution

The correct option is C All $$x<0$$
Since the angle between the vectors $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$ is acute and the angle between $$\overrightarrow{b}$$ and the $$y$$-axis is obtuse, therefore, $$\overrightarrow{a}.\overrightarrow{b}>0$$ and $$\overrightarrow{b}.\hat{j}<0$$
$$\Rightarrow 2{ x }^{ 2 }-3x+1>0$$ and $$x<0$$
$$\Rightarrow \left( 2x-1 \right) \left( x-1 \right) >0$$ and $$x<0$$
$$\displaystyle(x<\dfrac{1}{2}$$ or $$x>1)$$ and $$x<0\Rightarrow$$ all $$x<0$$

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