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# The set of all possible values of $'c'$ for which $\overrightarrow a=c \log_3 x\hat i-6\hat j+3\hat k$ and $\overrightarrow b=\log_3x\hat i+2\hat j+2c\log_3x\hat k$ making acute angle for $\forall x\in (0,\infty)$ is ?

$\begin{array}{1 1}\phi \\ (0,1) \\ (0,1] \\ (0,2) \end{array}$

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• A quadratic expression $ax^2+bx+c$ is positive if $a>0$ and $b^2-4ac<0$
Given: angle between $\overrightarrow a$ and $\overrightarrow b$ is acute.
$\Rightarrow\:\overrightarrow a.\overrightarrow b>0$
$\Rightarrow\:c(log_3x)^2-12+6c log_3x>0$ $\forall x\in (0,\infty)$
Let $y=log_3x$
$\Rightarrow\: cy^2+6cy-12>0$
A quadratic expression $ax^2+bx+c$ is positive if $a>0$ and $b^2-4ac<0$
$\therefore\:c>0$ and $36c^2+48c<0$
But if $c>0,$ then $36c^2+48c$ cannot be $<0$
$\Rightarrow$ The set of values of $c$ is $\phi$, the null set.