Verified Solution • 2025–26 Session moderate

QJEEM22Ma5401: JEE Main 2022 Physics - Class 11 JEEM22Ma5401 Solution: Let S be the set of all a for which the angle between the…

Let S be the set of all a $\in R$ for which the angle between the vectors $\vec{u}=a\left(\log _{e} b\right) \hat{i}-6 \hat{j}+3 \hat{k}$ and $\vec{v}=\left(\log _{e} b\right) \hat{i}+2 \hat{j}+2 a\left(\log _{e} b\right) \hat{k}$ , $(b>1)$ is acute. Then S is equal to :

Multiple Choice Options

A $\left(-\infty,-\frac{4}{3}\right)$
B $\Phi$
C $\left(-\frac{4}{3}, 0\right)$
D $\left(\frac{12}{7}, \infty\right)$

Step-by-Step Verified Solution

$\overrightarrow u = a({\log _e}b)\widehat i - 6\widehat j + 3\widehat k$

$\overrightarrow v = ({\log _e}b)\widehat i + 2\widehat j + 2a({\log _e}b)\widehat k$

For acute angle $\overrightarrow u \,.\,\overrightarrow v > 0$

$\Rightarrow a{({\log _e}b)^2} - 12 + 6a({\log _e}b) > 0$

$\because$ $b > 1$

Let ${\log _e}b = t \Rightarrow t > 0$ as $b > 1$

$a{t^2} + 6at - 12 > 0\,\,\,\,\,\,\,\forall t > 0$

$\Rightarrow a \in \phi$

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