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  • Let  be the set of all <img alt="\mathrm{a \in \mathbb{R}}" src="/latex-image/?%5Cmathrm%7Ba%20%5Cin%20%5Cmathbb%7BR%7D%7

Let \mathrm{S} be the set of all \mathrm{a \in \mathbb{R}} for which the angle between the vectors \mathrm{\overrightarrow{u}=a\left(\log _{e} b\right) \hat{i}-6 \hat{j}+3 \hat{k}} and \mathrm{\overrightarrow{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 \mathrm{S} is equal to:

Option: 1

\mathrm{\left(-\infty,-\frac{4}{3}\right)}


Option: 2

\Phi


Option: 3

\left(-\frac{4}{3}, 0\right)


Option: 4

\left(\frac{12}{7}, \infty\right)


Answers (1)

best_answer

\begin{aligned} &\mathrm{\vec{u} \cdot \vec{v}=a(\log b)^2-12+6 a(\log b)>0}\\ &\text{For } \mathrm{ b>1 \Rightarrow \log b>0}\\ &\text{Let }\mathrm{ \log b=t \Rightarrow t>0}\\ &\Rightarrow \mathrm{a t^2+6 a t-12>0} \text{ for all }\mathrm{ t>0}\text{ But for}\mathrm{ t=0 \Rightarrow f(t)=-12}\\ &\therefore \mathrm{a t^2+6 a t-12 } \text{ cannot be postive foer all } \mathrm{ t>0 } \text{ for any value of a} \\ &\therefore \text{option (B)} \end{aligned}

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Pankaj

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