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JEE Main & Advanced Sample Paper JEE Main Sample Paper-30

  • question_answer
    If \[{{p}^{th}},{{q}^{th}},{{r}^{th}}\] terms of a G.P. are the positive numbers \[a,b,c\] respectively then angle between the vectors\[(\log {{a}^{2}})\hat{i}+(\log {{b}^{2}})\hat{j}+(\log {{c}^{2}})\hat{k}\] and \[(q-r)\hat{i}+(r-p)\hat{j}+(p-q)\hat{k}\]

    A)  \[\frac{\pi }{3}\]                                

    B)  \[\frac{\pi }{2}\]

    C) \[{{\sin }^{-1}}\left( \frac{1}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}} \right)\]  

    D) \[{{\cos }^{-1}}\left( \frac{pqr}{\sqrt{{{p}^{2}}+{{q}^{2}}r}} \right)\]

    Correct Answer: B

    Solution :

    \[{{\vec{v}}_{1}}=(\log {{a}^{2}})\,\hat{i}+(\log {{b}^{2}})\hat{j}+(\log {{c}^{2}})\hat{k}\] \[{{\vec{v}}_{2}}=(q-r)\hat{i}+(r+p)\hat{j}+(p-q)\hat{k}\] \[{{\vec{v}}_{1}}\cdot {{\vec{v}}_{2}}=({{\log }^{2}})(q-r)+(\log {{b}^{2}})(r-p)+(\log {{c}^{2}})(p-q)\] \[\Rightarrow a=A{{R}^{p-1}},\,b=A{{R}^{q-1}},c=A{{R}^{r-1}}\] \[\therefore {{v}_{1}}\cdot {{v}_{2}}=\log ({{a}^{2(q-r)}}\cdot {{b}^{2(r-p)}}\cdot {{c}^{2(p-q)}}=0\]


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