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JCECE Engineering JCECE Engineering Solved Paper-2013

  • question_answer
    If\[\mathbf{a}=x\mathbf{i}+(x-1)\mathbf{j}+\mathbf{k}\]and\[\mathbf{b}=(x+1)\mathbf{i}+\mathbf{j}+a\mathbf{k}\] always make an acute angle with each other for every value of\[x\in R\], then

    A) \[a\in (-\infty ,\,\,2)\]                   

    B) \[a\in (2,\,\,\infty )\]

    C) \[a\in (-\infty ,\,\,1)\]                   

    D) \[a\in (1,\,\,\infty )\]

    Correct Answer: D

    Solution :

    \[\mathbf{a}\cdot \mathbf{b}=[x\mathbf{i}+(x-1)\mathbf{j}+\mathbf{k}]\cdot [(cx+1)\mathbf{i}+\mathbf{j}+a\mathbf{k}]\] \[=x(x+1)+x-1+a={{x}^{2}}+2x+a-1\] We must have\[\mathbf{a}\cdot \mathbf{b}>0,\,\,\forall x\in R\] \[\Rightarrow \]               \[{{x}^{2}}+2x+a-1>0,\,\,\forall x\in R\] \[\Rightarrow \]               \[4-4(a-1)<0\] \[\Rightarrow \]               \[a>2\]


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