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J & K CET Engineering J and K - CET Engineering Solved Paper-2005

  • question_answer
    If the vectors \[2\hat{i}+\hat{j}-\hat{k},-\hat{i}+2\hat{j}+\lambda \hat{k}\]  and \[-5\,\hat{i}+2\hat{j}-\hat{k}\]are coplanar, then the value of \[\lambda \] is equal to

    A)  \[-13\]          

    B)  \[13/9\]

    C)  \[-13/9\]        

    D)  \[-9/13\]

    Correct Answer: C

    Solution :

    Let \[\vec{a}=2\hat{i}+\hat{j}-\hat{k},\] \[\vec{b}=-\hat{i}+2\hat{j}+\lambda \hat{k}\] and \[\vec{c}=-5\hat{i}+2\hat{j}-\hat{k}\] are coplanar \[\therefore \]    \[[\vec{a}\vec{b}\vec{c}]=0\Rightarrow \left| \begin{matrix}    2 & 1 & -1  \\    -1 & 2 & \lambda   \\    -5 & 2 & -1  \\ \end{matrix} \right|=0\] \[\Rightarrow \]   \[2(-2-2\lambda )-1(1+5\lambda )-1(-2+10)=0\] \[\Rightarrow \] \[-4-4\lambda -1-5\lambda -8=0\] \[\Rightarrow \] \[-9\lambda =13\] \[\Rightarrow \] \[\lambda =\frac{-13}{9}\]


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