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JEE Main & Advanced Mathematics Vector Algebra Question Bank Modulus of vector Algebra of vectors

  • question_answer
     If \[(x,\,\,y,\,\,z)\ne (0,\,\,0,\,\,0)\] and \[(\mathbf{i}+\mathbf{j}+3\,\mathbf{k})\,x+(3\,\mathbf{i}-3\mathbf{j}+\mathbf{k})\,y\]\[+(-4\mathbf{i}+5\mathbf{j})\,z=\lambda \,(x\mathbf{i}+y\mathbf{j}+z\mathbf{k}),\] then the value of l will be                                                                                                                    [IIT 1982; RPET 1984]

    A) 2, 0        

    B) 0, ? 2

    C) 1, 0        

    D) 0, ? 1

    Correct Answer: D

    Solution :

    Comparing the coefficients of \[\mathbf{i},\,\,\mathbf{j}\] and \[\mathbf{k},\] the corresponding equations are \[x+3y-4z=\lambda x\] or \[(1-\lambda )x+3y-4z=0\]           ......(i) \[x-(\lambda +3)y+5z=0\]                                                 ......(ii) \[3x+y-\lambda z=0\]                                                         .....(iii) These equations (i), (ii) and (iii) have a non-trivial solution, if \[\left| \begin{matrix}    (1-\lambda ) & 3 & -4  \\    1 & -(\lambda +3) & 5  \\    3 & 1 & -\lambda   \\ \end{matrix} \right|=0\Rightarrow \lambda =0,\,\,-1.\]


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