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JEE Main & Advanced Mathematics Vector Algebra Question Bank Self Evaluation Test - Vector Algebra

  • question_answer
    Given that the vectors \[\overline{\alpha }\] and \[\overset{\to }{\mathop{\beta }}\,\] are non-collinear. The values of x and y for which \[\overset{\to }{\mathop{u}}\,-\overset{\to }{\mathop{v}}\,=\overset{\to }{\mathop{w}}\,\] holds true if   \[\overset{\to }{\mathop{u}}\,=2x\overset{\to }{\mathop{\alpha }}\,+y\overset{\to }{\mathop{\beta }}\,,\overset{\to }{\mathop{v}}\,=2\,y\overset{\to }{\mathop{\alpha }}\,+3x\overset{\to }{\mathop{\beta }}\,\] and \[\overset{\to }{\mathop{w}}\,=2\overset{\to }{\mathop{\alpha }}\,-5\overset{\to }{\mathop{\beta }}\,\]are

    A) \[x=2,y=1\]

    B) \[x=1,y=2\]

    C) \[x=-2,y=1\]

    D) \[x=-2,y=-1\]

    Correct Answer: A

    Solution :

    [a] \[\overset{\to }{\mathop{u}}\,-\overset{\to }{\mathop{v}}\,=\overset{\to }{\mathop{w}}\,\] \[\left( 2x\overset{\to }{\mathop{\alpha }}\,+y\overset{\to }{\mathop{\beta }}\, \right)-\left( 2y\overset{\to }{\mathop{\alpha }}\,+3x\overset{\to }{\mathop{\beta }}\, \right)=2\overset{\to }{\mathop{\alpha }}\,-5\overset{\to }{\mathop{\beta }}\,\] \[(2x-2y)\overset{\to }{\mathop{\alpha }}\,+(y-3x)\overset{\to }{\mathop{\beta }}\,=2\overset{\to }{\mathop{\alpha }}\,-5\overset{\to }{\mathop{\beta }}\,\] \[\therefore 2x-2y=2...(i)\] and       \[3x-y=-5...(ii)\] Solving equations (i) and (ii), we get \[x=2\] and \[y=1\]


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