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JEE Main & Advanced Mathematics Determinants & Matrices Question Bank Self Evaluation Test - Determinats

  • question_answer
    If in a triangle ABC, \[\left| \begin{matrix}    1 & \sin A & {{\sin }^{2}}A  \\    1 & \sin B & {{\sin }^{2}}B  \\    1 & \sin C & {{\sin }^{2}}C  \\ \end{matrix} \right|=0\] then the triangle is

    A) Equilateral or isosceles

    B) Equilateral or right-angled

    C) Right angled or isosceles

    D) None of these

    Correct Answer: A

    Solution :

    [a] \[\left| \begin{matrix}    1 & \sin A & {{\sin }^{2}}A  \\    1 & \sin B & {{\sin }^{2}}B  \\    1 & \sin C & {{\sin }^{2}}C  \\ \end{matrix} \right|=0\] \[\Rightarrow (sinA-sinB)(sinB-sinC)(sinC-sinA)=0\] \[\Rightarrow \sin A=\sin B\,\,or\,\,\sin B=\sin C\,\,or\,\,\sin C=\sin A\] \[\therefore \] at least two of A, B, C are equal. Hence the triangle is isosceles or equilateral.


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