11th Class Mathematics Trigonometric Identities Question Bank Critical Thinking

  • question_answer If \[a\,{{\cos }^{3}}\alpha +3a\,\cos \alpha \,{{\sin }^{2}}\alpha =m\] and\[a\,{{\sin }^{3}}\alpha +3a\,{{\cos }^{2}}\alpha \sin \alpha =n,\] then \[{{(m+n)}^{2/3}}+{{(m-n)}^{2/3}}\] is equal to

    A) \[2{{a}^{2}}\]

    B) \[2{{a}^{1/3}}\]

    C) \[2{{a}^{2/3}}\]

    D) \[2{{a}^{3}}\]

    Correct Answer: C

    Solution :

    Adding and subtracting the given relation, we get \[(m+n)=a{{\cos }^{3}}\alpha +3a\cos \alpha \,{{\sin }^{2}}\alpha \]\[+3a{{\cos }^{2}}\alpha .\sin \alpha +a{{\sin }^{3}}\alpha \] \[=a{{(\cos \alpha +\sin \alpha )}^{3}}\] and similarly \[(m-n)=a\,\,{{(\cos \alpha -\sin \alpha )}^{3}}\] Thus, \[{{(m+n)}^{2/3}}+{{(m-n)}^{2/3}}\] \[={{a}^{2/3}}{{\{\cos \alpha +\sin \alpha )}^{2}}+{{(\cos \alpha -\sin \alpha )}^{2}}\}\] \[={{a}^{2/3}}\{2({{\cos }^{2}}\alpha +{{\sin }^{2}}\alpha )\}=2{{a}^{2/3}}\].

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