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KVPY Sample Paper KVPY Stream-SX Model Paper-8

  • question_answer
    Solution (x, y) the system of equation \[x-y=\frac{1}{3}\] and \[{{\cos }^{2}}(\pi x)-{{\sin }^{2}}(\pi y)=\frac{1}{2}\] is given by and \[n\in I.\]

    A) \[\left( \frac{2}{3},\frac{1}{3} \right)\]                

    B) \[\left( \frac{7}{6},\frac{1}{6} \right)\]

    C) \[\left( \frac{13}{6},\frac{11}{6} \right)\]

    D) \[\left( \frac{1}{6},\frac{5}{6} \right)\]

    Correct Answer: C

    Solution :

    \[\cos \pi \,\,(x+y)\cos \pi \,\,(x-y)=\frac{1}{2}\]
    \[\cos \pi \,\,(x+y)=1\]
    \[\pi \,\,(x+y)=2n\pi \]
    \[x+y=2n\]
    \[x-y=\frac{1}{3}\]
    Hence, \[x=n+\frac{1}{6}\]
    \[y=n-\frac{1}{6}\]


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