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KVPY Sample Paper KVPY Stream-SX Model Paper-18

  • question_answer
    Let \[{{n}_{1}}=\sin 7+\cos 7,\] \[{{n}_{2}}=\sqrt{\sin 7}+\sqrt{\cos 7},\]\[{{n}_{3}}=\sqrt{1+\sin 14}\] and \[{{n}_{4}}=1.\] Then,

    A) \[{{n}_{2}}>{{n}_{3}}>{{n}_{4}}>{{n}_{1}}\]

    B) \[{{n}_{3}}>{{n}_{4}}>{{n}_{3}}>{{n}_{1}}\]

    C) \[{{n}_{3}}={{n}_{1}}>{{n}_{4}}>{{n}_{2}}\]

    D) \[{{n}_{2}}>{{n}_{1}}={{n}_{3}}>{{n}_{4}}\]

    Correct Answer: D

    Solution :

    Given, \[{{n}_{1}}=\sin 7+\cos 7\]
    \[{{n}_{2}}=\sqrt{\sin 7}+\sqrt{\cos 7}\]
    \[{{n}_{2}}=\sqrt{1+\sin 14}\]
    \[{{n}_{4}}=1\]
    \[n_{1}^{2}={{\sin }^{2}}7+{{\cos }^{2}}7+2\sin 7\cos 7\]
    \[n_{1}^{2}=1+\sin 14\]
    \[{{n}_{1}}=\sqrt{1+\sin 14}={{n}_{3}}\]
    \[\sqrt{1+\sin 14}>1\]
    \[\therefore \]\[{{n}_{3}}>{{n}_{4}}\]
    Hence, \[{{n}_{2}}>{{n}_{1}}={{n}_{3}}>{{n}_{4}}\]


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