KVPY Sample Paper KVPY Stream-SX Model Paper-25

  • question_answer
    The number of solutions of the equation \[{{\sin }^{3}}x\cos x+{{\sin }^{2}}x{{\cos }^{2}}x+\sin x{{\cos }^{3}}x=1,\] in the interval \[[0,2\pi ],\] is

    A) 4

    B) 2

    C) 1

    D) 0

    Correct Answer: D

    Solution :

    The given equation can be written as \[\sin x\cos x[{{\sin }^{2}}x+\sin x\cos x+{{\cos }^{2}}x]=1\] or \[\sin x\cos x\left[ 1+\sin x\cos x \right]=1\] or  \[\sin 2x\,[2+{{\sin }^{2}}x]=4\]\[\Rightarrow \]\[\sin 2x=\frac{-2\pm \sqrt{4+16}}{2}=-1\pm \sqrt{5}\]Which is not possible.

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