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JEE Main & Advanced Sample Paper JEE Main Sample Paper-2

  • question_answer
    Directions: Read the fallowing questions and choose: Statement 1: \[{{x}^{3}}+3{{x}^{2}}+9x+\sin x=0\] has exactly one real root in R. Statement 2: \[{{x}^{3}}+3{{x}^{2}}+9x+\sin x=0\] does not satisfy Rolle?s Theorem in any closed interval [a, b].

    A)  Both statements are True, Statement-2 explains Statement-1.

    B)  Both statements are True, Statement-2 does not explain Statement-1.

    C)  Statement-1 is True, Statement-2 is False.

    D)  Statement-1 is False, Statement-2 is true.

    Correct Answer: B

    Solution :

    \[f(x)={{x}^{3}}+6x+9x+\sin x\] \[f'(x)=3{{x}^{2}}+6x+9+\cos x>5\,\forall \,x\] as \[3{{x}^{2}}+6x+9\ge 6\,\forall \,x\] \[\Rightarrow \,\,f(x)\] is strictly increasing function, whose range is R, hence exactly one real root. Also, \[f(x)\] is one-one. \[\therefore \,\,\,\,f(a)\ne \,f(b)\,\forall \,a,\,b\,\in \,R,\,a\ne b\]


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