JEE Main & Advanced Sample Paper JEE Main Sample Paper-38

  • question_answer
    Statement-1: If \[n\] is an odd prime, then integral part of\[{{(\sqrt{5}+2)}^{n}}\]is divisible by\[n\].
    Statement-2: If \[n\] is prime, then\[^{n}{{c}_{1}},\,{{\,}^{n}}{{c}_{2}},\,{{\,}^{n}}{{c}_{3}},...{{,}^{n}}{{c}_{n-1}}\], must be divisible by\[n\].

    A)  Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

    B)  Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

    C)  Statement-1 is true, Statement-2 is false.

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

    Correct Answer: A

    Solution :

     Let\[{{\left( \sqrt{5}+2 \right)}^{n}}=I+f\], where\[I\]is an integer and\[0<f<1\] Let,\[{{\left( \sqrt{5}-2 \right)}^{n}}=f';\,\,0<f'<1\] \[\Rightarrow \]\[{{\left( \sqrt{5}+2 \right)}^{n}}-{{\left( \sqrt{5}-2 \right)}^{n}}=\]Integer\[(\because \,\,n\]is odd) \[\because \]\[{{\left( \sqrt{5}+2 \right)}^{n}}-{{\left( \sqrt{5}-2 \right)}^{n}}=2\left[ ^{n}{{c}_{1}}{{2.5}^{\frac{n-1}{2}}}{{+}^{n}}{{c}_{3}}{{2}^{3}}{{.5}^{\frac{n-3}{2}}}+... \right]\]\[\because \]\[(f=f')\] \[\Rightarrow \]\[I\]is divisible by \[20n\] on using statement\[-2.\]

adversite


You need to login to perform this action.
You will be redirected in 3 sec spinner

Free
Videos