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  • How to solve this problem- - Binomial theorem and its simple applications - JEE Main-2

If n is the degree of the polynomial, \left [ \frac{2}{\sqrt{5x^{3}+1}-\sqrt{5x^{3}-1}} \right ]^{8}+\left [ \frac{2}{\sqrt{5x^{3}+1}+\sqrt{5x^{3}-1}} \right ]^{8}  and m is the coefficient of xn in it, then the ordered pair (n, m) is equal to :

 

  • Option 1)

    \left ( 24,\left ( 10 \right )^{8} \right )

     

     

     

  • Option 2)

    \left ( 8,5\left ( 10 \right )^{4} \right )

  • Option 3)

    \left ( 12,\left (20 \right )^{4} \right )

  • Option 4)

    \left ( 12,8\left (10 \right )^{4} \right )

 

Answers (1)

best_answer

As we learned 

 

Properties of Binomial Theorem -

\left ( x+a \right )^{n}+\left ( x-a \right )^{n}= 2\left ( ^{n}c_{0} \, x^{n}+ ^{n}c_{2}\, x^{n-2}\, a^{2}+---\right )

- wherein

Sum of odd terms or even Binomial coefficients

 

 

Properties of Binomial Theorem -

\dpi{120} \left ( x+a \right )^{n}-\left ( x-a \right )^{n}= 2\left ( ^{n}c_{1} \, x^{n-1}a+ ^{n}c_{3}\, x^{n-3}\, a^{3}+---\right )

 

- wherein

Sum of even terms or odd Binomial coefficients.

 

 

2\times \frac{\sqrt{\left ( 5x^{3}+1 \right )}+\sqrt{\left ( 5x^{3} -1\right )^{8}}}{2}+\left ( \sqrt{5x^{3}+1} -\sqrt{5x^{3}-1}\right )^{8}

=2\left [ _{0}^{8}\textrm{C} \left ( 5x^{3}+1 \right )^{4}+_{2}^{8}\textrm{C}\left ( 5x^{3}+1 \right )^{3}\left ( 5x^{3}-1 \right )+_{4}^{8}\textrm{C}\left ( 5x^{3} +1\right )^{2}\left ( 5x^{3} -1\right )^{2}+_{6}^{8}\textrm{C}\left ( 5x^{3} +1\right )\left ( 5x^{3} -1\right )^{3}+_{8}^{8}\textrm{C}\left ( 5x^{3}-1 \right )^{4}\right ]

Thus degree = 12

Because of \left ( 5x^{3} +1\right )^{3}\left ( 5x^{3} -1\right )

9+3=12

Coeff of x12 = =2\left [ 8\times 5^{4}+28\times 5^{4} +70.5^{4}+28.5^{4}+5^{4}\right ]

=2\cdot 2^{7}\cdot 5^{4}=\left ( 20 \right )^{4}

 

 


Option 1)

\left ( 24,\left ( 10 \right )^{8} \right )

 

 

 

Option 2)

\left ( 8,5\left ( 10 \right )^{4} \right )

Option 3)

\left ( 12,\left (20 \right )^{4} \right )

Option 4)

\left ( 12,8\left (10 \right )^{4} \right )

Posted by

Himanshu

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