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Help me solve this - Binomial theorem and its simple applications - JEE Main

The sum of the co-efficients of all odddegree terms in the expansion of
\left ( x+\sqrt{x^{3-1}} \right )^{5} +\left ( x-\sqrt{x^{3-1}} \right )^{5} , \left ( x> 1 \right )
is :

  • Option 1)

    2

  • Option 2)

    -1

  • Option 3)

    0

  • Option 4)

    1

 
Answers (1)
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\left ( x+\sqrt{x^{3}-1} \right )^{5}+\left ( x-\sqrt{x^{3}-1} \right )^{5}

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

2x^{5}+ 20x^{6}-20x^{3}+10x^{7}-20x^{4}+10x

sum od odd degree terms 

= 2-20+10+10=2

 

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.

 

 


Option 1)

2

Option 2)

-1

Option 3)

0

Option 4)

1

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