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  • Confused! kindly explain, - Binomial theorem and its simple applications - JEE Main-3

The sum of the co-efficients of all even degree terms in x in the expansion of (x+\sqrt{x^{3}-1})^{6}+(x-\sqrt{x^{3}-1})^{6},(x>1) is equal to :
 

  • Option 1)

    32

  • Option 2)

    26

  • Option 3)

    24

     

  • Option 4)

    29

 

Answers (1)

best_answer

(x+\sqrt{x^{3}-1})^{6}+(x-\sqrt{x^{3}-1})^{6}\: \: \: \: \: x>1

=2\left ( ^{6}C_{o}\; x^{6}+ ^{6}C_{2}\; x^{2}(x^{3}-1)+^{6}C_{4}x^{A}(x^{3}-1)^{2}+^{6}C_{6}(x^{3}-1)^{3}\right )

\therefore 2\left ( ^{6}C_{o}\; x^{6}+ ^{6}C_{2}(x^{5}-x^{2})+^{6}C_{4}x^{4}(x^{6}-2x^{3}+1)^{2}+^{6}C_{6}(x^{9}-3x^{6}+3x^{3}-1)\right )

\therefore Sum of the coefficient of even powers

2(^{6}C_{o}-^{6}C_{2}+^{6}C_{4}+C_{4}-3-^{6}C_{6}-^{6}C_{6})

=2\left ( 1-15+15+15-3-1 \right )=24


Option 1)

32

Option 2)

26

Option 3)

24

 

Option 4)

29

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