Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main20244 Apr 2024Morning ShiftMathematicsBinomial TheoremActual

Let a=1+ ^2 C ₂ 3 ! + ^3 C ₂ 4 ! + ^4 C ₂ 5 ! + , b =1+ ^1 C ₀+ ^1 C ₁ 1 ! + ^2 C ₀+ ^2 C ₁+ ^2 C ₂ 2 ! + ^3 C ₀+ ^3 C ₁+ ^3 C ₂+ ^3 C ₃ 3 ! + Then 2 b a^2 is equal to

Correct answer

0

Step-by-step solution

aligned & f ( x )=1+ (1+ x ) 1 ! + (1+ x )^2 2 ! + (1+ x )^3 3 ! + . . & e ^ (1+ x ) 1+ x = 1 1+ x +1+ (1+ x ) 2 ! + (1+ x )^2 3 ! + (1+ x )^2 4 ! & coef x ^2 in RHS : 1+ ^2 C ₂ 3 + ^3 C ₂ 4 + = a aligned coeff. x^2 in L.H.S. e (1+x+ x^2 2 ! ) (1-x+ x^2 2 ! ) is e - e + e 2 ! = a aligned & b =1+ 2 1 ! + 2^2 2 ! + 2^3 3 ! + = e ^2 & 2 ~b a ^2 =8 aligned

Practice Binomial Theorem on Quantrex Academy →

More from Binomial Theorem

If 26 ( 2^3 3 12 2 + 2^5 5 12 4 + 2^7 7 12 6 + + 2¹³ 13 12 12 ) = 3¹³ - , then is equal to: 2026If (1 - x^3)¹⁰ = _ r=0 ¹⁰ a_r x^r (1-x)^ 30-2r , then 9a₉ a₁₀ is equal to __________. 2026If the coefficients of the middle terms in the binomial expansions of (1 + x)²⁶ and (1 - x)²⁸ , 0 , are equal, then the value of is: 2026The coefficient of x^2 in the expansion of (2x^2 + 1 x )¹⁰ , x 0 , is : 2026If the sum of the coefficients of x^7 and x¹⁴ in the expansion of ( 1 x^3 - x^4 )^n , x 0 , is zero, then the value of n is __________. 2026In the expansion of (9x- 1 3 x )¹⁸ , x>0 , if the term independent of x is (221)k , then k is equal to: 2026Let the smallest value of k N , for which the coefficient of x^3 in (1+x)^3 + (1+x)^4 + (1+x)^5 + + (1+x)⁹⁹ + (1+kx)¹⁰⁰ , x 0 , is (43n + 101 4 ) (¹⁰⁰C₃ ) for some n N , be p . The 2026If for 3 r 30 , 30 30-r + 3 30 31-r + 3 30 32-r + 30 33-r = m r , then m equals: 2026 Full Binomial Theorem list All JEE Main PYQs