Test: Binomial Theorem - 2 - Mathematics MCQ

Given expression

Given expansion 

Number of dissimilar  term = (2n + 1) + (2n -1) + ....(2n - 3) + .... + 5 + 3 + 1

Where ao = sum of all absolute terms   = 1 +¹⁰⁰ C₂ .2 + ....
Similarly  a₁, a₂ , ...a₁₀₀ and  b₁, b₂ ...b₁₀₀ are coefficients obtained after simplification.
∴ Total number of terms = 1 + 100 + 100 = 201

Subtract above equations, 


Coefficient of x⁵⁰ in S = coefficient of x⁵⁰ in 

 which is independent of x if 
Hence  required  coefficient = ¹⁰C₄ (-1)⁴ = 210

∴ I + f + F is integer.

I + f + F = 198 ⇒ I + 1 = 198 ⇒ I = 197

⇒ m² - m² + n - n - 2mn = -12
⇒ (m - n)² - (m + n) = -12 ⇒ m + n = 9 + 12 = 21      (2) using (1)
Solving (1) and (2), we get m = 12.

∴ (d) is correct 

coefficient of t²⁴ is (¹²C₁₂ +¹²C₆ + 1)
⇒ coefficient of t²⁴ is (2 +¹²C₆) .

^n+r-1C_r-1

Coefficient of a⁴b³c²d 

Substitute ‘n’ and verify the options.

(1 + a)²ⁿ = C_o + C₁a + C₂a² + … + C₂n a²ⁿ
Differentiate both sides w.r.t. ‘a’.

2ⁿ = 4096, find ⁿC_n/2

(n – 2r)² = n + 2

Put x = 1, x  = -1 and then add.

Substitute a positive integer for ‘n’ and verify.