Binomial Theorem questions
776 published questions with worked solutions.
- Let the smallest value of k ∈ ℕ, for which the coefficient of x³ in (1+x)³ + (1+x)⁴ + (1+x)⁵ + … + (1+x)⁹⁹ + (1+kx)¹⁰⁰, x ≠ 0, is...
- In the expansion of (9x - 1 3 x )^ 18 , x > 0, if the term independent of x is (221)k, then k is equal to:
- The coefficient of x² in the expansion of (2x^2+ 1x )^ 10 , x≠0, is:
- If the sum of the coefficients of x⁷ and x¹⁴ in the expansion of ( 1 x^3 -x^4 )^n, x≠0, is zero, then the value of n is _______.
- If the coefficients of the middle terms in the binomial expansions of (1+αx)²⁶ and (1−αx)²⁸, α≠0, are equal, then the value of α...
- If 26 ( 2^3 3 12 2+ 2^5 5 12 4+ 2^7 7 12 6+ + 2^ 13 13 12 12 )=3^ 13 - , then α is equal to:
- The sum of the coefficient of x^ 2/3 and x^ −2/5 in the binomial expansion of (x^ 2/3 + 12x^ -2/5 )^9 is :
- The coefficient of x⁷⁰ in x²(1 + x)⁹⁸ + x³(1 + x)⁹⁷ + x⁴(1 + x)⁹⁶ + .... + x⁵⁴(1 + x)⁴⁶ is ⁹⁹C_p − ⁴⁶C_q. Then a possible value...
- The coefficient of x²⁰¹² in the expansion of (1 – x)²⁰⁰⁸(1 + x + x²)²⁰⁰⁷ is equal to ____.
- If A denotes the sum of all the coefficients in the expansion of (1 - 3x + 10x^2)^n and B denotes the sum of all the coefficients...
- If ^ 11 C_1 2 + ^ 11 C_2 3 + + ^ 11 C_9 10 = nm with gcd(n, m) = 1, then n + m is equal to ________.
- Let = _ k=0 ^n nk^2 k+1 and = _ k=0 ^ n-1 nk n k+1 k+2 . If 5α = 6β, then n equals ________.
- Number of integral terms in the expansion of 7^ 12 + 11^ 16 ^ 824 is equal to ________.
- If for some m, n: ⁶Cₘ + 2(⁶Cₘ₊₁) + ⁶Cₘ₊₂ > ⁸C₃ and ⁿ⁻¹P₃ : ⁿP₄ = 1 : 8, then ⁿP_ m+1 + ⁿ⁺¹Cₘ is equal to
- Let the coefficient of xʳ in the expansion of (x + 3)ⁿ⁻¹ + (x + 3)ⁿ⁻²(x + 2) + (x + 3)ⁿ⁻³(x + 2)² + ....... (x + 2)ⁿ⁻¹ be αᵣ. If...
- In the expansion of (1 + x)(1 - x^2) (1 + 3x + 3 x^2 + 1 x^3 )^5, x ≠ 0, the sum of the coefficients of x³ and x⁻¹³ is equal to...
- If the sum of the coefficients in the expansion of (x+y)^n is 4096, then the greatest coefficient in the expansion is _________.
- If n is the number of irrational terms in the expansion of (3^ 1/4 +5^ 1/8 )^ 60 , then (n–1) is divisible by :
- Let [x] denote greatest integer less than or equal to x. If for n N, (1-x+x^3)^n= _ j=0 ^ 3n a_jx^j then _ j=0 ^ [ 3n 2 ] a_ 2j...
- The value of _ r=0 ^6( ^6C_r. ^6C_ 6-r ) is equal to :
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