PrepWiser › Learn › Mathematics › Binomial TheoremBinomial Theorem — JEE Main PYQs & Practice Questions Binomial Theorem is a Mathematics chapter in the JEE Main syllabus. This page brings together 212 previous-year and practice questions on Binomial Theorem, organised into 9 topics: Binomial Expansion, General Term, Middle Term, Applications of Binomial Theorem, General and Middle Term, Numerically Greatest Term, Coefficient Problems, Binomial Coefficient Identities and Multinomial Theorem. Most questions come with a step-by-step solution you can work through free in PrepWiser.
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Binomial Expansion The number of integral terms in the expansion of ( 512 + 718 )¹⁰¹⁶ is: (x + y)ⁿ = Σ C(n, r) x^(n-r) y^r. 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+… …… If the expansion of (a + b)ⁿ has 13 terms, then n = The remainder, when 19²⁰⁰+23²⁰⁰ is divided by 49 , is ___________. If the second, third and fourth terms in the expansion of (x+y)ⁿ are 135, 30 and 10/3… If α and β be the coefficients of x⁴ and x² respectively in the expansion of ( x + √(x²… The sum of all binomial coefficients in (a + b)ⁿ is x. If the co-efficient of x⁷ and x⁸ in the expansion of ( 2 + x/3 )ⁿ are equal, then the… If the constant term in the expansion of (⁵√3/x+(2 x)/∛5)¹², x ≠ 0, is α × 2⁸ × ⁵√3… If 2¹⁰ + 2⁹.3¹ + 2⁸ .3² +.....+ 2.3⁹ + 3¹⁰ = S - 2¹¹, then S is equal to : If the sum of the series (1/2-1/3)+(1/2²-1/(2 · 3)+1/3²)+(1/2³-1/(2² · 3)+1/(2 ·… If the coefficients of x and x² in (1+x)ᵖ(1-x)^q are 4 and -5 respectively, then 2 p+3 q… If the sum of the coefficients of all the positive powers of x, in the Binomial… The sum of coefficients in any binomial expansion equals 2ⁿ. For an integer n ≥ 2, if the arithmetic mean of all coefficients in the binomial… In the expansion of (1+x)(1-x²)(1+3/x+3/x²+1/x³)⁵, x ≠ 0, the sum of the coefficients of… If p denotes the fractional part of the number p, then { 3²⁰⁰/8 }, is equal to : If A denotes the sum of all the coefficients in the expansion of (1-3 x+10 x²)ⁿ and B… The sum of all rational terms in the expansion of (2+√3)⁸ is : General Term If ( 3⁶/4⁴ )k is the term, independent of x, in the binomial expansion of ( x/4 - 12/x²… If the term independent of x in the expansion of ( 3/2x² - 1/3x )⁹ is k, then 18 k is… If the coefficients of x⁷ in ( x² + 1/bx )¹¹ and x⁻⁷ in ( x - 1/bx² )¹¹, b ≠ 0, are… The sum of all those terms which are rational numbers in the expansion of (2 1/3 + 3 1/4… The number of rational terms in the binomial expansion of ( 4^(1/4) + 5^(1/6) )¹²⁰ is… If the constant term, in binomial expansion of ( 2xʳ + 1/x² )¹⁰ is 180, then r is equal… Let α>0, be the smallest number such that the expansion of (x^(2/3)+2/x³)³⁰ has a term β… The coefficient of x⁵ in (1 + x)¹⁰ is: The coefficient of x⁻⁶, in the expansion of ((4 x)/5+5/(2 x²))⁹, is The number of positive integers k such that the constant term in the binomial expansion… The coefficient of x⁵ in the expansion of (2 x³-1/(3 x²))⁵ is : Let [t] denote the greatest integer ≤ t. If the constant term in the expansion of (3… Number of integral terms in the expansion of {7^((1/2))+11^((1/6))}⁸²⁴ is equal to… Let the coefficients of third, fourth and fifth terms in the expansion of ( x + a/x²… The natural number m, for which the coefficient of x in the binomial expansion of ( xᵐ +… If the term independent of x in the expansion of (√a x²+1/(2 x³))¹⁰ is 105 , then a² is… In (2x + 3)⁹, the coefficient of x⁶ is: The coefficient of x¹⁸ in the expansion of (x⁴-1/x³)¹⁵ is __________. In the expansion of (9x-1/3√x)¹⁸,x>0, if the term independent of x is (221)k, then k is… If the coefficient of x¹⁰ in the binomial expansion of ( √x/5^(1/4) + √5/x^(1/3) )⁶⁰ is… Middle Term Applications of Binomial Theorem General and Middle Term Let the ratio of the fifth term from the beginning to the fifth term from the end in the… Let α > 0, β > 0 be such that α³ + β² = 4. If the maximum value of the term independent… Suppose A and B are the coefficients of 30ᵗʰ and 12ᵗʰ terms respectively in the binomial… The term independent of x in the expansion of [ (x + 1)/(x^(2/3) - x^(1/3) + 1) - (x -… If the coefficient of x¹⁵ in the expansion of (a x³+1/(b x^(1 / 3)))¹⁵ is equal to the… If n is the number of irrational terms in the expansion of ( 3^(1/4) + 5^(1/8) )⁶⁰, then… A possible value of 'x', for which the ninth term in the expansion of { 3^(log ₃√(25ˣ⁻¹… In the expansion of (∛2+1/∛3)ⁿ, n ∈ N, if the ratio of 15ᵗʰ term from the beginning to… If the 1011ᵗʰ term from the end in the binominal expansion of ((4 x)/5-5/(2 x))²⁰²² is… The number of integral terms in the expansion of (3^(1/2)+5^(1/4))⁶⁸⁰ is equal to… For two positive real numbers a and b such that 1/a² + 1/b³ = 4, then minimum value of… Let m and n be the coefficients of seventh and thirteenth terms respectively in the… The sum of all rational terms in the expansion of (2^(1/5)+5^(1/3))¹⁵ is equal to : The term independent of 'x' in the expansion of ( (x + 1)/(x^(2/3) - x^(1/3) + 1) - (x -… If the ratio of the fifth term from the beginning to the fifth term from the end in the… For a positive integer n, ( 1 + 1/x )ⁿ is expanded in increasing powers of x. If three… If the maximum value of the term independent of t in the expansion of (t²… The term independent of x in the expansion of (1 - x² + 3x³)( 5/2x³ - 1/5x² )¹¹, x ≠ 0… Let the coefficients of three consecutive terms Tᵣ, Tᵣ₊₁ and Tᵣ₊₂ in the binomial… If the fourth term in the expansion of (x + x^(log ₂x))⁷ is 4480, then the value of x… Numerically Greatest Term Coefficient Problems The coefficient of x⁴⁸ in (1+x)+2(1+x)²+3(1+x)³+… +100(1+x)¹⁰⁰ is equal to The coefficient of x⁴ in the expansion of (1 + x + x² + x³)⁶ in powers of x, is ______. Let [ x ] denote greatest integer less than or equal to x. If for n∈N, (1 - x + x³)ⁿ =… Let (a+b x+c x²)¹⁰=Σᵢ₌₀²⁰ pᵢ xⁱ, a, b, c ∈ ℕ. If p₁=20 and p₂=210, then 2(a+b+c) is… If the sum of the coefficients of all even powers of x in the product (1 + x + x² +… If aᵣ is the coefficient of x¹⁰⁻ʳ in the Binomial expansion of (1 + x)¹⁰, then Σᵣ₌₁¹⁰… If the sum of the co-efficient of all the positive even powers of x in the binomial… The sum of the coefficient of x^(2 / 3) and x^(-2 / 5) in the binomial expansion of… Let (1 + x + 2x²)²⁰ = a₀ + a₁x + a₂x² + .... + a₄₀x⁴⁰. Then a₁ + a₃ + a₅ + ..... + a₃₇… The coefficient of x²⁰¹² in the expansion of (1-x)²⁰⁰⁸(1+x+x²)²⁰⁰⁷ is equal to _________. If the coefficients of x and x² in the expansion of (1+x)ᵖ(1-x)^q, p, q ≤ 15, are -3 and… If for some positive integer n, the coefficients of three consecutive terms in the… The coefficient of x⁷⁰ in x²(1+x)⁹⁸+x³(1+x)⁹⁷+x⁴(1+x)⁹⁶+…+x⁵⁴(1+x)⁴⁶ is ⁹⁹ Cₚ- ⁴⁶ C_q… If the Coefficient of x³⁰ in the expansion of (1+1/x)⁶(1+x²)⁷(1-x³)⁸ ; x ≠ 0 is α, then… Let (1+x+x²)¹⁰=a₀+a₁ x+a₂ x²+…+a₂₀ x²⁰. If (a₁+a₃+a₅+…+a₁₉)-11 a₂=121 k, then k is equal… The coefficient of x⁴ is the expansion of (1 + x + x²)¹⁰ is _____. (2x - 3y)⁴ = ... The coefficient of x³y is: If a, b and c are the greatest value of ¹⁹Cₚ, ²⁰C q and ²¹Cᵣ respectively, then : If the coefficient of x⁷ in ( ax - 1/bx² )¹³ and the coefficient of x⁻⁵ in ( ax + 1/bx²… If the constant term in the binomial expansion of (x^(5/2)/2-4/xˡ)⁹ is -84 and the… Binomial Coefficient Identities The value of 1/(1 ! 50 !)+1/(3 ! 48 !)+1/(5 ! 46 !)+… .+1/(49 ! 2 !)+1/(51 ! 1 !) is : If for 3≤ r≤ 30, (³⁰C₃₀₋ᵣ)+3(³⁰C₃₁₋ᵣ)+3(³⁰C₃₂₋ᵣ)+(³⁰C₃₃₋ᵣ)=ᵐCᵣ, then m equals : If Σr=0⁵ (¹¹ C₂ᵣ₊₁)/(2 r+2)=m/n, gcd(m, n)=1, then m-n is equal to __________. The value of ¹⁰⁰C₅₀/51+¹⁰⁰C₅₁/52+….+¹⁰⁰C₁₀₀/101 is: If (1/¹⁵C₀+1/¹⁵C₁)(1/¹⁵C₁+1/¹⁵C₂)… (1/¹⁵C₁₂+1/¹⁵C₁₃)=α¹³/(¹⁴C₀¹⁴C₁… ¹⁴C₁₂), then 30α is… Let ⁿCᵣ denote the binomial coefficient of xʳ in the expansion of (1 + x)ⁿ. If Σₖ₌₀¹⁰… Let n∈N and [x] denote the greatest integer less than or equal to x. If the sum of (n +… Let α=Σr=0ⁿ(4 r²+2 r+1) ⁿ Cᵣ and β=(Σr=0ⁿ (ⁿ Cᵣ)/(r+1))+1/(n+1). If 140<(2 α)/β<281… For integers n and r, let ( n; r ) = { ⁿCᵣ,, if n ≥ r ≥ 0; 0,, otherwise The maximum… If Σᵣ₌₀¹⁰ ((10ʳ⁺¹ - 1)/10ʳ).¹¹Cᵣ₊₁ = (_α 11 - 11¹¹)/10¹⁰, then α is equal to: If ²⁰Cᵣ is the co-efficient of xʳ in the expansion of (1 + x)²⁰, then the value of… Σₖ₌₀²⁰ ( ²⁰Cₖ )² is equal to : Let S=1/25!+1/3!23!+1/5!21!+… up to 13 terms. If 13 S=2ᵏ/n!,k∈ N, then n+k is equal to The sum of the series 2 × 1 × ²⁰ C₄-3 × 2 × ²⁰ C₅+4 × 3 × ²⁰ C₆-5 × 4 × ²⁰ C₇+… …+18 ×… Σ_(i,j = 0; i ≠ j)ⁿ ⁿCᵢ ⁿCⱼ is equal to Suppose Σᵣ₌₀²⁰²³ r² ²⁰²³Cᵣ = 2023 × α × 2²⁰²² . Then the value of α is ___________ Let 0 ≤ r ≤ n. If ⁿ⁺¹ Cᵣ₊₁: ⁿ Cᵣ: ⁿ⁻¹ Cᵣ₋₁=55: 35: 21, then 2 n+5 r is equal to : If ( ⁴⁰C₀ ) + ( ⁴¹C₁ ) + ( ⁴²C₂ ) + ..... + ( ⁶⁰C₂₀ ) = m/n⁶⁰C₂₀ m and n are coprime… If 1² ·( ¹⁵ C₁)+2² ·( ¹⁵ C₂)+3² ·( ¹⁵ C₃)+…+15² ·( ¹⁵ C₁₅)=2ᵐ · 3ⁿ · 5ᵏ, where m, n, k ∈… Let Cᵣ denote the coefficient of xʳ in the binomial expansion of (1+x)ⁿ,n∈ N,0≤ r≤ n. If… Multinomial Theorem More Mathematics chapters Frequently asked questions How many Binomial Theorem questions are there for JEE Main? PrepWiser has 212 Binomial Theorem questions for JEE Main, grouped into 9 topics such as Binomial Expansion, General Term and Middle Term.
Are these Binomial Theorem questions from previous-year JEE Main papers? Yes — most are genuine previous-year questions from past JEE Main papers, and the rest are practice questions written to match the JEE Main pattern and difficulty.
Where can I see the answers and solutions to these Binomial Theorem questions? Open any Binomial Theorem question in PrepWiser (it is free) to attempt it; most come with a step-by-step solution.