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JEE Binomial Theorem PYQs with Solutions

550 previous year questions from Binomial Theorem (JEE Main & Advanced Mathematics) with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

550 questionsMathematics2010–2026Answers + solutions

About Binomial Theorem PYQs

Binomial Theorem has 550 previous year questions in the Quantrex Academy JEE Main & Advanced Mathematics bank. The 220 questions in this list come from JEE Main (210) and JEE Advanced (10), spanning 2010 to 2026. Each question opens on its own page with the options, the correct answer and the worked solution where available.

Questions by year (this list)

  • 2026: 18
  • 2025: 24
  • 2024: 24
  • 2023: 49
  • 2022: 27
  • 2021: 39
  • 2020: 21
  • 2019: 12
  • 2018: 1
  • 2016: 1
  • 2015: 1
  • 2014: 1

Binomial Theorem questions with solutions

1

If 26((2^3)/(3) 12 2 + (2^5)/(5) 12 4 + (2^7)/(7) 12 6 + … + 2^ 13 13 12 12 ) = 3^ 13 - α , then α is equal to:

JEE Main 2026 Solution View → 2

If (1 - x^3)^ 10 = ∑ _ r=0 ^ 10 a_r x^r (1-x)^ 30-2r , then 9a_9 a_ 10 is equal to __________.

JEE Main 2026 Solution View → 3

If the coefficients of the middle terms in the binomial expansions of (1 + α x)^ 26 and (1 - α x)^ 28 , α ≠ 0 , are equal, then the value of α is:

JEE Main 2026 Solution View → 4

The coefficient of x^2 in the expansion of (2x^2 + (1)/(x))^ 10 , x ≠ 0 , is :

JEE Main 2026 Solution View → 5

If the sum of the coefficients of x^7 and x^ 14 in the expansion of ((1)/(x^3) - x^4)^n , x ≠ 0 , is zero, then the value of n is __________.

JEE Main 2026 Solution View → 6

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:

JEE Main 2026 Solution View → 7

Let 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))(^ 100 C_3) for somen

JEE Main 2026 Solution View → 8

If for 3 ≤ r ≤ 30 , 30 30-r + 3 30 31-r + 3 30 32-r + 30 33-r = m r , then m equals:

JEE Main 2026 Solution View → 9

The number of elements in the set S = \ (r, k) : k ∈ ℤ and ^ 36 C_ r+1 = 6( ^ 35 C_r) (k^2 - 3) \ , is :

JEE Main 2026 Solution View → 10

Given below are two statements : Statement I: 25^ 13 +20^ 13 +8^ 13 +3^ 13 is divisible by 7. Statement II: The integral part of (7+4 3 )^ 25 is an odd number. In the light of the

JEE Main 2026 Solution View → 11

The sum of the coefficients of x^ 499 and x^ 500 in (1+x)^ 1000 +x(1+x)^ 999 +x^ 2 (1+x)^ 998 +…+x^ 1000 is :

JEE Main 2026 Solution View → 12

Let S=(1)/(25!)+(1)/(3!23!)+(1)/(5!21!)+… up to 13 terms. If 13 ~S= 2^ k n! , k ∈ ~N , then n+k is equal to

JEE Main 2026 Solution View → 13

The value of ^ 100 C_ 50 51 + ^ 100 C_ 51 52 +….+ ^ 100 C_ 100 101 is :

JEE Main 2026 Solution View → 14

The sum of all possible values of n ∈ N , so that the coefficients of x, x^ 2 and x^ 3 in the expansion of (1+x^ 2 )^ 2 (1+x)^ n , are in arithmetic progression is :

JEE Main 2026 Solution View → 15

Let C_ r denote the coefficient of x^ r in the binomial expansion of (1+x)^ n , n ∈ N, 0 ≤ r ≤ n . If P_ n =C_ 0 -C_ 1 + 2^ 2 3 C_ 2 - 2^ 3 4 C_ 3 +…. .+ (-2)^ n n+1 C_ n , then

JEE Main 2026 Solution View → 16

The coefficient of x^ 48 in (1+x)+2(1+x)^ 2 +3(1+x)^ 3 +…+100(1+x)^ 100 is equal to

JEE Main 2026 Solution View → 17

If ( 1 ^ 15 C_ 0 + 1 ^ 15 C_ 1 )( 1 ^ 15 C_ 1 + 1 ^ 15 C_ 2 ) ⋯( 1 ^ 15 C_ 12 + 1 ^ 15 C_ 13 )= α^ 13 ^ 14 C_ 0 ^ 14 C_ 1 ⋯ ^ 14 C_ 12 , then 30 α is equal to \_\_\_\_ .

JEE Main 2026 Solution View → 18

If the coefficient of x in the expansion of (a x^ 2 +b x+c)(1-2 x)^ 26 is -56 and the coefficients of x^ 2 and x^ 3 are both zero, then a+b+c is equal to :

JEE Main 2026 Solution View → 19

Let a_0, a_1, …, a_ 23 be real numbers such that (1+(2)/(5) x)^ 23 =∑_ i=0 ^ 23 a_i x^i for every real number x . let a_r be the largest among the numbers a_j for 0 ≤ j ≤ 23 . The

JEE Advanced 2025 Solution View → 20

The product of the last two digits of (1919)^ 1919 is

JEE Main 2025 Solution View → 21

The number of integral terms in the expansion of (5^ (1)/(2) +7^ (1)/(8) )^ 1016 is

JEE Main 2025 Solution View → 22

The sum of the series 2 × 1 × ^ 20 C_4-3 × 2 × ^ 20 C_5+4 × 3 × ^ 20 C_6-5 × 4 × ^ 20 C_7+…+18 × 17 × ^ 20 C_ 20 , is equal to

JEE Main 2025 Solution View → 23

The remainder when ((64)^ (64) )^ (64) is divided by 7 is equal to

JEE Main 2025 Solution View → 24

If \(1^2 ·( ^ 15 C_1)+2^2 ·( ^ 15 C_2)+3^2 ·( ^ 15 C_3)+…\) \(+15^2 ·( 15 15 )= 2^m · 3^n · 5^k\), where \(m, n, k ∈ N\), then \( m+n+k \) is equal to :

JEE Main 2025 Solution View → 25

For an integer n ≥ 2 , if the arithmetic mean of all coefficients in the binomial expansion of (x+y)^ 2 n-3 is 16 , then the distance of the point P(2 n-1, n^2-4 n) from the line

JEE Main 2025 Solution View → 26

In the expansion of ( [3] 2 + 1 [3] 3 )^n, n ∈ N , if the ratio of 15^ at term from the beginning to the 15^ th term from the end is (1)/(6) , then the value of ^n C_3 is:

JEE Main 2025 Solution View → 27

Let (1+x+x^2)^ 10 =a_0+a_1 x+a_2 x^2+… .+a_ 20 x^ 20 . If (a_1+a_3+a_5+… .+a_ 19 )-11 a_2=121 k , then k is equal to .

JEE Main 2025 Solution View → 28

The sum of all rational terms in the expansion of (2+ 3 )^8 is

JEE Main 2025 Solution View → 29

If ∑_ r=1 ^9( r+3 2^ r ) . ^9 C_ r =α((3)/(2))^9-β, α, β ∈ N, then (α+β)^2 is equal to

JEE Main 2025 Solution View → 30

If ∑_ r=0 ^ 10 ( 10^ r+1 -1 10^ r ) · ^ 11 C_ r+1 = α^ 11 -11^ 11 10^ 10 , then α is equal to :

JEE Main 2025 Solution View → 31

The term independent of x in the expansion of ( (x+1) (x^ 2 / 3 +1-x^ 1 / 3 ) - (x+1) (x-x^ 1 / 2 ) )^ 10 , x 1 is:

JEE Main 2025 Solution View → 32

The remainder, when 7^ 103 is divided by 23 , is equal to :

JEE Main 2025 Solution View → 33

The least value of n for which the number of integral terms in the Binomial expansion of \(( [3] 7 + [12] 11 )^ n \) is 183, is :

JEE Main 2025 Solution View → 34

Let the coefficients of three consecutive terms T_r, T_ r+1 and T_ r+2 in the binomial expansion of (a+b)^ 12 be in a G.P. and let p be the number of all possible values of r .

JEE Main 2025 Solution View → 35

^ If α=1+∑_ r=1 ^6(-3)^ r-1 ^ 12 C_ 2 r-1 , then the distance of the point (12, 3 ) from the line α x- 3 y+1=0 is _________.

JEE Main 2025 Solution View → 36

Suppose A and B are the coefficients of 30^ th and 12^ th terms respectively in the binomial expansion of (1+x)^ 2 n-1 . If 2 A=5 B , then n is equal to :

JEE Main 2025 Solution View → 37

For some n ≠ 10 , let the coefficients of the 5 th, 6 th and 7 th terms in the binomial expansion of (1+x)^ n+4 be in A.P. Then the largest coefficient in the expansion of (1+x)^

JEE Main 2025 Solution View → 38

If in the expansion of (1+x)^ p (1-x)^ q , the coefficients of x and x^2 are 1 and -2 , respectively, then p^2+q^2 is equal to :

JEE Main 2025 Solution View → 39

The sum of all rational terms in the expansion of (1+2^ 1 / 3 +3^ 1 / 2 )^6 is equal to

JEE Main 2025 Solution View → 40

Let α, β, γ and δ be the coefficients of x^7, x^5, x^3 and x respectively in the expansion of (x+ x^3-1 )^5+(x- x^3-1 )^5, x 1 . If u and v satisfy the equations aligned & α u+β

JEE Main 2025 Solution View → 41

If ∑_ r=1 ^ 30 r^2( ^ 30 C_r)^2 ^ 30 C_ r-1 =α × 2^ 29 , then α is equal to ______

JEE Main 2025 Solution View → 42

If ∑_ r=0 ^5 ^ 11 C_ 2r+1 2 r+2 = m n , gcd(m, n)=1 , then m-n is equal to _______

JEE Main 2025 Solution View → 43

The sum of the coefficient of x^ 2 / 3 and x^ -2 / 5 in the binomial expansion of (x^ 2 / 3 +(1)/(2) x^ -2 / 5 )^9 is

JEE Main 2024 Solution View → 44

The coefficient of x^ 70 in x^2(1+x)^ 98 +x^3(1+x)^ 97 +x^4(1+x)^ 96 +…+x^ 54 (1+x)^ 46 is ^ 99 C_ p - ^ 46 C_ q . Then a possible value of p+q is :

JEE Main 2024 Solution View → 45

The remainder when 428^ 2024 is divided by 21 is__________

JEE Main 2024 Solution View → 46

If the term independent of x in the expansion of ( a x^2+(1)/(2 x^3))^ 10 is 105 , then a^2 is equal to :

JEE Main 2024 Solution View → 47

Let α=∑_ r=0 ^n(4 r^2+2 r+1) ^n C_r and β=(∑_ r=0 ^n ^n C_r r+1 )+(1)/(n+1) . If 140 < (2 α)/(β) < 281 , then the value of n is _______

JEE Main 2024 Solution View → 48

If the second, third and fourth terms in the expansion of (x+y)^n are 135,30 and (10)/(3) , respectively, then 6(n^3+x^2+y) is equal to _______

JEE Main 2024 Solution View → 49

If the constant term in the expansion of ( [5] 3 x + 2 x [3] 5 )^ 12 , x ≠ 0 , is α × 2^8 × [5] 3 , then 25 α is equal to :

JEE Main 2024 Solution View → 50

If the constant term in the expansion of (1+2 x-3 x^3)((3)/(2) x^2-(1)/(3 x))^9 is p , then 108 p is equal to

JEE Main 2024 Solution View → 51

If the coefficients of x^4, x^5 and x^6 in the expansion of (1+x)^n are in the arithmetic progression, then the maximum value of n is:

JEE Main 2024 Solution View → 52

The sum of all rational terms in the expansion of (2^ (1)/(5) +5^ (1)/(3) )^ 15 is equal to :

JEE Main 2024 Solution View → 53

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

JEE Main 2024 Solution View → 54

Let mandn be the coefficients of seventh and thirteenth terms respectively in the expansion of ((1)/(3)x(1)/(3) + (1)/(2x(2)/(3))) ^ 18 . Then ((n)/(m))(1)/(3) is:

JEE Main 2024 Solution View → 55

If the coefficient of x ^ 30 in the expansion of (1 + (1)/(x)) ^ 6 (1 + x ^ 2 ) ^ 7 (1 - x ^ 3 ) ^ 8 ; x ≠ 0 is α , then |α| equals _________.

JEE Main 2024 Solution View → 56

If for some m, n; ^ 6 C _ m + 2( ^ 6 C _ m + 1 ) + ^ 6 C _ m + 2 > ^ 8 C _ 3 and ^ n - 1 P _ 3 : ^ n P _ 4 = 1: 8 , then ^ n P _ m + 1 + ^ n + 1 C _ m is equal to

JEE Main 2024 Solution View → 57

Let the coefficient of x ^ r 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 α _ r . If ∑ _ r = 0 ^ n α _

JEE Main 2024 Solution View → 58

In the expansion of (1 + x)(1 - x ^ 2 ) (1 + (3)/(x) + 3 x ^ 2 + 1 x ^ 3 ) ^ 5 , x ≠ 0 , the sum of the coefficient of x ^ 3 and x ^ - 13 is equal to ______

JEE Main 2024 Solution View → 59

Suppose 28 - p, p, 70 - α, α are the coefficient of four consecutive terms in the expansion of ( 1 + x ) ^ n . Then the value of 2α - 3p equals

JEE Main 2024 Solution View → 60

Let α = ∑ _ k = 0 ^ n ( (Ck n) ^ 2 k + 1 ) and β = ∑ _ k = 0 ^ n - 1 ((Ck n Ck + 1 n)/(k + 2)) . If 5α = 6β , then n equals

JEE Main 2024 Solution View → 61

Number of integral terms in the expansion of \ 7 ^ ((1)/(2)) + 11 ^ ((1)/(6)) \ ^ 824 is equal to ______.

JEE Main 2024 Solution View → 62

Remainder when 64 ^ 32 ^ 32 is divided by 9 is equal to _____.

JEE Main 2024 Solution View → 63

If (C1 11)/(2) + (C2 11)/(3) + … . . + (C9 11)/(10) = (n)/(m) with gcd ( n, m ) = 1 , then n + m is equal to

JEE Main 2024 Solution View → 64

The coefficient of x ^ 2012 in the expansion of (1 - x) ^ 2008 (1 + x + x ^ 2 ) ^ 2007 is equal to _____.

JEE Main 2024 Solution View → 65

If A denotes the sum of all the coefficients in the expansion of (1 - 3x + 10 x ^ 2 ) ^ n and B denotes the sum of all the coefficients in the expansion of (1 + x ^ 2 ) ^ n , then

JEE Main 2024 Solution View → 66

Cr n - 1 = ( k ^ 2 - 8) Cr + 1 n if and only if :

JEE Main 2024 Solution View → 67

Let a and b be two nonzero real numbers. If the coefficient of x ^ 5 in the expansion of (a x ^ 2 + (70)/(27bx)) ^ 4 is equal to the coefficient of x ^ - 5 in the expansion of (ax

JEE Advanced 2023 Solution View → 68

Let (a + bx + c x ^ 2 ) ^ 10 = ∑ _ i = 10 ^ 20 p _ i x ^ i , a, b, c ∈ ℕ . If p _ 1 = 20 and p _ 2 = 210 , then 2(a + b + c) is equal to

JEE Main 2023 Solution View → 69

The coefficient of x ^ 5 in the expansion of (2 x ^ 3 - 1 3 x ^ 2 ) ^ 5 is

JEE Main 2023 Solution View → 70

The remainder, when 7 ^ 103 is divided by 17, is

JEE Main 2023 Solution View → 71

Fractional part of the number 4 ^ 2022 15 is equal to

JEE Main 2023 Solution View → 72

Let α be the constant term in the binomial expansion of ( x - (6)/(x(3)/(2))) ^ n , n ≤ 15 . If the sum of the coefficients of the remaining terms in the expansion is 649 and the

JEE Main 2023 Solution View → 73

If (1)/(n + 1) ^ n C _ n + (1)/(n) ^ n C _ n - 1 + . . . + (1)/(2) ^ n C _ 1 + ^ n C _ 0 = (1023)/(10) then n is equal to

JEE Main 2023 Solution View → 74

The sum, of the coefficients of the first 50 terms in the binomial expansion of (1 - x) ^ 100 , is equal to

JEE Main 2023 Solution View → 75

If the 1011 ^ th term from the end in the binomial expansion of ((4x)/(5) - (5)/(2x)) ^ 2022 is 1024 times 1011 ^ th term from the beginning, then 32|x| is equal to

JEE Main 2023 Solution View → 76

The sum of the coefficients of three consecutive terms in the binomial expansion of (1 + x) ^ n + 2 , which are in the ratio 1: 3: 5 , is equal to

JEE Main 2023 Solution View → 77

The number of integral terms in the expansion of (3(1)/(2) + 5(1)/(4)) ^ 680 is equal to

JEE Main 2023 Solution View → 78

The mean of the coefficients of x, x ^ 2 , … … , x ^ 7 in the binomial expression of ( 2 + x ) ^ 9 is _________

JEE Main 2023 Solution View → 79

Let the number ( 22 ) ^ 2022 + ( 2022 ) ^ 22 leave the remainder α when divided by 3 and β when divided by 7 . Then ( α ^ 2 + β ^ 2 ) is equal to

JEE Main 2023 Solution View → 80

If the coefficients of x and x ^ 2 in ( 1 + x ) ^ p ( 1 - x ) ^ q are 4 and - 5 respectively, then 2p + 3q is equal to

JEE Main 2023 Solution View → 81

If the coefficient of x ^ 7 in (ax - 1 b x ^ 2 ) ^ 13 and the coefficient of x ^ - 5 in (ax + 1 b x ^ 2 ) ^ 13 are equal, then a ^ 4 b ^ 4 is equal to:

JEE Main 2023 Solution View → 82

The coefficient of x ^ 7 in (1 - x + 2 x ^ 3 ) ^ 10 is __________ .

JEE Main 2023 Solution View → 83

The absolute difference of the coefficients of x ^ 10 and x ^ 7 in the expansion of (2 x ^ 2 + (1)/(2x)) ^ 11 is equal to

JEE Main 2023 Solution View → 84

25 ^ 190 - 19 ^ 190 - 8 ^ 190 + 2 ^ 190 is divisible by

JEE Main 2023 Solution View → 85

if the coefficients of three consecutive terms in the expansion of (1 + x) ^ n are the ratio 1: 5: 20 then the coefficient of the fourth term is

JEE Main 2023 Solution View → 86

Let [t] denote the greatest integer ≤ t . if the constant term in the expansion of (3 x ^ 2 - 1 2 x ^ 5 ) ^ 7 is α then [α] is equal to _____

JEE Main 2023 Solution View → 87

If the coefficients of x ^ 7 in (a x ^ 2 + (1)/(2bx)) ^ 11 and x ^ - 7 in (ax - 1 3b x ^ 2 ) ^ 11 are equal, then

JEE Main 2023 Solution View → 88

Among the statements : ( S1 ) : 2023 ^ 2022 - 1999 ^ 2022 is divisible by 8 . ( S2 ) : 13 ( 13 ) ^ n - 11n - 13 is divisible by 144 for infinitely many n ∈ ℕ

JEE Main 2023 Solution View → 89

If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of (24 + (1)/(34)) ^ n is 6 : 1 , then the third term from the beginning is:

JEE Main 2023 Solution View → 90

If C3 2n : C3 n = 10: 1, then the ratio ( n ^ 2 + 3n): ( n ^ 2 - 3n + 4) is

JEE Main 2023 Solution View → 91

The coefficient of x ^ 18 in the expansion of ( x ^ 4 - 1 x ^ 3 ) ^ 15 is ____________

JEE Main 2023 Solution View → 92

If the term without x in the expansion of (x(2)/(3) + α x ^ 3 ) ^ 22 is 7315 , then |α| is equal to _____ .

JEE Main 2023 Solution View → 93

Let the sixth term in the binomial expansion of ( 2 ^ _ 2 (10 - 3 ^ x ) + 2 ^ ( x - 2 ) _ 2 3 5) ^ m powers of 2 ^ (x - 2) _ 2 3 , be 21 . If the binomial coefficients of the

JEE Main 2023 Solution View → 94

The value of (1)/(1 ! 50 !) + (1)/(3 ! 48 !) + (1)/(5 ! 46 !) + … . + (1)/(49 ! 2 !) + (1)/(51 ! 1 !) is

JEE Main 2023 Solution View → 95

The remainder when 19 ^ 200 + 23 ^ 200 is divided by 49 , is _____ .

JEE Main 2023 Solution View → 96

The coefficient of x ^ - 6 , in the expansion of ((4x)/(5) + 5 2 x ^ 2 ) ^ 9 , is

JEE Main 2023 Solution View → 97

If the constant term in the binomial expansion of ((x(5)/(2))/(2) - 4 x ^ l ) ^ 9 is - 84 and the coefficient of x ^ - 3l is 2 ^ α β where β < 0 is an odd number, then | - β| is

JEE Main 2023 Solution View → 98

If Pn - 1 2n + 1 : Pn 2n - 1 = 11: 21 , then n ^ 2 + n + 15 is equal to :

JEE Main 2023 Solution View → 99

Let α > 0 , be the smallest number such that the expansion of (x(2)/(3) + 2 x ^ 3 ) ^ 30 has a term β x ^ - α , β ∈ N . Then α is equal to _____ .

JEE Main 2023 Solution View → 100

The remainder on dividing 5 ^ 99 by 11 is _____ .

JEE Main 2023 Solution View → 101

Let x = (8 3 + 13) ^ 13 and y = (7 2 + 9) ^ 9 . If [t] denotes the greatest integer ≤ t , then

JEE Main 2023 Solution View → 102

50 ^ th root of a number x is 12 and 50 ^ th root of another number y is 18 . Then the remainder obtained on dividing ( x + y ) by 25 is ________.

JEE Main 2023 Solution View → 103

If the coefficient of x ^ 15 in the expansion of (a x ^ 3 + (1)/(bx(1)/(3))) ^ 15 is equal to the coefficient of x ^ - 15 in the expansion of (ax(1)/(3) - 1 b x ^ 3 ) ^ 15 , where

JEE Main 2023 Solution View → 104

The coefficient of x ^ 301 in (1 + x) ^ 500 + x (1 + x) ^ 499 + x ^ 2 (1 + x) ^ 498 + … . . + x ^ 500 is:

JEE Main 2023 Solution View → 105

Let K be the sum of the coefficients of the odd powers of x in the expansion of (1 + x) ^ 99 . Let a be the middle term in the expansion of (2 + 1 2 ) ^ 200 . If (C99 200 K)/(a) =

JEE Main 2023 Solution View → 106

Let the coefficients of three consecutive terms in the binomial expansion of ( 1 + 2x ) ^ n be in the ratio 2: 5: 8 . Then the coefficient of the term, which is in the middle of

JEE Main 2023 Solution View → 107

If the co-efficient of x ^ 9 in (α x ^ 3 + (1)/( )) ^ 11 and the co-efficient of x ^ - 9 in ( - 1 β x ^ 3 ) ^ 11 are equal, then ( αβ ) ^ 2 is equal to

JEE Main 2023 Solution View → 108

∑ _ k = 0 ^ 6 C3 51 - k is equal to

JEE Main 2023 Solution View → 109

The remainder when (2023) ^ 2023 is divided by 35 is

JEE Main 2023 Solution View → 110

If a _ r is the coefficient of x ^ 10 - r in the Binomial expansion of (1 + x) ^ 10 , then ∑ _ r = 1 ^ 10 r ^ 3 ( a _ r a _ r - 1 ) ^ 2 is equal to

JEE Main 2023 Solution View → 111

The constant term in the expansion of (2x + 1 x ^ 7 + 3 x ^ 2 ) ^ 5 is _____ .

JEE Main 2023 Solution View → 112

If (C1 30) ^ 2 + 2 (C2 30) ^ 2 + 3 (C3 30) ^ 2 . . . . . . . . . . 30 (C30 30) ^ 2 = α60 ! (30 !) ^ 2 , then α is equal to

JEE Main 2023 Solution View → 113

Let he sum of the coefficient of first three terms in the expansion of (x - 3 x ^ 2 ) ^ n ; x = 0, n ∈ N be 376 . Then, the coefficient of x ^ 4 is equal to:

JEE Main 2023 Solution View → 114

Suppose ∑ _ r = 0 ^ 2023 r ^ 2 · Cr 2023 = 2023 × α × 2 ^ 2022 , then the value of α is

JEE Main 2023 Solution View → 115

The value of ∑ _ r = 0 ^ 22 Cr 22 · Cr 23 is

JEE Main 2023 Solution View → 116

If ∑ _ k = 1 ^ 10 K ^ 2 ( 10 _ C _ K ) ^ 2 = 22000L , then L is equal to _____.

JEE Main 2022 Solution View → 117

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of (24 + (1)/(34)) ^ n , in the increasing powers of (1)/(34) be 64 : 1

JEE Main 2022 Solution View → 118

Let the coefficients of the middle terms in the expansion of ( 1 6 + ) ^ 4 , (1 - 3 ) ^ 2 and (1 - (β)/(2)x) ^ 6 , β > 0 , respectively form the first three terms of an A.P. If d

JEE Main 2022 Solution View → 119

If 1 + (2 + C1 49 + C2 49 + … . + C49 49)(C2 50 + C4 50 + … . . + C50 50) is equal to 2 ^ n . m , where m is odd, then n + m is equal to _____ .

JEE Main 2022 Solution View → 120

The remainder when 7 ^ 2022 + 3 ^ 2022 is divided by 5 is

JEE Main 2022 Solution View → 121

Let for the 9 ^ th term in the binomial expansion of (3 + 6x) ^ n , in the increasing powers of 6x , to be the greatest for x = (3)/(2) , the least value of n is n _ 0 . If k is

JEE Main 2022 Solution View → 122

The remainder when (2021) ^ 2022 + (2022) ^ 2021 is divided by 7 is

JEE Main 2022 Solution View → 123

∑ _ i, j = 0 _ i ≠ j ^ n Ci n Cj n is equal to

JEE Main 2022 Solution View → 124

If the coefficients of x and x ^ 2 in the expansion of (1 + x) ^ p (1 - x) ^ q , p, q ≤ 15 , are - 3 and - 5 respectively, then the coefficient of x ^ 3 is equal to ______.

JEE Main 2022 Solution View → 125

The remainder when (11) ^ 1011 + (1011) ^ 11 is divided by 9 is _____ .

JEE Main 2022 Solution View → 126

If the maximum value of the term independent of t in the expansion of ( t ^ 2 x(1)/(5) + (1 - x) ^ (1)/(10) t ) ^ 15 , x ≥ 0 , is K , then 8K is equal to _____ .

JEE Main 2022 Solution View → 127

Let n ≥ 5 be an integer. If 9 ^ n - 8n - 1 = 64α and 6 ^ n - 5n - 1 = 25β , then α - β is equal to:

JEE Main 2022 Solution View → 128

Let the coefficients of x ^ - 1 and x ^ - 3 in the expansion of (2x(1)/(5) - (1)/(x(1)/(5))) ^ 15 , x > 0 , be m and n respectively. If r is a positive integer such m n ^ 2 = Cr .

JEE Main 2022 Solution View → 129

If the constant term in the expansion of (3 x ^ 3 - 2 x ^ 2 + 5 x ^ 5 ) ^ 10 is 2 ^ k . l , where l is an odd integer, then the value of k is equal to

JEE Main 2022 Solution View → 130

The term independent of x in the expression of (1 - x ^ 2 + 3 x ^ 3 ) ((5)/(2) x ^ 3 - 1 5 x ^ 2 ) ^ 11 , x ≠ 0 is

JEE Main 2022 Solution View → 131

If ∑ _ k = 1 ^ 31 (Ck 31)(Ck - 1 31) - ∑ _ k = 1 ^ 30 (Ck 30)(Ck - 1 30) = (α(60 !))/((30 !)(31 !)) , where α ∈ R , then the value of 16α is equal to

JEE Main 2022 Solution View → 132

The number of positive integers k such that the constant term in the binomial expansion of (2 x ^ 3 + 3 x ^ k ) ^ 12 , x ≠ 0 is 2 ^ 8 · l , where l is an odd integer, is ______.

JEE Main 2022 Solution View → 133

If the sum of the coefficients of all the positive powers of x , in the binomial expansion of ( x ^ n + 2 x ^ 5 ) ^ 7 is 939 , then the sum of all the possible integral values of

JEE Main 2022 Solution View → 134

If the coefficient of x ^ 10 in the binomial expansion of ( x 5(1)/(4) + 5 x(1)/(3) ) ^ 60 is 5 ^ k l , where l, k ∈ N and l is coprime to 5 , then k is equal to ______.

JEE Main 2022 Solution View → 135

If C0 40 + C1 41 + C2 42 + ⋯ + C20 60 = (m)/(n) × C20 60 where m & n are co-prime, then m + n is equal to

JEE Main 2022 Solution View → 136

The remainder when (2021) ^ 2023 is divided by 7 is

JEE Main 2022 Solution View → 137

The coefficient of x ^ 101 in the expression (5 + x) ^ 500 + x (5 + x) ^ 499 + x ^ 2 (5 + x) ^ 498 + … … + x ^ 500 , x > 0 is

JEE Main 2022 Solution View → 138

If the sum of the co-efficients of all the positive even powers of x in the binomial expansion of (2 x ^ 3 + (3)/(x)) ^ 10 is 5 ^ 10 - β · 3 ^ 9 , then β is equal to _____.

JEE Main 2022 Solution View → 139

If 1 2 · 3 ^ 10 + 1 2 ^ 2 · 3 ^ 9 + … + 1 2 ^ 10 · 3 = K 2 ^ 10 · 3 ^ 10 , then the remainder when K is divided by 6 is

JEE Main 2022 Solution View → 140

Let C _ r denote the binomial coefficient of x ^ r in the expansion of (1 + x) ^ 10 . If for α, β ∈ R , C _ 1 + 3 · 2 C _ 2 + 5 · 3 C _ 3 + … upto 10 terms = α × 2 ^ 11 2 ^ β - 1

JEE Main 2022 Solution View → 141

The remainder on dividing 1 + 3 + 3 ^ 2 + 3 ^ 3 + … + 3 ^ 2021 by 50 is _____.

JEE Main 2022 Solution View → 142

The remainder when 3 ^ 2022 is divided by 5 is

JEE Main 2022 Solution View → 143

If the sum of the coefficients in the expansion of ( x + y ) ^ n is 4096, then the greatest coefficient in the expansion is _____.

JEE Main 2021 Solution View → 144

If the coefficient of a ^ 7 b ^ 8 in the expansion of ( a + 2b + 4ab ) ^ 10 is K · 2 ^ 16 , then K is equal to

JEE Main 2021 Solution View → 145

If ( 3 ^ 6 4 ^ 4 )k is the term, independent of x, in the binomial expansion of ((x)/(4) - 12 x ^ 2 ) ^ 12 , then k is equal to

JEE Main 2021 Solution View → 146

3 × 7 ^ 22 + 2 × 10 ^ 22 - 44 when divided by 18 leaves the remainder

JEE Main 2021 Solution View → 147

∑ _ k = 0 ^ 20 (Ck 20) ^ 2 is equal to

JEE Main 2021 Solution View → 148

Let ( array ll n k array ) denote Ck n and [ array ll n k array ] = \ array ll n k array , if0 ≤ k ≤ n 0, otherwise . If A _ k = ∑ _ i = 0 ^ 9 ( array ll 9 i array )[ array ll 12

JEE Main 2021 Solution View → 149

If Cr 20 is the co-efficient of x ^ r in the expansion of ( 1 + x ) ^ 20 , then the value of ∑ _ r = 0 ^ 20 r ^ 2 (Cr 20) is equal to:

JEE Main 2021 Solution View → 150

A possible value of x, for which the ninth term in the expansion of \ 3 ^ _ 3 25 ^ x - 1 + 7 + 3 ^ (- (1)/(8)) _ 3 ( 5 ^ x - 1 + 1) \ ^ 10 in the increasing powers of 3 ^ (

JEE Main 2021 Solution View → 151

If the coefficients of x ^ 7 in ( x ^ 2 + (1)/(bx)) ^ 11 and x ^ - 7 in (x - 1 b x ^ 2 ) ^ 11 , b ≠ 0, are equal, then the value of b is equal to:

JEE Main 2021 Solution View → 152

The sum of all those terms which are rational numbers in the expansion of (2(1)/(3) + 3(1)/(4)) ^ 12 is:

JEE Main 2021 Solution View → 153

If the greatest value of the term independent of x in the expansion of (xsinα + a(cosα)/(x)) ^ 10 is 10 ! (5 !) ^ 2 , then the value of a is equal to:

JEE Main 2021 Solution View → 154

The lowest integer which is greater than (1 + 1 10 ^ 100 ) ^ 10 ^ 100 is

JEE Main 2021 Solution View → 155

If Pr n = Pr + 1 n and Cr n = Cr - 1 n , then the value of r is equal to:

JEE Main 2021 Solution View → 156

Let n ∈ N and [x] denote the greatest integer less than or equal to x . If the sum of (n + 1) terms of C0 n , 3 · C1 n , 5 · C2 n , 7 · C3 n , … is equal to 2 ^ 100 · 101, then

JEE Main 2021 Solution View → 157

If the co-efficient of x ^ 7 and x ^ 8 in the expansion of (2 + (x)/(3)) ^ n are equal, then the value of n is equal to :

JEE Main 2021 Solution View → 158

If b is very small as compared to the value of a , so that the cube and other higher powers of (b)/(a) can be neglected in the identity (1)/(a - b) + (1)/(a - 2b) + (1)/(a - 3b) +

JEE Main 2021 Solution View → 159

The ratio of the coefficient of the middle term in the expansion of ( 1 + x ) ^ 20 and the sum of the coefficients of two middle terms in expansion of ( 1 + x ) ^ 19 is .

JEE Main 2021 Solution View → 160

The term independent of x in the expansion of ( x + 1 x ^ 2 / 3 - x ^ 1 / 3 + 1 - x - 1 x - x ^ 1 / 2 ) ^ 10 , where x ≠ 0, 1 is equal to

JEE Main 2021 Solution View → 161

If the constant term, in binomial expansion of (2 x ^ r + 1 x ^ 2 ) ^ 10 is 180, then r is equal to ____________.

JEE Main 2021 Solution View → 162

The number of elements in the set n ∈ 1, 2, 3, … , 100 ∣ ( 11 ) ^ n > ( 10 ) ^ n + ( 9 ) ^ n \ is ___________.

JEE Main 2021 Solution View → 163

For the natural numbers m, n, if ( 1 - y ) ^ m ( 1 + y ) ^ n = 1 + a _ 1 y + a _ 2 y ^ 2 + … . + a _ m + n y ^ m + n and a _ 1 = a _ 2 = 10, then the value of m + n, is equal to:

JEE Main 2021 Solution View → 164

The coefficient of x ^ 256 in the expansion of (1 - x) ^ 101 ( x ^ 2 + x + 1) ^ 100 is:

JEE Main 2021 Solution View → 165

The number of rational terms in the binomial expansion of (4(1)/(4) + 5(1)/(6)) ^ 120 is_______.

JEE Main 2021 Solution View → 166

The term independent of x in the expansion of [ x + 1 x ^ 2 / 3 - x ^ 1 / 3 + 1 - x - 1 x - x ^ 1 / 2 ] ^ 10 , x ≠ 1 , is equal to ___.

JEE Main 2021 Solution View → 167

Let Cr n denote the binomial coefficient of x ^ r in the expansion of ( 1 + x ) ^ n . If ∑ _ k = 0 ^ 10 ( 2 ^ 2 + 3k) Ck n = α . 3 ^ 10 + β · 2 ^ 10 , α, β ∈ R , then α + β is

JEE Main 2021 Solution View → 168

Let (1 + x + 2 x ^ 2 ) ^ 20 = a _ 0 + a _ 1 x + a _ 2 x ^ 2 + … + a _ 40 x ^ 40 , then a _ 1 + a _ 3 + a _ 5 + … + a _ 37 is equal to

JEE Main 2021 Solution View → 169

The value of ∑ _ r = 0 ^ 6 (Cr 6 · C6 - r 6) is equal to :

JEE Main 2021 Solution View → 170

Let the coefficients of third, fourth and fifth terms in the expansion of (x + a x ^ 2 ) ^ n , x ≠ 0, be in the ratio 12: 8: 3 . Then the term independent of x in the expansion

JEE Main 2021 Solution View → 171

If the fourth term in the expansion of (x + x ^ _ 2 x ) ^ 7 is 4480, then the value of x where x ∈ N is equal to:

JEE Main 2021 Solution View → 172

If ( 2021 ) ^ 3762 is divided by 17, then the remainder is _______.

JEE Main 2021 Solution View → 173

Let n be a positive integer. Let A = ∑ _ k = 0 ^ n (- 1) ^ k × Ck n [ ((1)/(2)) ^ k + ((3)/(4)) ^ k + ((7)/(8)) ^ k + ((15)/(16)) ^ k + ((31)/(32)) ^ k ] . If 63A = 1 - 1 2 ^ 30

JEE Main 2021 Solution View → 174

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 :

JEE Main 2021 Solution View → 175

Let [x] denote greatest integer less than or equal to x . If for n ∈ N, (1 - x + x ^ 3 ) ^ n = ∑ _ j = 0 ^ 3n a _ j x ^ j , then ∑ _ j = 0 ^ [(3n)/(2)] a _ 2j + 4 ∑ _ j = 0 ^ [(3n

JEE Main 2021 Solution View → 176

The maximum value of the term independent of t in the expansion of (tx(1)/(5) + ((1 - x)(1)/(10))/(t)) ^ 10 where x ∈ (0, 1) is:

JEE Main 2021 Solution View → 177

Let m, n ∈ N and gcd(2, n) = 1 . If 30( array ll 30 0 array ) + 29( array ll 30 1 array ) + … . . + 2( array ll 30 28 array ) + 1( array ll 30 29 array ) = n . 2 ^ m , then n + m

JEE Main 2021 Solution View → 178

If the remainder when x is divided by 4 is 3 , then the remainder when (2020 + x) ^ 2022 is divided by 8 is ___ .

JEE Main 2021 Solution View → 179

If n ⩾ 2 is a positive integer, then the sum of the series C2 n + 1 + 2(C2 2 + C2 3 + C2 4 + … + C2 n) is

JEE Main 2021 Solution View → 180

For integers n and r, let ( array ll n r array ) = \ cases Cr n , & ifn ≥ r ≥ 0 0, & otherwise cases ). The maximum value of k for which the sum ∑ _ i = 0 ^ k ( array ll 10 i

JEE Main 2021 Solution View → 181

The value of - C1 15 + 2 · C2 15 - 3 · ^ 15 C _ 3 + . . . . . - 15 · C15 15 + C1 14 + C3 14 + C5 14 + . . . . + C11 14 is equal to

JEE Main 2021 Solution View → 182

For nonnegative integers s and r , let ( array ll s r array ) = \ cases (s !)/(r ! (s - r) !) & ifr ≤ s 0 & ifr > s cases ). For positive integers m and n, let g(m, n) = ∑ _ p = 0

JEE Advanced 2020 Solution View → 183

If the constant term in the binomial expansion of ( x - k x ^ 2 ) ^ 10 is 405 , then |k| equals :

JEE Main 2020 Solution View → 184

If p denotes the fractional part of the number p, then \ 3 ^ 200 8 \ is equal to

JEE Main 2020 Solution View → 185

The coefficient of x ^ 4 in the expansion of (1 + x + x ^ 2 + x ^ 3 ) ^ 6 in powers of x, is … . .

JEE Main 2020 Solution View → 186

The natural number m , for which the coefficient of x in the binomial expansion of ( x ^ m + 1 x ^ 2 ) ^ 22 is 1540, is

JEE Main 2020 Solution View → 187

If for some positive integer n , the coefficients of three consecutive terms in the binomial expansion of (1 + x) ^ n + 5 are in the ratio 5: 10: 14 , then the largest coefficient

JEE Main 2020 Solution View → 188

The value of ∑ _ r = 0 ^ 20 C6 50 - r is equal to:

JEE Main 2020 Solution View → 189

Let (2 x ^ 2 + 3x + 4) ^ 10 = ∑ _ r = 0 ^ 20 a _ r x ^ r . Then a _ 7 a _ 13 is equal to ______

JEE Main 2020 Solution View → 190

If the term independent of x in the expansion of ((3)/(2) x ^ 2 - (1)/(3x)) ^ 9 is k , then 18k is equal to:

JEE Main 2020 Solution View → 191

The value of ( 2 ⋅ ^ 1 P _ 0 - 3 ⋅ ^ 2 P _ 1 + 4 ⋅ ^ 3 P _ 2 - . . . . . . . . up to 51 ^ th term) + ( 1 ! - 2 ! + 3 ! - . . . . . . . up to 51 ^ th term) is equal to

JEE Main 2020 Solution View → 192

If the number of integral terms in the expansion of (3(1)/(2) + 5(1)/(8)) ^ n is exactly 33 , then the least value of n is

JEE Main 2020 Solution View → 193

For a positive integer n, (1 + (1)/(x)) ^ n is expanded in increasing powers of x . If three consecutive coefficients in this expansion are in the ratio, 2: 5: 12, then n is equal

JEE Main 2020 Solution View → 194

Let α > 0, β > 0 be such that α ^ 3 + β ^ 2 = 4 . If the maximum value of the term independent of x in the binomial expansion of ( (1)/(9) + β x ^ - (1)/(6) ) ^ 10 is 10k , then k

JEE Main 2020 Solution View → 195

In the expansion of ((x)/(cos ⁡ θ) + (1)/(xsin ⁡ θ)) ^ 16 , if l _ 1 is the least value of the term independent of x when (π)/(8) ≤ θ ≤ (π)/(4) and l _ 2 is the least value of the

JEE Main 2020 Solution View → 196

If C _ r ≡ C _ r 25 and C _ 0 + 5 ∙ C _ 1 + 9 ∙ C _ 2 + … + (101) ∙ C _ 25 = 2 ^ 25 ∙ k, then k is equal to ____________.

JEE Main 2020 Solution View → 197

The coefficient of x ^ 4 in the expansion of (1 + x + x ^ 2 ) ^ 10 is ________

JEE Main 2020 Solution View → 198

If α and β , be the coefficients of x ^ 4 and x ^ 2 , respectively in the expansion of (x + x ^ 2 - 1 ) ^ 6 + (x - x ^ 2 - 1 ) ^ 6 , then

JEE Main 2020 Solution View → 199

If a, b and c are the greatest values of C _ p 19 , C _ q 20 and C _ r 21 respectively, then:

JEE Main 2020 Solution View → 200

The coefficient of x ^ 7 in the expression (1 + x) ^ 10 + x (1 + x) ^ 9 + x ^ 2 (1 + x) ^ 8 + . . . . + x ^ 10 , is

JEE Main 2020 Solution View →
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Related JEE Mathematics chapters

Application of Derivatives (795)Three Dimensional Geometry (756)Sequences and Series (748)Functions (700)Definite Integration (692)Vector Algebra (616)Differential Equations (586)Continuity and Differentiability (576)Probability (561)Complex Number (551)Quadratic Equation (517)Determinants (508)

Binomial Theorem in other exams

NDA Binomial Theorem (79)BITSAT Binomial Theorem (34)