Quantrex Academy Quantrex AcademyPYQs with solutions · JEE · NEET · NDA

Home · JEE Main & Advanced · Mathematics · Binomial Theorem

Binomial Theorem — JEE Main & Advanced Mathematics PYQs

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

550 questionsMathematicsSolutions on every page
1

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:

JEE Main 2026 Solution View → 2

If (1 - x^3)¹⁰ = _ r=0 ¹⁰ a_r x^r (1-x)^ 30-2r , then 9a₉ a₁₀ is equal to __________.

JEE Main 2026 Solution View → 3

If 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:

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 x¹⁴ 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 ) (¹⁰⁰C₃ ) for some n N , be p . The

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 r+1 = 6 ( ³⁵C_r ) (k^2 - 3) , is :

JEE Main 2026 Solution View → 10

Given below are two statements : Statement I: 25¹³+20¹³+8¹³+3¹³ is divisible by 7. Statement II: The integral part of (7+4 3 )²⁵ is an odd number. In the light of the above stateme

JEE Main 2026 Solution View → 11

The sum of the coefficients of x⁴⁹⁹ and x⁵⁰⁰ in (1+x)¹⁰⁰⁰+x(1+x)⁹⁹⁹+x²(1+x)⁹⁹⁸+ +x¹⁰⁰⁰ 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, x² and x³ in the expansion of (1+x² )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 =C₀-C₁+ 2² 3 C₂- 2³ 4 C₃+ . .+ (-2)^ n n+1 C_ n , then the value of _ n=1 ²⁵

JEE Main 2026 Solution View → 16

The coefficient of x⁴⁸ 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 )= ¹³ ¹⁴ C ₀ ¹⁴ C ₁ ¹⁴ C ₁₂ , then 30 is equal to _ _ _ _ .

JEE Main 2026 Solution View → 18

If the coefficient of x in the expansion of (a x²+b x+c )(1-2 x)²⁶ is -56 and the coefficients of x² and x³ are both zero, then a + b + c is equal to :

JEE Main 2026 Solution View → 19

Let a₀, a₁, , a₂₃ 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 the value of r i

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 C₁ )+2^2 ( ¹⁵ C₂ )+3^2 ( ¹⁵ C₃ )+ ) (+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 x

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 C₃ is:

JEE Main 2025 Solution View → 27

Let (1+x+x^2 )¹⁰=a₀+a₁ x+a₂ x^2+ .+a₂₀ x²⁰ . If (a₁+a₃+a₅+ .+a₁₉ )-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 = ( 3 2 )^9- , , N , then ( + )^2 is equal to

JEE Main 2025 Solution View → 30

If _ r =0 ¹⁰ ( 10^ r +1 -1 10^ r ) ¹¹ C _ r +1 = ¹¹-11¹¹ 10¹⁰ , then 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 7¹⁰³ 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 . Let q

JEE Main 2025 Solution View → 35

^ If =1+ _ r=1 ^6(-3)^ r-1 ¹² C _ 2 r-1 , then the distance of the point (12, 3 ) from the line 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 , , and 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 & u+ v=18 & u+

JEE Main 2025 Solution View → 41

If _ r=1 ³⁰ r^2 ( ³⁰ C_r )^2 30 r-1 = 2²⁹ , then is equal to ______

JEE Main 2025 Solution View → 42

If _ r=0 ^5 11 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 x⁷⁰ in x^2(1+x)⁹⁸+x^3(1+x)⁹⁷+x^4(1+x)⁹⁶+ +x⁵⁴(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 428²⁰²⁴ 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 = _ r=0 ^n (4 r^2+2 r+1 ) ^n C_r and = ( _ r=0 ^n ^n C_r r+1 )+ 1 n+1 . If 140 < 2 < 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 2^8 [5] 3 , then 25 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 m and n be the coefficients of seventh and thirteenth terms respectively in the expansion of 1 3 x 1 3 + 1 2 x 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 r = β n − γ n , β , γ ∈ N , then

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 2 α - 3 p equals

JEE Main 2024 Solution View → 60

Let α = ∑ k = 0 n C k n 2 k + 1 and β = ∑ k = 0 n - 1 C k n C k + 1 n k + 2 . If 5 α = 6 β , 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 C 1 11 2 + C 2 11 3 + … . . + C 9 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 - 3 x + 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

C r n - 1 = k 2 - 8 C r + 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 27 b x 4 is equal to the coefficient of x - 5 in the expansion of a x - 1 b x 2 7

JEE Advanced 2023 Solution View → 68

Let a + b x + c x 2 10 = &#8721; i = 10 20 p i x i , a , b , c &#8712; &#8469; . 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 &#945; be the constant term in the binomial expansion of x - 6 x 3 2 n , n &#8804; 15 . If the sum of the coefficients of the remaining terms in the expansion is 649 and the co

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 4 x 5 - 5 2 x 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 , &#8230; &#8230; , 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 &#945; when divided by 3 and &#946; when divided by 7 . Then ( &#945; 2 + &#946; 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 2 p + 3 q is equal to

JEE Main 2023 Solution View → 81

If the coefficient of x 7 in a x - 1 b x 2 13 and the coefficient of x - 5 in a x + 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 2 x 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 &#8804; t . if the constant term in the expansion of 3 x 2 - 1 2 x 5 7 is &#945; then &#945; is equal to _ _ _ _ _

JEE Main 2023 Solution View → 87

If the coefficients of x 7 in a x 2 + 1 2 b x 11 and x - 7 in a x - 1 3 b x 2 11 are equal, then

JEE Main 2023 Solution View → 88

Among the statements : ( S 1 ) : 2023 2022 - 1999 2022 is divisible by 8 . ( S 2 ) : 13 ( 13 ) n - 11 n - 13 is divisible by 144 for infinitely many n &#8712; &#8469;

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 2 4 + 1 3 4 n is 6 : 1 , then the third term from the beginning is:

JEE Main 2023 Solution View → 90

If C 3 2 n : C 3 n = 10 : 1 , then the ratio n 2 + 3 n : n 2 - 3 n + 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 + &#945; x 3 22 is 7315 , then &#945; is equal to _____ .

JEE Main 2023 Solution View → 93

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

JEE Main 2023 Solution View → 94

The value of 1 1 ! 50 ! + 1 3 ! 48 ! + 1 5 ! 46 ! + &#8230; . + 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 4 x 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 - 3 l is 2 &#945; &#946; where &#946; &#60; 0 is an odd number, then &#945; l -

JEE Main 2023 Solution View → 98

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

JEE Main 2023 Solution View → 99

Let &#945; &#62; 0 , be the smallest number such that the expansion of x 2 3 + 2 x 3 30 has a term &#946; x - &#945; , &#946; &#8712; N . Then &#945; 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 &#8804; 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 b x 1 3 15 is equal to the coefficient of x - 15 in the expansion of a x 1 3 - 1 b x 3 15 , where a and b are positive real

JEE Main 2023 Solution View → 104

The coefficient of x 301 in 1 + x 500 + x 1 + x 499 + x 2 1 + x 498 + &#8230; . . + 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 C 99 200 K a = 2 l m n , wh

JEE Main 2023 Solution View → 106

Let the coefficients of three consecutive terms in the binomial expansion of ( 1 + 2 x ) n be in the ratio 2 : &#160; 5 : &#160; 8 . Then the coefficient of the term, which is in t

JEE Main 2023 Solution View → 107

If the co-efficient of x 9 in &#945; x 3 + 1 &#946; x 11 and the co-efficient of x - 9 in &#945; x - 1 &#946; x 3 11 are equal, then ( &#945; &#946; ) 2 is equal to

JEE Main 2023 Solution View → 108

&#8721; k = 0 6 C 3 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 &#8721; 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 2 x + 1 x 7 + 3 x 2 5 is _____ .

JEE Main 2023 Solution View → 112

If C 1 30 2 + 2 C 2 30 2 + 3 C 3 30 2 . . . . . . . . . . 30 C 30 30 2 = &#945; 60 ! 30 ! 2 , then &#945; 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 &#8712; N be 376 . Then, the coefficient of x 4 is equal to:

JEE Main 2023 Solution View → 114

Suppose &#8721; r = 0 2023 r 2 &#183; C r 2023 = 2023 &#215; &#945; &#215; 2 2022 , then the value of &#945; is

JEE Main 2023 Solution View → 115

The value of &#8721; r = 0 22 C r 22 &#183; C r 23 is

JEE Main 2023 Solution View → 116

If &#8721; k = 1 10 K 2 10 C K 2 = 22000 L , 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 2 4 + 1 3 4 n , in the increasing powers of 1 3 4 be 6 4 : 1 . If the

JEE Main 2022 Solution View → 118

Let the coefficients of the middle terms in the expansion of 1 6 + &#946; x 4 , 1 - 3 &#946; x 2 and 1 - &#946; 2 x 6 , &#946; &#62; 0 , respectively form the first three terms of

JEE Main 2022 Solution View → 119

If 1 + 2 + C 1 49 + C 2 49 + &#8230; . + C 49 49 C 2 50 + C 4 50 + &#8230; . . + C 50 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 →
Practice all Binomial Theorem questions in the app →