How many questions from Binomial Theorem and Its Simple Applications appear in JEE?

Binomial Theorem and Its Simple Applications has appeared in 158 of the last 186 JEE Main and JEE Advanced papers — about 85% of them — contributing 211 questions in total across those papers.

Is Binomial Theorem and Its Simple Applications an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 85% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Binomial Theorem and Its Simple Applications questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Binomial Theorem and Its Simple Applications JEE Previous Year Questions with Answers

Q1·MathematicsMultiple correctJEE Advanced 2026

Let

\mathbb{R}

denote the set of all real numbers. Consider the polynomial function

f : \mathbb{R} \to \mathbb{R}

defined by

f(x) = \dfrac{d^{10}}{dx^{10}}\left((x^{2} - 1)^{10}\right)

, for all

x \in \mathbb{R}

. Here

\dfrac{d^{10}}{dx^{10}}\left((x^{2} - 1)^{10}\right)

is the 10th order derivative of the function

(x^{2} - 1)^{10}

. Then which of the following statements is (are) TRUE ?

(A)The coefficient of $x^{8}$ in the polynomial $f(x)$ is $(-10)\left(\dfrac{18!}{8!}\right)$
(B)The value of $f(1) + f(-1)$ is equal to $10!\, 2^{11}$
(C)The degree of the polynomial $f(x)$ is 10
(D)The constant term of the polynomial $f(x)$ is $-\left(\dfrac{10!}{5!}\right)$

Q2·MathematicsSingle correctJEE Main 2026

If

26\left(\frac{2^3}{3}\binom{12}{2} + \frac{2^5}{5}\binom{12}{4} + \frac{2^7}{7}\binom{12}{6} + \ldots + \frac{2^{13}}{13}\binom{12}{12}\right) = 3^{13} - \alpha

, then

\alpha

is equal to:

(A)45
(B)48
(C)51
(D)54

Q3·MathematicsNumericalJEE Main 2026

If

(1 - x^{3})^{10} = \sum_{r=0}^{10} a_{r} x^{r} (1 - x)^{30-2r}

, then

\frac{9a_{9}}{a_{10}}

is equal to __________.

Q4·MathematicsSingle correctJEE Main 2026

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 is:

(A)1
(B) $\frac{14}{13}$
(C) $\frac{27}{7}$
(D) $\frac{7}{27}$

Q5·MathematicsSingle correctJEE Main 2026

The coefficient of

x^2

in the expansion of

\left( 2x^2 + \frac{1}{x} \right)^{10}

,

x \neq 0

, is :

(A)3240
(B)3360
(C)3480
(D)3600

Q6·MathematicsNumericalJEE Main 2026

If the sum of the coefficients of

x^7

and

x^{14}

in the expansion of

\left(\frac{1}{x^3} - x^4\right)^n

,

x \neq 0

, is zero, then the value of

n

is __________.

Q7·MathematicsSingle correctJEE Main 2026

In the expansion of

\left(9x - \frac{1}{3\sqrt{x}}\right)^{18}

,

x > 0

, if the term independent of

x

is

(221)k

, then

k

is equal to:

(A)84
(B)78
(C)168
(D)198

Q8·MathematicsSingle correctJEE Main 2026

Let the smallest value of

k \in \mathbb{N}

, for which the coefficient of

x^3

in

(1 + x)^3 + (1 + x)^4 + (1 + x)^5 + \ldots + (1 + x)^{99} + (1 + kx)^{100}

,

x \neq 0

, is

\left(43n + \frac{101}{4}\right)\left({}^{100}C_3\right)

for some

n \in \mathbb{N}

, be

p

. Then the value of

p + n

is:

(A) $10$
(B) $11$
(C) $12$
(D) $13$

Q9·MathematicsSingle correctJEE Main 2026

The number of elements in the set

S=\left\{(r,k):k\in\mathbb{Z}\ \text{and}\ {}^{36}C_{r+1}=\frac{6\left({}^{35}C_{r}\right)}{(k^{2}-3)}\right\}

, is :

(A)2
(B)4
(C)8
(D)16

Q10·MathematicsSingle correctJEE Main 2026

If for

3 \leq r \leq 30

,

\binom{30}{30-r} + 3\binom{30}{31-r} + 3\binom{30}{32-r} + \binom{30}{33-r} = \binom{m}{r}

, then

m

equals:

(A) $31$
(B) $32$
(C) $33$
(D) $34$

Q11·MathematicsSingle correctJEE Main 2026

The sum of the coefficients of

x^{499}

and

x^{500}

in

(1 + x)^{1000} + x(1 + x)^{999} + x^{2}(1 + x)^{998} + \ldots\ldots + x^{1000}

is

(A) $^{1001}C_{501}$
(B) $^{1002}C_{500}$
(C) $^{1002}C_{501}$
(D) $^{1000}C_{501}$

Q12·MathematicsSingle correctJEE Main 2026

Given below two statements : Statement I :

25^{13} + 20^{13} + 8^{13} + 3^{13}

is divisible by 7. Statement II : The integral part of

\left(7+4\sqrt{3}\right)^{25}

is an odd number. In the light of the above statements, choose the correct answer from the options given below :

(A)Both Statement I and Statement II are false.
(B)Both Statement I and Statement II are true.
(C)Statement I is false but Statement II is true.
(D)Statement I is true but Statement II is false.

Q13·MathematicsSingle correctJEE Main 2026

Let S

= \frac{1}{25!} + \frac{1}{3!23!} + \frac{1}{5!21!} + ...

up to 13 terms. If

13\mathrm{S} = \frac{2^{k}}{n!}

, k∈N, then n + k is equal to

(A)51
(B)52
(C)49
(D)50

Q14·MathematicsSingle correctJEE Main 2026

The value of

\frac{^{100}C_{50}}{51} + \frac{^{100}C_{51}}{52} + \ldots + \frac{^{100}C_{100}}{101}

is :

(A) $\frac{2^{101}}{100}$
(B) $\frac{2^{100}}{100}$
(C) $\frac{2^{101}}{101}$
(D) $\frac{2^{100}}{101}$

Q15·MathematicsSingle correctJEE Main 2026

The sum of all possible values of

n \in 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 :

(A)3
(B)7
(C)12
(D)9

Q16·MathematicsSingle correctJEE Main 2026

Let

C_r

denote the coefficient of

x^r

in the binomial expansion of

(1 + x)^n

,

n \in \mathbb{N}

,

0 \le r \le n

. If

P_n = C_0 - C_1 + \frac{2^2}{3}C_2 - \frac{2^3}{4}C_3 + ..... + \frac{(-2)^n}{n+1}C_n

, then the value of

\sum_{n=1}^{25} \frac{1}{P_{2n}}

equals.

(A)580
(B)525
(C)650
(D)675

Q17·MathematicsNumericalJEE Main 2026

If

\left(\frac{1}{{}^{15}C_{0}} + \frac{1}{{}^{15}C_{1}}\right)\left(\frac{1}{{}^{15}C_{1}} + \frac{1}{{}^{15}C_{2}}\right)\cdots\left(\frac{1}{{}^{15}C_{12}} + \frac{1}{{}^{15}C_{13}}\right) = \frac{\alpha^{13}}{{}^{14}C_{0}\,{}^{14}C_{1}\cdots{}^{14}C_{12}}

, then

30\alpha

is equal to _______.

Q18·MathematicsSingle correctJEE Main 2026

If the coefficient of x in the expansion of

(ax^{2}+bx+c)(1-2x)^{26}

is

-56

and the coefficients of

x^{2}

and

x^{3}

are both zero, then a + b + c is equal to

(A)1300
(B)1500
(C)1403
(D)1483

Q19·MathematicsNumericalJEE Advanced 2025

Let

a_0, a_1, \ldots\ldots, a_{23}

be real numbers such that

\left(1 + \frac{2}{5}x\right)^{23} = \sum_{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 \leq j \leq 23

. Then the value of r is ______

Q20·MathematicsIntegerJEE Main 2025

The product of the last two digits of

(1919)^{1919}

is ___

Q21·MathematicsSingle correctJEE Main 2025

The number of integral terms in the expansion of

\left(5^{1/2}+7^{1/8}\right)^{1016}

is:

(A) $127$
(B) $130$
(C) $129$
(D) $128$

Q22·MathematicsSingle correctJEE Main 2025

The remainder when

\left((64)^{(64)}\right)^{(64)}

is divided by 7 is equal to

(A)4
(B)1
(C)3
(D)6

Q23·MathematicsIntegerJEE Main 2025

The sum of the series

2\times 1\times{}^{20}C_4-3\times 2\times{}^{20}C_5+4\times 3\times{}^{20}C_6-5\times 4\times{}^{20}C_7+\dots+18\times 17\times{}^{20}C_{20}

, is equal to __________.

Q24·MathematicsSingle correctJEE Main 2025

If

1^2\binom{15}{1}+2^2\binom{15}{2}+3^2\binom{15}{3}+\dots+15^2\binom{15}{15}=2^m\cdot3^n\cdot5^k

, where

m,n,k\in\mathbb{N}

, then

m+n+k

is equal to:

(A)19
(B)21
(C)18
(D)15

Q25·MathematicsSingle correctJEE Main 2025

For an integer

n\ge2

, if the arithmetic mean of all coefficients in the binomial expansion of

(x+y)^{2n-3}

is 16, then the distance of the point

P(2n-1, n^2-4n)

from the line

x+y=8

is:

(A) $\sqrt{2}$
(B) $2\sqrt{2}$
(C) $5\sqrt{2}$
(D) $3\sqrt{2}$

Q26·MathematicsSingle correctJEE Main 2025

In the expansion of

\left(\sqrt[3]{2}+\dfrac{1}{\sqrt[3]{3}}\right)^n

,

n\in\mathbb{N}

, if the ratio of

15^{\text{th}}

term from the beginning to the

15^{\text{th}}

term from the end is

\dfrac{1}{6}

, then the value of

{}^nC_3

is:

(A)4060
(B)1040
(C)2300
(D)4960

Q27·MathematicsSingle correctJEE Main 2025

The sum of all rational terms in the expansion of

\left(2+\sqrt3\right)^8

is:

(A)16923
(B)3763
(C)33845
(D)18817

Q28·MathematicsSingle correctJEE Main 2025

If

\displaystyle\sum_{r=1}^{9}\dfrac{r+3}{2^r}\,{}^9C_r=\alpha\left(\dfrac{3}{2}\right)^9-\beta

,

\alpha,\beta\in\mathbb{N}

, then

(\alpha+\beta)^2

is equal to:

(A)27
(B)9
(C)81
(D)18

Q29·MathematicsIntegerJEE Main 2025

Let

(1+x+x^2)^{10}=a_0+a_1x+a_2x^2+\ldots+a_{20}x^{20}

. If

(a_1+a_3+a_5+\ldots+a_{19})-11a_2=121k

, then

k

is equal to __________.

Q30·MathematicsSingle correctJEE Main 2025

If

\displaystyle\sum_{r=0}^{10}\left(\dfrac{10^{r+1}-1}{10^r}\right)\cdot {}^{11}C_{r+1}=\dfrac{\alpha^{11}-11^{11}}{10^{10}}

, then

\alpha

is equal to:

(A)15
(B)11
(C)24
(D)20

Q31·MathematicsSingle correctJEE Main 2025

The term independent of

x

in the expansion of

\left(\dfrac{(x+1)}{x^{2/3}+1-x^{1/3}}-\dfrac{(x+1)}{x-x^{1/2}}\right)^{10}

,

x>1

is:

(A)210
(B)150
(C)240
(D)120

Q32·MathematicsSingle correctJEE Main 2025

The remainder, when

7^{103}

is divided by 23, is equal to:

(A)14
(B)9
(C)17
(D)6

Q33·MathematicsSingle correctJEE Main 2025

The least value of n for which the number of integral terms in the Binomial expansion of

\left(\sqrt[3]{7}+\sqrt[12]{11}\right)^n

is 183, is:

(A)2184
(B)2148
(C)2172
(D)2196

Q34·MathematicsSingle correctJEE Main 2025

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 be the sum of all rational terms in the binomial expansion of

\left(\sqrt[4]{3}+\sqrt[3]{4}\right)^{12}

. Then p+q is equal to:

(A)283
(B)295
(C)287
(D)299

Q35·MathematicsIntegerJEE Main 2025

If

\alpha=1+\sum_{r=1}^{6}(-3)^{r-1}\,^{12}C_{2r-1}

, then the distance of the point

(12, \sqrt{3})

from the line

\alpha x-\sqrt{3}y+1=0

is ____.

Q36·MathematicsSingle correctJEE Main 2025

For some

n\ne 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)^{n+4}

is :

(A) $70$
(B) $35$
(C) $20$
(D) $10$

Q37·MathematicsSingle correctJEE Main 2025

Suppose A and B are the coefficients of

30^{\text{th}}

and

12^{\text{th}}

terms respectively in the binomial expansion of

(1+x)^{2n-1}

. If

2A=5B

, then

n

is equal to:

(A)22
(B)21
(C)20
(D)19

Q38·MathematicsSingle correctJEE Main 2025

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 :

(A)8
(B)18
(C)13
(D)20

Q39·MathematicsIntegerJEE Main 2025

The sum of all rational terms in the expansion of

(1+2^{1/3}+3^{1/2})^6

is equal to _______

Q40·MathematicsIntegerJEE Main 2025

If

\displaystyle\sum_{r=1}^{30}\dfrac{r^2\left(^{30}C_r\right)^2}{^{30}C_{r-1}}=\alpha\times2^{29}

, then

\alpha

is equal to ______.

Q41·MathematicsSingle correctJEE Main 2025

Let

\alpha,\beta,\gamma

and

\delta

be the coefficients of

x^7,x^5,x^3

and

x

respectively in the expansion of

\left(x+\sqrt{x^3-1}\right)^5+\left(x-\sqrt{x^3-1}\right)^5

,

x>1

. If

u

and

v

satisfy the equations

\alpha u+\beta v=18

,

\gamma u+\delta v=20

, then

u+v

equals:

(A)5
(B)4
(C)3
(D)8

Q42·MathematicsNumericalJEE Main 2024

The remainder when

428^{2024}

is divided by

21

is ________.

Q43·MathematicsSingle correctJEE Main 2024

The coefficient of

x^{70}

in

x^2(1 + x)^{98} + x^3(1 + x)^{97} + x^4(1 + x)^{96} + \ldots + x^{54}(1 + x)^{46}

is

{}^{99}C_p - {}^{46}C_q

. Then a possible value to

p + q

is:

(A)55
(B)61
(C)68
(D)83

Q44·MathematicsSingle correctJEE Main 2024

The sum of the coefficient of

x^{2/3}

and

x^{-2/5}

in the binomial expansion of

\left(x^{2/3}+\dfrac{1}{2}x^{-2/5}\right)^{9}

is:

(A) $\dfrac{21}{4}$
(B) $\dfrac{69}{16}$
(C) $\dfrac{63}{16}$
(D) $\dfrac{19}{4}$

Q45·MathematicsNumericalJEE Main 2024

Let

\alpha=\displaystyle\sum_{r=0}^{n}(4r^2+2r+1)\,{}^nC_r

and

\beta=\displaystyle\sum_{r=0}^{n}\dfrac{{}^nC_r}{r+1}

. If

140<\dfrac{2\alpha}{\beta}<281

, then the value of

n

is ___

Q46·MathematicsSingle correctJEE Main 2024

If the term independent of

x

in the expansion of

\left(\sqrt{a}\,x^2+\dfrac{1}{2x^3}\right)^{10}

is

105

, then

a^2

is equal to

(A) $4$
(B) $9$
(C) $6$
(D) $2$

Q47·MathematicsSingle correctJEE Main 2024

Let

0 \le r \le n

. If

^{n+1}C_{r+1} : {}^{n}C_{r} : {}^{n-1}C_{r-1} = 55 : 35 : 21

, then

2n + 5r

is equal to:

(A)60
(B)62
(C)50
(D)55

Q48·MathematicsNumericalJEE Main 2024

If

S(x)=(1+x)+2(1+x)^{2}+3(1+x)^{3}+\dots+60(1+x)^{60}

,

x\neq 0

, and

(60)^{2}S(60)=a(b)^{b}+b

, where

a,b\in N

, then

(a+b)

equal to ______.

Q49·MathematicsNumericalJEE Main 2024

If the second, third and fourth terms in the expansion of

(x+y)^{n}

are 135, 30 and

\dfrac{10}{3}

, respectively, then

6(n^{3}+x^{2}+y)

is equal to _______.

Q50·MathematicsSingle correctJEE Main 2024

If the constant term in the expansion of

\left(\dfrac{\sqrt[5]{3}}{x}+\dfrac{2x}{\sqrt[3]{5}}\right)^{12}

,

x\ne 0

, is

\alpha\times 2^8\times\sqrt[5]{3}

, then

25\alpha

is equal to:

(A)639
(B)724
(C)693
(D)742

Q51·MathematicsNumericalJEE Main 2024

If the constant term in the expansion of

\left(1 + 2x - 3x^3\right)\left(\dfrac{3}{2}x^2 - \dfrac{1}{3x}\right)^9

is

p

, then

108p

is equal to ___.

Q52·MathematicsSingle correctJEE Main 2024

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

(A)14
(B)21
(C)28
(D)7

Q53·MathematicsSingle correctJEE Main 2024

The sum of all rational terms in the expansion of

\left(2^{1/5}+5^{1/3}\right)^{15}

is equal to:

(A) $3133$
(B) $633$
(C) $931$
(D) $6131$

Q54·MathematicsNumericalJEE Main 2024

If the coefficient of

x^{30}

in the expansion of

\left(1+\dfrac1x\right)^6(1+x^2)^7(1-x^3)^8

;

x\ne0

is

|\alpha|

, then

|\alpha|

equals ___

Q55·MathematicsSingle correctJEE Main 2024

Let

m

and

n

be the coefficients of seventh and thirteenth terms respectively in the expansion of

\left(\dfrac{1}{3}x^{1/3}+\dfrac{1}{2x^{2/3}}\right)^{18}

. Then

\left(\dfrac{n}{m}\right)^{1/3}

is:

(A) $\dfrac{4}{9}$
(B) $\dfrac{1}{9}$
(C) $\dfrac{1}{4}$
(D) $\dfrac{9}{4}$

Q56·MathematicsNumericalJEE Main 2024

In the expansion of

(1+x)\left(1-x^2\right)\left(1+\dfrac{3}{x}+\dfrac{3}{x^2}+\dfrac{1}{x^3}\right)^5

,

x\ne 0

, the sum of the coefficient of

x^3

and

x^{-13}

is equal to ______.

Q57·MathematicsNumericalJEE Main 2024

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+\ldots+(x+2)^{n-1}

be

a_r

. If

\displaystyle\sum_{r=0}^{n}a_r=\beta^n-\alpha^n

,

\beta,\alpha\in N

, then the value of

\beta^2+\alpha^2

equals ______.

Q58·MathematicsNumericalJEE Main 2024

Let

\alpha=\displaystyle\sum_{k=0}^{n}\dfrac{\left({}^nC_k\right)^2}{k+1}

and

\beta=\displaystyle\sum_{k=0}^{n-1}\dfrac{{}^nC_k\,{}^nC_{k+1}}{k+2}

. If

5\alpha=6\beta

, then

n

equals ______.

Q59·MathematicsNumericalJEE Main 2024

Number of integral terms in the expansion of

\left\{7^{(1/2)}+11^{(1/6)}\right\}^{824}

is equal to ___

Q60·MathematicsNumericalJEE Main 2024

If

\dfrac{^{11}C_1}{2}+\dfrac{^{11}C_2}{3}+\cdots+\dfrac{^{11}C_9}{10}=\dfrac{n}{m}

with

\gcd(n,m)=1

, then

n+m

is equal to ______.

Q61·MathematicsNumericalJEE Main 2024

Remainder when

64^{32^{32}}

is divided by

9

is equal to ___.

Q62·MathematicsNumericalJEE Main 2024

The coefficient of

x^{2012}

in the expansion of

(1-x)^{2008}(1+x+x^2)^{2007}

is equal to __________.

Q63·MathematicsSingle correctJEE Main 2024

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 in the expansion of

(1+x^2)^n

, then:

(A) $A=B^3$
(B) $3A=B$
(C) $B=A^3$
(D) $A=3B$

Q64·MathematicsNumericalJEE Advanced 2023

Let a and b be two nonzero real numbers. If the coefficient of

x^5

in the expansion of

\left(ax^2 + \frac{70}{27bx}\right)^4

is equal to the coefficient of

x^{-5}

in the expansion of

\left(ax - \frac{1}{bx^2}\right)^7

, then the value of 2b is

Q65·MathematicsSingle correctJEE Main 2023

Let

(a+bx+cx^2)^{10}=\sum_{i=0}^{20}p_i x^i

,

a,b,c\in\mathbb{N}

. If

p_1=20

and

p_2=210

, then

2(a+b+c)

is equal to

(A)8
(B)12
(C)15
(D)6

Q66·MathematicsNumericalJEE Main 2023

Let

\alpha

be the constant term in the binomial expansion of

\left(\sqrt x-\dfrac{6}{x^{3/2}}\right)^{n},\,n\le 15

. If the sum of the coefficients of the remaining terms in the expansion is

649

and the coefficient of

x^{-n}

is

\lambda\alpha

, then

\lambda

is equal to _____.

Q67·MathematicsNumericalJEE Main 2023

The remainder, when

7^{103}

is divided by 17, is

Q68·MathematicsSingle correctJEE Main 2023

The coefficient of

x^5

in the expansion of

\left(2x^3 - \dfrac{1}{3x^2}\right)^5

is

(A)8
(B)9
(C) $\dfrac{80}{9}$
(D) $\dfrac{26}{3}$

Q69·MathematicsSingle correctJEE Main 2023

Fractional part of the number

\dfrac{4^{2022}}{15}

is equal to:

(A) $\dfrac{4}{15}$
(B) $\dfrac{1}{15}$
(C) $\dfrac{14}{15}$
(D) $\dfrac{8}{15}$

Q70·MathematicsSingle correctJEE Main 2023

If

\dfrac{1}{n+1}\,{}^nC_n+\dfrac{1}{n}\,{}^nC_{n-1}+\cdots+\dfrac{1}{2}\,{}^nC_1+{}^nC_0=\dfrac{1023}{10}

then n is equal to

(A)6
(B)9
(C)8
(D)7

Q71·MathematicsSingle correctJEE Main 2023

The sum of the coefficients of the first 50 terms in the binomial expansion of

(1-x)^{100}

, is equal to

(A) $-{}^{100}C_{50}$
(B) ${}^{99}C_{49}$
(C) $-{}^{99}C_{49}$
(D) ${}^{100}C_{50}$

Q72·MathematicsSingle correctJEE Main 2023

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

(A)25
(B)63
(C)41
(D)92

Q73·MathematicsNumericalJEE Main 2023

The mean of the coefficients of

x, x^2, \ldots, x^7

in the binomial expansion of

(2 + x)^9

is _______ .

Q74·MathematicsSingle correctJEE Main 2023

If the

1011^{\text{th}}

term from the end in the binomial expansion of

\left(\dfrac{4x}{5}-\dfrac{5}{2x}\right)^{2022}

is

1024

times the

1011^{\text{th}}

term from the beginning, then

|x|

is equal to

(A) $12$
(B) $8$
(C) $10$
(D) $15$

Q75·MathematicsNumericalJEE Main 2023

The number of integral terms in the expansion of

\left(3^{\frac{1}{2}} + 5^{\frac{1}{4}}\right)^{680}

is equal to _______ .

Q76·MathematicsSingle correctJEE Main 2023

If

(22)^{2022}+(2022)^{22}

leaves the remainder

\alpha

when divided by

3

and

\beta

when divided by

7

, then

(\alpha^2+\beta^2)

is equal to:

(A) $10$
(B) $5$
(C) $20$
(D) $13$

Q77·MathematicsSingle correctJEE Main 2023

If the coefficient of

x^7

in

\left(ax-\dfrac{1}{bx^2}\right)^{13}

and the coefficient of

x^{-5}

in

\left(ax+\dfrac{1}{bx^2}\right)^{13}

are equal, then

a^4 b^4

is equal to:

(A) $44$
(B) $22$
(C) $11$
(D) $33$

Q78·MathematicsSingle correctJEE Main 2023

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:

(A) $63$
(B) $69$
(C) $66$
(D) $60$

Q79·MathematicsNumericalJEE Main 2023

The coefficient of

x^7

in

\left(1-x+2x^3\right)^{10}

is _________.

Q80·MathematicsSingle correctJEE Main 2023

The absolute difference of the coefficients of

x^{10}

and

x^{7}

in the expansion of

\left(2x^{2}+\dfrac{1}{2x}\right)^{11}

is equal to

(A) $12^{3}-12$
(B) $11^{3}-11$
(C) $10^{3}-10$
(D) $13^{3}-13$

Q81·MathematicsNumericalJEE Main 2023

Let

[t]

denotes the greatest integer

\le t

. If the constant term in the expansion of

\left(3x^{2}-\frac{1}{2x^{5}}\right)^{7}

is

\alpha

, then

[\alpha]

is equal to

Q82·MathematicsSingle correctJEE Main 2023

25^{190}-19^{190}-8^{190}+2^{190}

is divisible by

(A)34 but not by 14
(B)both 14 and 34
(C)neither 14 nor 34
(D)14 but not by 34

Q83·MathematicsSingle correctJEE Main 2023

If the coefficients of the three consecutive terms in the expansion of

(1+x)^{n}

are in the ratio

1:5:20

, then the coefficient of the fourth term is

(A)3654
(B)1827
(C)5481
(D)2436

Q84·MathematicsNumericalJEE Main 2023

The coefficient of

x^{18}

in the expansion of

\left(x^{4}-\dfrac{1}{x^{3}}\right)^{15}

is _____.

Q85·MathematicsSingle correctJEE Main 2023

If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of

\left(\sqrt[4]{2}+\dfrac{1}{\sqrt[4]{3}}\right)^{n}

is

\sqrt6:1

, then the third term from the beginning is:

(A) $60\sqrt2$
(B) $60\sqrt3$
(C) $30\sqrt2$
(D) $30\sqrt3$

Q86·MathematicsSingle correctJEE Main 2023

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\in \mathbb{N}

.

(A)both (S1) and (S2) are incorrect
(B)only (S2) is correct
(C)both (S1) and (S2) are correct
(D)only (S1) is correct

Q87·MathematicsSingle correctJEE Main 2023

If the coefficients of

x^7

in

\left(ax^2+\frac{1}{2bx}\right)^{11}

and

x^{-7}

in

\left(ax-\frac{1}{3bx^2}\right)^{11}

are equal, then

(A) $64ab=243$
(B) $729ab=32$
(C) $243ab=64$
(D) $32ab=729$

Q88·MathematicsSingle correctJEE Main 2023

If

^{2n}C_3:^nC_3=10:1

, then the ratio

(n^2+3n):(n^2-3n+4)

is:

(A) $35:16$
(B) $65:37$
(C) $27:11$
(D) $2:1$

Q89·MathematicsNumericalJEE Main 2023

If the term without x in the expansion of

\left(x^{\frac23}+\dfrac{\alpha}{x^3}\right)^{22}

is 7315, then

|\alpha|

is equal to

Q90·MathematicsSingle correctJEE Main 2023

The value of

\frac{1}{1!\,50!}+\frac{1}{3!\,48!}+\frac{1}{5!\,46!}+\cdots+\frac{1}{49!\,2!}+\frac{1}{51!\,1!}

is

(A) $\frac{2^{50}}{51!}$
(B) $\frac{2^{51}}{50!}$
(C) $\frac{2^{50}}{50!}$
(D) $\frac{2^{51}}{51!}$

Q91·MathematicsNumericalJEE Main 2023

Let the sixth term in the binomial expansion of

\left(\sqrt{2^{\log_2(10-3^x)}}+\sqrt[5]{2^{(x-2)\log_2 3}}\right)^m

, in the increasing powers of

2^{(x-2)\log_2 3}

, be 21. If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of x is

Q92·MathematicsNumericalJEE Main 2023

The remainder, when

19^{200}+23^{200}

is divided by

49

, is _______ .

Q93·MathematicsNumericalJEE Main 2023

Let

\alpha > 0

, be the smallest number such that the expansion of

\left(x^{\frac{2}{3}} + \frac{2}{x^3}\right)^{30}

has a term

\beta x^{-\alpha},\ \beta \in \mathbb{N}

. Then

\alpha

is equal to

Q94·MathematicsNumericalJEE Main 2023

If the constant term in the binomial expansion of

\left(\dfrac{x^{5/2}}{2}-\dfrac{4}{x^{l}}\right)^{9}

is

-84

and the coefficient of

x^{-3l}

is

2^{\alpha}\beta

, where

\beta<0

is an odd number, then

|\alpha l-\beta|

is equal to

Q95·MathematicsNumericalJEE Main 2023

The coefficient of

x^{-6}

, in the expansion of

\left(\dfrac{4x}{5}+\dfrac{5}{2x^2}\right)^{9}

, is

Q96·MathematicsNumericalJEE Main 2023

The remainder on dividing

5^{99}

by 11 is :

Q97·MathematicsNumericalJEE Main 2023

The

50^{th}

root of a number

x

is

12

and the

50^{th}

root of another number

y

is

18

. Then the remainder obtained on dividing

(x + y)

by

25

is _______.

Q98·MathematicsSingle correctJEE Main 2023

If the coefficient of

x^{15}

in the expansion of

\left(ax^3+\dfrac{1}{bx^{1/3}}\right)^{15}

is equal to the coefficient of

x^{-15}

in the expansion of

\left(ax^{1/3}-\dfrac{1}{bx^3}\right)^{15}

, where

a

and

b

are positive real numbers, then for each such ordered pair

(a,b)

:

(A) $a=3$
(B) $ab=1$
(C) $a=b$
(D) $a=3b$

Q99·MathematicsSingle correctJEE Main 2023

x=(8\sqrt{3} + 13)^{13}

and

y=(7\sqrt{2} + 9)^{9}

. If

[t]

denotes the greatest integer

\le t

, then:

(A) $[x]$ is odd but $[y]$ is even
(B) $[x] + [y]$ is even
(C) $[x]$ and $[y]$ are both odd
(D) $[x]$ is even but $[y]$ is odd

Q100·MathematicsSingle correctJEE Main 2023

The coefficient of

x^{301}

in

(1+x)^{500}+x(1+x)^{499}+x^2(1+x)^{498}+\cdots+x^{500}

is:

(A) $^{500}C_{300}$
(B) $^{501}C_{200}$
(C) $^{501}C_{302}$
(D) $^{500}C_{301}$

Q101·MathematicsSingle correctJEE Main 2023

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

\left(2+\dfrac{1}{\sqrt2}\right)^{200}

. If

\dfrac{{}^{200}C_{99}\cdot K}{a}=\dfrac{2^{\ell}\cdot m}{n}

, where

m

and

n

are odd numbers, then the ordered pair

(\ell,n)

is equal to:

(A) $(50,51)$
(B) $(50,101)$
(C) $(51,99)$
(D) $(51,101)$

Q102·MathematicsNumericalJEE Main 2023

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 these three terms, is ________ .

Q103·MathematicsNumericalJEE Main 2023

If the co-efficient of

x^9

in

\left(\alpha x^3 + \dfrac{1}{\beta x}\right)^{11}

and the co-efficient of

x^{-9}

in

\left(\alpha x - \dfrac{1}{\beta x^3}\right)^{11}

are equal, then

(\alpha\beta)^2

is equal to ________ .

Q104·MathematicsSingle correctJEE Main 2023

If

a_r

is the coefficient of

x^{10-r}

in the Binomial expansion of

(1+x)^{10}

, then

\displaystyle\sum_{r=1}^{10}r^3\left(\dfrac{a_r}{a_{r-1}}\right)^2

is equal to:

(A) $5445$
(B) $3025$
(C) $4895$
(D) $1210$

Q105·MathematicsSingle correctJEE Main 2023

\displaystyle\sum_{k=0}^{6} {}^{51-k}C_3

is equal to

(A) ${}^{51}C_4 - {}^{45}C_4$
(B) ${}^{52}C_3 - {}^{45}C_3$
(C) ${}^{52}C_4 - {}^{45}C_4$
(D) ${}^{51}C_3 - {}^{45}C_3$

Q106·MathematicsNumericalJEE Main 2023

The remainder when

(2023)^{2023}

is divided by

35

is

Q107·MathematicsNumericalJEE Main 2023

The constant term in the expansion of

\left(2x+\dfrac{1}{x^7}+3x^2\right)^5

is _______.

Q108·MathematicsSingle correctJEE Main 2023

If

\left(^{30}C_1\right)^2+2\left(^{30}C_2\right)^2+3\left(^{30}C_3\right)^2+\cdots+30\left(^{30}C_{30}\right)^2=\dfrac{\alpha\,60!}{(30!)^2}

then

\alpha

is equal to:

(A) $30$
(B) $10$
(C) $60$
(D) $15$

Q109·MathematicsNumericalJEE Main 2023

Let the sum of the coefficients of the first three terms in the expansion of

\left(x-\dfrac{3}{x^2}\right)^n,\ x\neq0,\ n\in\mathbb{N}

, be

376

. Then the coefficient of

x^4

is

Q110·MathematicsNumericalJEE Main 2023

Suppose

\sum\limits_{r=0}^{2023} r^2\,{}^{2023}C_r=2023\times\alpha\times2^{2022}

. Then the value of

\alpha

is _______ .

Q111·MathematicsSingle correctJEE Main 2023

The value of

\sum\limits_{r=0}^{22}{}^{22}C_r\,{}^{23}C_r

is

(A) ${}^{44}C_{23}$
(B) ${}^{45}C_{23}$
(C) ${}^{44}C_{22}$
(D) ${}^{45}C_{24}$

Q112·MathematicsNumericalJEE Main 2022

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of

\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^{n}

, in the increasing powers of

\frac{1}{\sqrt[4]{3}}

be

\sqrt[4]{6} : 1

. If the sixth term from the beginning is

\frac{\alpha}{\sqrt[4]{3}}

, then

\alpha

is equal to __________.

Q113·MathematicsNumericalJEE Main 2022

If

\sum_{k=1}^{10} K^2 \left({}^{10}C_K\right)^2 = 22000L

, then L is equal to ___.

Q114·MathematicsNumericalJEE Main 2022

If

1+ \left(2 + {}^{49}C_{1} + {}^{49}C_{2} + \ldots + {}^{49}C_{49}\right)\left({}^{50}C_{2} + {}^{50}C_{4} + \ldots + {}^{50}C_{50}\right)

is equal to

2^{n}.m

, where m is odd, then

n + m

is equal to _____

Q115·MathematicsNumericalJEE Main 2022

Let the coefficients of the middle terms in the expansion of

\left(\frac{1}{\sqrt{6}}+\beta x\right)^{4}

,

\left(1-3\beta x\right)^{2}

and

\left(1-\frac{\beta}{2}x\right)^{6}

,

\beta>0

, respectively form the first three terms of an A.P. If d is the common difference of this A.P., then

50-\frac{2d}{\beta^{2}}

is equal to _____

Q116·MathematicsSingle correctJEE Main 2022

The remainder when

7^{2022} + 3^{2022}

is divided by 5 is:

(A)0
(B)2
(C)3
(D)4

Q117·MathematicsNumericalJEE Main 2022

Let for the

9^{\text{th}}

term in the binomial expansion of

(3 + 6x)^{n}

, in the increasing powers of

6x

, to be the greatest for

x = \frac{3}{2}

, the least value of n is

n_0

. If k is the ratio of the coefficient of

x^{6}

to the coefficient of

x^{3}

, then

k + n_0

is equal to:

Q118·MathematicsSingle correctJEE Main 2022

The remainder when

(2021)^{2022} + (2022)^{2021}

is divided by 7 is

(A)0
(B)1
(C)2
(D)6

Q119·MathematicsNumericalJEE Main 2022

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______________.

Q120·MathematicsSingle correctJEE Main 2022

\sum\limits_{\substack{i,j=0 \\ i\neq j}}^{n}{}^{n}C_{i}\;{}^{n}C_{j}

is equal to

(A) $2^{2n}-{}^{2n}C_{n}$
(B) $2^{2n-1}-{}^{2n-1}C_{n-1}$
(C) $2^{2n}-\frac{1}{2}\,{}^{2n}C_{n}$
(D) $2^{n-1}+{}^{2n-1}C_{n}$

Q121·MathematicsNumericalJEE Main 2022

If the maximum value of the term independent of t in the expansion of

\left( t^{2}x^{\frac{1}{5}} + \frac{(1-x)^{\frac{1}{10}}}{t} \right)^{15}

,

x \ge 0

, is K, then 8K is equal to ________.

Q122·MathematicsNumericalJEE Main 2022

Let the coefficients of

x^{-1}

and

x^{-3}

in the expansion of

\left(2x^{\frac{1}{5}} - \frac{1}{x^{\frac{1}{5}}}\right)^{15}

,

x > 0

, be

m

and

n

respectively. If r is a positive integer such

mn^2 = {}^{15}C_r . 2^r

, then the value of r is equal to____.

Q123·MathematicsSingle correctJEE Main 2022

If the constant term in the expansion of

\left(3x^3 - 2x^2 + \frac{5}{x^5}\right)^{10}

is

2^k \cdot l

, where

l

is an odd integer, then the value of k is equal to :

(A)6
(B)7
(C)8
(D)9

Q124·MathematicsSingle correctJEE Main 2022

Let

n \geq 5

be an integer. If

9^n - 8n - 1 = 64\alpha

and

6^n - 5n - 1 = 25\beta

, then

\alpha - \beta

is equal to:

(A) $1 + {}^nC_2 (8-5) + {}^nC_3 (8^2 - 5^2) + ... + {}^nC_n (8^{n-1} - 5^{n-1})$
(B) $1 + {}^nC_3 (8-5) + {}^nC_4 (8^2 - 5^2) + ... + {}^nC_n (8^{n-2} - 5^{n-2})$
(C) ${}^nC_3 (8-5) + {}^nC_4 (8^2 - 5^2) + ... + {}^nC_n (8^{n-2} - 5^{n-2})$
(D) ${}^nC_4 (8-5) + {}^nC_5 (8^2 - 5^2) + ... + {}^nC_n (8^{n-3} - 5^{n-3})$

Q125·MathematicsSingle correctJEE Main 2022

If

\sum_{k=1}^{31}\left({}^{31}\mathrm{C}_{k}\right)\left({}^{31}\mathrm{C}_{k-1}\right)-\sum_{k=1}^{30}\left({}^{30}\mathrm{C}_{k}\right)\left({}^{30}\mathrm{C}_{k-1}\right)=\frac{\alpha\,(60!)}{(30!)(31!)},

Where

\alpha \in \mathrm{R},

then the value of

16\alpha

is equal to

(A)1411
(B)1320
(C)1615
(D)1855

Q126·MathematicsNumericalJEE Main 2022

The number of positive integers k such that the constant term in the binomial expansion of

\left(2x^3 + \frac{3}{x^k}\right)^{12}

,

x \neq 0

is

2^8 \cdot \ell,

where

\ell

is an odd integer, is ______.

Q127·MathematicsSingle correctJEE Main 2022

The term independent of x in the expression of

\left(1 - x^{2} + 3x^{3}\right)\left(\frac{5}{2}x^{3} - \frac{1}{5x^{2}}\right)^{11}

,

x \neq 0

is

(A) $\frac{7}{40}$
(B) $\frac{33}{200}$
(C) $\frac{39}{200}$
(D) $\frac{11}{50}$

Q128·MathematicsNumericalJEE Main 2022

If the sum of the coefficients of all the positive powers of x, in the binomial expansion of

\left(x^{n}+\dfrac{2}{x^{5}}\right)^{7}

is 939, then the sum of all the possible integral values of n is :

Q129·MathematicsNumericalJEE Main 2022

If the coefficient of

x^{10}

in the binomial expansion of

\left(\frac{\sqrt{x}}{5^{\frac{1}{4}}}+\frac{\sqrt{5}}{x^{\frac{1}{3}}}\right)^{60}

is

5^{k}l

, where

l

, k

\in

N and

l

is co-prime to 5, then k is equal to _____.

Q130·MathematicsNumericalJEE Main 2022

If

\left(^{40}C_{0}\right)+\left(^{41}C_{1}\right)+\left(^{42}C_{2}\right)+...+\left(^{60}C_{20}\right)=\frac{m}{n}\,^{60}C_{20}

, m and n are coprime, then

m+n

is equal to _____ .

Q131·MathematicsSingle correctJEE Main 2022

The remainder when

(2021)^{2023}

is divided by 7 is :

(A)1
(B)2
(C)5
(D)6

Q132·MathematicsNumericalJEE Main 2022

Let

C_r

denote the binomial coefficient of

x^r

in the expansion of

(1 + x)^{10}

. If α, β ∈ R.

C_1 + 3\cdot 2C_2 + 5\cdot 3C_3 + \ldots

upto 10 terms

= \frac{\alpha \times 2^{11}}{2^{\beta} - 1}\left(C_0 + \frac{C_1}{2} + \frac{C_2}{3} + \ldots\ \text{upto 10 terms}\right)

then the value of α + β is equal to

Q133·MathematicsSingle correctJEE Main 2022

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

(A) $^{501}C_{101}(5)^{399}$
(B) $^{501}C_{101}(5)^{400}$
(C) $^{501}C_{100}(5)^{400}$
(D) $^{500}C_{101}(5)^{399}$

Q134·MathematicsNumericalJEE Main 2022

If the sum of the coefficients of all the positive even powers of x in the binomial expansion of

\left(2x^{3} + \frac{3}{x}\right)^{10}

is

5^{10} - \beta \cdot 3^{9}

, then

\beta

is equal to ______

Q135·MathematicsSingle correctJEE Main 2022

The remainder when

3^{2022}

is divided by 5 is

(A)1
(B)2
(C)3
(D)4

Q136·MathematicsNumericalJEE Main 2022

The remainder on dividing

1 + 3 + 3^{2} + 3^{3} + ... + 3^{2021}

by 50 is ______.

Q137·MathematicsNumericalJEE Main 2021

If the sum of the coefficients in the expansion of

(x + y)^{n}

is 4096, then the greatest coefficient in the expansion is ________ .

Q138·MathematicsNumericalJEE Main 2021

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 ______.

Q139·MathematicsNumericalJEE Main 2021

If

\left( \frac{3^{6}}{4^{4}} \right) k

is the term, independent of x, in the binomial expansion of

\left( \frac{x}{4} - \frac{12}{x^{2}} \right)^{12}

, then k is equal to ______.

Q140·MathematicsSingle correctJEE Main 2021

\sum_{k=0}^{20} \left({}^{20}C_k\right)^2

is equal to :

(A) ${}^{40}C_{21}$
(B) ${}^{40}C_{19}$
(C) ${}^{40}C_{20}$
(D) ${}^{41}C_{20}$

Q141·MathematicsNumericalJEE Main 2021

3 \times 7^{22} + 2 \times 10^{22} - 44

when divided by 18 leaves the remainder __________ .

Q142·MathematicsSingle correctJEE Main 2021

If

^{20}C_r

is the co-efficient of

x^r

in the expansion of

(1 + x)^{20}

, then the value of

\sum_{r=0}^{20} r^2\, ^{20}C_r

is equal to :

(A) $420 \times 2^{19}$
(B) $380 \times 2^{19}$
(C) $380 \times 2^{18}$
(D) $420 \times 2^{18}$

Q143·MathematicsNumericalJEE Main 2021

Let

\begin{pmatrix} n \\ k \end{pmatrix}

denotes

^{n}C_{k}

and

\begin{bmatrix} n \\ k \end{bmatrix} = \begin{cases} \begin{pmatrix} n \\ k \end{pmatrix}, & \text{if } 0 \le k \le n \\ 0, & \text{otherwise} \end{cases}

If

A_{k} = \sum_{i=0}^{9} \begin{pmatrix} 9 \\ i \end{pmatrix} \begin{bmatrix} 12 \\ 12-k+i \end{bmatrix} + \sum_{i=0}^{8} \begin{pmatrix} 8 \\ i \end{pmatrix} \begin{bmatrix} 13 \\ 13-k+i \end{bmatrix}

and

A_{4} - A_{3} = 190\,p

, then p is equal to :

Q144·MathematicsSingle correctJEE Main 2021

If the coefficients of

x^7

in

\left(x^2 + \frac{1}{bx}\right)^{11}

and

x^{-7}

in

\left(x - \frac{1}{bx^2}\right)^{11}

,

b \ne 0

are equal, then the value of b is equal to :

(A) $-1$
(B)1
(C)2
(D) $-2$

Q145·MathematicsSingle correctJEE Main 2021

A possible value of 'x', for which the ninth term in the expansion of

\left\{ 3^{\log_3 \sqrt{25^{x-1} + 7}} + 3^{\left( -\frac{1}{8} \right) \log_3 \left( 5^{x-1} + 1 \right)} \right\}^{10}

in the increasing powers of

3^{\left( -\frac{1}{8} \right) \log_3 \left( 5^{x-1} + 1 \right)}

is equal to 180, is :

(A)1
(B)2
(C)0
(D)−1

Q146·MathematicsNumericalJEE Main 2021

Let

n \in N

and [x] denote the greatest integer less than or equal to x. If the sum of (n+1) terms

^{n}C_{0}, 3.^{n}C_{1}, 5.^{n}C_{2}, 7.^{n}C_{3},.....

is equal to

2^{100}.101

, then

2\left[\dfrac{n-1}{2}\right]

is equal to

Q147·MathematicsSingle correctJEE Main 2021

If the greatest value of the term independent of 'x' in the expansion of

\left(x\sin\alpha + a\frac{\cos\alpha}{x}\right)^{10}

is

\frac{10!}{\left(5!\right)^2}

, then the value of 'a' is equal to :

(A)-1
(B)2
(C)-2
(D)1

Q148·MathematicsNumericalJEE Main 2021

The ratio of the coefficient of the middle term in the expansion of

\left( 1+x \right)^{20}

and the sum of the coefficients of two middle terms in expansion of

\left( 1+x \right)^{19}

is.......

Q149·MathematicsNumericalJEE Main 2021

If the co-efficients of

x^{7}

and

x^{8}

in the expansion of

\left(2 + \dfrac{x}{3}\right)^{n}

are equal, then the value of n is equal to .........

Q150·MathematicsSingle correctJEE Main 2021

If b is very small as compared to the value of a, so that the cube and other higher powers of

\frac{b}{a}

can be neglected in the identity

\frac{1}{a-b}+\frac{1}{a-2b}+\frac{1}{a-3b}+....+\frac{1}{a-nb}=\alpha n+\beta n^{2}+\gamma n^{3}

,

(A) $\frac{a+b}{3a^{2}}$
(B) $\frac{a^{2}+b}{3a^{3}}$
(C) $\frac{b^{2}}{3a^{3}}$
(D) $\frac{a+b^{2}}{3a^{3}}$

Q151·MathematicsSingle correctJEE Main 2021

The sum of all those terms which are rational numbers in the expansion of

\left(2^{\frac{1}{3}} + 3^{\frac{1}{4}}\right)^{12}

is :

(A)89
(B)35
(C)27
(D)43

Q152·MathematicsNumericalJEE Main 2021

the term independent of 'x' in the expression of

\left( \frac{x+1}{x^{2/3} - x^{1/3} + 1} - \frac{x-1}{x - x^{1/2}} \right)^{10}

, where

x \neq 0, 1

is equal to.......

Q153·MathematicsNumericalJEE Main 2021

If the constant term, in binomial expansion of

\left(2x^{r} + \dfrac{1}{x^{2}}\right)^{10}

is 180, then r is equal to.......

Q154·MathematicsNumericalJEE Main 2021

The number of elements in the set

\left\{n \in \{1,2,3,.... 100\} \middle| (11)^n > (10)^n + (9)^n\right\}

is..........

Q155·MathematicsSingle correctJEE Main 2021

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 :

(A) $64$
(B) $88$
(C) $80$
(D) $100$

Q156·MathematicsNumericalJEE Main 2021

The number of rational terms in the binomial expansion of

\left(4^{\frac{1}{4}} + 5^{\frac{1}{6}}\right)^{120}

is.....

Q157·MathematicsSingle correctJEE Main 2021

The coefficient of

x^{256}

in the expansion of

(1-x)^{101}\left(x^2+x+1\right)^{100}

is :

(A) $-{}^{100}C_{15}$
(B) ${}^{100}C_{16}$
(C) $-{}^{100}C_{16}$
(D) ${}^{100}C_{15}$

Q158·MathematicsNumericalJEE Main 2021

The term independent of x in the expansion of

\left[\frac{x+1}{x^{2/3} - x^{1/3} + 1} - \frac{x-1}{x - x^{1/2}}\right]^{10}

, x ≠ 1, is equal to ________ .

Q159·MathematicsSingle correctJEE Main 2021

Let

(1 + x + 2x^{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

(A) $2^{20}(2^{20} - 21)$
(B) $2^{19}(2^{20} - 21)$
(C) $2^{19}(2^{20} + 21)$
(D) $2^{20}(2^{20} + 21)$

Q160·MathematicsSingle correctJEE Main 2021

If the fourth term in the expansion of

\left(x + x^{\log_2 x}\right)^7

is 4480, then the value of x where

x \in N

is equal to :

(A)2
(B)4
(C)3
(D)1

Q161·MathematicsNumericalJEE Main 2021

If

(2021)^{3762}

is divided by 17, then the remainder is _______ .

Q162·MathematicsNumericalJEE Main 2021

Let the coefficients of third, fourth and fifth terms in the expansion of

\left(x+\frac{a}{x^{2}}\right)^{n}

,

x \neq 0

, be in the ratio

12 : 8 : 3

. Then the term independent of

x

in the expansion, is equal to ________.

Q163·MathematicsSingle correctJEE Main 2021

The value of

\sum_{r=0}^{6}\left({}^{6}C_{r}\cdot{}^{6}C_{6-r}\right)

is equal to :

(A) $1124$
(B) $1324$
(C) $1024$
(D) $924$

Q164·MathematicsSingle correctJEE Main 2021

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 :

(A)26
(B)30
(C)8
(D)7

Q165·MathematicsNumericalJEE Main 2021

Let n be a positive integer. Let

A=\sum\limits_{k=0}^{n}(-1)^{k}\ {}^{n}C_{k}\left[\left(\frac{1}{2}\right)^{k}+\left(\frac{3}{4}\right)^{k}+\left(\frac{7}{8}\right)^{k}+\left(\frac{15}{16}\right)^{k}+\left(\frac{31}{32}\right)^{k}\right]

If

63A = 1-\frac{1}{2^{30}}

, then n is equal to _______.

Q166·MathematicsSingle correctJEE Main 2021

The maximum value of the term independent of ‘t’ in the expansion of

\left( tx^{\frac{1}{5}} + \frac{(1-x)^{\frac{1}{10}}}{t} \right)^{10}

where x ∈ (0,1) is:

(A) $\frac{10!}{\sqrt{3}(5!)^{2}}$
(B) $\frac{2.10!}{3(5!)^{2}}$
(C) $\frac{10!}{3(5!)^{2}}$
(D) $\frac{2.10!}{3\sqrt{3}(5!)^{2}}$

Q167·MathematicsNumericalJEE Main 2021

Let m, n ∈ N and gcd (2, n) = 1. If

30\binom{30}{0}+29\binom{30}{1}+....+2\binom{30}{28}+1\binom{30}{29}=n\cdot 2^{m}

, then n + m is equal to _______.

Q168·MathematicsNumericalJEE Main 2021

The total number of two digit numbers ‘n’, such that 3

^{n}

+7

^{n}

is a multiple of 10, is ______.

Q169·MathematicsNumericalJEE Main 2021

If the remainder when x is divided by 4 is 3, then the remainder when (2020+x)

^{2022}

is divided by 8 is ______.

Q170·MathematicsSingle correctJEE Main 2021

If

n\ge 2

is a positive integer, then the sum of the series

^{n+1}C_{2}+2\left(^{2}C_{2}+^{3}C_{2}+^{4}C_{2}+....+^{n}C_{2}\right)

is :

(A) $\frac{n(n+1)^{2}(n+2)}{12}$
(B) $\frac{n(n-1)(2n+1)}{6}$
(C) $\frac{n(n+1)(2n+1)}{6}$
(D) $\frac{n(2n+1)(3n+1)}{6}$

Q171·MathematicsNumericalJEE Main 2021

For integers n and r, let

\binom{n}{r}=\begin{cases}{}^{n}C_{r}, & \text{if } n\ge r\ge 0 \\ 0, & \text{otherwise}\end{cases}

The maximum value of k for which the sum

\sum_{i=0}^{k}\binom{10}{i}\binom{15}{k-i}+\sum_{i=0}^{k+1}\binom{12}{i}\binom{13}{k+1-i}

exists, is equal to________.

Q172·MathematicsSingle correctJEE Main 2021

The value of

-{}^{15}C_1 + 2.{}^{15}C_2 - 3.{}^{15}C_3 + .... -15.{}^{15}C_{15} + {}^{14}C_1 + {}^{14}C_3 + {}^{14}C_5 + ....+ {}^{14}C_{11}

is:

(A) $2^{14}$
(B) $2^{13} - 13$
(C) $2^{16} - 1$
(D) $2^{13} - 14$

Q173·MathematicsMultiple correctJEE Advanced 2020

For non-negative integers

s

and

r

, let

\binom{s}{r} = \begin{cases} \frac{s!}{r!(s-r)!} & \text{if } r \le s, \\ 0 & \text{if } r > s. \end{cases}

For positive integers

m

and

n

, let

g(m,n) = \sum_{p=0}^{m+n} \frac{f(m,n,p)}{\binom{n+p}{p}}

where for any nonnegative integer

p

,

f(m,n,p) = \sum_{i=0}^{p} \binom{m}{i} \binom{n+i}{p} \binom{p+n}{p-i}

Then which of the following statements is/are TRUE?

(A) $g(m,n) = g(n,m)$ for all positive integers $m, n$
(B) $g(m,n+1) = g(m+1,n)$ for all positive integers $m, n$
(C) $g(2m,2n) = 2g(m,n)$ for all positive integers $m, n$
(D) $g(2m,2n) = (g(m,n))^{2}$ for all positive integers $m, n$

Q174·MathematicsSingle correctJEE Main 2020

If {p} denotes the fractional part of the number p, then

\left\{\dfrac{3^{200}}{8}\right\}

, is equal to

(A) $\dfrac{5}{8}$
(B) $\dfrac{7}{8}$
(C) $\dfrac{3}{8}$
(D) $\dfrac{1}{8}$

Q175·MathematicsSingle correctJEE Main 2020

If the constant term in the binomial expansion of

\left(\sqrt{x} - \frac{k}{x^2}\right)^{10}

is 405, then

|k|

equals:

(A) $9$
(B) $1$
(C) $3$
(D) $2$

Q176·MathematicsNumericalJEE Main 2020

The natural number m, for which the coefficient of x in the binomial expansion of

\left(x^{m}+\dfrac{1}{x^{2}}\right)^{22}

is 1540, is ____

Q177·MathematicsNumericalJEE Main 2020

The coefficient of

x^4

in the expansion of

(1 + x + x^2 + x^3)^6

in powers of x, is __________.

Q178·MathematicsSingle correctJEE Main 2020

The value of

\sum_{r=0}^{20} {}^{50-r}C_{6}

is equal to:

(A) ${}^{50}C_{6} - {}^{30}C_{6}$
(B) ${}^{51}C_{7} + {}^{30}C_{7}$
(C) ${}^{51}C_{7} - {}^{30}C_{7}$
(D) ${}^{50}C_{7} - {}^{30}C_{7}$

Q179·MathematicsSingle correctJEE Main 2020

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 in the expansion is:

(A)252
(B)330
(C)792
(D)462

Q180·MathematicsNumericalJEE Main 2020

Let

(2x^{2} + 3x + 4)^{10} = \sum_{r=0}^{20} a_{r} x^{r}

. Then

\frac{a_{7}}{a_{13}}

is equal to __________.

Q181·MathematicsSingle correctJEE Main 2020

If the term independent of x in the expansion of

\left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^9

is k, then 18k is equal to:

(A)9
(B)5
(C)7
(D)11

Q182·MathematicsSingle correctJEE Main 2020

If the number of integral terms in the expansion of

\left(3^{\frac{1}{2}} + 5^{\frac{1}{8}}\right)^{n}

is exactly 33, then the least value of 'n' is

(A)264
(B)248
(C)256
(D)128

Q183·MathematicsNumericalJEE Main 2020

If

C_{r} = {}^{25}C_{r}

and

C_{0} + 5.C_{1} + 9.C_{2} + .......... + (101).C_{25} = 2^{25}.k

, then k is equal to ______

Q184·MathematicsNumericalJEE Main 2020

The coefficient of

x^{4}

in the expansion of

(1+x+x^{2})^{10}

is ______.

Q185·MathematicsSingle correctJEE Main 2020

If a, b and c are the greatest values of

^{19}C_{p}

,

^{20}C_{q}

and

^{21}C_{r}

respectively, then:

(A) $\dfrac{a}{10}=\dfrac{b}{11}=\dfrac{c}{42}$
(B) $\dfrac{a}{11}=\dfrac{b}{22}=\dfrac{c}{21}$
(C) $\dfrac{a}{11}=\dfrac{b}{22}=\dfrac{c}{42}$
(D) $\dfrac{a}{10}=\dfrac{b}{11}=\dfrac{c}{21}$

Q186·MathematicsSingle correctJEE Main 2020

If

\alpha

and

\beta

be the coefficients of

x^{4}

and

x^{2}

respectively in the expansion of

\left(x+\sqrt{x^{2}-1}\right)^{6}+\left(x-\sqrt{x^{2}-1}\right)^{6}

, then:

(A) $\alpha+\beta=-30$
(B) $\alpha-\beta=60$
(C) $\alpha-\beta=-132$
(D) $\alpha+\beta=60$

Q187·MathematicsNumericalJEE Main 2020

If the sum of the coefficients of all even powers of x in the product

(1+x+x^{2}+......+x^{2n})(1-x+x^{2}-x^{3}+.....+x^{2n})

is 61, then n is equal to

Q188·MathematicsNumericalJEE Advanced 2019

Suppose

\det \begin{bmatrix} \displaystyle\sum_{k=0}^{n} k & \displaystyle\sum_{k=0}^{n} {}^{n}C_k k^{2} \\ \displaystyle\sum_{k=0}^{n} {}^{n}C_k k & \displaystyle\sum_{k=0}^{n} {}^{n}C_k 3^{k} \end{bmatrix} = 0

holds for some positive integer n. Then

\displaystyle\sum_{k=0}^{n} \dfrac{{}^{n}C_k}{k+1}

equals. ____

Q189·MathematicsSingle correctJEE Main 2019

The coefficient of

x^{18}

in the product

(1+x)(1-x)^{10}(1+x+x^{2})^{9}

is :

(A)84
(B)126
(C) $-126$
(D) $-84$

Q190·MathematicsSingle correctJEE Main 2019

The term independent of x in the expansion of

\left(\dfrac{1}{60} - \dfrac{x^{8}}{81}\right)\cdot\left(2x^{2} - \dfrac{3}{x^{2}}\right)^{6}

is equal to:

(A)36
(B) $-36$
(C) $-108$
(D) $-72$

Q191·MathematicsSingle correctJEE Main 2019

The smallest natural number n, such that the coefficient of x in the expansion of

\left(x^{2} + \dfrac{1}{x^{3}}\right)^{n}

is

^{n}C_{23}

is

(A)38
(B)58
(C)23
(D)35

Q192·MathematicsSingle correctJEE Main 2019

If the coefficients of x

^{2}

and x

^{3}

are both zero, in the expansion of the expression (1 + ax + bx

^{2}

) (1 − 3x)

^{15}

in powers of x, then the ordered pair (a, b) is equal to

(A)(−54, 315)
(B)(28, 861)
(C)(28, 315)
(D)(−21, 714)

Q193·MathematicsSingle correctJEE Main 2019

If some three consecutive in the binomial expansion of

(x+1)^{n}

in powers of x are in the ratio

2 : 15 : 70

, then the average of these three coefficient is:

(A) $964$
(B) $625$
(C) $227$
(D) $232$

Q194·MathematicsSingle correctJEE Main 2019

If the fourth term in the Binomial expansion of

\left(\dfrac{2}{x}+x^{\log_{8}x}\right)^{6}(x>0)

is

20\times 8^{7}

, then a value of x is:

(A) $8^{3}$
(B) $8^{-2}$
(C) $8$
(D) $8^{2}$

Q195·MathematicsSingle correctJEE Main 2019

2\cdot {}^{20}C_{0}+5\cdot {}^{20}C_{1}+8\cdot {}^{20}C_{2}+11\cdot {}^{20}C_{3}+....+62\cdot {}^{20}C_{20}

is equal to

(A) $2^{23}$
(B) $2^{26}$
(C) $2^{24}$
(D) $2^{25}$

Q196·MathematicsSingle correctJEE Main 2019

The sum of the co-efficient of all even degree terms in x in the expansion of

\left(x+\sqrt{x^{3}-1}\right)^{6}+\left(x-\sqrt{x^{3}-1}\right)^{6},(x>1)

is equal to:

(A)26
(B)24
(C)32
(D)29

Q197·MathematicsSingle correctJEE Main 2019

If

^{n}C_{4}, {}^{n}C_{5}

and

^{n}C_{6}

are in A.P., then n can be :

(A)9
(B)14
(C)11
(D)12

Q198·MathematicsSingle correctJEE Main 2019

The total number or irrational terms in the binomial expansion of

\left(7^{1/5} - 3^{1/10}\right)^{60}

is :

(A)55
(B)49
(C)48
(D)54

Q199·MathematicsSingle correctJEE Main 2019

The value of r for which

^{20}C_{r}\,^{20}C_{0}+^{20}C_{r-1}\,^{20}C_{1}+^{20}C_{r-2}\,^{20}C_{2}+\ldots+^{20}C_{0}\,^{20}C_{r}

is maximum is:

(A) $15$
(B) $20$
(C) $11$
(D) $10$

Q200·MathematicsSingle correctJEE Main 2019

Let

S_{n}=1+q+q^{2}+\ldots+q^{n}

and

T_{n}=1+\left(\dfrac{q+1}{2}\right)+\left(\dfrac{q+1}{2}\right)^{2}+\ldots+\left(\dfrac{q+1}{2}\right)^{n}

where q is a real number and

q\neq 1

. If

{}^{101}C_{1}+{}^{101}C_{2}\cdot S_{1}+\ldots+{}^{101}C_{101}\cdot S_{100}=\alpha T_{100}

then α is equal to:

(A) $2^{99}$
(B)202
(C)200
(D) $2^{100}$

Q201·MathematicsSingle correctJEE Main 2019

The sum of the real values of x for which the middle term in the binomial expansion of

\left(\frac{x^{3}}{3}+\frac{3}{x}\right)^{8}

equals 5670 is:

(A)0
(B)6
(C)4
(D)8

Q202·MathematicsSingle correctJEE Main 2019

Let

(x+10)^{50}+(x-10)^{50}=a_0+a_1x+a_2x^{2}+...+a_{50}x^{50}

, for

x\in R

; then

\frac{a_2}{a_0}

is equal to:

(A)12.50
(B)12.00
(C)12.25
(D)12.75

Q203·MathematicsSingle correctJEE Main 2019

If

\sum_{i=1}^{20}\left(\frac{^{20}C_{i-1}}{^{20}C_i+^{20}C_{i-1}}\right)^{3}=\frac{k}{21}

, then k equals:

(A)400
(B)50
(C)200
(D)100

Q204·MathematicsSingle correctJEE Main 2019

If the third term in the binomial expansion of

\left(1+x^{\log_{2}x}\right)^{5}

equals 2560, then a possible value of x is:

(A) $\frac{1}{4}$
(B) $4\sqrt{2}$
(C) $\frac{1}{8}$
(D) $2\sqrt{2}$

Q205·MathematicsSingle correctJEE Main 2019

If the fractional part of the number

\dfrac{2^{403}}{15}

is

\dfrac{k}{15}

, then k is equal to:

(A)6
(B)8
(C)4
(D)14

Q206·MathematicsSingle correctJEE Main 2019

The coefficient of

t^{4}

in the expansion of

\left(\frac{1-t^{6}}{1-t}\right)^{3}

is

(A)12
(B)15
(C)10
(D)14

Q207·MathematicsNumericalJEE Advanced 2018

Let

X = ({}^{10}C_{1})^{2} + 2({}^{10}C_{2})^{2} + 3({}^{10}C_{3})^{2} + .... + 10({}^{10}C_{10})^{2}

, where

{}^{10}C_{r}

,

r \in \{1, 2, ....., 10\}

denote binomial coefficients. Then the value of

\frac{1}{1430}X

is ______ .

Q208·MathematicsIntegerJEE Advanced 2016

Let

m

be the smallest positive integer such that the coefficient of

x^{2}

in the expansion of

(1 + x)^{2} + (1 + x)^{3} + \ldots + (1 + x)^{49} + (1 + mx)^{50}

is

(3n + 1)\,{}^{51}C_3

for some positive integer

n

. Then the value of

n

is

Q209·MathematicsIntegerJEE Advanced 2015

The coefficient of

x^{9}

in the expansion of

(1 + x)(1 + x^{2})(1 + x^{3}) \ldots (1 + x^{100})

is

Q210·MathematicsSingle correctJEE Advanced 2014

Coefficient of

x^{11}

in the expansion of

(1 + x^{2})^{4} (1 + x^{3})^{7} (1 + x^{4})^{12}

is

(A)1051
(B)1106
(C)1113
(D)1120

Q211·MathematicsIntegerJEE Advanced 2013

The coefficients of three consecutive terms of

(1+x)^{n+5}

are in the ratio

5:10:14

. Then n = _______

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