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Binomial Theorem & its Simple Applications: JEE Mains Previous Year Questions (2021-2024) | Mathematics (Maths) for JEE Main & Advanced PDF Download

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 Page 1


JEE Mains Previous Year Questions 
(2021-2024): Binomial Theorem & its 
Simple Applications 
2024 
Q1 - 2024 (01 Feb Shift 1) 
If the Coefficient of ?? 30
 in the expansion of (1+
1
?? )
6
(1+?? 2
)
7
(1-?? 3
)
8
;?? ?0 is ?? , then 
|?? | equals 
Q2 - 2024 (01 Feb Shift 2) 
Let ?? and ?? be the coefficients of seventh and thirteenth terms respectively in the 
expansion of (
1
3
?? 1
3
+
1
2?? 2
3
)
18
. Then (
n
m
)
1
3
 is : 
(1) 
4
9
 
(2) 
1
9
 
(3) 
1
4
 
(4) 
9
4
 
Q3 - 2024 (27 Jan Shift 1) 
 
?? -1
?? ?? =(?? 2
-8)
?? ?? ?? +1
 if and only if : 
(1) 2v2<k=3 
(2) 2v3<k=3v2 
(3) 2v3<k<3v3 
(4) 2v2<k<2v3 
Q4 - 2024 (27 Jan Shift 1) 
Page 2


JEE Mains Previous Year Questions 
(2021-2024): Binomial Theorem & its 
Simple Applications 
2024 
Q1 - 2024 (01 Feb Shift 1) 
If the Coefficient of ?? 30
 in the expansion of (1+
1
?? )
6
(1+?? 2
)
7
(1-?? 3
)
8
;?? ?0 is ?? , then 
|?? | equals 
Q2 - 2024 (01 Feb Shift 2) 
Let ?? and ?? be the coefficients of seventh and thirteenth terms respectively in the 
expansion of (
1
3
?? 1
3
+
1
2?? 2
3
)
18
. Then (
n
m
)
1
3
 is : 
(1) 
4
9
 
(2) 
1
9
 
(3) 
1
4
 
(4) 
9
4
 
Q3 - 2024 (27 Jan Shift 1) 
 
?? -1
?? ?? =(?? 2
-8)
?? ?? ?? +1
 if and only if : 
(1) 2v2<k=3 
(2) 2v3<k=3v2 
(3) 2v3<k<3v3 
(4) 2v2<k<2v3 
Q4 - 2024 (27 Jan Shift 1) 
If ?? denotes the sum of all the coefficients in the expansion of (1-3?? +10?? 2
)
?? and ?? 
denotes the sum of all the coefficients in the expansion of (1+?? 2
)
?? , then : 
(1) ?? =?? 3
 
(2) 3 A=B 
(3) ?? =?? 3
 
(4) ?? =3?? 
Q5 - 2024 (27 Jan Shift 2) 
The coefficient of ?? 2012
 in the expansion of (1-?? )
2008
(1+?? +?? 2
)
2007
 is equal to 
Q6 - 2024 (29 Jan Shift 1) 
If 
 
11
C
1
2
+
 
11
C
2
3
+?+
 
11
C
9
10
=
n
m
 with gcd (n,m)=1, then n+m is equal to 
Q7 - 2024 (29 Jan Shift 2) 
Remainder when 64
32
32
 is divided by 9 is equal to 
Q8 - 2024 (30 Jan Shift 1) 
Number of integral terms in the expansion of {7
(
1
2
)
+11
(
1
6
)
}
824
 is equal to 
Q9 - 2024 (30 Jan Shift 2) 
Suppose 2-?? ,?? ,2-?? ,?? are the coefficient of four consecutive terms in the expansion 
of (1+?? )
?? . Then the value of ?? 2
-?? 2
+6?? +2?? equals 
[We changed options. In official NTA paper no option was correct.] 
(1) 4 
(2) 10 
(3) 8 
(4) Data Inconsistent 
Q10 - 2024 (30 Jan Shift 2) 
Let ?? =?
k=0
n
?(
( 
n
C
k
)
2
k+1
) and ?? =?
k=0
n-1
?(
 
n
C
k
 
n
C
k+1
k+2
) 
If 5?? =6?? , then n equals 
Q11 - 2024 (31 Jan Shift 1) 
Page 3


JEE Mains Previous Year Questions 
(2021-2024): Binomial Theorem & its 
Simple Applications 
2024 
Q1 - 2024 (01 Feb Shift 1) 
If the Coefficient of ?? 30
 in the expansion of (1+
1
?? )
6
(1+?? 2
)
7
(1-?? 3
)
8
;?? ?0 is ?? , then 
|?? | equals 
Q2 - 2024 (01 Feb Shift 2) 
Let ?? and ?? be the coefficients of seventh and thirteenth terms respectively in the 
expansion of (
1
3
?? 1
3
+
1
2?? 2
3
)
18
. Then (
n
m
)
1
3
 is : 
(1) 
4
9
 
(2) 
1
9
 
(3) 
1
4
 
(4) 
9
4
 
Q3 - 2024 (27 Jan Shift 1) 
 
?? -1
?? ?? =(?? 2
-8)
?? ?? ?? +1
 if and only if : 
(1) 2v2<k=3 
(2) 2v3<k=3v2 
(3) 2v3<k<3v3 
(4) 2v2<k<2v3 
Q4 - 2024 (27 Jan Shift 1) 
If ?? denotes the sum of all the coefficients in the expansion of (1-3?? +10?? 2
)
?? and ?? 
denotes the sum of all the coefficients in the expansion of (1+?? 2
)
?? , then : 
(1) ?? =?? 3
 
(2) 3 A=B 
(3) ?? =?? 3
 
(4) ?? =3?? 
Q5 - 2024 (27 Jan Shift 2) 
The coefficient of ?? 2012
 in the expansion of (1-?? )
2008
(1+?? +?? 2
)
2007
 is equal to 
Q6 - 2024 (29 Jan Shift 1) 
If 
 
11
C
1
2
+
 
11
C
2
3
+?+
 
11
C
9
10
=
n
m
 with gcd (n,m)=1, then n+m is equal to 
Q7 - 2024 (29 Jan Shift 2) 
Remainder when 64
32
32
 is divided by 9 is equal to 
Q8 - 2024 (30 Jan Shift 1) 
Number of integral terms in the expansion of {7
(
1
2
)
+11
(
1
6
)
}
824
 is equal to 
Q9 - 2024 (30 Jan Shift 2) 
Suppose 2-?? ,?? ,2-?? ,?? are the coefficient of four consecutive terms in the expansion 
of (1+?? )
?? . Then the value of ?? 2
-?? 2
+6?? +2?? equals 
[We changed options. In official NTA paper no option was correct.] 
(1) 4 
(2) 10 
(3) 8 
(4) Data Inconsistent 
Q10 - 2024 (30 Jan Shift 2) 
Let ?? =?
k=0
n
?(
( 
n
C
k
)
2
k+1
) and ?? =?
k=0
n-1
?(
 
n
C
k
 
n
C
k+1
k+2
) 
If 5?? =6?? , then n equals 
Q11 - 2024 (31 Jan Shift 1) 
In the expansion of (1+?? )(1-?? 2
)(1+
3
?? +
3
?? 2
+
1
?? 3
)
5
,?? ?0, the sum of the coefficient of 
?? 3
 and ?? -13
 is equal to 
Q12 - 2024 (31 Jan Shift 2) 
If for some m,n; 
6
C
m
+2( 
6
C
m+1
)+ 
6
C
m+2
> 
8
C
3
 and  
?? -1
?? 3
: 
?? ?? 4
=1:8, then  
?? ?? ?? +1
+
 
?? +1
?? ?? is equal to 
(1) 380 
(2) 376 
(3) 384 
(4) 372 
Q13 - 2024 (31 Jan Shift 2) 
Let the coefficient of ?? ?? 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 ?? ?? . If ?
?? =0
?? ??? ?? =?? ?? -?? ?? ,?? ,?? ??? , then the value of 
?? 2
+?? 2
 equals 
Answer Key 
Q1 (678)  Q2(4) Q3 (1)  Q4(1) 
Q5 (0)  Q6 (2041) Q7 (1) Q8 (138) 
Q9 (4) Q10 (10) Q11 (118) Q12 (4) 
Q13 (25)    
 
Solutions 
Q1 
coeff of ?? 30
 in 
(?? +1)
6
(1+?? 2
)
7
(1-?? 3
)
8
?? 6
 coeff. of ?? 36
 in (1+?? )
6
(1+?? 2
)
7
(1-?? 3
)
8
 
General term 
 
6
C
r
1
 
7
C
r
2
 
8
C
r
3
(-1)
r
3
X
r
1
+2r
2
+3r
3
 
r
1
+2r
2
+3r
3
=36 
 
Page 4


JEE Mains Previous Year Questions 
(2021-2024): Binomial Theorem & its 
Simple Applications 
2024 
Q1 - 2024 (01 Feb Shift 1) 
If the Coefficient of ?? 30
 in the expansion of (1+
1
?? )
6
(1+?? 2
)
7
(1-?? 3
)
8
;?? ?0 is ?? , then 
|?? | equals 
Q2 - 2024 (01 Feb Shift 2) 
Let ?? and ?? be the coefficients of seventh and thirteenth terms respectively in the 
expansion of (
1
3
?? 1
3
+
1
2?? 2
3
)
18
. Then (
n
m
)
1
3
 is : 
(1) 
4
9
 
(2) 
1
9
 
(3) 
1
4
 
(4) 
9
4
 
Q3 - 2024 (27 Jan Shift 1) 
 
?? -1
?? ?? =(?? 2
-8)
?? ?? ?? +1
 if and only if : 
(1) 2v2<k=3 
(2) 2v3<k=3v2 
(3) 2v3<k<3v3 
(4) 2v2<k<2v3 
Q4 - 2024 (27 Jan Shift 1) 
If ?? denotes the sum of all the coefficients in the expansion of (1-3?? +10?? 2
)
?? and ?? 
denotes the sum of all the coefficients in the expansion of (1+?? 2
)
?? , then : 
(1) ?? =?? 3
 
(2) 3 A=B 
(3) ?? =?? 3
 
(4) ?? =3?? 
Q5 - 2024 (27 Jan Shift 2) 
The coefficient of ?? 2012
 in the expansion of (1-?? )
2008
(1+?? +?? 2
)
2007
 is equal to 
Q6 - 2024 (29 Jan Shift 1) 
If 
 
11
C
1
2
+
 
11
C
2
3
+?+
 
11
C
9
10
=
n
m
 with gcd (n,m)=1, then n+m is equal to 
Q7 - 2024 (29 Jan Shift 2) 
Remainder when 64
32
32
 is divided by 9 is equal to 
Q8 - 2024 (30 Jan Shift 1) 
Number of integral terms in the expansion of {7
(
1
2
)
+11
(
1
6
)
}
824
 is equal to 
Q9 - 2024 (30 Jan Shift 2) 
Suppose 2-?? ,?? ,2-?? ,?? are the coefficient of four consecutive terms in the expansion 
of (1+?? )
?? . Then the value of ?? 2
-?? 2
+6?? +2?? equals 
[We changed options. In official NTA paper no option was correct.] 
(1) 4 
(2) 10 
(3) 8 
(4) Data Inconsistent 
Q10 - 2024 (30 Jan Shift 2) 
Let ?? =?
k=0
n
?(
( 
n
C
k
)
2
k+1
) and ?? =?
k=0
n-1
?(
 
n
C
k
 
n
C
k+1
k+2
) 
If 5?? =6?? , then n equals 
Q11 - 2024 (31 Jan Shift 1) 
In the expansion of (1+?? )(1-?? 2
)(1+
3
?? +
3
?? 2
+
1
?? 3
)
5
,?? ?0, the sum of the coefficient of 
?? 3
 and ?? -13
 is equal to 
Q12 - 2024 (31 Jan Shift 2) 
If for some m,n; 
6
C
m
+2( 
6
C
m+1
)+ 
6
C
m+2
> 
8
C
3
 and  
?? -1
?? 3
: 
?? ?? 4
=1:8, then  
?? ?? ?? +1
+
 
?? +1
?? ?? is equal to 
(1) 380 
(2) 376 
(3) 384 
(4) 372 
Q13 - 2024 (31 Jan Shift 2) 
Let the coefficient of ?? ?? 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 ?? ?? . If ?
?? =0
?? ??? ?? =?? ?? -?? ?? ,?? ,?? ??? , then the value of 
?? 2
+?? 2
 equals 
Answer Key 
Q1 (678)  Q2(4) Q3 (1)  Q4(1) 
Q5 (0)  Q6 (2041) Q7 (1) Q8 (138) 
Q9 (4) Q10 (10) Q11 (118) Q12 (4) 
Q13 (25)    
 
Solutions 
Q1 
coeff of ?? 30
 in 
(?? +1)
6
(1+?? 2
)
7
(1-?? 3
)
8
?? 6
 coeff. of ?? 36
 in (1+?? )
6
(1+?? 2
)
7
(1-?? 3
)
8
 
General term 
 
6
C
r
1
 
7
C
r
2
 
8
C
r
3
(-1)
r
3
X
r
1
+2r
2
+3r
3
 
r
1
+2r
2
+3r
3
=36 
 
 
 
 
 
Case-I : 
?? 1
 ?? 2
 ?? 3
 
0 6 8 
2 5 8 
4 4 8 
6 3 8 
 
?? 1
+2?? 2
=12Taking ?? 3
=8) 
Case II. 
?? 1
 ?? 2
 ?? 3
 
1 7 7 
3 6 7 
5 5 7 
 
?? 1
+2?? 2
=15 Taking ?? 3
=7) 
?? 1
 ?? 2
 ?? 3
 
4 7 6 
6 6 6 
 
Case-III : 
?? 1
+2?? 2
=18 (Taking ?? 3
=6) 
Page 5


JEE Mains Previous Year Questions 
(2021-2024): Binomial Theorem & its 
Simple Applications 
2024 
Q1 - 2024 (01 Feb Shift 1) 
If the Coefficient of ?? 30
 in the expansion of (1+
1
?? )
6
(1+?? 2
)
7
(1-?? 3
)
8
;?? ?0 is ?? , then 
|?? | equals 
Q2 - 2024 (01 Feb Shift 2) 
Let ?? and ?? be the coefficients of seventh and thirteenth terms respectively in the 
expansion of (
1
3
?? 1
3
+
1
2?? 2
3
)
18
. Then (
n
m
)
1
3
 is : 
(1) 
4
9
 
(2) 
1
9
 
(3) 
1
4
 
(4) 
9
4
 
Q3 - 2024 (27 Jan Shift 1) 
 
?? -1
?? ?? =(?? 2
-8)
?? ?? ?? +1
 if and only if : 
(1) 2v2<k=3 
(2) 2v3<k=3v2 
(3) 2v3<k<3v3 
(4) 2v2<k<2v3 
Q4 - 2024 (27 Jan Shift 1) 
If ?? denotes the sum of all the coefficients in the expansion of (1-3?? +10?? 2
)
?? and ?? 
denotes the sum of all the coefficients in the expansion of (1+?? 2
)
?? , then : 
(1) ?? =?? 3
 
(2) 3 A=B 
(3) ?? =?? 3
 
(4) ?? =3?? 
Q5 - 2024 (27 Jan Shift 2) 
The coefficient of ?? 2012
 in the expansion of (1-?? )
2008
(1+?? +?? 2
)
2007
 is equal to 
Q6 - 2024 (29 Jan Shift 1) 
If 
 
11
C
1
2
+
 
11
C
2
3
+?+
 
11
C
9
10
=
n
m
 with gcd (n,m)=1, then n+m is equal to 
Q7 - 2024 (29 Jan Shift 2) 
Remainder when 64
32
32
 is divided by 9 is equal to 
Q8 - 2024 (30 Jan Shift 1) 
Number of integral terms in the expansion of {7
(
1
2
)
+11
(
1
6
)
}
824
 is equal to 
Q9 - 2024 (30 Jan Shift 2) 
Suppose 2-?? ,?? ,2-?? ,?? are the coefficient of four consecutive terms in the expansion 
of (1+?? )
?? . Then the value of ?? 2
-?? 2
+6?? +2?? equals 
[We changed options. In official NTA paper no option was correct.] 
(1) 4 
(2) 10 
(3) 8 
(4) Data Inconsistent 
Q10 - 2024 (30 Jan Shift 2) 
Let ?? =?
k=0
n
?(
( 
n
C
k
)
2
k+1
) and ?? =?
k=0
n-1
?(
 
n
C
k
 
n
C
k+1
k+2
) 
If 5?? =6?? , then n equals 
Q11 - 2024 (31 Jan Shift 1) 
In the expansion of (1+?? )(1-?? 2
)(1+
3
?? +
3
?? 2
+
1
?? 3
)
5
,?? ?0, the sum of the coefficient of 
?? 3
 and ?? -13
 is equal to 
Q12 - 2024 (31 Jan Shift 2) 
If for some m,n; 
6
C
m
+2( 
6
C
m+1
)+ 
6
C
m+2
> 
8
C
3
 and  
?? -1
?? 3
: 
?? ?? 4
=1:8, then  
?? ?? ?? +1
+
 
?? +1
?? ?? is equal to 
(1) 380 
(2) 376 
(3) 384 
(4) 372 
Q13 - 2024 (31 Jan Shift 2) 
Let the coefficient of ?? ?? 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 ?? ?? . If ?
?? =0
?? ??? ?? =?? ?? -?? ?? ,?? ,?? ??? , then the value of 
?? 2
+?? 2
 equals 
Answer Key 
Q1 (678)  Q2(4) Q3 (1)  Q4(1) 
Q5 (0)  Q6 (2041) Q7 (1) Q8 (138) 
Q9 (4) Q10 (10) Q11 (118) Q12 (4) 
Q13 (25)    
 
Solutions 
Q1 
coeff of ?? 30
 in 
(?? +1)
6
(1+?? 2
)
7
(1-?? 3
)
8
?? 6
 coeff. of ?? 36
 in (1+?? )
6
(1+?? 2
)
7
(1-?? 3
)
8
 
General term 
 
6
C
r
1
 
7
C
r
2
 
8
C
r
3
(-1)
r
3
X
r
1
+2r
2
+3r
3
 
r
1
+2r
2
+3r
3
=36 
 
 
 
 
 
Case-I : 
?? 1
 ?? 2
 ?? 3
 
0 6 8 
2 5 8 
4 4 8 
6 3 8 
 
?? 1
+2?? 2
=12Taking ?? 3
=8) 
Case II. 
?? 1
 ?? 2
 ?? 3
 
1 7 7 
3 6 7 
5 5 7 
 
?? 1
+2?? 2
=15 Taking ?? 3
=7) 
?? 1
 ?? 2
 ?? 3
 
4 7 6 
6 6 6 
 
Case-III : 
?? 1
+2?? 2
=18 (Taking ?? 3
=6) 
Coeff. =7+(15×21)+(15×35)+(35) 
-(6×8)-(20×7×8)-(6×21×8)+(15×28) 
+(7×28)=-678=?? 
|?? |=678 
 
Q2 
(
?? 1
3
3
+
?? -2
3
18
)
18
 
t
7
= 
18
c
6
(
x
1
3
3
)
12
(
x
-2
3
2
)
6
= 
18
c
6
1
(3)
12
·
1
2
6
 
?? 13
= 
18
?? 12
(
?? 1
3
3
)
6
(
?? -2
3
2
)
12
= 
18
?? 12
1
(3)
6
·
1
2
12
·?? -6
 
m= 
18
c
6
·3
-12
·2
-6
:n= 
18
c
12
·2
-12
·3
-6
 
(
n
m
)
1
3
=(
2
-12
·3
-6
3
-12
·2
-6
)
1
3
=(
3
2
)
2
=
9
4
 
Q3 
 
?? -1
?? ?? =(?? 2
-8) 
?? ?? ?? +1
?? +1=0,?? =0 ?          
?? =0
 
?? ?? ?? 
?? ?? ?? +1
=?? 2
-8
?? +1
?? =?? 2
-8
 ??? 2
-8>0
 
k?(-8,-2v2)?(2v2,8) 
?n=r+1,
r+1
n
=1 
?k
2
-8=1 
k
2
-9=0#(???? )
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Document Description: Binomial Theorem & its Simple Applications: JEE Mains Previous Year Questions (2021-2024) for JEE 2024 is part of Mathematics (Maths) for JEE Main & Advanced preparation. The notes and questions for Binomial Theorem & its Simple Applications: JEE Mains Previous Year Questions (2021-2024) have been prepared according to the JEE exam syllabus. Information about Binomial Theorem & its Simple Applications: JEE Mains Previous Year Questions (2021-2024) covers topics like and Binomial Theorem & its Simple Applications: JEE Mains Previous Year Questions (2021-2024) Example, for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Binomial Theorem & its Simple Applications: JEE Mains Previous Year Questions (2021-2024).
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Additional Information about Binomial Theorem & its Simple Applications: JEE Mains Previous Year Questions (2021-2024) for JEE Preparation

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