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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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Binomial Theorem & its Simple Applications: JEE Mains Previous Year Questions (2021-2024) Notes
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The content of Binomial Theorem & its Simple Applications: JEE Mains Previous Year Questions (2021-2024) is prepared as per the latest JEE syllabus.
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