Page 1
JEE Mains Previous Year Questions
(20212024): Complex Number and
Quadratic Equations
2024
Q1  2024 (01 Feb Shift 1)
Let S = {?? ? ?? : ??  1 = 1 and ( v 2  1) ( ?? + ?? ?) ?? ( ??  ?? ?)= 2v 2}. Let z
1
, z
2
? ?? be such
that
?? 1
 = max
?? ??? ???  and ?? 2
 = min
?? ??? ??? . Then v 2?? 1
 ?? 2

2
equals :
(1) 1
(2) 4
(3) 3
(4) 2
Q2  2024 (01 Feb Shift 1)
Let P = {z ? C: z + 2  3i = 1} and ?? = {?? ? C: ?? ( 1 + ?? )+ ?? ?( 1  ?? )= 8}. Let in P n
Q, z  3 + 2i be maximum and minimum at ?? 1
and ?? 2
respectively. If ?? 1

2
+ 2?? 
2
= ?? +
?? v 2, where ?? , ?? are integers, then ?? + ?? equals
Q3  2024 (01 Feb Shift 2)
If ?? is a complex number such that ??  = 1, then the minimum value of ?? +
1
2
( 3 + 4?? )  is:
[We changed options. In official NTA paper no option was correct.]
(1)
5
2
(2) 2
(3) 3
(4) 0
Q4  2024 (27 Jan Shift 1)
Page 2
JEE Mains Previous Year Questions
(20212024): Complex Number and
Quadratic Equations
2024
Q1  2024 (01 Feb Shift 1)
Let S = {?? ? ?? : ??  1 = 1 and ( v 2  1) ( ?? + ?? ?) ?? ( ??  ?? ?)= 2v 2}. Let z
1
, z
2
? ?? be such
that
?? 1
 = max
?? ??? ???  and ?? 2
 = min
?? ??? ??? . Then v 2?? 1
 ?? 2

2
equals :
(1) 1
(2) 4
(3) 3
(4) 2
Q2  2024 (01 Feb Shift 1)
Let P = {z ? C: z + 2  3i = 1} and ?? = {?? ? C: ?? ( 1 + ?? )+ ?? ?( 1  ?? )= 8}. Let in P n
Q, z  3 + 2i be maximum and minimum at ?? 1
and ?? 2
respectively. If ?? 1

2
+ 2?? 
2
= ?? +
?? v 2, where ?? , ?? are integers, then ?? + ?? equals
Q3  2024 (01 Feb Shift 2)
If ?? is a complex number such that ??  = 1, then the minimum value of ?? +
1
2
( 3 + 4?? )  is:
[We changed options. In official NTA paper no option was correct.]
(1)
5
2
(2) 2
(3) 3
(4) 0
Q4  2024 (27 Jan Shift 1)
If ?? = {?? ? ?? : ??  ??  = ?? + ??  = ??  1}, then, ?? ( ?? ) is:
(1) 1
(2) 0
(3) 3
(4) 2
Q5  2024 (27 Jan Shift 1)
If ?? satisfies the equation ?? 2
+ ?? + 1 = 0 and ( 1 + ?? )
7
= A + B?? + C
2
, A, B, C = 0, then
5( 3 A  2 B  C) is equal to
Q6  2024 (27 Jan Shift 2)
Let the complex numbers ?? and
1
?? ?
lie on the circles ??  ?? 0

2
= 4 and ??  ?? 0

2
= 16
respectively, where z
0
= 1 + i. Then, the value of 100?? 
2
is.
Q7  2024 (29 Jan Shift 1)
If ?? =
1
2
 2?? , is such that ?? + 1 = ???? + ?? ( 1 + ?? ) , ?? = v 1 and ?? , ?? ? ?? , then ?? + ?? is
equal to
(1) 4
(2) 3
(3) 2
(4) 1
Q8  2024 (29 Jan Shift 1)
Let ?? , ?? be the roots of the equation ?? 2
 ?? + 2 = 0 with Im ( ?? )> Im ( ?? ) . Then ?? 6
+ ?? 4
+
?? 4
 5?? 2
is equal to
Q9  2024 (29 Jan Shift 2)
Let ?? and ?? respectively be the modulus and amplitude of the complex number ?? = 2 
?? ( 2tan
5?? 8
) , then ( ?? , ?? ) is equal to
(1) ( 2sec
3?? 8
,
3?? 8
)
Page 3
JEE Mains Previous Year Questions
(20212024): Complex Number and
Quadratic Equations
2024
Q1  2024 (01 Feb Shift 1)
Let S = {?? ? ?? : ??  1 = 1 and ( v 2  1) ( ?? + ?? ?) ?? ( ??  ?? ?)= 2v 2}. Let z
1
, z
2
? ?? be such
that
?? 1
 = max
?? ??? ???  and ?? 2
 = min
?? ??? ??? . Then v 2?? 1
 ?? 2

2
equals :
(1) 1
(2) 4
(3) 3
(4) 2
Q2  2024 (01 Feb Shift 1)
Let P = {z ? C: z + 2  3i = 1} and ?? = {?? ? C: ?? ( 1 + ?? )+ ?? ?( 1  ?? )= 8}. Let in P n
Q, z  3 + 2i be maximum and minimum at ?? 1
and ?? 2
respectively. If ?? 1

2
+ 2?? 
2
= ?? +
?? v 2, where ?? , ?? are integers, then ?? + ?? equals
Q3  2024 (01 Feb Shift 2)
If ?? is a complex number such that ??  = 1, then the minimum value of ?? +
1
2
( 3 + 4?? )  is:
[We changed options. In official NTA paper no option was correct.]
(1)
5
2
(2) 2
(3) 3
(4) 0
Q4  2024 (27 Jan Shift 1)
If ?? = {?? ? ?? : ??  ??  = ?? + ??  = ??  1}, then, ?? ( ?? ) is:
(1) 1
(2) 0
(3) 3
(4) 2
Q5  2024 (27 Jan Shift 1)
If ?? satisfies the equation ?? 2
+ ?? + 1 = 0 and ( 1 + ?? )
7
= A + B?? + C
2
, A, B, C = 0, then
5( 3 A  2 B  C) is equal to
Q6  2024 (27 Jan Shift 2)
Let the complex numbers ?? and
1
?? ?
lie on the circles ??  ?? 0

2
= 4 and ??  ?? 0

2
= 16
respectively, where z
0
= 1 + i. Then, the value of 100?? 
2
is.
Q7  2024 (29 Jan Shift 1)
If ?? =
1
2
 2?? , is such that ?? + 1 = ???? + ?? ( 1 + ?? ) , ?? = v 1 and ?? , ?? ? ?? , then ?? + ?? is
equal to
(1) 4
(2) 3
(3) 2
(4) 1
Q8  2024 (29 Jan Shift 1)
Let ?? , ?? be the roots of the equation ?? 2
 ?? + 2 = 0 with Im ( ?? )> Im ( ?? ) . Then ?? 6
+ ?? 4
+
?? 4
 5?? 2
is equal to
Q9  2024 (29 Jan Shift 2)
Let ?? and ?? respectively be the modulus and amplitude of the complex number ?? = 2 
?? ( 2tan
5?? 8
) , then ( ?? , ?? ) is equal to
(1) ( 2sec
3?? 8
,
3?? 8
)
(2) ( 2sec
3?? 8
,
5?? 8
)
(3) ( 2sec
5?? 8
,
3?? 8
)
(4) ( 2sec
11?? 8
,
11?? 8
)
Q10  2024 (29 Jan Shift 2)
Let ?? , ?? be the roots of the equation ?? 2
 v 6?? + 3 = 0 such that Im ( ?? )> Im ( ?? ) . Let ?? , ??
be integers not divisible by 3 and ?? be a natural number such that
?? 99
?? + ?? 98
= 3
n
( a +
ib) , i = v 1. Then n + a + b is equal to
Q11  2024 (30 Jan Shift 1)
If ?? = ?? + ???? , ???? ? 0, satisfies the equation ?? 2
+ ?? ?? ? = 0, then ?? 2
 is equal to :
(1) 9
(2) 1
(3) 4
(4)
1
4
Q12  2024 (30 Jan Shift 2)
If z is a complex number, then the number of common roots of the equation ?? 1985
+
?? 100
+ 1 = 0 and ?? 3
+ 2?? 2
+ 2?? + 1 = 0, is equal to :
(1) 1
(2) 2
(3) 0
(4) 3
Q13  2024 (31 Jan Shift 1)
If ?? denotes the number of solutions of 1  ?? 
?? = 2
?? and ?? = (
?? 
arg ( ?? )
) , where ?? =
?? 4
( 1 +
?? )
4
(
1v ?? i
v ?? +i
+
v ?? i
1+v ?? i
), i = v1, then the distance of the point ( ?? , ?? ) from the line 4??  3?? =
7 is
Q14  2024 (31 Jan Shift 2)
Page 4
JEE Mains Previous Year Questions
(20212024): Complex Number and
Quadratic Equations
2024
Q1  2024 (01 Feb Shift 1)
Let S = {?? ? ?? : ??  1 = 1 and ( v 2  1) ( ?? + ?? ?) ?? ( ??  ?? ?)= 2v 2}. Let z
1
, z
2
? ?? be such
that
?? 1
 = max
?? ??? ???  and ?? 2
 = min
?? ??? ??? . Then v 2?? 1
 ?? 2

2
equals :
(1) 1
(2) 4
(3) 3
(4) 2
Q2  2024 (01 Feb Shift 1)
Let P = {z ? C: z + 2  3i = 1} and ?? = {?? ? C: ?? ( 1 + ?? )+ ?? ?( 1  ?? )= 8}. Let in P n
Q, z  3 + 2i be maximum and minimum at ?? 1
and ?? 2
respectively. If ?? 1

2
+ 2?? 
2
= ?? +
?? v 2, where ?? , ?? are integers, then ?? + ?? equals
Q3  2024 (01 Feb Shift 2)
If ?? is a complex number such that ??  = 1, then the minimum value of ?? +
1
2
( 3 + 4?? )  is:
[We changed options. In official NTA paper no option was correct.]
(1)
5
2
(2) 2
(3) 3
(4) 0
Q4  2024 (27 Jan Shift 1)
If ?? = {?? ? ?? : ??  ??  = ?? + ??  = ??  1}, then, ?? ( ?? ) is:
(1) 1
(2) 0
(3) 3
(4) 2
Q5  2024 (27 Jan Shift 1)
If ?? satisfies the equation ?? 2
+ ?? + 1 = 0 and ( 1 + ?? )
7
= A + B?? + C
2
, A, B, C = 0, then
5( 3 A  2 B  C) is equal to
Q6  2024 (27 Jan Shift 2)
Let the complex numbers ?? and
1
?? ?
lie on the circles ??  ?? 0

2
= 4 and ??  ?? 0

2
= 16
respectively, where z
0
= 1 + i. Then, the value of 100?? 
2
is.
Q7  2024 (29 Jan Shift 1)
If ?? =
1
2
 2?? , is such that ?? + 1 = ???? + ?? ( 1 + ?? ) , ?? = v 1 and ?? , ?? ? ?? , then ?? + ?? is
equal to
(1) 4
(2) 3
(3) 2
(4) 1
Q8  2024 (29 Jan Shift 1)
Let ?? , ?? be the roots of the equation ?? 2
 ?? + 2 = 0 with Im ( ?? )> Im ( ?? ) . Then ?? 6
+ ?? 4
+
?? 4
 5?? 2
is equal to
Q9  2024 (29 Jan Shift 2)
Let ?? and ?? respectively be the modulus and amplitude of the complex number ?? = 2 
?? ( 2tan
5?? 8
) , then ( ?? , ?? ) is equal to
(1) ( 2sec
3?? 8
,
3?? 8
)
(2) ( 2sec
3?? 8
,
5?? 8
)
(3) ( 2sec
5?? 8
,
3?? 8
)
(4) ( 2sec
11?? 8
,
11?? 8
)
Q10  2024 (29 Jan Shift 2)
Let ?? , ?? be the roots of the equation ?? 2
 v 6?? + 3 = 0 such that Im ( ?? )> Im ( ?? ) . Let ?? , ??
be integers not divisible by 3 and ?? be a natural number such that
?? 99
?? + ?? 98
= 3
n
( a +
ib) , i = v 1. Then n + a + b is equal to
Q11  2024 (30 Jan Shift 1)
If ?? = ?? + ???? , ???? ? 0, satisfies the equation ?? 2
+ ?? ?? ? = 0, then ?? 2
 is equal to :
(1) 9
(2) 1
(3) 4
(4)
1
4
Q12  2024 (30 Jan Shift 2)
If z is a complex number, then the number of common roots of the equation ?? 1985
+
?? 100
+ 1 = 0 and ?? 3
+ 2?? 2
+ 2?? + 1 = 0, is equal to :
(1) 1
(2) 2
(3) 0
(4) 3
Q13  2024 (31 Jan Shift 1)
If ?? denotes the number of solutions of 1  ?? 
?? = 2
?? and ?? = (
?? 
arg ( ?? )
) , where ?? =
?? 4
( 1 +
?? )
4
(
1v ?? i
v ?? +i
+
v ?? i
1+v ?? i
), i = v1, then the distance of the point ( ?? , ?? ) from the line 4??  3?? =
7 is
Q14  2024 (31 Jan Shift 2)
Let ?? 1
and ?? 2
be two complex number such that ?? 1
+ ?? 2
= 5 and ?? 1
3
+ ?? 2
3
= 20 + 15?? .
Then ?? 1
4
+ ?? 2
4
 equals
(1) 30v 3
(2) 75
(3) 15v 15
(4) 25v 3
Q15  2024 (01 Feb Shift 1)
Let ?? = {?? ? ?? : ( v 3 + v 2)
?? + ( v 3  v 2)
?? = 10}.
Then the number of elements in S is :
(1) 4
(2) 0
(3) 2
(4) 1
Q16  2024 (01 Feb Shift 2)
Let ?? and ?? be the roots of the equation px
2
+ qx  ?? = 0, where ?? ? 0. If ?? , ?? and ?? be
the consecutive terms of a nonconstant G.P and
1
?? +
1
?? =
3
4
, then the value of ( ??  ?? )
2
is
:
(1)
80
9
(2) 9
(3)
20
3
(4) 8
Q17  2024 (27 Jan Shift 2)
If ?? , ?? are the roots of the equation, ?? 2
 ??  1 = 0 and ?? ?? = 2023?? ?? + 2024?? ?? , then
(1) 2 S
12
= S
11
+ S
10
(2) S
12
= S
11
+ S
10
(3) 2 S
11
= S
12
+ S
10
Page 5
JEE Mains Previous Year Questions
(20212024): Complex Number and
Quadratic Equations
2024
Q1  2024 (01 Feb Shift 1)
Let S = {?? ? ?? : ??  1 = 1 and ( v 2  1) ( ?? + ?? ?) ?? ( ??  ?? ?)= 2v 2}. Let z
1
, z
2
? ?? be such
that
?? 1
 = max
?? ??? ???  and ?? 2
 = min
?? ??? ??? . Then v 2?? 1
 ?? 2

2
equals :
(1) 1
(2) 4
(3) 3
(4) 2
Q2  2024 (01 Feb Shift 1)
Let P = {z ? C: z + 2  3i = 1} and ?? = {?? ? C: ?? ( 1 + ?? )+ ?? ?( 1  ?? )= 8}. Let in P n
Q, z  3 + 2i be maximum and minimum at ?? 1
and ?? 2
respectively. If ?? 1

2
+ 2?? 
2
= ?? +
?? v 2, where ?? , ?? are integers, then ?? + ?? equals
Q3  2024 (01 Feb Shift 2)
If ?? is a complex number such that ??  = 1, then the minimum value of ?? +
1
2
( 3 + 4?? )  is:
[We changed options. In official NTA paper no option was correct.]
(1)
5
2
(2) 2
(3) 3
(4) 0
Q4  2024 (27 Jan Shift 1)
If ?? = {?? ? ?? : ??  ??  = ?? + ??  = ??  1}, then, ?? ( ?? ) is:
(1) 1
(2) 0
(3) 3
(4) 2
Q5  2024 (27 Jan Shift 1)
If ?? satisfies the equation ?? 2
+ ?? + 1 = 0 and ( 1 + ?? )
7
= A + B?? + C
2
, A, B, C = 0, then
5( 3 A  2 B  C) is equal to
Q6  2024 (27 Jan Shift 2)
Let the complex numbers ?? and
1
?? ?
lie on the circles ??  ?? 0

2
= 4 and ??  ?? 0

2
= 16
respectively, where z
0
= 1 + i. Then, the value of 100?? 
2
is.
Q7  2024 (29 Jan Shift 1)
If ?? =
1
2
 2?? , is such that ?? + 1 = ???? + ?? ( 1 + ?? ) , ?? = v 1 and ?? , ?? ? ?? , then ?? + ?? is
equal to
(1) 4
(2) 3
(3) 2
(4) 1
Q8  2024 (29 Jan Shift 1)
Let ?? , ?? be the roots of the equation ?? 2
 ?? + 2 = 0 with Im ( ?? )> Im ( ?? ) . Then ?? 6
+ ?? 4
+
?? 4
 5?? 2
is equal to
Q9  2024 (29 Jan Shift 2)
Let ?? and ?? respectively be the modulus and amplitude of the complex number ?? = 2 
?? ( 2tan
5?? 8
) , then ( ?? , ?? ) is equal to
(1) ( 2sec
3?? 8
,
3?? 8
)
(2) ( 2sec
3?? 8
,
5?? 8
)
(3) ( 2sec
5?? 8
,
3?? 8
)
(4) ( 2sec
11?? 8
,
11?? 8
)
Q10  2024 (29 Jan Shift 2)
Let ?? , ?? be the roots of the equation ?? 2
 v 6?? + 3 = 0 such that Im ( ?? )> Im ( ?? ) . Let ?? , ??
be integers not divisible by 3 and ?? be a natural number such that
?? 99
?? + ?? 98
= 3
n
( a +
ib) , i = v 1. Then n + a + b is equal to
Q11  2024 (30 Jan Shift 1)
If ?? = ?? + ???? , ???? ? 0, satisfies the equation ?? 2
+ ?? ?? ? = 0, then ?? 2
 is equal to :
(1) 9
(2) 1
(3) 4
(4)
1
4
Q12  2024 (30 Jan Shift 2)
If z is a complex number, then the number of common roots of the equation ?? 1985
+
?? 100
+ 1 = 0 and ?? 3
+ 2?? 2
+ 2?? + 1 = 0, is equal to :
(1) 1
(2) 2
(3) 0
(4) 3
Q13  2024 (31 Jan Shift 1)
If ?? denotes the number of solutions of 1  ?? 
?? = 2
?? and ?? = (
?? 
arg ( ?? )
) , where ?? =
?? 4
( 1 +
?? )
4
(
1v ?? i
v ?? +i
+
v ?? i
1+v ?? i
), i = v1, then the distance of the point ( ?? , ?? ) from the line 4??  3?? =
7 is
Q14  2024 (31 Jan Shift 2)
Let ?? 1
and ?? 2
be two complex number such that ?? 1
+ ?? 2
= 5 and ?? 1
3
+ ?? 2
3
= 20 + 15?? .
Then ?? 1
4
+ ?? 2
4
 equals
(1) 30v 3
(2) 75
(3) 15v 15
(4) 25v 3
Q15  2024 (01 Feb Shift 1)
Let ?? = {?? ? ?? : ( v 3 + v 2)
?? + ( v 3  v 2)
?? = 10}.
Then the number of elements in S is :
(1) 4
(2) 0
(3) 2
(4) 1
Q16  2024 (01 Feb Shift 2)
Let ?? and ?? be the roots of the equation px
2
+ qx  ?? = 0, where ?? ? 0. If ?? , ?? and ?? be
the consecutive terms of a nonconstant G.P and
1
?? +
1
?? =
3
4
, then the value of ( ??  ?? )
2
is
:
(1)
80
9
(2) 9
(3)
20
3
(4) 8
Q17  2024 (27 Jan Shift 2)
If ?? , ?? are the roots of the equation, ?? 2
 ??  1 = 0 and ?? ?? = 2023?? ?? + 2024?? ?? , then
(1) 2 S
12
= S
11
+ S
10
(2) S
12
= S
11
+ S
10
(3) 2 S
11
= S
12
+ S
10
(4) ?? 11
= ?? 10
+ ?? 12
Q18  2024 (29 Jan Shift 2)
Let the set ?? = {( ?? , ?? ) ?? 2
 2
?? = 2023 , ?? , ?? ? N}. Then ?
( ?? ,?? ) =?? ?( ?? + ?? ) is equal to
Q19  2024 (30 Jan Shift 1)
Let ?? , ?? ? N be roots of equation x
2
 70x + ?? = 0, where
?? 2
,
?? 3
? N. If ?? assumes the
minimum possible
value, then
( v ?? 1+v?? 1) ( ?? +35)
?? ?? 
is equal to
Q20  2024 (30 Jan Shift 2)
The number of real solutions of the equation ?? ( ?? 2
+ 3??  + 5??  1 + 6??  2)= 0 is
Q21  2024 (31 Jan Shift 1)
For 0 < c < b < a, let ( a + b  2c) x
2
+ ( b + c  2a) x + ( ?? + ??  2?? )= 0 and ?? ? 1 be one
of its root.
Then, among the two statements
(I) If ?? ? ( 1,0) , then b cannot be the geometric mean of a and c
(II) If ?? ? ( 0,1) , then ?? may be the geometric mean of a and c
(1) Both (I) and (II) are true
(2) Neither (I) nor (II) is true
(3) Only (II) is true
(4) Only (I) is true
Q22  2024 (31 Jan Shift 1)
Let ?? be the set of positive integral values of a for which
?? 2
+2( ?? +1) ?? +9?? +4
?? 2
8x+32
< 0, ??? ? R.
Then, the number of elements in S is :
(1) 1
(2) 0
(3) 8
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