Complex Numbers and Quadratic Equation: JEE Mains Previous Year Questions (2021-2024)

``` Page 1

JEE Mains Previous Year Questions
(2021-2024): Complex Number and
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
(2021-2024): Complex Number and
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
(2021-2024): Complex Number and
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
(
1-v ?? i
v ?? +i
+
v ?? -i
1+v ?? i
), i = v-1, 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
(2021-2024): Complex Number and
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
(
1-v ?? i
v ?? +i
+
v ?? -i
1+v ?? i
), i = v-1, 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 non-constant 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
(2021-2024): Complex Number and
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
(
1-v ?? i
v ?? +i
+
v ?? -i
1+v ?? i
), i = v-1, 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 non-constant 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
```

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

## FAQs on Complex Numbers and Quadratic Equation: JEE Mains Previous Year Questions (2021-2024) - Mathematics (Maths) for JEE Main & Advanced

 1. How can complex numbers be represented geometrically on the complex plane?
Ans. Complex numbers can be represented geometrically on the complex plane by associating the real part of the complex number with the x-coordinate and the imaginary part with the y-coordinate. This representation allows for the visualization of complex numbers as points in a plane.
 2. What is the significance of the modulus and argument of a complex number?
Ans. The modulus of a complex number represents its distance from the origin on the complex plane, while the argument represents the angle the complex number makes with the positive real axis. These values provide important information about the magnitude and direction of the complex number.
 3. How are complex numbers used in solving quadratic equations?
Ans. Complex numbers are used in solving quadratic equations when the roots are non-real or complex conjugates. By using the quadratic formula with complex numbers, we can find the roots of the equation even when they involve imaginary numbers.
 4. Can quadratic equations with real coefficients have complex roots?
Ans. Yes, quadratic equations with real coefficients can have complex roots. This occurs when the discriminant of the quadratic equation is negative, resulting in complex conjugate roots. The use of complex numbers allows us to find these roots accurately.
 5. How do complex numbers relate to trigonometry and polar coordinates?
Ans. Complex numbers can be represented in polar form, where the modulus and argument correspond to the radius and angle in polar coordinates. This representation is closely related to trigonometry, as the complex exponential function can be used to simplify trigonometric operations.

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

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