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Page 1
Solved Examples on Complex
Numbers
JEE Mains
Q1. If ?? ?? = -?? , then the value of ?
?? =?? ??????
??? ?? is
(a) 50
(b) -50
(c) ??
(d) 100
Ans: (c) ?
?? =1
200
??? ?? = ?? + ?? 2
+ ?? 3
+ ? . . +?? 200
=
?? ( 1-?? 200
)
1-?? (since G.P.) =
?? ( 1-1)
1-?? = 0.
Q2. If (
?? +?? ?? -?? )
?? = ?? then
(a) ?? = ?? ?? , where ?? is any positive integer
(b) ?? = ?? ?? , where ?? is any positive integer
(c) ?? = ?? ?? + ?? , where ?? is any positive integer
(d) ?? = ?? ?? + ?? , where ?? is any positive integer
Ans: (a)
1+?? 1-?? =
1+?? 1-?? ·
1+?? 1+?? ? (
1+?? 1-?? )
?? = 1 ? ?? ?? = 1 ? ?? = 4?? , ?? ? ?? +
.
Q3. (
?? ?? -?? ?? +
?? ?? +?? )(
?? +?? ?? ?? -?? ?? )
(a)
?? ?? +
?? ?? ??
(b)
?? ?? -
?? ?? ??
(c)
?? ?? -
?? ?? ??
(d)
?? ?? +
?? ?? ??
Ans: (d)
(
1
1 - 2?? +
3
1 + ?? )(
3 + 4?? 2 - 4?? )= [
1 + 2?? 1
2
+ 2
2
+
3 - 3?? 1
2
+ 1
2
] [
6 - 16 + 12?? + 8?? 2
2
+ 4
2
] = (
2 + 4?? + 15 - 15?? 10
)(
-1 + 2?? 2
)
=
( 17 - 11?? ) ( -1 + 2?? )
20
=
5 + 45?? 20
=
1
4
+
9
4
?? .
Q4. The real values of ?? and ?? for which the equation ( ?? ?? + ?? ???? )- ( ?? ?? ?? + ???? )= ( ?? -
?? ?? )+ ( ?? + ?? ???? ) is satisfied, are
(a) ?? = ?? , ?? = ??
(b) ?? = -?? , ?? =
?? ??
(c) Both (a) and (b)
(d) None of these
Ans: (c) Given equation ( ?? 4
+ 2???? )- ( 3?? 2
+ ???? )= ( 3 - 5?? )+ ( 1 + 2???? )? ( ?? 4
- 3?? 2
)+
?? ( 2?? - 3?? )= 4 - 5?? Equating real and imaginary parts, we get
?? 4
- 3?? 2
= 4
(i) and
Page 2
Solved Examples on Complex
Numbers
JEE Mains
Q1. If ?? ?? = -?? , then the value of ?
?? =?? ??????
??? ?? is
(a) 50
(b) -50
(c) ??
(d) 100
Ans: (c) ?
?? =1
200
??? ?? = ?? + ?? 2
+ ?? 3
+ ? . . +?? 200
=
?? ( 1-?? 200
)
1-?? (since G.P.) =
?? ( 1-1)
1-?? = 0.
Q2. If (
?? +?? ?? -?? )
?? = ?? then
(a) ?? = ?? ?? , where ?? is any positive integer
(b) ?? = ?? ?? , where ?? is any positive integer
(c) ?? = ?? ?? + ?? , where ?? is any positive integer
(d) ?? = ?? ?? + ?? , where ?? is any positive integer
Ans: (a)
1+?? 1-?? =
1+?? 1-?? ·
1+?? 1+?? ? (
1+?? 1-?? )
?? = 1 ? ?? ?? = 1 ? ?? = 4?? , ?? ? ?? +
.
Q3. (
?? ?? -?? ?? +
?? ?? +?? )(
?? +?? ?? ?? -?? ?? )
(a)
?? ?? +
?? ?? ??
(b)
?? ?? -
?? ?? ??
(c)
?? ?? -
?? ?? ??
(d)
?? ?? +
?? ?? ??
Ans: (d)
(
1
1 - 2?? +
3
1 + ?? )(
3 + 4?? 2 - 4?? )= [
1 + 2?? 1
2
+ 2
2
+
3 - 3?? 1
2
+ 1
2
] [
6 - 16 + 12?? + 8?? 2
2
+ 4
2
] = (
2 + 4?? + 15 - 15?? 10
)(
-1 + 2?? 2
)
=
( 17 - 11?? ) ( -1 + 2?? )
20
=
5 + 45?? 20
=
1
4
+
9
4
?? .
Q4. The real values of ?? and ?? for which the equation ( ?? ?? + ?? ???? )- ( ?? ?? ?? + ???? )= ( ?? -
?? ?? )+ ( ?? + ?? ???? ) is satisfied, are
(a) ?? = ?? , ?? = ??
(b) ?? = -?? , ?? =
?? ??
(c) Both (a) and (b)
(d) None of these
Ans: (c) Given equation ( ?? 4
+ 2???? )- ( 3?? 2
+ ???? )= ( 3 - 5?? )+ ( 1 + 2???? )? ( ?? 4
- 3?? 2
)+
?? ( 2?? - 3?? )= 4 - 5?? Equating real and imaginary parts, we get
?? 4
- 3?? 2
= 4
(i) and
2?? - 3?? = -5
Form (i) and (ii), we get ?? = ±2 and ?? = 3,
1
3
.
Trick: Put ?? = 2, ?? = 3 and then ?? = -2, ?? =
1
3
, we see that they both satisfy the given
equation.
Q5. The complex numbers ?????? ?? + ?? ?????? ?? ?? and ?????? ?? - ?? ?????? ?? ?? are conjugate to each
other for
(a) ?? = ????
(b) ?? = ( ?? +
?? ?? )??
(c) ?? = ??
(d) No value of ??
Ans: (d) sin ?? + ?? cos 2?? and cos ?? - ?? sin 2?? are conjugate to each other if sin ?? = cos ?? and
cos 2?? = sin 2??
or tan ?? = 1 ? ?? =
?? 4
,
5?? 4
,
9?? 4
, … ….
(i) and tan 2?? = 1 ? 2?? =
?? 4
,
5?? 4
,
9?? 4
, or ?? =
?? 8
,
5?? 8
,
9?? 8
, … …
(ii) There exists no value of ?? common in (i) and (ii). Therefore there is no value of ?? for
which the given complex numbers are conjugate.
Q6. The real part of ( ?? - ?????? ?? + ?? ?? ?????? ?? )
-?? is
(a)
?? ?? +???????? ??
(b)
?? ?? -???????? ??
(c)
?? ?? -???????? ??
(d)
?? ?? +???????? ??
A ns : (c) {( 1 - cos ?? )+ ?? . 2sin ?? }
-1
= {2sin
2
?? 2
+ ?? . 4sin
?? 2
cos
?? 2
}
-1
= ( 2sin
?? 2
)
-1
{sin
?? 2
+ ?? . 2cos
?? 2
}
-1
= ( 2sin
?? 2
)
-1
·
1
sin
?? 2
+ ?? · 2cos
?? 2
×
sin
?? 2
- ?? · 2cos
?? 2
sin
?? 2
- ?? · 2cos
?? 2
=
sin
?? 2
- ?? · 2cos
?? 2
2sin
?? 2
( sin
2
?? 2
+ 4cos
2
?? 2
)
Hence, real part =
sin
?? 2
2sin
?? 2
( 1 + 3cos
2
?? 2
)
=
1
2 ( 1 + 3cos
2
?? 2
)
=
1
5 + 3cos ?? .
Q7. For ?? ?? , ?? ?? , ?? ?? , ?? ?? ? ?? , if ?? < ?? ?? < ?? ?? , ?? ?? = ?? ?? and ?? ?? = ?? ?? + ?? ?? ?? , ?? ?? = ?? ?? + ?? ?? ?? , and
?? ?? =
?? ?? ( ?? ?? + ?? ?? ) , then ?? ?? , ?? ?? and ?? ?? satisfy
(a) |?? ?? | = |?? ?? | = |?? ?? |
(b) |?? ?? | ? ?? ?? |< ?? ?? |
(c) |?? ?? | > |?? ?? | > |?? ?? |
(d) |?? ?? | < ?? ?? | < |?? ?? |
Ans: (d) 0 < ?? 1
< ?? 2
, ?? 1
= ?? 2
(Given)
Page 3
Solved Examples on Complex
Numbers
JEE Mains
Q1. If ?? ?? = -?? , then the value of ?
?? =?? ??????
??? ?? is
(a) 50
(b) -50
(c) ??
(d) 100
Ans: (c) ?
?? =1
200
??? ?? = ?? + ?? 2
+ ?? 3
+ ? . . +?? 200
=
?? ( 1-?? 200
)
1-?? (since G.P.) =
?? ( 1-1)
1-?? = 0.
Q2. If (
?? +?? ?? -?? )
?? = ?? then
(a) ?? = ?? ?? , where ?? is any positive integer
(b) ?? = ?? ?? , where ?? is any positive integer
(c) ?? = ?? ?? + ?? , where ?? is any positive integer
(d) ?? = ?? ?? + ?? , where ?? is any positive integer
Ans: (a)
1+?? 1-?? =
1+?? 1-?? ·
1+?? 1+?? ? (
1+?? 1-?? )
?? = 1 ? ?? ?? = 1 ? ?? = 4?? , ?? ? ?? +
.
Q3. (
?? ?? -?? ?? +
?? ?? +?? )(
?? +?? ?? ?? -?? ?? )
(a)
?? ?? +
?? ?? ??
(b)
?? ?? -
?? ?? ??
(c)
?? ?? -
?? ?? ??
(d)
?? ?? +
?? ?? ??
Ans: (d)
(
1
1 - 2?? +
3
1 + ?? )(
3 + 4?? 2 - 4?? )= [
1 + 2?? 1
2
+ 2
2
+
3 - 3?? 1
2
+ 1
2
] [
6 - 16 + 12?? + 8?? 2
2
+ 4
2
] = (
2 + 4?? + 15 - 15?? 10
)(
-1 + 2?? 2
)
=
( 17 - 11?? ) ( -1 + 2?? )
20
=
5 + 45?? 20
=
1
4
+
9
4
?? .
Q4. The real values of ?? and ?? for which the equation ( ?? ?? + ?? ???? )- ( ?? ?? ?? + ???? )= ( ?? -
?? ?? )+ ( ?? + ?? ???? ) is satisfied, are
(a) ?? = ?? , ?? = ??
(b) ?? = -?? , ?? =
?? ??
(c) Both (a) and (b)
(d) None of these
Ans: (c) Given equation ( ?? 4
+ 2???? )- ( 3?? 2
+ ???? )= ( 3 - 5?? )+ ( 1 + 2???? )? ( ?? 4
- 3?? 2
)+
?? ( 2?? - 3?? )= 4 - 5?? Equating real and imaginary parts, we get
?? 4
- 3?? 2
= 4
(i) and
2?? - 3?? = -5
Form (i) and (ii), we get ?? = ±2 and ?? = 3,
1
3
.
Trick: Put ?? = 2, ?? = 3 and then ?? = -2, ?? =
1
3
, we see that they both satisfy the given
equation.
Q5. The complex numbers ?????? ?? + ?? ?????? ?? ?? and ?????? ?? - ?? ?????? ?? ?? are conjugate to each
other for
(a) ?? = ????
(b) ?? = ( ?? +
?? ?? )??
(c) ?? = ??
(d) No value of ??
Ans: (d) sin ?? + ?? cos 2?? and cos ?? - ?? sin 2?? are conjugate to each other if sin ?? = cos ?? and
cos 2?? = sin 2??
or tan ?? = 1 ? ?? =
?? 4
,
5?? 4
,
9?? 4
, … ….
(i) and tan 2?? = 1 ? 2?? =
?? 4
,
5?? 4
,
9?? 4
, or ?? =
?? 8
,
5?? 8
,
9?? 8
, … …
(ii) There exists no value of ?? common in (i) and (ii). Therefore there is no value of ?? for
which the given complex numbers are conjugate.
Q6. The real part of ( ?? - ?????? ?? + ?? ?? ?????? ?? )
-?? is
(a)
?? ?? +???????? ??
(b)
?? ?? -???????? ??
(c)
?? ?? -???????? ??
(d)
?? ?? +???????? ??
A ns : (c) {( 1 - cos ?? )+ ?? . 2sin ?? }
-1
= {2sin
2
?? 2
+ ?? . 4sin
?? 2
cos
?? 2
}
-1
= ( 2sin
?? 2
)
-1
{sin
?? 2
+ ?? . 2cos
?? 2
}
-1
= ( 2sin
?? 2
)
-1
·
1
sin
?? 2
+ ?? · 2cos
?? 2
×
sin
?? 2
- ?? · 2cos
?? 2
sin
?? 2
- ?? · 2cos
?? 2
=
sin
?? 2
- ?? · 2cos
?? 2
2sin
?? 2
( sin
2
?? 2
+ 4cos
2
?? 2
)
Hence, real part =
sin
?? 2
2sin
?? 2
( 1 + 3cos
2
?? 2
)
=
1
2 ( 1 + 3cos
2
?? 2
)
=
1
5 + 3cos ?? .
Q7. For ?? ?? , ?? ?? , ?? ?? , ?? ?? ? ?? , if ?? < ?? ?? < ?? ?? , ?? ?? = ?? ?? and ?? ?? = ?? ?? + ?? ?? ?? , ?? ?? = ?? ?? + ?? ?? ?? , and
?? ?? =
?? ?? ( ?? ?? + ?? ?? ) , then ?? ?? , ?? ?? and ?? ?? satisfy
(a) |?? ?? | = |?? ?? | = |?? ?? |
(b) |?? ?? | ? ?? ?? |< ?? ?? |
(c) |?? ?? | > |?? ?? | > |?? ?? |
(d) |?? ?? | < ?? ?? | < |?? ?? |
Ans: (d) 0 < ?? 1
< ?? 2
, ?? 1
= ?? 2
(Given)
|?? 1
| = v?? 1
2
+ ?? 1
2
, |?? 2
| = v?? 2
2
+ ?? 2
2
? |?? 2
| > |?? 1
| ? |?? 3
| =
|?? 1
+ ?? 2
|
2
=
v
(
?? 1
+ ?? 2
2
)
2
+ (
?? 1
+ ?? 2
2
)
2
v
(
?? 1
+ ?? 2
2
)
2
+ ?? 1
2
< |?? 2
| > |?? 1
|. Hence, |?? 1
| < |?? 3
| < ?? 2
|
Q8. The amplitude of ??????
?? ?? + ?? ( ?? - ??????
?? ?? )
(a) ?? /??
(b) ?? ?? /??
(c) ?? /????
(d) ?? /????
Ans: (c) sin
?? 5
+ ?? ( 1 - cos
?? 5
)= 2sin
?? 10
· cos
?? 10
+ ?? 2sin
2
?? 10
= 2sin
?? 10
( cos
?? 10
+ ?? sin
?? 10
)
For amplitude, tan ?? =
sin
?? 10
cos
?? 10
= tan
?? 10
? ?? =
?? 10
.
Q9. If ?? and ?? are to non-zero complex numbers such that |???? | = ?? and ?????? ( ?? )-
?????? ( ?? )=
?? ?? , then ????? is equal to
(a) 1
(b) -1
(c) ??
(d) -??
Ans: (d) |?? ? ?? | = 1
.(i) and arg (
?? ?? )=
?? 2
?
?? ?? = ?? ? |
?? ?? | = 1
From equation (i) and (ii),
|?? | = |?? | = 1 and
?? ?? +
?? ?
?? ?
= 0; ?? ?? ? + ?? ??? = 0 ? ?? ??? = -?? ?? ? =
-?? ?? ?? ? ?? ? ?? ??? = -?? |?? |
2
= -?? .
Q10. If -?? + v -?? = ?? ?? ????
, then ?? is equal to
(a)
?? ??
(b) -
?? ??
(c)
?? ?? ??
(d) -
?? ?? ??
Ans: (c) Here -1 + v -3 = ?? ?? ????
? -1 + ?? v 3 = ?? ?? ????
= ?? cos ?? + ???? sin ??
Equating real and imaginary part, we get ?? cos ?? = -1 and ?? sin ?? = v 3
Hence tan ?? = -v 3 ? tan ?? = tan
2?? 3
? ?? =
2?? 3
.
Q11. Real part of ?? ?? ????
is
(a) ?? ?????? ?? [?????? ( ?????? ?? ) ]
(b) ?? ?????? ?? [?????? ( ?????? ?? ) ]
(c) ?? ?????? ?? [?????? ( ?????? ?? ) ]
(d) ?? ?????? ?? [?????? ( ?????? ?? ) ]
Ans: (a) ?? ?? ????
= ?? ( cos ?? +?? sin ?? )
= ?? cos ?? · ?? ?? sin ?? = ?? cos ?? [cos ( ?? sin ?? )+ ?? sin ( sin ?? ) ]
? Real part of ?? ?? ????
is ?? cos ?? [cos ( sin ?? ) ].
Page 4
Solved Examples on Complex
Numbers
JEE Mains
Q1. If ?? ?? = -?? , then the value of ?
?? =?? ??????
??? ?? is
(a) 50
(b) -50
(c) ??
(d) 100
Ans: (c) ?
?? =1
200
??? ?? = ?? + ?? 2
+ ?? 3
+ ? . . +?? 200
=
?? ( 1-?? 200
)
1-?? (since G.P.) =
?? ( 1-1)
1-?? = 0.
Q2. If (
?? +?? ?? -?? )
?? = ?? then
(a) ?? = ?? ?? , where ?? is any positive integer
(b) ?? = ?? ?? , where ?? is any positive integer
(c) ?? = ?? ?? + ?? , where ?? is any positive integer
(d) ?? = ?? ?? + ?? , where ?? is any positive integer
Ans: (a)
1+?? 1-?? =
1+?? 1-?? ·
1+?? 1+?? ? (
1+?? 1-?? )
?? = 1 ? ?? ?? = 1 ? ?? = 4?? , ?? ? ?? +
.
Q3. (
?? ?? -?? ?? +
?? ?? +?? )(
?? +?? ?? ?? -?? ?? )
(a)
?? ?? +
?? ?? ??
(b)
?? ?? -
?? ?? ??
(c)
?? ?? -
?? ?? ??
(d)
?? ?? +
?? ?? ??
Ans: (d)
(
1
1 - 2?? +
3
1 + ?? )(
3 + 4?? 2 - 4?? )= [
1 + 2?? 1
2
+ 2
2
+
3 - 3?? 1
2
+ 1
2
] [
6 - 16 + 12?? + 8?? 2
2
+ 4
2
] = (
2 + 4?? + 15 - 15?? 10
)(
-1 + 2?? 2
)
=
( 17 - 11?? ) ( -1 + 2?? )
20
=
5 + 45?? 20
=
1
4
+
9
4
?? .
Q4. The real values of ?? and ?? for which the equation ( ?? ?? + ?? ???? )- ( ?? ?? ?? + ???? )= ( ?? -
?? ?? )+ ( ?? + ?? ???? ) is satisfied, are
(a) ?? = ?? , ?? = ??
(b) ?? = -?? , ?? =
?? ??
(c) Both (a) and (b)
(d) None of these
Ans: (c) Given equation ( ?? 4
+ 2???? )- ( 3?? 2
+ ???? )= ( 3 - 5?? )+ ( 1 + 2???? )? ( ?? 4
- 3?? 2
)+
?? ( 2?? - 3?? )= 4 - 5?? Equating real and imaginary parts, we get
?? 4
- 3?? 2
= 4
(i) and
2?? - 3?? = -5
Form (i) and (ii), we get ?? = ±2 and ?? = 3,
1
3
.
Trick: Put ?? = 2, ?? = 3 and then ?? = -2, ?? =
1
3
, we see that they both satisfy the given
equation.
Q5. The complex numbers ?????? ?? + ?? ?????? ?? ?? and ?????? ?? - ?? ?????? ?? ?? are conjugate to each
other for
(a) ?? = ????
(b) ?? = ( ?? +
?? ?? )??
(c) ?? = ??
(d) No value of ??
Ans: (d) sin ?? + ?? cos 2?? and cos ?? - ?? sin 2?? are conjugate to each other if sin ?? = cos ?? and
cos 2?? = sin 2??
or tan ?? = 1 ? ?? =
?? 4
,
5?? 4
,
9?? 4
, … ….
(i) and tan 2?? = 1 ? 2?? =
?? 4
,
5?? 4
,
9?? 4
, or ?? =
?? 8
,
5?? 8
,
9?? 8
, … …
(ii) There exists no value of ?? common in (i) and (ii). Therefore there is no value of ?? for
which the given complex numbers are conjugate.
Q6. The real part of ( ?? - ?????? ?? + ?? ?? ?????? ?? )
-?? is
(a)
?? ?? +???????? ??
(b)
?? ?? -???????? ??
(c)
?? ?? -???????? ??
(d)
?? ?? +???????? ??
A ns : (c) {( 1 - cos ?? )+ ?? . 2sin ?? }
-1
= {2sin
2
?? 2
+ ?? . 4sin
?? 2
cos
?? 2
}
-1
= ( 2sin
?? 2
)
-1
{sin
?? 2
+ ?? . 2cos
?? 2
}
-1
= ( 2sin
?? 2
)
-1
·
1
sin
?? 2
+ ?? · 2cos
?? 2
×
sin
?? 2
- ?? · 2cos
?? 2
sin
?? 2
- ?? · 2cos
?? 2
=
sin
?? 2
- ?? · 2cos
?? 2
2sin
?? 2
( sin
2
?? 2
+ 4cos
2
?? 2
)
Hence, real part =
sin
?? 2
2sin
?? 2
( 1 + 3cos
2
?? 2
)
=
1
2 ( 1 + 3cos
2
?? 2
)
=
1
5 + 3cos ?? .
Q7. For ?? ?? , ?? ?? , ?? ?? , ?? ?? ? ?? , if ?? < ?? ?? < ?? ?? , ?? ?? = ?? ?? and ?? ?? = ?? ?? + ?? ?? ?? , ?? ?? = ?? ?? + ?? ?? ?? , and
?? ?? =
?? ?? ( ?? ?? + ?? ?? ) , then ?? ?? , ?? ?? and ?? ?? satisfy
(a) |?? ?? | = |?? ?? | = |?? ?? |
(b) |?? ?? | ? ?? ?? |< ?? ?? |
(c) |?? ?? | > |?? ?? | > |?? ?? |
(d) |?? ?? | < ?? ?? | < |?? ?? |
Ans: (d) 0 < ?? 1
< ?? 2
, ?? 1
= ?? 2
(Given)
|?? 1
| = v?? 1
2
+ ?? 1
2
, |?? 2
| = v?? 2
2
+ ?? 2
2
? |?? 2
| > |?? 1
| ? |?? 3
| =
|?? 1
+ ?? 2
|
2
=
v
(
?? 1
+ ?? 2
2
)
2
+ (
?? 1
+ ?? 2
2
)
2
v
(
?? 1
+ ?? 2
2
)
2
+ ?? 1
2
< |?? 2
| > |?? 1
|. Hence, |?? 1
| < |?? 3
| < ?? 2
|
Q8. The amplitude of ??????
?? ?? + ?? ( ?? - ??????
?? ?? )
(a) ?? /??
(b) ?? ?? /??
(c) ?? /????
(d) ?? /????
Ans: (c) sin
?? 5
+ ?? ( 1 - cos
?? 5
)= 2sin
?? 10
· cos
?? 10
+ ?? 2sin
2
?? 10
= 2sin
?? 10
( cos
?? 10
+ ?? sin
?? 10
)
For amplitude, tan ?? =
sin
?? 10
cos
?? 10
= tan
?? 10
? ?? =
?? 10
.
Q9. If ?? and ?? are to non-zero complex numbers such that |???? | = ?? and ?????? ( ?? )-
?????? ( ?? )=
?? ?? , then ????? is equal to
(a) 1
(b) -1
(c) ??
(d) -??
Ans: (d) |?? ? ?? | = 1
.(i) and arg (
?? ?? )=
?? 2
?
?? ?? = ?? ? |
?? ?? | = 1
From equation (i) and (ii),
|?? | = |?? | = 1 and
?? ?? +
?? ?
?? ?
= 0; ?? ?? ? + ?? ??? = 0 ? ?? ??? = -?? ?? ? =
-?? ?? ?? ? ?? ? ?? ??? = -?? |?? |
2
= -?? .
Q10. If -?? + v -?? = ?? ?? ????
, then ?? is equal to
(a)
?? ??
(b) -
?? ??
(c)
?? ?? ??
(d) -
?? ?? ??
Ans: (c) Here -1 + v -3 = ?? ?? ????
? -1 + ?? v 3 = ?? ?? ????
= ?? cos ?? + ???? sin ??
Equating real and imaginary part, we get ?? cos ?? = -1 and ?? sin ?? = v 3
Hence tan ?? = -v 3 ? tan ?? = tan
2?? 3
? ?? =
2?? 3
.
Q11. Real part of ?? ?? ????
is
(a) ?? ?????? ?? [?????? ( ?????? ?? ) ]
(b) ?? ?????? ?? [?????? ( ?????? ?? ) ]
(c) ?? ?????? ?? [?????? ( ?????? ?? ) ]
(d) ?? ?????? ?? [?????? ( ?????? ?? ) ]
Ans: (a) ?? ?? ????
= ?? ( cos ?? +?? sin ?? )
= ?? cos ?? · ?? ?? sin ?? = ?? cos ?? [cos ( ?? sin ?? )+ ?? sin ( sin ?? ) ]
? Real part of ?? ?? ????
is ?? cos ?? [cos ( sin ?? ) ].
Q12. The complex numbers ?? ?? , ?? ?? and ?? ?? satisfying
?? ?? -?? ?? ?? ?? -?? ?? =
?? -?? v ?? ?? are the vertices of a
triangle which is
(a) Of area zero
(b) Right-angled isosceles
(c)Equilateral
(d) Parabola
Ans: (b) Taking mod of both sides of given relation |
?? 1
-?? 3
?? 2
-?? 3
| = |
1
2
- ?? v 3
2
| = v
1
4
+
3
4
= 1.
So, |?? 1
- ?? 3
| = |?? 2
- ?? 3
|. Also, amp (
?? 1
-?? 3
?? 2
-?? 3
)= tan
-1
( -v 3)= -
?? 3
or amp (
?? 2
-?? 3
?? 1
-?? 3
)=
?? 3
or
??? 2
?? 3
?? 1
= 60
°
? The triangle has two sides equal and the angle between the equal sides = 60
°
. So it is
equilateral.
Q13. Let ?? ?? , ?? ?? , ?? ?? be three vertices of an equilateral triangle circumscribing the circle
|?? | =
?? ?? . If ?? ?? =
?? ?? +
v ?? ?? ?? and ?? ?? , ?? ?? , ?? ?? are in anticlockwise sense then ?? ?? is
(a) ?? + v ?? ??
(b) ?? - v ?? ??
(c) 1
(d) -1
Ans: (d) ?? 2
= ?? 1
?? ?? 2?? /3
= (
1
2
+
v 3
2
?? )( cos
2?? 3
+ ?? sin
2?? 3
)= (
1
2
+
v 3
2
?? )(
-1
2
+
v 3
2
?? )=
-3
4
-
1
4
= -1.
Q14. If ?? = ?? + ???? , then area of the triangle whose vertices are points ?? , ???? and ?? + ???? is
(a) ?? |?? |
??
(b)
?? ?? |?? |
??
(c) |?? |
??
(d)
?? ?? |?? |
??
Ans (b) Let ?? = ?? + ???? , ?? + ???? = ( ?? - ?? )+ ?? ( ?? + ?? ) and ???? = -?? + ????
If A denotes the area of the triangle formed by ?? , ?? + ???? and ???? , then
?? =
1
2
|
?? ?? 1
?? - ?? ?? + ?? 1
-?? ?? 1
| (Applying transforn We get ?? =
1
2
|
?? ?? 1
0 0 -1
-?? ?? 0
| =
1
2
( ?? 2
+ ?? 2
)=
1
2
|?? |
2
.
Q15. |?? ?? + ?? ?? | = |?? ?? | + |?? ?? | is possible if
(a) ?? ?? = ???
??
(b) ?? ?? =
?? ?? ??
(c) ?????? ( ?? ?? )= ?????? ( ?? ?? )
(d) |?? ?? | = |?? ?? |
Ans: (c) Squaring both sides, we get
|?? 1
|
2
+ |?? 2
|
2
+ 2|?? 1
?
??? 2
|cos ( ?? 1
- ?? 2
)? ?? 1
|
2
+ |?? 2
|
2
+ 2|?? 1
? ?? 2
|
? 2|?? 1
? ??? 2
|cos ( ?? 1
- ?? 2
)= 2|?? 1
? ??? 2
| ? cos ( ?? 1
- ?? 2
)= 1 ? ?? 1
- ?? 2
= 0
°
? ?? 1
= ?? 2
Page 5
Solved Examples on Complex
Numbers
JEE Mains
Q1. If ?? ?? = -?? , then the value of ?
?? =?? ??????
??? ?? is
(a) 50
(b) -50
(c) ??
(d) 100
Ans: (c) ?
?? =1
200
??? ?? = ?? + ?? 2
+ ?? 3
+ ? . . +?? 200
=
?? ( 1-?? 200
)
1-?? (since G.P.) =
?? ( 1-1)
1-?? = 0.
Q2. If (
?? +?? ?? -?? )
?? = ?? then
(a) ?? = ?? ?? , where ?? is any positive integer
(b) ?? = ?? ?? , where ?? is any positive integer
(c) ?? = ?? ?? + ?? , where ?? is any positive integer
(d) ?? = ?? ?? + ?? , where ?? is any positive integer
Ans: (a)
1+?? 1-?? =
1+?? 1-?? ·
1+?? 1+?? ? (
1+?? 1-?? )
?? = 1 ? ?? ?? = 1 ? ?? = 4?? , ?? ? ?? +
.
Q3. (
?? ?? -?? ?? +
?? ?? +?? )(
?? +?? ?? ?? -?? ?? )
(a)
?? ?? +
?? ?? ??
(b)
?? ?? -
?? ?? ??
(c)
?? ?? -
?? ?? ??
(d)
?? ?? +
?? ?? ??
Ans: (d)
(
1
1 - 2?? +
3
1 + ?? )(
3 + 4?? 2 - 4?? )= [
1 + 2?? 1
2
+ 2
2
+
3 - 3?? 1
2
+ 1
2
] [
6 - 16 + 12?? + 8?? 2
2
+ 4
2
] = (
2 + 4?? + 15 - 15?? 10
)(
-1 + 2?? 2
)
=
( 17 - 11?? ) ( -1 + 2?? )
20
=
5 + 45?? 20
=
1
4
+
9
4
?? .
Q4. The real values of ?? and ?? for which the equation ( ?? ?? + ?? ???? )- ( ?? ?? ?? + ???? )= ( ?? -
?? ?? )+ ( ?? + ?? ???? ) is satisfied, are
(a) ?? = ?? , ?? = ??
(b) ?? = -?? , ?? =
?? ??
(c) Both (a) and (b)
(d) None of these
Ans: (c) Given equation ( ?? 4
+ 2???? )- ( 3?? 2
+ ???? )= ( 3 - 5?? )+ ( 1 + 2???? )? ( ?? 4
- 3?? 2
)+
?? ( 2?? - 3?? )= 4 - 5?? Equating real and imaginary parts, we get
?? 4
- 3?? 2
= 4
(i) and
2?? - 3?? = -5
Form (i) and (ii), we get ?? = ±2 and ?? = 3,
1
3
.
Trick: Put ?? = 2, ?? = 3 and then ?? = -2, ?? =
1
3
, we see that they both satisfy the given
equation.
Q5. The complex numbers ?????? ?? + ?? ?????? ?? ?? and ?????? ?? - ?? ?????? ?? ?? are conjugate to each
other for
(a) ?? = ????
(b) ?? = ( ?? +
?? ?? )??
(c) ?? = ??
(d) No value of ??
Ans: (d) sin ?? + ?? cos 2?? and cos ?? - ?? sin 2?? are conjugate to each other if sin ?? = cos ?? and
cos 2?? = sin 2??
or tan ?? = 1 ? ?? =
?? 4
,
5?? 4
,
9?? 4
, … ….
(i) and tan 2?? = 1 ? 2?? =
?? 4
,
5?? 4
,
9?? 4
, or ?? =
?? 8
,
5?? 8
,
9?? 8
, … …
(ii) There exists no value of ?? common in (i) and (ii). Therefore there is no value of ?? for
which the given complex numbers are conjugate.
Q6. The real part of ( ?? - ?????? ?? + ?? ?? ?????? ?? )
-?? is
(a)
?? ?? +???????? ??
(b)
?? ?? -???????? ??
(c)
?? ?? -???????? ??
(d)
?? ?? +???????? ??
A ns : (c) {( 1 - cos ?? )+ ?? . 2sin ?? }
-1
= {2sin
2
?? 2
+ ?? . 4sin
?? 2
cos
?? 2
}
-1
= ( 2sin
?? 2
)
-1
{sin
?? 2
+ ?? . 2cos
?? 2
}
-1
= ( 2sin
?? 2
)
-1
·
1
sin
?? 2
+ ?? · 2cos
?? 2
×
sin
?? 2
- ?? · 2cos
?? 2
sin
?? 2
- ?? · 2cos
?? 2
=
sin
?? 2
- ?? · 2cos
?? 2
2sin
?? 2
( sin
2
?? 2
+ 4cos
2
?? 2
)
Hence, real part =
sin
?? 2
2sin
?? 2
( 1 + 3cos
2
?? 2
)
=
1
2 ( 1 + 3cos
2
?? 2
)
=
1
5 + 3cos ?? .
Q7. For ?? ?? , ?? ?? , ?? ?? , ?? ?? ? ?? , if ?? < ?? ?? < ?? ?? , ?? ?? = ?? ?? and ?? ?? = ?? ?? + ?? ?? ?? , ?? ?? = ?? ?? + ?? ?? ?? , and
?? ?? =
?? ?? ( ?? ?? + ?? ?? ) , then ?? ?? , ?? ?? and ?? ?? satisfy
(a) |?? ?? | = |?? ?? | = |?? ?? |
(b) |?? ?? | ? ?? ?? |< ?? ?? |
(c) |?? ?? | > |?? ?? | > |?? ?? |
(d) |?? ?? | < ?? ?? | < |?? ?? |
Ans: (d) 0 < ?? 1
< ?? 2
, ?? 1
= ?? 2
(Given)
|?? 1
| = v?? 1
2
+ ?? 1
2
, |?? 2
| = v?? 2
2
+ ?? 2
2
? |?? 2
| > |?? 1
| ? |?? 3
| =
|?? 1
+ ?? 2
|
2
=
v
(
?? 1
+ ?? 2
2
)
2
+ (
?? 1
+ ?? 2
2
)
2
v
(
?? 1
+ ?? 2
2
)
2
+ ?? 1
2
< |?? 2
| > |?? 1
|. Hence, |?? 1
| < |?? 3
| < ?? 2
|
Q8. The amplitude of ??????
?? ?? + ?? ( ?? - ??????
?? ?? )
(a) ?? /??
(b) ?? ?? /??
(c) ?? /????
(d) ?? /????
Ans: (c) sin
?? 5
+ ?? ( 1 - cos
?? 5
)= 2sin
?? 10
· cos
?? 10
+ ?? 2sin
2
?? 10
= 2sin
?? 10
( cos
?? 10
+ ?? sin
?? 10
)
For amplitude, tan ?? =
sin
?? 10
cos
?? 10
= tan
?? 10
? ?? =
?? 10
.
Q9. If ?? and ?? are to non-zero complex numbers such that |???? | = ?? and ?????? ( ?? )-
?????? ( ?? )=
?? ?? , then ????? is equal to
(a) 1
(b) -1
(c) ??
(d) -??
Ans: (d) |?? ? ?? | = 1
.(i) and arg (
?? ?? )=
?? 2
?
?? ?? = ?? ? |
?? ?? | = 1
From equation (i) and (ii),
|?? | = |?? | = 1 and
?? ?? +
?? ?
?? ?
= 0; ?? ?? ? + ?? ??? = 0 ? ?? ??? = -?? ?? ? =
-?? ?? ?? ? ?? ? ?? ??? = -?? |?? |
2
= -?? .
Q10. If -?? + v -?? = ?? ?? ????
, then ?? is equal to
(a)
?? ??
(b) -
?? ??
(c)
?? ?? ??
(d) -
?? ?? ??
Ans: (c) Here -1 + v -3 = ?? ?? ????
? -1 + ?? v 3 = ?? ?? ????
= ?? cos ?? + ???? sin ??
Equating real and imaginary part, we get ?? cos ?? = -1 and ?? sin ?? = v 3
Hence tan ?? = -v 3 ? tan ?? = tan
2?? 3
? ?? =
2?? 3
.
Q11. Real part of ?? ?? ????
is
(a) ?? ?????? ?? [?????? ( ?????? ?? ) ]
(b) ?? ?????? ?? [?????? ( ?????? ?? ) ]
(c) ?? ?????? ?? [?????? ( ?????? ?? ) ]
(d) ?? ?????? ?? [?????? ( ?????? ?? ) ]
Ans: (a) ?? ?? ????
= ?? ( cos ?? +?? sin ?? )
= ?? cos ?? · ?? ?? sin ?? = ?? cos ?? [cos ( ?? sin ?? )+ ?? sin ( sin ?? ) ]
? Real part of ?? ?? ????
is ?? cos ?? [cos ( sin ?? ) ].
Q12. The complex numbers ?? ?? , ?? ?? and ?? ?? satisfying
?? ?? -?? ?? ?? ?? -?? ?? =
?? -?? v ?? ?? are the vertices of a
triangle which is
(a) Of area zero
(b) Right-angled isosceles
(c)Equilateral
(d) Parabola
Ans: (b) Taking mod of both sides of given relation |
?? 1
-?? 3
?? 2
-?? 3
| = |
1
2
- ?? v 3
2
| = v
1
4
+
3
4
= 1.
So, |?? 1
- ?? 3
| = |?? 2
- ?? 3
|. Also, amp (
?? 1
-?? 3
?? 2
-?? 3
)= tan
-1
( -v 3)= -
?? 3
or amp (
?? 2
-?? 3
?? 1
-?? 3
)=
?? 3
or
??? 2
?? 3
?? 1
= 60
°
? The triangle has two sides equal and the angle between the equal sides = 60
°
. So it is
equilateral.
Q13. Let ?? ?? , ?? ?? , ?? ?? be three vertices of an equilateral triangle circumscribing the circle
|?? | =
?? ?? . If ?? ?? =
?? ?? +
v ?? ?? ?? and ?? ?? , ?? ?? , ?? ?? are in anticlockwise sense then ?? ?? is
(a) ?? + v ?? ??
(b) ?? - v ?? ??
(c) 1
(d) -1
Ans: (d) ?? 2
= ?? 1
?? ?? 2?? /3
= (
1
2
+
v 3
2
?? )( cos
2?? 3
+ ?? sin
2?? 3
)= (
1
2
+
v 3
2
?? )(
-1
2
+
v 3
2
?? )=
-3
4
-
1
4
= -1.
Q14. If ?? = ?? + ???? , then area of the triangle whose vertices are points ?? , ???? and ?? + ???? is
(a) ?? |?? |
??
(b)
?? ?? |?? |
??
(c) |?? |
??
(d)
?? ?? |?? |
??
Ans (b) Let ?? = ?? + ???? , ?? + ???? = ( ?? - ?? )+ ?? ( ?? + ?? ) and ???? = -?? + ????
If A denotes the area of the triangle formed by ?? , ?? + ???? and ???? , then
?? =
1
2
|
?? ?? 1
?? - ?? ?? + ?? 1
-?? ?? 1
| (Applying transforn We get ?? =
1
2
|
?? ?? 1
0 0 -1
-?? ?? 0
| =
1
2
( ?? 2
+ ?? 2
)=
1
2
|?? |
2
.
Q15. |?? ?? + ?? ?? | = |?? ?? | + |?? ?? | is possible if
(a) ?? ?? = ???
??
(b) ?? ?? =
?? ?? ??
(c) ?????? ( ?? ?? )= ?????? ( ?? ?? )
(d) |?? ?? | = |?? ?? |
Ans: (c) Squaring both sides, we get
|?? 1
|
2
+ |?? 2
|
2
+ 2|?? 1
?
??? 2
|cos ( ?? 1
- ?? 2
)? ?? 1
|
2
+ |?? 2
|
2
+ 2|?? 1
? ?? 2
|
? 2|?? 1
? ??? 2
|cos ( ?? 1
- ?? 2
)= 2|?? 1
? ??? 2
| ? cos ( ?? 1
- ?? 2
)= 1 ? ?? 1
- ?? 2
= 0
°
? ?? 1
= ?? 2
Hence arg ( ?? 1
)= arg ( ?? 2
)
Trick: Let ?? 1
and ?? 2
are the two sides of a triangle. By applying triangle inequality ( ?? 1
+ ?? 2
) is
the third side. Equality holds only when ?? 1
= ?? 2
i.e., ?? 1
and ?? 2
are parallel.
Q16. If |?? ?? - ?? | = |?? |
?? + ?? , then ?? lies on
(a) An ellipse
(b) The imaginary axis
(c) A circle
(d) The real axis
Ans: (b)
|?? 2
- 1| = |?? |
2
+ 1
? |?? - 1|
2
|?? + 1|
2
= ( ?? ?? ? + 1)
2
? ( ?? - 1) ( ?? ? - 1) ( ?? + 1) ( ?? ? + 1)= ( ?? ?? ? + 1)
2
? ?? + ?? ? = 0
? ?? lies on imaginary axis.
Q17. The locus of the point ?? satisfying ?????? (
?? -?? ?? +?? )= ?? . (where ?? is non-zero) is
(a) Circle with centre on ?? -axis
(b)
Circle with centre on ?? -axis
(b) A straight line parallel to ?? -axis
(d) A straight line making an angle ????
°
with ?? -axis
Ans: (a)
arg (
?? - 1
?? + 1
) = ?? ? arg [
( ?? - 1)+ ????
( ?? + 1)+ ????
] = ?? ? arg [( ?? - 1)+ ???? ] - arg [( ?? + 1)+ ???? ] = ?? ? tan
-1
(
?? ?? - 1
)- tan
-1
(
?? ?? + 1
) = ?? ? tan
-1
[
?? ?? - 1
-
?? ?? + 1
1 +
?? 2
?? 2
- 1
] = ?? ? tan ?? =
?? ( ?? + 1)- ?? ( ?? - 1)
?? 2
+ ?? 2
- 1
=
2?? ?? 2
+ ?? 2
- 1
?
2?? tan ?? = ?? 2
+ ?? 2
- 1 ? ?? 2
+ ?? 2
-
2?? tan ?? - 1 = 0
It is an equation of circle whose centre is ( -?? , -?? )= ( 0, cot ?? ) on ?? -axis.
Q18. If ?? ?? = ??????
????
?? ?? + ?? ??????
????
?? ?? , where ?? = ?? , ?? , ?? , … … , ?? , then ?????? ?? ?8
??? ?? ?? ?? ?? ?? … . . ?? ?? is equal
to
(a) ?????? ?? + ?? ?????? ??
(b) ?????? ( ?? /?? )- ?? ?????? ( ?? /?? )
(c) ?? ???? /??
(d) v ?? ????
??
Ans: (c) ?? ?? = cos
????
?? 2
+ ?? sin
????
?? 2
? ?? 1
= cos
?? ?? 2
+ ?? sin
?? ?? 2
;
?? 2
= cos
2?? ?? 2
+ ?? sin
2?? ?? 2
; … … … … … … . . . . .
? ?? ?? = cos
????
?? 2
+ ?? sin
????
?? 2
? lim
?? ?8
?( ?? 1
, ?? 2
, ?? 3
, … … ?? ?? )= lim
?? ?8
? [cos {
?? ?? 2
( 1 + 2 + 3 + ? . . +?? ) } + ?? sin {
?? ?? 2
( 1 + 2 + 3 + ? . . +?? ) }]
= lim
?? ?8
? [cos
???? ( ?? + 1)
2?? 2
+ ?? sin
???? ( ?? + 1)
2?? 2
] = cos
?? 2
+ ?? sin
?? 2
= ?? ???? /2
.
Read More
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209 videos|443 docs|143 tests
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FAQs on Complex Number Solved Examples - Mathematics (Maths) for JEE Main & Advanced
| 1. What are complex numbers? |  |
Ans. Complex numbers are numbers that can be expressed in the form a + bi, where "a" and "b" are real numbers and "i" is the imaginary unit (i.e., √(-1)).
| 2. How do you add complex numbers? | |
Ans. To add complex numbers, simply add the real parts together and the imaginary parts together. For example, (3 + 4i) + (2 + 5i) = 5 + 9i.
| 3. How do you multiply complex numbers? | |
Ans. To multiply complex numbers, use the distributive property and remember that i^2 = -1. For example, (3 + 4i) * (2 + 5i) = 6 + 15i + 8i + 20i^2 = -14 + 23i.
| 4. What is the conjugate of a complex number? | |
Ans. The conjugate of a complex number a + bi is a - bi. It involves changing the sign of the imaginary part.
| 5. How do you divide complex numbers? | |
Ans. To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. Then, simplify the expression to obtain the quotient.
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