Complex Numbers Advanced provides a comprehensive exploration of complex number theory, including operations, properties, and applications. This study guide is ideal for students preparing for advanced mathematics exams, such as JEE or other competitive assessments. Key topics include the geometric interpretation of complex numbers, polar forms, and the use of complex numbers in calculus and algebra. With numerous examples and practice problems, learners can enhance their understanding and problem-solving skills in complex analysis.

Key Points

  • Explores the fundamental concepts of complex numbers, including addition, subtraction, multiplication, and division.
  • Covers polar representation and the geometric interpretation of complex numbers on the Argand plane.
  • Includes detailed examples and practice problems to reinforce understanding and application of complex number theory.
  • Discusses the significance of complex numbers in calculus, algebra, and engineering applications.
aadesh partap
29 pages
Language:English
Type:Study Guide
Complex Numbers
aadesh partap
29 pages
Language:English
Type:Study Guide
Complex Numbers
276
/ 29
Q1. Single Correct
Let be a complex number satisfying . then maximum values must be
(1) (2)
(3) (4)
Q2. Single Correct
If and are three complex numbers and and are three positive real numbers, such that
then the value of is
(1) 0 (2) 1
(3) 2 (4) 3
Q3. Single Correct
Angular hexagon is inscribed in a circle of radius 1 . A point is taken on the circle of radius 2 having
same centre. is equal to
(1) 18 (2) 12
(3) 24 (4) 30
Q4. Single Correct
If and and then is equal to
(1) (2)
(3) (4)
Q5. Single Correct
If all the three roots of have negative real part , then
(1) (2)
(3) (4) None of these
Q6. Single Correct
Suppose A be a complex number and , such that , then the least value of is
(1) 3 (2) 6
(3) 9 (4) 12
Q7. Single Correct
If are the th roots of unity and is a non-real complex cube root of unity, then the product
cannot be equal to
(1) 0 (2) 1
(3) -1 (4)
Q8. Single Correct
If are the roots of the equation then the value of , is
(1) -85 (2) -25
(3) 25 (4) 75
Q9. Single Correct
Complex Numbers
Advanced DPP
JEE Advanced
MathonGo
z z
2
+ 2
z
cos
α
≤ 1
∣ ∣
(
α
∈
R
) |
z
|
√
2 + 1
√
3 − 1
√
3 + 1
√
6
z
1
,
z
2
z
3
a
,
b c
a
|
z
2
−
z
3
|
=
b
|
z
3
−
z
1
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=
c
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z
1
−
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a
2
z
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2
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2
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PB
2
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2
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z
n
−1
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n
−1
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r
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(
ω
−
Z
r
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1 +
ω
z
r
,
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∑
50
r
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(
z
)
r
= 0,
∑
50
r
=1
1
z
r
−1
Consider the circle and two points and on it such that and . A
tangent is drawn at to the circle, which cuts the real axis at , then is equal to
(1) 3 (2)
(3) (4)
Q10. Single Correct
If the complex numbers and are the solutions of the equation , then the value of the expression
, is
(1) 1 (2) 2
(3) -200 (4) -2
Q11. Multiple Correct
Let be any point on the circle with as diameter being origin). The points and are an the same side of
the diameter such that . Also be are on the numbers and respectively such
that then
(1) (2)
(3) (4)
Q12. Multiple Correct
One vertex of the triangle of maximum area that can be inscribed in the curve , is remaining
vertices is/are
(1) (2)
(3) (4)
Q13. Multiple Correct
If is a comnlex number which simultaneously satify the equations and ,
then the can be
(1) 15 (2) 16
(3) 17 (4) 8
Q14. Multiple Correct
If satisfy the inequality , then
(1) (2)
(3) (4)
Q15. Multiple Correct
If is a complex number then the equation is satisfied by
(1) (2) where is non-negative real
(3) where is positive real (4)
Q16. Multiple Correct
Let the complex number iy where satisfy then
can be
Complex Numbers
Advanced DPP
JEE Advanced
MathonGo
|
z
− 5
i
| = 3
z
1
z
2
|
z
1
| < |
z
2
| arg (
z
1
) = arg (
z
2
) =
π
3
z
2
z
3
|
z
3
|
√
3
√
7
√
11
z
1
z
2
z
2
−
z
+ 1 = 0
E
=
(
z
4
1
−
z
3
1
+ 2
z
2
1
− 2
z
1
+ 1
)
2013
+
(
z
4
2
−
z
3
2
+ 2
z
2
2
− 2
z
2
+ 1
)
2013
P OP
(
O Q R
∠
POQ
= ∠
QOR
=
θ P
,
Q
,
R z
1
,
z
2
z
3
√
2
√
3
z
2
2
= (1 +
√
3)
z
1
z
3
,
θ
π
8
π
12
π
18
π
10
|
z
− 2
i
| = 2 2 + 2
i
−1 +
i
(2 +
√
3) −1 −
i
(2 +
√
3)
−1 +
i
(2 −
√
3) −1 −
i
(2 −
√
3)
z
3|
z
− 12| = 5|
z
− 8
i
| |
z
− 4| = |
z
− 8|
lm(
z
)
z
|
z
− 1 − 2
i
| ≤ 1
min{arg(
z
)} = tan
−1
(
3
4
)
max{arg(
z
)} =
π
2
min(|
z
|) =
√
5 − 1 max(|
z
|) =
√
5 + 1
z z
2
+
z
|
z
| + |
z
|
2
= 0
z
=
kw
,
k
∈
R z
=
kw
2
k
z
=
kw
,
k z
=
kω
2
,
k
∈
R
z
=
x
+ (
x
,
y
∈
R
) cot
−1
(log
3
|2
z
+ 1|) > cot
−1
(log
3
|2
z
− 1|),
x
(1) -2 (2) -1
(3) 1 (4) 2
Q17. Multiple Correct
If then possible value of can be
(1) 1 (2) 2
(3) 3 (4) 4
Q18. Multiple Correct
Let and be the coordinates of the vertices of the then which of the following statements are
equivalent?
(1) is an equilateral triangle (2) where
is a cube root of unity
(3) (4)
Q19. Multiple Correct
If are the imaginary th roots of unity, then the product (where, ) can
take the value(s)
(1) 0 (2) 1
(3) i (4)
Q20. Multiple Correct
If then which of the following relation(s) represents a circle on an argand diagram?
(1) (2)
(3) (4)
Q21. Multiple Correct
Let and be three complex numbers such that and Then,
can take the value equals
(1) 1 (2) 2
(3) 3 (4) 4
Q22. Multiple Correct
Let and be two distinct points denoting the complex numbers and , respectively A complex number lies
between and such that . Which of the following relation(s) hold good?
(1) (2) There exists a positive real number such that
(3) (4)
Q23. Multiple Correct
Complex Numbers
Advanced DPP
JEE Advanced
MathonGo
|
z
− 2| = min{|
z
|, |
z
− 4|}, (
s
) Re(
z
)
z
1
,
z
2
z
3
Δ
A
1
A
2
A
3
,
A
1
A
2
A
3
(
z
1
+
ωz
2
+
ω
2
z
3
) (
z
1
+
ω
2
z
2
+
ωz
3
)
= 0,
ω
z
2
−
z
1
z
3
−
z
2
=
z
3
−
z
2
z
1
−
z
3
= 0
∣
1 1 1
z
1
z
2
z
3
z
2
z
3
z
1
∣
α
1
,
α
2
, … ,
α
n
−1
n
∏
n
−1
r
=1
(
i
−
α
r
)
i
=
√
−1
(1 +
i
)
z
∈
c
,
|
z
− 1| + |
z
+ 1| = 3 (
z
− 3 +
i
)(¯
z
− 3 −
i
) = 5
3|
z
− 2 +
i
| = 7 |
z
− 3| = 2
z
1
,
z
2
z
3
|
z
1
| = |
z
2
| = |
z
3
| = 1
z
2
1
z
2
z
3
+
z
2
2
z
1
z
3
+
z
2
3
z
1
z
2
+ 1 = 0.
|
z
1
+
z
2
+
z
3
|
A B α β z
A B z
≠
α
,
z
≠
β
|
α
−
z
| + |
z
−
β
| = |
α
−
β
|
t
z
= (1 −
t
)
α
+
tβ
= 0
∣
z
−
α
¯
z
− ¯
α
β
−
α
¯
β
− ¯
α
∣
= 0
∣
z
¯
z
1
α
¯
α
1
β
¯
β
1
∣
/ 29
End of Document
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FAQs

What are the key operations involving complex numbers?
Key operations with complex numbers include addition, subtraction, multiplication, and division. Addition and subtraction involve combining the real and imaginary parts separately. For multiplication, the distributive property is used, and for division, complex conjugates are applied to simplify the expression. Understanding these operations is crucial for solving equations and performing calculations in advanced mathematics.
How are complex numbers represented geometrically?
Complex numbers can be represented geometrically on the Argand plane, where the x-axis represents the real part and the y-axis represents the imaginary part. Each complex number corresponds to a point in this plane, allowing for visual interpretation of operations such as addition and multiplication. This geometric approach helps in understanding concepts like modulus and argument, which are essential in complex analysis.
What is the significance of polar form in complex numbers?
The polar form of a complex number expresses it in terms of its magnitude and angle, making it easier to perform multiplication and division. In polar form, a complex number is represented as r(cos θ + i sin θ), where r is the modulus and θ is the argument. This representation is particularly useful in calculus and engineering, where complex numbers are used to model oscillations and waves.
What applications do complex numbers have in real-world scenarios?
Complex numbers have numerous applications in engineering, physics, and applied mathematics. They are used in electrical engineering to analyze AC circuits, in fluid dynamics to describe wave patterns, and in control theory for system stability analysis. Additionally, complex numbers play a vital role in signal processing and quantum mechanics, showcasing their importance across various scientific fields.

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