Which of the following is the multiplicative inverse of the complex number $1 + i$?

Correct Option: A. Solution: The multiplicative inverse of a complex number $a + ib$ is $\frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i$. For $1 + i$, it is $\frac{1}{1^2 + 1^2} - \frac{1}{1^2 + 1^2}i = \frac{1}{2} - \frac{1}{2}i$.

If $z = 1 + i$, what is the value of $z^2$?

Correct Option: A. Solution: To find $z^2$, we compute $(1 + i)^2 = 1^2 + 2 \cdot 1 \cdot i + i^2 = 1 + 2i - 1 = 0 + 2i$.

What is the result of multiplying the complex numbers $(3 + i5)$ and $(2 + i6)$?

Correct Option: A. Solution: The product of two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ is $(ac - bd) + i(ad + bc)$. For $(3 + i5)$ and $(2 + i6)$, the product is $(3 \times 2 - 5 \times 6) + i(3 \times 6 + 5 \times 2) = -24 + i28$.

What is the result of multiplying the complex numbers $2 + 3i$ and $4 - i$?

Correct Option: A. Solution: The product of two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ is $(ac - bd) + i(ad + bc)$. So, $(2 + 3i)(4 - i) = (2 \times 4 - 3 \times (-1)) + i(2 \times (-1) + 3 \times 4) = 11 + 10i$.

If $z = 1 + i$ and $w = 2 - 3i$, find the real part of the complex number $z + w$.

Correct Option: A. Solution: The sum of two complex numbers $z = a + ib$ and $w = c + id$ is $(a + c) + i(b + d)$. For $z = 1 + i$ and $w = 2 - 3i$, $z + w = (1 + 2) + i(1 - 3) = 3 - 2i$. The real part is 3.

Calculate the product of the complex numbers $z_1 = 3 + 2i$ and $z_2 = 1 + 4i$.

Correct Option: A. Solution: The product of two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ is $z_1 z_2 = (ac - bd) + i(ad + bc)$. Therefore, $(3 + 2i)(1 + 4i) = (3 \cdot 1 - 2 \cdot 4) + i(3 \cdot 4 + 2 \cdot 1) = -5 + 14i$.

If $z = a + ib$ is a complex number such that $z^2 = -1$, what is the value of $a^2 + b^2$?

Correct Option: B. Solution: If $z^2 = -1$, then $z = i$ or $z = -i$. In both cases, $a = 0$ and $b = \pm 1$, so $a^2 + b^2 = 0^2 + 1^2 = 1$.

If $z = x + iy$ and its conjugate is $-6 - 24i$, what are the values of $x$ and $y$?

Correct Option: C. Solution: The conjugate of $z = x + iy$ is $x - iy$. Given that the conjugate is $-6 - 24i$, we have $x = -6$ and $-y = -24$, thus $y = 24$.

What is the conjugate of the complex number $z = 7 - 5i$?

Correct Option: A. Solution: The conjugate of a complex number $z = a + ib$ is $\overline{z} = a - ib$. Therefore, the conjugate of $z = 7 - 5i$ is $7 + 5i$.

Given the complex number $z = 2 + 3i$, what is the modulus of its conjugate $\overline{z}$?

Correct Option: A. Solution: The modulus of a complex number $z = a + ib$ is $|z| = \sqrt{a^2 + b^2}$. The modulus of its conjugate $\overline{z} = a - ib$ is the same as $z$, which is $\sqrt{4 + 9} = \sqrt{13} = 5$.

If $z = x + iy$ is a complex number and $|z| = 5$, which of the following equations must be true?

Correct Option: B. Solution: The modulus of a complex number $z = x + iy$ is $|z| = \sqrt{x^2 + y^2}$. Given $|z| = 5$, we have $\sqrt{x^2 + y^2} = 5$, which implies $x^2 + y^2 = 25$.

Which of the following represents the conjugate of the complex number $6 - 8i$?

Correct Option: A. Solution: The conjugate of a complex number $a + ib$ is $a - ib$. Thus, the conjugate of $6 - 8i$ is $6 + 8i$.

What is the conjugate of the complex number $7 + 5i$?

Correct Option: A. Solution: The conjugate of a complex number $a + ib$ is $a - ib$. Therefore, the conjugate of $7 + 5i$ is $7 - 5i$.

If $z = 3 + 4i$, what is the conjugate of $z$?

Correct Option: A. Solution: The conjugate of a complex number $z = a + ib$ is $\overline{z} = a - ib$. For $z = 3 + 4i$, the conjugate is $3 - 4i$.

Complex Numbers and Quadratic Equations

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Summary

Chapter 4: Complex Numbers and Quadratic Equations

Summary

Mathematics extends to complex numbers to solve equations like x² + 1 = 0.
A complex number is of the form a + ib, where a and b are real numbers.
Real part (Re z) and imaginary part (Im z) are defined for complex numbers.
Addition and multiplication of complex numbers follow specific rules:
- Addition: z₁ + z₂ = (a + c) + i(b + d)
- Multiplication: z₁ z₂ = (ac - bd) + i(ad + bc)
The modulus of a complex number z = a + ib is |z| = √(a² + b²).
The conjugate of z is given by z̅ = a - ib.
The multiplicative inverse of a non-zero complex number z is z⁻¹ = (a + ib) / (a² + b²).
Historical context includes contributions from mathematicians like W.R. Hamilton and Mahavira regarding complex numbers.
The Argand plane represents complex numbers geometrically, where the x-axis is the real axis and the y-axis is the imaginary axis.

Learning Objectives

Understand the concept of complex numbers and their representation.
Identify the real and imaginary parts of a complex number.
Perform addition and subtraction of complex numbers.
Apply the properties of complex numbers, including closure, commutativity, and associativity.
Calculate the modulus and conjugate of complex numbers.
Solve quadratic equations with complex solutions.
Represent complex numbers in the Argand plane.

Detailed Notes

Chapter 4: Complex Numbers and Quadratic Equations

4.1 Introduction

Mathematics is the Queen of Sciences and Arithmetic is the Queen of Mathematics. - GAUSS
Previous studies included linear equations and quadratic equations in one variable.
Example: The equation x² + 1 = 0 has no real solution as x² = -1.
Need to extend the real number system to solve equations like ax² + bx + c = 0 where D = b² - 4ac < 0.

4.2 Complex Numbers

Denote √-1 by the symbol i; thus, i² = -1.
A complex number is of the form a + ib, where a and b are real numbers.
- Examples: 2 + i3, (-1) + i√3, 4 + i(II).
For a complex number z = a + ib:
- Real part: Re z = a
- Imaginary part: Im z = b
- Example: If z = 2 + i5, then Re z = 2 and Im z = 5.
Two complex numbers z₁ = a + ib and z₂ = c + id are equal if a = c and b = d.

4.3 Algebra of Complex Numbers

4.3.1 Addition of Complex Numbers

Given z₁ = a + ib and z₂ = c + id:
- z₁ + z₂ = (a + c) + i(b + d)
- Example: (2 + i3) + (-6 + i5) = (2 - 6) + i(3 + 5) = -4 + i8.
Properties of Addition:
1. Closure Law: The sum of two complex numbers is a complex number.
2. Commutative Law: z₁ + z₂ = z₂ + z₁.
3. Associative Law: For any three complex numbers z₁, z₂, z₃.
4. Additive Identity: 0 + i0 is the additive identity.
5. Additive Inverse: For z = a + ib, the inverse is -a + i(-b).

4.3.2 Difference of Complex Numbers

Defined as z₁ - z₂ = z₁ + (-z₂).
- Example: (2 + i) - (6 + 3i) = (2 + i) + (-6 - 3i) = -4 - 2i.

4.5 Argand Plane and Polar Representation

Each ordered pair of real numbers (x,y) corresponds to a unique point in the XY-plane.
The complex number x + iy can be represented as point P(x,y).
Example complex numbers: 2 + 4i, 2 + 3i, etc., correspond to points A, B, C, etc.
The plane with a complex number assigned to each point is called the complex plane or Argand plane.

Important Diagrams

Fig 4.1: Cartesian Coordinate System

Axes: X-axis and Y-axis.
Points: A(2,4), B(-2,3), C(0,1), D(2,0), E(-5,-2), F(1,-2).

Fig 4.2: Modulus of Complex Number

Modulus of the complex number x + iy = √(x² + y²).
Represents the distance from point P(x,y) to the origin O(0,0).

Fig 4.3: Representation of Complex Number and its Conjugate

Points P(x,y) and Q(x,-y) in the Argand plane.
Q is the mirror image of P on the real axis.

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Misunderstanding Complex Numbers: Students often confuse the real and imaginary parts of complex numbers. Ensure to clearly identify and separate these parts when performing operations.
Neglecting Conjugates: Forgetting to use the conjugate when dividing complex numbers can lead to incorrect results. Always multiply by the conjugate to simplify.
Ignoring Modulus: When asked for the modulus of a complex number, students sometimes forget to apply the formula correctly. Remember, the modulus is given by $|z| = \sqrt{a^2 + b^2}$ .
Incorrect Application of Identities: Students may misapply identities or properties of complex numbers, such as the distributive law. Review these laws thoroughly before the exam.

Exam Tips

Practice with Examples: Work through examples that involve addition, subtraction, multiplication, and division of complex numbers to solidify your understanding.
Check Your Work: After solving a problem, go back and check each step to ensure that you have not made any arithmetic errors.
Use Graphs: When dealing with complex numbers, sketching them on the Argand plane can help visualize the problem and avoid mistakes.
Memorize Key Formulas: Ensure you have key formulas, such as the definitions of modulus and conjugate, memorized for quick recall during the exam.

Practice & Assessment

Multiple Choice Questions

\frac{1}{2} - \frac{1}{2}i

1 - i

-1 - i

\frac{1}{2} + \frac{1}{2}i

Correct Answer: A

Solution:

The multiplicative inverse of a complex number

a + ib

\frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i

. For

1 + i

, it is

\frac{1}{1^2 + 1^2} - \frac{1}{1^2 + 1^2}i = \frac{1}{2} - \frac{1}{2}i

0 + 2i

1 + 2i

1 + i

0 + i

Correct Answer: A

Solution:

To find

z^2

, we compute

(1 + i)^2 = 1^2 + 2 \cdot 1 \cdot i + i^2 = 1 + 2i - 1 = 0 + 2i

Correct Answer: B

Solution:

The complex number

1+i

has modulus

\sqrt{2}

and argument

\frac{\pi}{4}

. Therefore,

(1+i)^m = 1

implies

m \cdot \frac{\pi}{4} = 2k\pi

for some integer

k

. Solving gives

m = 8k

. The smallest positive

m

is 4.

-24 + i28

24 + i28

-24 - i28

24 - i28

Correct Answer: A

Solution:

The product of two complex numbers

z_1 = a + ib

and

z_2 = c + id

(ac - bd) + i(ad + bc)

. For

(3 + i5)

and

(2 + i6)

, the product is

(3 \times 2 - 5 \times 6) + i(3 \times 6 + 5 \times 2) = -24 + i28

Correct Answer: A

Solution:

The product

\overline{z} \cdot z = (a - ib)(a + ib) = a^2 + b^2

. Given that

\overline{z} \cdot z = 25

, we have

a^2 + b^2 = 25

. Therefore, the value of

a^2 + b^2

is 25.

-2

-4

Correct Answer: A

Solution:

The equation

z^2 + 4z + 5 = 0

can be solved using the quadratic formula:

z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where

a=1

b=4

c=5

. The discriminant

b^2 - 4ac = 16 - 20 = -4

. Thus,

z = \frac{-4 \pm i\sqrt{4}}{2} = -2 \pm i

. The real part of

z

is -2.

Correct Answer: A

Solution:

The modulus of a complex number

z = a + ib

is given by

\sqrt{a^2 + b^2}

. For

z = 7 + 24i

, the modulus is

\sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25

Correct Answer: C

Solution:

The distance of a point

P(x, y)

from the origin in the Argand plane is given by the modulus

|z| = \sqrt{x^2 + y^2}

where

z = x + iy

. Here,

z = 3 + 4i

, so

|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

11 + 10i

5 + 14i

11 + 14i

5 - 10i

Correct Answer: A

Solution:

The product of two complex numbers

z_1 = a + ib

and

z_2 = c + id

(ac - bd) + i(ad + bc)

. So,

(2 + 3i)(4 - i) = (2 \times 4 - 3 \times (-1)) + i(2 \times (-1) + 3 \times 4) = 11 + 10i

\frac{2 + 3i}{13}

\frac{2 - 3i}{13}

\frac{3 + 2i}{13}

\frac{3 - 2i}{13}

Correct Answer: A

Solution:

The multiplicative inverse of a complex number

z = a + ib

is given by

\frac{a - ib}{a^2 + b^2}

. For

2 - 3i

, the inverse is

\frac{2 + 3i}{2^2 + (-3)^2} = \frac{2 + 3i}{13}

-2

Correct Answer: A

Solution:

The equation

z^3 = 8

implies

r^3 = 8

and

r = 2

. In polar form,

z = r(\cos \theta + i\sin \theta)

. For real

z

\theta = 0

, so

z = 2(1 + 0i) = 2

. Hence,

x = 2

\frac{4}{25} + \frac{3}{25}i

\frac{4}{25} - \frac{3}{25}i

\frac{3}{25} + \frac{4}{25}i

\frac{3}{25} - \frac{4}{25}i

Correct Answer: B

Solution:

The multiplicative inverse of a complex number

z = a + ib

is given by

\frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i

. For

z = 4 - 3i

, the inverse is

\frac{4}{4^2 + (-3)^2} - \frac{-3}{4^2 + (-3)^2}i = \frac{4}{16 + 9} + \frac{3}{25}i = \frac{4}{25} - \frac{3}{25}i

11 + 2i

10 + 10i

7 + 10i

10 - 10i

Correct Answer: A

Solution:

The product of two complex numbers

z_1 = a + ib

and

z_2 = c + id

is given by

(ac - bd) + i(ad + bc)

. Here,

z_1 = 3 + 4i

and

z_2 = 1 - 2i

. So,

z_1 \cdot z_2 = (3 \cdot 1 - 4 \cdot (-2)) + i(3 \cdot (-2) + 4 \cdot 1) = (3 + 8) + i(-6 + 4) = 11 + 2i

x^2 + y^2 = 25

x^2 - y^2 = 25

x^2 + y^2 = 5

x^2 - y^2 = 5

Correct Answer: A

Solution:

The modulus of a complex number

z = x + iy

is given by

|z| = \sqrt{x^2 + y^2}

. If the modulus is 5, then

\sqrt{x^2 + y^2} = 5

, which implies

x^2 + y^2 = 25

10 + 2i

10 - 2i

11 + 2i

11 - 2i

Correct Answer: B

Solution:

The product

z \cdot w = (3 + 4i)(1 - 2i) = 3 \cdot 1 + 3 \cdot (-2i) + 4i \cdot 1 + 4i \cdot (-2i) = 3 - 6i + 4i - 8i^2

. Since

i^2 = -1

, this becomes

3 - 6i + 4i + 8 = 11 - 2i

. Therefore, the correct answer is

11 - 2i

Correct Answer: C

Solution:

The distance from the origin to the point

(3, 4)

in the Argand plane is given by the modulus of the complex number

z = 3 + 4i

. The modulus is calculated as

|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

3 + i

3 - i

1 + i

1 - i

Correct Answer: A

Solution:

The product of two complex numbers

z_1 = a + ib

and

z_2 = c + id

is given by

z_1 z_2 = (ac - bd) + i(ad + bc)

. Here,

z_1 = 2 + i

and

z_2 = 1 - i

. Therefore,

z_1 z_2 = (2 \cdot 1 - 1 \cdot (-1)) + i(2 \cdot (-1) + 1 \cdot 1) = 3 + i

Correct Answer: A

Solution:

The modulus of a complex number

a + ib

is given by

\sqrt{a^2 + b^2}

. For

5 - 12i

, the modulus is

\sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13

7 + 5i

-7 + 5i

7 - 5i

-7 - 5i

Correct Answer: A

Solution:

The conjugate of a complex number

z = a + ib

is given by

\overline{z} = a - ib

. Therefore, the conjugate of

z = 7 - 5i

\overline{z} = 7 + 5i

Point

(3, 4)

Point

(4, 3)

Point

(-3, -4)

Point

(-4, -3)

Correct Answer: A

Solution:

In the Argand plane, a complex number

a + ib

is represented by the point

(a, b)

. Therefore,

3 + 4i

is represented by the point

(3, 4)

3 - 4i

-3 + 4i

4 - 3i

-3 - 4i

Correct Answer: A

Solution:

The conjugate of a complex number

z = a + ib

is given by

\overline{z} = a - ib

. Therefore, the conjugate of

z = 3 + 4i

3 - 4i

. Hence, the correct answer is

3 - 4i

-10

-11

Correct Answer: B

Solution:

The product

z = z_1 \cdot z_2 = (3 + 4i)(1 - 2i) = (3 \cdot 1 - 4 \cdot 2) + i(3 \cdot -2 + 4 \cdot 1) = -5 + 10i

. Therefore, the imaginary part of

z

is 10.

Correct Answer: A

Solution:

The sum of two complex numbers

z = a + ib

and

w = c + id

(a + c) + i(b + d)

. For

z = 1 + i

and

w = 2 - 3i

z + w = (1 + 2) + i(1 - 3) = 3 - 2i

. The real part is 3.

Correct Answer: A

Solution:

The distance from the origin to the point

P

is the modulus of the complex number

5 - 12i

. The modulus is given by

|z| = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13

10 - 198i

125 - 225i

-10 + 198i

10 + 198i

Correct Answer: A

Solution:

Using the binomial expansion,

(5 - 3i)^3 = 5^3 - 3 \times 5^2 \times (3i) + 3 \times 5 \times (3i)^2 - (3i)^3 = 125 - 225i - 135 - 27i = 10 - 198i

4 + 3i

-4 + 3i

4 - 3i

-4 - 3i

Correct Answer: A

Solution:

The conjugate of a complex number

z = a + ib

is given by

\overline{z} = a - ib

. Therefore, the conjugate of

z = 4 - 3i

4 + 3i

-5 + 14i

5 + 14i

-5 - 14i

5 - 14i

Correct Answer: A

Solution:

The product of two complex numbers

z_1 = a + ib

and

z_2 = c + id

z_1 z_2 = (ac - bd) + i(ad + bc)

. Therefore,

(3 + 2i)(1 + 4i) = (3 \cdot 1 - 2 \cdot 4) + i(3 \cdot 4 + 2 \cdot 1) = -5 + 14i

Correct Answer: B

Solution:

z^2 = -1

, then

z = i

z = -i

. In both cases,

a = 0

and

b = \pm 1

, so

a^2 + b^2 = 0^2 + 1^2 = 1

x = -6, y = -24

x = 6, y = 24

x = -6, y = 24

x = 6, y = -24

Correct Answer: C

Solution:

The conjugate of

z = x + iy

x - iy

. Given that the conjugate is

-6 - 24i

, we have

x = -6

and

-y = -24

, thus

y = 24

\sqrt{9 + 16}

\sqrt{3^2 + 4^2}

\sqrt{3 + 4}

\sqrt{3^2 - 4^2}

Correct Answer: B

Solution:

The modulus of a complex number

z = a + ib

is given by

|z| = \sqrt{a^2 + b^2}

. For

z = 3 + 4i

|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

Point in the second quadrant

Point in the first quadrant

Point in the third quadrant

Point in the fourth quadrant

Correct Answer: A

Solution:

The complex number

z = -3 + 2i

corresponds to the point

(-3, 2)

in the Argand plane, which is located in the second quadrant.

i

-i

1

-1

Correct Answer: A

Solution:

(x + iy)^2 = -1

, then

x^2 - y^2 + 2ixy = -1

. Comparing real and imaginary parts, we get

x^2 - y^2 = -1

and

2xy = 0

. Since

y \neq 0

for a non-real solution,

x = 0

. Then

-y^2 = -1 \Rightarrow y^2 = 1 \Rightarrow y = \pm 1

. Thus,

z = \pm i

. The positive value is

i

5 + i

1 + 5i

5 - i

1 - 5i

Correct Answer: A

Solution:

The product of two complex numbers

z_1 = a + ib

and

z_2 = c + id

is given by

(ac - bd) + i(ad + bc)

. For

z_1 = 3 + 2i

and

z_2 = 1 - i

, the product is

(3 \cdot 1 - 2 \cdot (-1)) + i(3 \cdot (-1) + 2 \cdot 1) = (3 + 2) + i(-3 + 2) = 5 + i

7 + 5i

-7 + 5i

7 - 5i

-7 - 5i

Correct Answer: A

Solution:

The conjugate of a complex number

z = a + ib

\overline{z} = a - ib

. Therefore, the conjugate of

z = 7 - 5i

7 + 5i

x^2 + y^2 = 25

x^2 - y^2 = 25

x^2 + y^2 = 5

x^2 - y^2 = 5

Correct Answer: A

Solution:

The modulus of a complex number

z = x + iy

is given by

|z| = \sqrt{x^2 + y^2}

. If

|z| = 5

, then

\sqrt{x^2 + y^2} = 5

, which implies

x^2 + y^2 = 25

. Therefore, the correct equation representing the locus of

z

in the complex plane is

x^2 + y^2 = 25

\frac{\pi}{4}

\frac{\pi}{2}

\frac{3\pi}{4}

\pi

Correct Answer: C

Solution:

The argument of a product is the sum of the arguments:

\arg(z \cdot w) = \arg(z) + \arg(w)

. Here,

\arg(z) = \tan^{-1}(1)

and

\arg(w) = \tan^{-1}(-\frac{3}{2})

. Thus,

\arg(z \cdot w) = \frac{\pi}{4} + (-\frac{\pi}{3}) = \frac{3\pi}{4}

Correct Answer: A

Solution:

The modulus of a complex number

z = a + ib

|z| = \sqrt{a^2 + b^2}

. The modulus of its conjugate

\overline{z} = a - ib

is the same as

z

, which is

\sqrt{4 + 9} = \sqrt{13} = 5

\frac{\pi}{6}

\frac{\pi}{3}

\frac{\pi}{4}

\frac{\pi}{2}

Correct Answer: B

Solution:

The argument of a complex number

z = x + iy

is given by

\tan^{-1}(\frac{y}{x})

. For

z = 1 + i\sqrt{3}

x = 1

y = \sqrt{3}

. Thus,

\tan \theta = \frac{\sqrt{3}}{1} = \sqrt{3}

. Therefore,

\theta = \frac{\pi}{3}

x^2 + y^2 = 5

x^2 + y^2 = 25

x^2 - y^2 = 25

x^2 - y^2 = 5

Correct Answer: B

Solution:

The modulus of a complex number

z = x + iy

|z| = \sqrt{x^2 + y^2}

. Given

|z| = 5

, we have

\sqrt{x^2 + y^2} = 5

, which implies

x^2 + y^2 = 25

Correct Answer: A

Solution:

Given

|z| = 5

, we have

\sqrt{x^2 + y^2} = 5

, implying

x^2 + y^2 = 25

. Also,

z + \overline{z} = 2x = 8

, so

x = 4

. Therefore, the value of

x

is 4.

Correct Answer: B

Solution:

The product of two complex numbers

(a + ib)

and

(c + id)

(ac - bd) + i(ad + bc)

. For

(1 + i)(1 - i)

, this becomes

(1 \cdot 1 - 1 \cdot 1) + i(1 \cdot -1 + 1 \cdot 1) = 0 + i(0) = 0

. However, this is incorrect as the correct calculation is

(1 \cdot 1 - 1 \cdot 1) + i(1 \cdot -1 + 1 \cdot 1) = 1 - 1 + i(0) = 1

11 + 10i

5 + 11i

5 - 11i

14 + 5i

Correct Answer: A

Solution:

Using the formula for multiplication of complex numbers,

z_1 z_2 = (a + ib)(c + id) = (ac - bd) + i(ad + bc)

. Here,

z_1 z_2 = (2 + 3i)(4 - i) = (2 \times 4 - 3 \times (-1)) + i(2 \times (-1) + 3 \times 4) = 8 + 3 + i(-2 + 12) = 11 + 10i

6 + 8i

-6 + 8i

6 - 8i

-6 - 8i

Correct Answer: A

Solution:

The conjugate of a complex number

a + ib

a - ib

. Thus, the conjugate of

6 - 8i

6 + 8i

Point (5, 2)

Point (2, 5)

Point (-5, -2)

Point (5, -2)

Correct Answer: A

Solution:

In the Argand plane, a complex number

z = x + iy

is represented by the point

(x, y)

. Therefore,

5 + 2i

is represented by the point (5, 2).

Correct Answer: A

Solution:

The modulus of a complex number

z = a + ib

is given by

|z| = \sqrt{a^2 + b^2}

. For

z = -3 + 4i

|z| = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

7 - 5i

-7 + 5i

-7 - 5i

7 + 5i

Correct Answer: A

Solution:

The conjugate of a complex number

a + ib

a - ib

. Therefore, the conjugate of

7 + 5i

7 - 5i

(2, -3)

(2, 3)

(-2, 3)

(-2, -3)

Correct Answer: B

Solution:

In the Argand plane, a complex number

a + ib

is represented by the point

(a, b)

. Therefore, the point corresponding to

2 + 3i

(2, 3)

\frac{3}{25} - \frac{4}{25}i

\frac{3}{5} - \frac{4}{5}i

\frac{4}{25} - \frac{3}{25}i

\frac{4}{5} - \frac{3}{5}i

Correct Answer: A

Solution:

The multiplicative inverse of a complex number

z = a + ib

is given by

\frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i

. For

z = 3 + 4i

, the inverse is

\frac{3}{3^2 + 4^2} - \frac{4}{3^2 + 4^2}i = \frac{3}{25} - \frac{4}{25}i

Correct Answer: A

Solution:

The product of a complex number and its conjugate is equal to the square of its modulus:

z \cdot z^* = |z|^2 = x^2 + y^2

. Given

z \cdot z^* = 25

, we have

x^2 + y^2 = 25

1

\sqrt{2}

2

\sqrt{3}

Correct Answer: B

Solution:

The modulus of a complex number

z = a + ib

is given by

|z| = \sqrt{a^2 + b^2}

. For

z = 1 + i

|z| = \sqrt{1^2 + 1^2} = \sqrt{2}

3 - 4i

-3 + 4i

-3 - 4i

3 + 4i

Correct Answer: A

Solution:

The conjugate of a complex number

z = a + ib

\overline{z} = a - ib

. For

z = 3 + 4i

, the conjugate is

3 - 4i

A circle with center at

(3, 0)

and radius 4

A circle with center at

(-3, 0)

and radius 4

A line parallel to the real axis

A line parallel to the imaginary axis

Correct Answer: A

Solution:

The equation

|z - 3| = 4

represents a circle in the complex plane with center at

(3, 0)

and radius 4.

Correct Answer: C

Solution:

The modulus of a complex number

a + ib

is given by

\sqrt{a^2 + b^2}

. For

3 + 4i

, it is

\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

\frac{4}{25} - \frac{3}{25}i

\frac{4}{5} - \frac{3}{5}i

\frac{3}{5} + \frac{4}{5}i

\frac{3}{25} + \frac{4}{25}i

Correct Answer: A

Solution:

The multiplicative inverse of a complex number

z = a + ib

is given by

z^{-1} = \frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i

. For

z = 4 + 3i

a = 4

and

b = 3

, so

a^2 + b^2 = 16 + 9 = 25

. Thus,

z^{-1} = \frac{4}{25} - \frac{3}{25}i

Correct Answer: B

Solution:

The equation

z^2 + 4z + 5 = 0

can be solved using the quadratic formula

z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

. Here,

a = 1

b = 4

c = 5

. The discriminant

b^2 - 4ac = 16 - 20 = -4

. Thus,

z = \frac{-4 \pm i\sqrt{4}}{2} = -2 \pm i

. The modulus

|z| = \sqrt{(-2)^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}

. However, the question asks for the closest integer modulus, which is approximately 2.

\frac{1}{\sqrt{13}}

\frac{\sqrt{13}}{5}

\frac{5}{\sqrt{13}}

\sqrt{13}

Correct Answer: C

Solution:

The modulus of a complex number

z = a + ib

|z| = \sqrt{a^2 + b^2}

. The modulus of

z = 1 + i

\sqrt{1^2 + 1^2} = \sqrt{2}

, and the modulus of

w = 2 - 3i

\sqrt{2^2 + (-3)^2} = \sqrt{13}

. The modulus of the quotient

\frac{z}{w}

\frac{|z|}{|w|} = \frac{\sqrt{2}}{\sqrt{13}} = \frac{\sqrt{13}}{5}

. Therefore, the correct answer is

\frac{\sqrt{13}}{5}

3 + 0i

0 + 4i

-5 + 0i

2 - 2i

Correct Answer: B

Solution:

A complex number lies on the imaginary axis if its real part is zero. The complex number

0 + 4i

has a real part of zero, so it lies on the imaginary axis.

\frac{2}{13} - \frac{3}{13}i

\frac{2}{13} + \frac{3}{13}i

\frac{3}{13} - \frac{2}{13}i

\frac{3}{13} + \frac{2}{13}i

Correct Answer: B

Solution:

The multiplicative inverse of a complex number

z = a + ib

is given by

\frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i

. For

z = 2 + 3i

, the multiplicative inverse is

\frac{2}{2^2 + 3^2} + \frac{3}{2^2 + 3^2}i = \frac{2}{13} + \frac{3}{13}i

-1

Correct Answer: A

Solution:

The equation

z^2 + 1 = 0

implies

z^2 = -1

. If

z = x + iy

, then

z^2 = (x + iy)^2 = x^2 - y^2 + 2xyi

. For

z^2 = -1

, we must have

x^2 - y^2 = -1

and

2xy = 0

. The second equation implies either

x = 0

y = 0

. If

x = 0

, then

-y^2 = -1

, giving

y^2 = 1

. If

y = 0

, then

x^2 = -1

, which is not possible for real

x

. Hence,

x^2 + y^2 = 0 + 1 = 1

\tan^{-1}\left(\frac{5}{11}\right)

\tan^{-1}\left(\frac{11}{5}\right)

\tan^{-1}\left(\frac{5}{13}\right)

\tan^{-1}\left(\frac{13}{5}\right)

Correct Answer: A

Solution:

The product

z_1 \cdot z_2 = (4 + 3i)(2 - i) = 8 - 4i + 6i - 3i^2 = 11 + 2i

. The argument of a complex number

a + ib

is given by

\tan^{-1}\left(\frac{b}{a}\right)

. Therefore, the argument of

11 + 2i

\tan^{-1}\left(\frac{2}{11}\right)

. Hence, the correct answer is

\tan^{-1}\left(\frac{2}{11}\right)

Correct Answer: B

Solution:

The modulus of a product is the product of the moduli:

|z_1 \cdot z_2| = |z_1| \cdot |z_2|

. Here,

|z_1| = \sqrt{4^2 + 3^2} = 5

and

|z_2| = \sqrt{2^2 + (-1)^2} = \sqrt{5}

. Thus,

|z_1 \cdot z_2| = 5 \cdot \sqrt{5} = 5\sqrt{5} \approx 20

Point on the x-axis

Point on the y-axis

Point in the first quadrant

Point in the fourth quadrant

Correct Answer: B

Solution:

The complex number

z = 0 + 5i

has no real part and is purely imaginary, so it lies on the y-axis in the Argand plane.

-4

Correct Answer: C

Solution:

Using the binomial theorem,

(1 + i)^4 = (1 + i)(1 + i)(1 + i)(1 + i) = 2i \cdot 2i = 4i^2 = 4(-1) = -4

4 + 2i

2 + 6i

4 + 6i

2 + 2i

Correct Answer: A

Solution:

The sum of two complex numbers

z_1 = a + ib

and

z_2 = c + id

is given by

z_1 + z_2 = (a + c) + i(b + d)

. Here,

z_1 = 3 + 4i

and

z_2 = 1 - 2i

. Therefore,

z_1 + z_2 = (3 + 1) + i(4 - 2) = 4 + 2i

True or False

Correct Answer: False

Solution:

The plane having a complex number assigned to each of its points is called the Argand plane.

Correct Answer: True

Solution:

In the Argand plane, the complex number

z = x + iy

is represented by the point

(x, y)

, and its conjugate

\overline{z} = x - iy

is represented by the point

(x, -y)

Correct Answer: True

Solution:

In the Argand plane, the complex number

z = x + iy

corresponds to the point

(x, y)

Correct Answer: True

Solution:

In the Argand plane, complex numbers of the form

a + i0

lie on the x-axis, as they have no imaginary component.

Correct Answer: False

Solution:

The product of two complex numbers

z_1 = a + ib

and

z_2 = c + id

z_1 z_2 = (ac - bd) + i(ad + bc)

, which is generally a complex number unless

ad + bc = 0

Correct Answer: False

Solution:

In a complex number

a + ib

a

is the real part and

b

is the imaginary part.

Correct Answer: True

Solution:

The multiplicative identity for complex numbers is

1 + i0

, which is equivalent to the real number 1. It satisfies

z \cdot 1 = z

for any complex number

z

Correct Answer: False

Solution:

The equation

x^2 + 1 = 0

has no real solution because the square of a real number is always non-negative, and

x^2 = -1

is not possible in the real number system.

Correct Answer: False

Solution:

The equation

x^2 + 1 = 0

has no real solution because the square of a real number is always non-negative, and thus cannot equal

-1

Correct Answer: True

Solution:

The multiplicative identity for complex numbers is indeed

1 + i0

, which is equivalent to the real number 1.

Correct Answer: True

Solution:

Albert Girard accepted the concept of square roots of negative numbers, which was a step towards understanding complex numbers and their role in polynomial equations.

Correct Answer: True

Solution:

In the Argand plane, the point

(x, -y)

is indeed the mirror image of the point

(x, y)

across the real axis, reflecting the concept of conjugation in complex numbers.

Correct Answer: True

Solution:

The Argand plane is a geometric representation of complex numbers where each point corresponds to a complex number.

Correct Answer: True

Solution:

Euler was the first to introduce the symbol 'i' for

\sqrt{-1}

Correct Answer: True

Solution:

W.R. Hamilton regarded the complex number

a + ib

as an ordered pair of real numbers

(a, b)

Correct Answer: True

Solution:

Albert Girard accepted the square root of negative numbers around 1625, which allowed for finding as many roots as the degree of the polynomial equation.

Correct Answer: True

Solution:

The modulus of a complex number

z = a + ib

is indeed given by

\sqrt{a^2 + b^2}

, which represents the distance from the origin to the point

(a, b)

in the Argand plane.

Correct Answer: False

Solution:

The product of two complex numbers is generally another complex number, not necessarily a real number.

Correct Answer: True

Solution:

The commutative law for multiplication of complex numbers states that the order of multiplication does not affect the result, i.e.,

z_1 z_2 = z_2 z_1

for any complex numbers

z_1

and

z_2

Correct Answer: True

Solution:

The equation

x^2 + 1 = 0

implies

x^2 = -1

, which has no solution in the real number system since the square of a real number cannot be negative.

Correct Answer: True

Solution:

A complex number

z = a + ib

is represented as the ordered pair

(a, b)

in the Argand plane, where

a

is the real part and

b

is the imaginary part.

Correct Answer: True

Solution:

The equation

x^2 + 1 = 0

implies

x^2 = -1

, which has no solution in the real number system as the square of any real number is always non-negative.

Correct Answer: True

Solution:

The Argand plane is a two-dimensional plane where each point corresponds to a complex number, with the x-axis representing the real part and the y-axis representing the imaginary part.

Correct Answer: True

Solution:

The modulus of a complex number

x + iy

is given by

\sqrt{x^2 + y^2}

, which represents the distance from the origin to the point

(x, y)

in the Argand plane.

Correct Answer: True

Solution:

The conjugate of a complex number

z = a + ib

is defined as

z = a - ib

. This operation reflects the point across the real axis in the Argand plane.

Correct Answer: False

Solution:

The product of two complex numbers is generally another complex number, not necessarily a real number.

Correct Answer: False

Solution:

In the Argand plane, the x-axis is called the real axis, and the y-axis is called the imaginary axis.

Correct Answer: True

Solution:

In the real number system, the square root of a negative number is not defined. This limitation led to the development of complex numbers, where

i

is defined as the square root of

-1

Correct Answer: False

Solution:

In the Argand plane, the x-axis is called the real axis, and the y-axis is called the imaginary axis.

Correct Answer: True

Solution:

W.R. Hamilton, around 1830, regarded the complex number

a + ib

as an ordered pair of real numbers

(a, b)

, providing a purely mathematical definition.

Correct Answer: True

Solution:

The multiplication of two complex numbers follows the distributive property, resulting in

z_1 z_2 = (ac - bd) + i(ad + bc)

. This is a standard result in complex number multiplication.

Correct Answer: True

Solution:

Euler introduced the symbol

i

to represent the square root of

-1

, which is a fundamental concept in complex numbers.

Correct Answer: True

Solution:

W.R. Hamilton regarded the complex number

a + ib

as an ordered pair of real numbers

(a, b)

, giving it a purely mathematical definition.

Correct Answer: True

Solution:

The plane where each point corresponds to a complex number is known as the Argand plane.

Correct Answer: True

Solution:

The modulus of a complex number

z = x + iy

is indeed

\sqrt{x^2 + y^2}

, representing the distance from the origin to the point

(x, y)

in the Argand plane.

Correct Answer: True

Solution:

W.R. Hamilton, around 1830, defined the complex number

a + ib

as an ordered pair of real numbers

(a, b)

, providing a mathematical definition for complex numbers.

Correct Answer: True

Solution:

The modulus of

z = x + iy

\sqrt{x^2 + y^2}

, which represents the distance from the origin

(0, 0)

to the point

(x, y)

in the Argand plane.

Correct Answer: True

Solution:

The product of two complex numbers

z_1 = a + ib

and

z_2 = c + id

is calculated as

(ac - bd) + i(ad + bc)

, which is consistent with the definition of complex number multiplication.

Correct Answer: True

Solution:

Albert Girard accepted the square root of negative numbers around 1625, which allowed for finding as many roots as the degree of a polynomial equation.

Correct Answer: True

Solution:

The multiplicative inverse of a non-zero complex number

z = a + ib

is calculated as

\frac{a}{a^2 + b^2} - i\frac{b}{a^2 + b^2}

, which when multiplied by

z

gives the multiplicative identity

1 + i0

Correct Answer: False

Solution:

In the Argand plane, the x-axis is called the real axis, and the y-axis is called the imaginary axis.

Correct Answer: True

Solution:

In the Argand plane, the modulus of the complex number

z = x + iy

is given by

\sqrt{x^2 + y^2}

, which represents the distance from the origin to the point

(x, y)

Correct Answer: True

Solution:

The conjugate of a complex number

z = a + ib

is indeed

z = a - ib

, which reflects the point across the real axis in the Argand plane.

Correct Answer: True

Solution:

The modulus of

z = a + ib

is defined as

\sqrt{a^2 + b^2}

Correct Answer: True

Solution:

Euler introduced the symbol

i

to represent the square root of

-1

, which is a fundamental concept in complex numbers.

Correct Answer: True

Solution:

The multiplicative inverse of a non-zero complex number

z = a + ib

is indeed given by

\frac{a}{a^2 + b^2} - i\frac{b}{a^2 + b^2}

Correct Answer: True

Solution:

The modulus of a complex number

z = x + iy

is indeed calculated as

|z| = \sqrt{x^2 + y^2}

, representing the distance from the origin to the point

(x, y)

in the Argand plane.

Correct Answer: False

Solution:

In a complex number

a + ib

a

is the real part and

b

is the imaginary part.

Complex Numbers and Quadratic Equations

AI Learning Assistant

Summary

Chapter 4: Complex Numbers and Quadratic Equations

Summary

Learning Objectives

Detailed Notes

Chapter 4: Complex Numbers and Quadratic Equations

4.1 Introduction

4.2 Complex Numbers

4.3 Algebra of Complex Numbers

4.3.1 Addition of Complex Numbers

4.3.2 Difference of Complex Numbers

4.5 Argand Plane and Polar Representation

Important Diagrams

Fig 4.1: Cartesian Coordinate System

Fig 4.2: Modulus of Complex Number

Fig 4.3: Representation of Complex Number and its Conjugate

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Exam Tips

Practice & Assessment

Multiple Choice Questions

Q1.Which of the following is the multiplicative inverse of the complex number 1+i1 + i1+i?

Correct Answer: A

Q2.If z=1+iz = 1 + iz=1+i, what is the value of z2z^2z2?

Correct Answer: A

Q3.What is the least positive integral value of mmm for which (1+i)m=1(1+i)^m = 1(1+i)m=1?

Correct Answer: B

Q4.What is the result of multiplying the complex numbers (3+i5)(3 + i5)(3+i5) and (2+i6)(2 + i6)(2+i6)?

Correct Answer: A

Q5.If z=a+ibz = a + ibz=a+ib is a complex number such that its conjugate z‾=a−ib\overline{z} = a - ibz=a−ib satisfies z‾⋅z=25\overline{z} \cdot z = 25z⋅z=25, what is the value of a2+b2a^2 + b^2a2+b2?

Correct Answer: A

Q6.If z=x+iyz = x + iyz=x+iy is a complex number such that z2+4z+5=0z^2 + 4z + 5 = 0z2+4z+5=0, what is the real part of zzz?

Correct Answer: A

Q7.What is the modulus of the complex number z=7+24iz = 7 + 24iz=7+24i?

Correct Answer: A

Q8.In the Argand plane, if the point PPP represents the complex number 3+4i3 + 4i3+4i, what is the distance of PPP from the origin?

Correct Answer: C

Q9.What is the result of multiplying the complex numbers 2+3i2 + 3i2+3i and 4−i4 - i4−i?

Correct Answer: A

Q10.What is the multiplicative inverse of the complex number 2−3i2 - 3i2−3i?

Correct Answer: A

Q11.If z=x+iyz = x + iyz=x+iy is a complex number such that z3=8z^3 = 8z3=8, what is the value of xxx when zzz is expressed in polar form as z=r(cos⁡θ+isin⁡θ)z = r(\cos \theta + i\sin \theta)z=r(cosθ+isinθ)?

Correct Answer: A

Q12.If z=4−3iz = 4 - 3iz=4−3i, what is the multiplicative inverse of zzz?

Correct Answer: B

Q13.Consider the complex numbers z1=3+4iz_1 = 3 + 4iz1​=3+4i and z2=1−2iz_2 = 1 - 2iz2​=1−2i. Calculate the product z1⋅z2z_1 \cdot z_2z1​⋅z2​ and express it in the form a+iba + iba+ib.

Correct Answer: A

Q14.If a complex number z=x+iyz = x + iyz=x+iy has a modulus of 5, which of the following equations must be true?

Correct Answer: A

Q15.Let z=3+4iz = 3 + 4iz=3+4i and w=1−2iw = 1 - 2iw=1−2i. Find the product z⋅wz \cdot wz⋅w and express it in the form a+iba + iba+ib.

Correct Answer: B

Q16.If the complex number z=3+4iz = 3 + 4iz=3+4i is represented as a point in the Argand plane, what is the distance from the origin to this point?

Correct Answer: C

Q17.If z1=2+iz_1 = 2 + iz1​=2+i and z2=1−iz_2 = 1 - iz2​=1−i, find the product z1⋅z2z_1 \cdot z_2z1​⋅z2​.

Correct Answer: A

Q18.What is the modulus of the complex number 5−12i5 - 12i5−12i?

Correct Answer: A

Q19.What is the conjugate of the complex number z=7−5iz = 7 - 5iz=7−5i?

Correct Answer: A

Q20.Which of the following is the correct representation of the complex number 3+4i3 + 4i3+4i in the Argand plane?

Correct Answer: A

Q21.If z=3+4iz = 3 + 4iz=3+4i, what is the conjugate of the complex number zzz?

Correct Answer: A

Q22.Consider the complex numbers z1=3+4iz_1 = 3 + 4iz1​=3+4i and z2=1−2iz_2 = 1 - 2iz2​=1−2i. If z=z1⋅z2z = z_1 \cdot z_2z=z1​⋅z2​, what is the imaginary part of zzz?

Correct Answer: B

Q23.If z=1+iz = 1 + iz=1+i and w=2−3iw = 2 - 3iw=2−3i, find the real part of the complex number z+wz + wz+w.

Correct Answer: A

Q24.In the Argand plane, if the point PPP represents the complex number 5−12i5 - 12i5−12i, what is the distance of PPP from the origin?

Correct Answer: A

Q25.Express (5−3i)3(5 - 3i)^3(5−3i)3 in the form a+iba + iba+ib.

Correct Answer: A

Q26.Which of the following is the correct expression for the conjugate of the complex number z=4−3iz = 4 - 3iz=4−3i?

Correct Answer: A

Q27.Calculate the product of the complex numbers z1=3+2iz_1 = 3 + 2iz1​=3+2i and z2=1+4iz_2 = 1 + 4iz2​=1+4i.

Correct Answer: A

Q28.If z=a+ibz = a + ibz=a+ib is a complex number such that z2=−1z^2 = -1z2=−1, what is the value of a2+b2a^2 + b^2a2+b2?

Correct Answer: B