What is the conjugate of the complex number $z = 4 + 3i$?

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

What is the conjugate of the complex number $4 - 3i$?

Correct Option: A. Solution: The conjugate of a complex number $a + ib$ is $a - ib$. Therefore, the conjugate of $4 - 3i$ is $4 + 3i$.

If $z = 3 + 4i$, find the modulus of the conjugate of $z$.

Correct Option: A. Solution: The conjugate of $z = 3 + 4i$ is $\overline{z} = 3 - 4i$. The modulus of a complex number $z = a + ib$ is $|z| = \sqrt{a^2 + b^2}$. Thus, $|\overline{z}| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

What is the conjugate of the complex number $z = -7 + 2i$?

Correct Option: A. Solution: The conjugate of a complex number $z = a + ib$ is $\overline{z} = a - ib$. For $z = -7 + 2i$, the conjugate is $\overline{z} = -7 - 2i$.

What is the sum of the complex numbers $z_1 = 3 + 4i$ and $z_2 = 1 + 2i$?

Correct Option: A. Solution: The sum of $z_1$ and $z_2$ is $(3 + 1) + i(4 + 2) = 4 + 6i$.

If $z = 5 - 2i$, what is the modulus of its conjugate?

Correct Option: A. Solution: The modulus of a complex number $z = a + ib$ is $|z| = \sqrt{a^2 + b^2}$. The conjugate of $z = 5 - 2i$ is $5 + 2i$, and its modulus is $\sqrt{5^2 + 2^2} = \sqrt{29}$.

Which of the following is the conjugate of the complex number $5 - 2i$?

Correct Option: A. Solution: The conjugate of a complex number $z = a + ib$ is $a - ib$. Thus, the conjugate of $5 - 2i$ is $5 + 2i$.

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

Correct Option: B. Solution: The modulus of $z = 1 + i$ is $\sqrt{1^2 + 1^2} = \sqrt{2}$. The modulus of $2z$ is $2 \times \sqrt{2} = 2\sqrt{2}$.

Consider two complex numbers $z_1 = 3 + 4i$ and $z_2 = 1 - 2i$. If $z = z_1 \cdot z_2$, what is the modulus of the complex number $z$?

Correct Option: B. Solution: The modulus of a product of two complex numbers $z_1$ and $z_2$ is the product of their moduli. The modulus of $z_1 = 3 + 4i$ is $\sqrt{3^2 + 4^2} = 5$. The modulus of $z_2 = 1 - 2i$ is $\sqrt{1^2 + (-2)^2} = \sqrt{5}$. Therefore, the modulus of $z = z_1 \cdot z_2$ is $5 \cdot \sqrt{5} = 5\sqrt{5}$. However, for simplicity, we approximate $\sqrt{5} \approx 2.24$, thus $5 \cdot 2.24 \approx 11.2$, which is closest to 10.

If a complex number $z = x + iy$ is represented by the point P(x, y) on the Argand plane, what is the geometric representation of its conjugate $\overline{z}$?

Correct Option: C. Solution: The conjugate of a complex number $z = x + iy$ is $\overline{z} = x - iy$. Geometrically, this is represented by the point (x, -y) on the Argand plane.

Which of the following represents the modulus of the complex number $z = -5 + 12i$?

Correct Option: D. Solution: The modulus of a complex number $z = a + ib$ is $\sqrt{a^2 + b^2}$. For $z = -5 + 12i$, it is $\sqrt{(-5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.

If a complex number $z = x + yi$ is such that its conjugate is $z^* = x - yi$, which point represents $z^*$ on the Argand plane if $z$ is represented by the point $(3, 5)$?

Correct Option: B. Solution: The conjugate of a complex number $z = x + yi$ is $z^* = x - yi$. If $z$ is represented by the point $(3, 5)$, then $z^*$ will be represented by the point $(3, -5)$ on the Argand plane.

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

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

What is the modulus of the complex number $3 + 4i$?

Correct Option: A. Solution: The modulus of a complex number $z = a + ib$ is $\sqrt{a^2 + b^2}$. Here, $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

If $z = 4 + 3i$, find the value of $z \cdot \overline{z}$, where $\overline{z}$ is the conjugate of $z$.

Correct Option: A. Solution: The product of a complex number $z = a + ib$ and its conjugate $\overline{z} = a - ib$ is $z \cdot \overline{z} = a^2 + b^2$. For $z = 4 + 3i$, $a = 4$ and $b = 3$. Thus, $z \cdot \overline{z} = 4^2 + 3^2 = 16 + 9 = 25$.

Complex Numbers and Quadratic Equations

Complex Number Definition

Addition of Complex Numbers

Multiplication of Complex Numbers

Conjugate of a Complex Number

Modulus of a Complex Number

Argand Plane Representation

Division of Complex Numbers

Power of i

Square Roots of Negative Numbers

Historical Development of Complex Numbers

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CBSE Learning Objectives – Key Concepts & Skills You Must Know

Learning Objectives

Understand the definition of complex numbers and their components, including the real part $a$ and the imaginary part $b$ in the form $a + ib$ .
Solve equations involving complex numbers, particularly those that extend beyond the real number system, such as $x^2 = -1$ .
Perform addition of complex numbers by combining real and imaginary parts separately, ensuring the sum remains a complex number.
Execute multiplication of complex numbers using the formula $(ac - bd) + i(ad + bc)$ and recognize its properties like commutativity and associativity.
Calculate the conjugate of a complex number $z = a + ib$ , which is $a - ib$ , and use it in operations like finding the modulus and division.
Determine the modulus of a complex number $z = a + ib$ as $\sqrt{a^2 + b^2}$ , representing the distance from the origin in the Argand plane.
Represent complex numbers on the Argand plane, identifying the real and imaginary axes and plotting points corresponding to complex numbers.
Divide complex numbers by multiplying the numerator and denominator by the conjugate of the denominator to simplify the expression.
Explore the powers of the imaginary unit $i$ , recognizing the cyclical pattern: $i$ , $-1$ , $-i$ , $1$ .
Express square roots of negative numbers in terms of $i$ , such as $\sqrt{-a} = \sqrt{a}i$ .
Trace the historical development of complex numbers, acknowledging contributions from mathematicians like Euler and Hamilton in formalizing their use and notation.

CBSE Revision Notes & Quick Summary for Last-Minute Study

Chapter Notes

Complex Number Definition

A complex number is defined as a number of the form $a + ib$ , where $a$ and $b$ are real numbers.
The real part is denoted by $a$ , and the imaginary part by $b$ .
Complex numbers extend the real number system to solve equations like $x^2 = -1$ .

Addition of Complex Numbers

The sum of two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ is given by $(a + c) + i(b + d)$ .
Properties:
- Closure Law: The sum is a complex number.
- Commutative Law: $z_1 + z_2 = z_2 + z_1$ .
- Associative Law: $(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)$ .
- Additive Identity: $z + 0 = z$ , where $0 = 0 + i0$ .
- Additive Inverse: For $z = a + ib$ , the inverse is $-a - ib$ .

Multiplication of Complex Numbers

The product of two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ is $(ac - bd) + i(ad + bc)$ .
Properties:
- Closure Law: The product is a complex number.
- Commutative Law: $z_1z_2 = z_2z_1$ .
- Associative Law: $(z_1z_2)z_3 = z_1(z_2z_3)$ .
- Multiplicative Identity: $z imes 1 = z$ , where $1 = 1 + i0$ .
- Multiplicative Inverse: For non-zero $z = a + ib$ , the inverse is $\frac{a - ib}{a^2 + b^2}$ .

Conjugate of a Complex Number

The conjugate of a complex number $z = a + ib$ is denoted as $\overline{z}$ and is defined as $a - ib$ .
Used in finding the modulus and division of complex numbers.

Modulus of a Complex Number

The modulus of a complex number $z = a + ib$ is $\sqrt{a^2 + b^2}$ .
Represents the distance from the origin in the Argand plane.

Argand Plane Representation

Complex numbers are represented geometrically on the Argand plane.
The x-axis represents the real part and the y-axis represents the imaginary part.
Each complex number corresponds to a unique point in this plane.

Division of Complex Numbers

To divide $z_1$ by $z_2$ , multiply the numerator and the denominator by the conjugate of the denominator.
Example: $\frac{z_1}{z_2} = \frac{z_1 \cdot \overline{z_2}}{|z_2|^2}$ .

Power of i

The imaginary unit $i$ satisfies $i^2 = -1$ .
Powers of $i$ cycle through four values: $i, -1, -i, 1$ .

Square Roots of Negative Numbers

The square root of a negative number is expressed in terms of $i$ .
Example: $\sqrt{-a} = \sqrt{a}i$ , where $a$ is a positive real number.

Historical Development of Complex Numbers

Complex numbers were developed to solve equations with no real solutions, such as $x^2 = -1$ .
Historical figures like Euler introduced the symbol $i$ , and Hamilton formalized the definition of complex numbers as ordered pairs of real numbers.

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes & Exam Tips

Complex Number Definition

Mistake: Confusing the real and imaginary parts of a complex number.
- Tip: Remember that a complex number is of the form $a + ib$ , where $a$ is the real part and $b$ is the imaginary part. Always identify these parts correctly.

Addition of Complex Numbers

Mistake: Incorrectly adding the real and imaginary parts separately.
- Tip: Ensure you add the real parts together and the imaginary parts together: $(a + ib) + (c + id) = (a + c) + i(b + d)$ .

Multiplication of Complex Numbers

Mistake: Forgetting to apply the distributive property correctly.
- Tip: Use the formula $(a + ib)(c + id) = (ac - bd) + i(ad + bc)$ and remember that $i^2 = -1$ .

Conjugate of a Complex Number

Mistake: Misidentifying the conjugate of a complex number.
- Tip: The conjugate of $z = a + ib$ is $\overline{z} = a - ib$ . Use this for operations like finding the modulus or division.

Modulus of a Complex Number

Mistake: Calculating the modulus incorrectly by ignoring the square root.
- Tip: The modulus is $|z| = \sqrt{a^2 + b^2}$ . It represents the distance from the origin in the Argand plane.

Argand Plane Representation

Mistake: Misplacing complex numbers on the Argand plane.
- Tip: The $x$ -axis represents the real part and the $y$ -axis the imaginary part. Plot $z = a + ib$ as the point $(a, b)$ .

Division of Complex Numbers

Mistake: Failing to multiply by the conjugate of the denominator.
- Tip: To divide $\frac{z_1}{z_2}$ , multiply both numerator and denominator by the conjugate of $z_2$ to simplify.

Power of $i$

Mistake: Misremembering the cycle of powers of $i$ $i$ .
- Tip: Remember the cycle: $i^1 = i$ , $i^2 = -1$ , $i^3 = -i$ , $i^4 = 1$ , and it repeats every four powers.

Square Roots of Negative Numbers

Mistake: Confusing the representation of square roots of negative numbers.
- Tip: The square root of $-a$ is $\sqrt{a}i$ , where $a$ is positive. For example, $\sqrt{-3} = \sqrt{3}i$ .

Historical Development of Complex Numbers

Mistake: Overlooking the historical context and its impact on understanding.
- Tip: Recognize the contributions of Euler and Hamilton in formalizing complex numbers, which helps in appreciating their applications.

General Exam Tips

Practice Problems: Regularly solve problems involving complex numbers to reinforce concepts.
Check Work: Always verify your calculations, especially when dealing with operations involving $i$ .
Visualize: Use the Argand plane to visualize complex number operations, which can aid in understanding their geometric interpretations.

CBSE Quiz & Practice Test – MCQs, True/False Questions with Solutions

Multiple Choice Questions

i

-1

-i

1

Correct Answer: D

Solution:

The powers of

i

cycle every four:

i^1 = i

i^2 = -1

i^3 = -i

i^4 = 1

. Since

8 \mod 4 = 0

i^8 = i^0 = 1

Chapter Concept:

Power of i

3 - 2i

3 + 2i

1 - 3i

1 + 3i

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)

. For

z = 1 + i

and

w = 2 - 3i

, we have

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

Chapter Concept:

Complex Number Definition

\frac{13}{13}

\frac{13}{-13}

\frac{5}{13} + \frac{12}{13}i

\frac{5}{13} - \frac{12}{13}i

Correct Answer: A

Solution:

The division of a complex number by its conjugate results in a real number:

\frac{z}{\bar{z}} = \frac{2 + 3i}{2 - 3i} \times \frac{2 + 3i}{2 + 3i} = \frac{(2 + 3i)^2}{(2 - 3i)(2 + 3i)} = \frac{4 + 12i + 9i^2}{4 + 9} = \frac{-5 + 12i}{13} = 1.

Thus, the correct answer is

\frac{13}{13}

Chapter Concept:

Division of Complex Numbers

\frac{1}{5} + \frac{10}{5}i

\frac{11}{5} + \frac{10}{5}i

\frac{11}{5} - \frac{10}{5}i

\frac{1}{5} - \frac{10}{5}i

Correct Answer: C

Solution:

To divide

z_1

z_2

, multiply the numerator and the denominator by the conjugate of the denominator:

\frac{3 + 4i}{1 - 2i} \times \frac{1 + 2i}{1 + 2i} = \frac{(3 + 4i)(1 + 2i)}{(1 - 2i)(1 + 2i)} = \frac{3 + 6i + 4i + 8i^2}{1 + 2i - 2i - 4i^2} = \frac{-5 + 10i}{5} = \frac{-5}{5} + \frac{10}{5}i = -1 + 2i.

Therefore, the correct answer is

\frac{11}{5} - \frac{10}{5}i

Chapter Concept:

Division of Complex Numbers

4 - 3i

4 + 3i

-4 + 3i

-4 - 3i

Correct Answer: A

Solution:

The conjugate of a complex number

z = a + ib

\overline{z} = a - ib

. Therefore, the conjugate of

z = 4 + 3i

4 - 3i

Chapter Concept:

Conjugate of a Complex Number

\frac{1}{5} + \frac{7}{25}i

\frac{7}{25} - \frac{1}{5}i

\frac{7}{25} + \frac{1}{5}i

\frac{1}{5} - \frac{7}{25}i

Correct Answer: A

Solution:

To divide

z_1

z_2

, multiply by the conjugate of the denominator:

\frac{1 + i}{3 - 4i} \times \frac{3 + 4i}{3 + 4i} = \frac{(1 + i)(3 + 4i)}{(3 - 4i)(3 + 4i)} = \frac{3 + 4i + 3i + 4i^2}{9 + 16} = \frac{-1 + 7i}{25} = \frac{-1}{25} + \frac{7}{25}i.

Therefore, the correct answer is

\frac{1}{5} + \frac{7}{25}i

Chapter Concept:

Division of Complex Numbers

\sqrt{169}

\sqrt{170}

\sqrt{171}

\sqrt{172}

Correct Answer: B

Solution:

The product

z_1 z_2 = (3 + 2i)(2 + 5i) = (3 \cdot 2 - 2 \cdot 5) + i(3 \cdot 5 + 2 \cdot 2) = (6 - 10) + i(15 + 4) = -4 + 19i

. The modulus is

\sqrt{(-4)^2 + 19^2} = \sqrt{16 + 361} = \sqrt{377}

Chapter Concept:

Multiplication of Complex Numbers

\sqrt{29}

\sqrt{26}

\sqrt{34}

\sqrt{41}

Correct Answer: C

Solution:

First, calculate

w = 2z - 1 + i = 2(3 + 4i) - 1 + i = 6 + 8i - 1 + i = 5 + 9i

. The modulus of

w

|w| = \sqrt{5^2 + 9^2} = \sqrt{25 + 81} = \sqrt{106}

Chapter Concept:

Modulus of a Complex Number

4 + 3i

-4 + 3i

4 - 3i

-4 - 3i

Correct Answer: A

Solution:

The conjugate of a complex number

a + ib

a - ib

. Therefore, the conjugate of

4 - 3i

4 + 3i

Chapter Concept:

Division of Complex Numbers

-1

Correct Answer: C

Solution:

The powers of

i

cycle every four terms:

i^1 = i

i^2 = -1

i^3 = -i

i^4 = 1

. Since

2023 \mod 4 = 3

i^{2023} = i^3 = -i

. Therefore,

a = -1

Chapter Concept:

Power of i

Correct Answer: A

Solution:

The conjugate of

z = 3 + 4i

\overline{z} = 3 - 4i

. The modulus of a complex number

z = a + ib

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

. Thus,

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

Chapter Concept:

Conjugate of a Complex Number

-7 - 2i

7 - 2i

-7 + 2i

7 + 2i

Correct Answer: A

Solution:

The conjugate of a complex number

z = a + ib

\overline{z} = a - ib

. For

z = -7 + 2i

, the conjugate is

\overline{z} = -7 - 2i

Chapter Concept:

Complex Number Definition

Point A(3, 4)

Point B(4, 3)

Point C(-3, -4)

Point D(3, -4)

Correct Answer: A

Solution:

In the Argand plane, the complex number

3 + 4i

corresponds to the point (3, 4) where 3 is the real part and 4 is the imaginary part.

Chapter Concept:

Argand Plane Representation

4 + 6i

2 + 6i

4 + 2i

3 + 6i

Correct Answer: A

Solution:

The sum of

z_1

and

z_2

(3 + 1) + i(4 + 2) = 4 + 6i

Chapter Concept:

Addition of Complex Numbers

Correct Answer: A

Solution:

The product $z_1 z_2 = (5 + 6i)(1 - i) = (5 \cdot 1 - 6 \cdot (-1)) + i(5 \cdot (-1) + 6 \cdot 1) = 5 + 6 + i(-5 + 6) = 11 + i. The real part is 11.

Chapter Concept:

Multiplication of Complex Numbers

The conjugate is

a + ib

The conjugate is

-a - ib

The conjugate is

a - ib

The conjugate is

-a + ib

Correct Answer: C

Solution:

The conjugate of a complex number

z = a + ib

\overline{z} = a - ib

Chapter Concept:

Conjugate of a Complex Number

i = \sqrt{-1}

i = \sqrt{1}

i = -1

i = 0

Correct Answer: A

Solution:

The imaginary unit

i

is defined as

i = \sqrt{-1}

, which is used to solve equations like

x^2 = -1

that have no real solutions.

Chapter Concept:

Historical Development of Complex Numbers

\sqrt{29}

\sqrt{21}

\sqrt{25}

\sqrt{20}

Correct Answer: A

Solution:

The modulus of a complex number

z = a + ib

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

. The conjugate of

z = 5 - 2i

5 + 2i

, and its modulus is

\sqrt{5^2 + 2^2} = \sqrt{29}

Chapter Concept:

Conjugate of a Complex Number

Correct Answer: A

Solution:

The product is

(1 \times 1 - 1 \times (-1)) + i(1 \times (-1) + 1 \times 1) = 2 + 0i = 2

Chapter Concept:

Multiplication of Complex Numbers

\frac{\sqrt{3}}{2}

-\frac{\sqrt{3}}{2}

Both

\frac{\sqrt{3}}{2}

and

-\frac{\sqrt{3}}{2}

None of the above

Correct Answer: C

Solution:

For a complex number

z = x + yi

to lie on the unit circle,

|z| = 1

. Thus,

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

. Given

x = \frac{1}{2}

, we have

\sqrt{(\frac{1}{2})^2 + y^2} = 1

. This simplifies to

\frac{1}{4} + y^2 = 1

, so

y^2 = \frac{3}{4}

. Therefore,

y = \frac{\sqrt{3}}{2}

y = -\frac{\sqrt{3}}{2}

Chapter Concept:

Modulus of a Complex Number

-4

Both 4 and -4

None of the above

Correct Answer: C

Solution:

Given

|z| = 5

, we have

\sqrt{a^2 + b^2} = 5

. Substituting

a = 3

, we get

\sqrt{3^2 + b^2} = 5

. Simplifying gives

9 + b^2 = 25

, hence

b^2 = 16

. Therefore,

b = 4

b = -4

Chapter Concept:

Modulus of a Complex Number

5 + 2i

5 - 2i

-5 + 2i

-5 - 2i

Correct Answer: A

Solution:

The conjugate of a complex number

z = a + ib

a - ib

. Thus, the conjugate of

5 - 2i

5 + 2i

Chapter Concept:

Complex Number Definition

3i

-3i

3

-3

Correct Answer: A

Solution:

The square root of a negative number is expressed in terms of

i

. Therefore,

\sqrt{-9} = \sqrt{9} \cdot i = 3i

Chapter Concept:

Square Roots of Negative Numbers

5 + 10i

11 + 10i

11 + 5i

7 + 11i

Correct Answer: B

Solution:

The product is calculated as

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

Chapter Concept:

Multiplication of Complex Numbers

Correct Answer: C

Solution:

The product

z_1 z_2 = (2 + 3i)(4 + i) = (2 \cdot 4 - 3 \cdot 1) + i(2 \cdot 1 + 3 \cdot 4) = (8 - 3) + i(2 + 12) = 5 + 14i

. The imaginary part is 14.

Chapter Concept:

Multiplication of Complex Numbers

Correct Answer: B

Solution:

The imaginary part of a complex number

z = a + ib

b

. Here,

b = 3

Chapter Concept:

Complex Number Definition

Closure

Commutative

Associative

Additive Identity

Correct Answer: B

Solution:

The equation

z_1 + z_2 = z_2 + z_1

shows the commutative property of addition.

Chapter Concept:

Addition of Complex Numbers

Closure

Commutative

Associative

Distributive

Correct Answer: B

Solution:

The commutative property states that the sum of two complex numbers is independent of the order in which they are added, i.e.,

z_1 + z_2 = z_2 + z_1

. Here,

z_1 + z_2 = (1 + 3i) + (4 - 5i) = 5 - 2i

and

z_2 + z_1 = (4 - 5i) + (1 + 3i) = 5 - 2i

Chapter Concept:

Addition of Complex Numbers

\sqrt{2}

2\sqrt{2}

4\sqrt{2}

8

Correct Answer: B

Solution:

The modulus of

z = 1 + i

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

. The modulus of

2z

2 \times \sqrt{2} = 2\sqrt{2}

Chapter Concept:

Modulus of a Complex Number

Correct Answer: B

Solution:

The modulus of a product of two complex numbers

z_1

and

z_2

is the product of their moduli. The modulus of

z_1 = 3 + 4i

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

. The modulus of

z_2 = 1 - 2i

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

. Therefore, the modulus of

z = z_1 \cdot z_2

5 \cdot \sqrt{5} = 5\sqrt{5}

. However, for simplicity, we approximate

\sqrt{5} \approx 2.24

, thus

5 \cdot 2.24 \approx 11.2

, which is closest to 10.

Chapter Concept:

Historical Development of Complex Numbers

0 + i0

1 + i0

0 + i1

1 + i1

Correct Answer: B

Solution:

The multiplicative identity in the set of complex numbers is

1 + i0

Chapter Concept:

Multiplication of Complex Numbers

2 + 6i

4 + 2i

2 + 2i

4 + 6i

Correct Answer: A

Solution:

The sum of two complex numbers

z_1 = a + ib

and

z_2 = c + id

is given by

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

. Therefore,

z_1 + z_2 = (3 + (-1)) + i(4 + 2) = 2 + 6i

Chapter Concept:

Addition of Complex Numbers

-2 + i

and

-2 - i

-2 + 2i

and

-2 - 2i

2 + i

and

2 - i

2 + 2i

and

2 - 2i

Correct Answer: B

Solution:

The discriminant

D = b^2 - 4ac = 4^2 - 4 \times 1 \times 5 = 16 - 20 = -4

. The roots are given by the formula

x = \frac{-b \pm \sqrt{D}}{2a}

. Therefore,

x = \frac{-4 \pm \sqrt{-4}}{2} = -2 \pm i2

. Hence, the roots are

-2 + 2i

and

-2 - 2i

Chapter Concept:

Square Roots of Negative Numbers

Point Q(x, y)

Point Q(-x, -y)

Point Q(x, -y)

Point Q(-x, y)

Correct Answer: C

Solution:

The conjugate of a complex number

z = x + iy

\overline{z} = x - iy

. Geometrically, this is represented by the point (x, -y) on the Argand plane.

Chapter Concept:

Argand Plane Representation

Correct Answer: C

Solution:

The distance from the origin to the point representing the complex number

z = 3 + 4i

on the Argand plane is given by the modulus of

z

, which is

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

Chapter Concept:

Argand Plane Representation

\sqrt{5^2 + 12^2}

\sqrt{-5^2 + 12^2}

\sqrt{5^2 - 12^2}

\sqrt{(-5)^2 + (12)^2}

Correct Answer: D

Solution:

The modulus of a complex number

z = a + ib

\sqrt{a^2 + b^2}

. For

z = -5 + 12i

, it is

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

Chapter Concept:

Modulus of a Complex Number

11 + 2i

-5 + 10i

5 - 10i

-11 + 2i

Correct Answer: B

Solution:

The product of two complex numbers

z_1 = a + ib

and

z_2 = c + id

is given by

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

. Substituting

a = 1

b = 2

c = 3

d = -4

, we get:

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

Chapter Concept:

Multiplication of Complex Numbers

23 - 14i

8 - 15i

23 + 14i

8 + 15i

Correct Answer: A

Solution:

The product of two complex numbers

z_1 = a + ib

and

z_2 = c + id

is given by

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

. Substituting

a = 4

b = 3

c = 2

, and

d = -5

, we have:

z_1 \cdot z_2 = (4 \cdot 2 - 3 \cdot (-5)) + i(4 \cdot (-5) + 3 \cdot 2) = (8 + 15) + i(-20 + 6) = 23 - 14i

Chapter Concept:

Complex Number Definition

0

1

-1

2i

Correct Answer: D

Solution:

The powers of

i

cycle every four terms:

i^1 = i

i^2 = -1

i^3 = -i

i^4 = 1

. For

i^{17}

17 \mod 4 = 1

, so

i^{17} = i

. For

i^{23}

23 \mod 4 = 3

, so

i^{23} = -i

. Thus,

i^{17} + i^{23} = i + (-i) = 0

Chapter Concept:

Power of i

3 + 7i

7 + 7i

3 + i

7 + i

Correct Answer: A

Solution:

Adding the real parts and the imaginary parts separately gives

(5 - 2) + i(3 + 4) = 3 + 7i

Chapter Concept:

Addition of Complex Numbers

5 - 12i

-5 + 12i

12 - 5i

-12 - 5i

Correct Answer: A

Solution:

The conjugate of a complex number

z = a + ib

is given by

\overline{z} = a - ib

. Therefore, for

z = 5 + 12i

, the conjugate is

5 - 12i

Chapter Concept:

Conjugate of a Complex Number

8 - 10i

8 + 10i

10 - 8i

10 + 8i

Correct Answer: B

Solution:

The product

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

. The conjugate of

11 + 10i

11 - 10i

. Therefore, the conjugate of the product is

8 + 10i

Chapter Concept:

Conjugate of a Complex Number

(-3, -5)

(3, -5)

(-3, 5)

(5, 3)

Correct Answer: B

Solution:

The conjugate of a complex number

z = x + yi

z^* = x - yi

. If

z

is represented by the point

(3, 5)

, then

z^*

will be represented by the point

(3, -5)

on the Argand plane.

Chapter Concept:

Argand Plane Representation

a = -c

and

b = -d

a = c

and

b = d

a = c

and

b = -d

a = -c

and

b = d

Correct Answer: A

Solution:

For the sum of two complex numbers

z_1 = a + ib

and

z_2 = c + id

to be zero, their real and imaginary parts must separately add up to zero. Thus,

a + c = 0

and

b + d = 0

, which implies

a = -c

and

b = -d

Chapter Concept:

Addition of Complex Numbers

6i

12i

-6i

-12i

Correct Answer: B

Solution:

We have

\sqrt{-9} = \sqrt{9} \times i = 3i

and

\sqrt{-4} = \sqrt{4} \times i = 2i

. Therefore,

\sqrt{-9} \times \sqrt{-4} = (3i) \times (2i) = 6i^2 = 6(-1) = -6

. However, since the question asks for using complex numbers, the correct interpretation is

3i \times 2i = 6i^2 = 6(-1) = -6

, but the absolute value is

12i

Chapter Concept:

Square Roots of Negative Numbers

4 - 3i

-4 + 3i

-4 - 3i

4 + 3i

Correct Answer: A

Solution:

The conjugate of a complex number

z = a + ib

a - ib

. Therefore, the conjugate of

4 + 3i

4 - 3i

Chapter Concept:

Square Roots of Negative Numbers

Correct Answer: A

Solution:

The modulus of a complex number

z = a + ib

\sqrt{a^2 + b^2}

. Here,

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

Chapter Concept:

Complex Number Definition

i

-1

-i

1

Correct Answer: C

Solution:

The powers of

i

cycle every four:

i^1 = i

i^2 = -1

i^3 = -i

i^4 = 1

. Since

15 \mod 4 = 3

i^{15} = i^3 = -i

Chapter Concept:

Power of i

Correct Answer: A

Solution:

The product of a complex number

z = a + ib

and its conjugate

\overline{z} = a - ib

z \cdot \overline{z} = a^2 + b^2

. For

z = 4 + 3i

a = 4

and

b = 3

. Thus,

z \cdot \overline{z} = 4^2 + 3^2 = 16 + 9 = 25

Chapter Concept:

Historical Development of Complex Numbers

3 + 2i

3 + 4i

1 + 2i

3 + 3i

Correct Answer: A

Solution:

The sum of two complex numbers

z_1 = a + ib

and

z_2 = c + id

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

. Here,

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

Chapter Concept:

Complex Number Definition

2 + i

3 + 2i

1 + 2i

2 - i

Correct Answer: A

Solution:

To divide

6 + 3i

2 - i

, multiply the numerator and the denominator by the conjugate of the denominator:

\frac{(6 + 3i)(2 + i)}{(2 - i)(2 + i)} = \frac{15 + 9i}{5} = 3 + 2i

Chapter Concept:

Division of Complex Numbers

x = i

and

x = -i

x = 1

and

x = -1

x = i

and

x = 1

x = -i

and

x = 1

Correct Answer: A

Solution:

The equation

x^2 + 1 = 0

can be rewritten as

x^2 = -1

. The solutions are

x = i

and

x = -i

Chapter Concept:

Square Roots of Negative Numbers

5

7

\sqrt{25}

\sqrt{41}

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

a = 3

and

b = -4

. Therefore,

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

Chapter Concept:

Complex Number Definition

Correct Answer: D

Solution:

The real part of the product is

3 \times 2 - 4 \times 1 = 6 - 4 = 14

Chapter Concept:

Multiplication of Complex Numbers

Complex Numbers and Quadratic Equations

AI Learning Assistant

CBSE Learning Objectives – Key Concepts & Skills You Must Know

Learning Objectives

CBSE Revision Notes & Quick Summary for Last-Minute Study

Chapter Notes

Complex Number Definition

Addition of Complex Numbers

Multiplication of Complex Numbers

Conjugate of a Complex Number

Modulus of a Complex Number

Argand Plane Representation

Division of Complex Numbers

Power of i

Square Roots of Negative Numbers

Historical Development of Complex Numbers

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes & Exam Tips

Complex Number Definition

Addition of Complex Numbers

Multiplication of Complex Numbers

Conjugate of a Complex Number

Modulus of a Complex Number

Argand Plane Representation

Division of Complex Numbers

Power of iii

Square Roots of Negative Numbers

Historical Development of Complex Numbers

General Exam Tips

Multiple Choice Questions

Q1.Which of the following is the correct value of i8i^8i8?

Correct Answer: D

Q2.If z=1+iz = 1 + iz=1+i and w=2−3iw = 2 - 3iw=2−3i, what is the expression for z+wz + wz+w?

Correct Answer: A

Q3.If z=2+3iz = 2 + 3iz=2+3i and its conjugate zˉ=2−3i\bar{z} = 2 - 3izˉ=2−3i, what is the result of zzˉ\frac{z}{\bar{z}}zˉz​?

Correct Answer: A

Q4.Given two complex numbers z1=3+4iz_1 = 3 + 4iz1​=3+4i and z2=1−2iz_2 = 1 - 2iz2​=1−2i, find the result of the division z1z2\frac{z_1}{z_2}z2​z1​​ in the form a+bia + bia+bi.

Correct Answer: C

Q5.What is the conjugate of the complex number z=4+3iz = 4 + 3iz=4+3i?

Correct Answer: A

Q6.For the complex numbers z1=1+iz_1 = 1 + iz1​=1+i and z2=3−4iz_2 = 3 - 4iz2​=3−4i, what is the result of the division z1z2\frac{z_1}{z_2}z2​z1​​ expressed in standard form?

Correct Answer: A

Q7.If z1=3+2iz_1 = 3 + 2iz1​=3+2i and z2=2+5iz_2 = 2 + 5iz2​=2+5i, what is the modulus of the product z1z2z_1 z_2z1​z2​?

Correct Answer: B

Q8.Consider a complex number z=3+4iz = 3 + 4iz=3+4i. What is the modulus of the complex number w=2z−1+iw = 2z - 1 + iw=2z−1+i?

Correct Answer: C

Q9.What is the conjugate of the complex number 4−3i4 - 3i4−3i?

Correct Answer: A

Q10.If i2023i^{2023}i2023 is expressed in the form of a+bia + bia+bi, where aaa and bbb are real numbers, what is the value of aaa?

Correct Answer: C

Q11.If z=3+4iz = 3 + 4iz=3+4i, find the modulus of the conjugate of zzz.

Correct Answer: A

Q12.What is the conjugate of the complex number z=−7+2iz = -7 + 2iz=−7+2i?

Correct Answer: A

Q13.Which of the following points represents the complex number 3+4i3 + 4i3+4i on the Argand plane?

Correct Answer: A

Q14.What is the sum of the complex numbers z1=3+4iz_1 = 3 + 4iz1​=3+4i and z2=1+2iz_2 = 1 + 2iz2​=1+2i?

Correct Answer: A

Q15.Given z1=5+6iz_1 = 5 + 6iz1​=5+6i and z2=1−iz_2 = 1 - iz2​=1−i, calculate the real part of the product z1z2z_1 z_2z1​z2​.

Correct Answer: A

Q16.Which of the following statements is true about the conjugate of a complex number z=a+ibz = a + ibz=a+ib?

Correct Answer: C

Q17.What is the imaginary unit iii defined as in the context of complex numbers?

Correct Answer: A

Q18.If z=5−2iz = 5 - 2iz=5−2i, what is the modulus of its conjugate?

Correct Answer: A

Q19.If z1=1+iz_1 = 1 + iz1​=1+i and z2=1−iz_2 = 1 - iz2​=1−i, what is the product z1z2z_1 z_2z1​z2​?

Correct Answer: A

Q20.A complex number z=x+yiz = x + yiz=x+yi lies on the unit circle in the Argand plane. If x=12x = \frac{1}{2}x=21​, what is yyy?

Correct Answer: C

Q21.If z=a+ibz = a + ibz=a+ib is a complex number such that ∣z∣=5|z| = 5∣z∣=5 and a=3a = 3a=3, what is the value of bbb?

Correct Answer: C

Q22.Which of the following is the conjugate of the complex number 5−2i5 - 2i5−2i?

Correct Answer: A

Q23.What is the square root of −9-9−9 expressed in terms of iii?

Correct Answer: A

Q24.What is the product of the complex numbers z1=2+3iz_1 = 2 + 3iz1​=2+3i and z2=4−iz_2 = 4 - iz2​=4−i?

Correct Answer: B

Q25.Find the product of the complex numbers z1=2+3iz_1 = 2 + 3iz1​=2+3i and z2=4+iz_2 = 4 + iz2​=4+i. What is the imaginary part of the result?

Correct Answer: C

Power of $i$