Understanding Complex Numbers is a vital step for Class 11 students pursuing mathematics. Chapter 13 in RD Sharma's textbook introduces students to this essential algebraic concept with clear examples and graded exercises. Whether you're a CBSE board student or preparing for competitive exams like JEE, mastering this chapter with RD Sharma Solutions can help you build a solid foundation.
This guide will walk you through the key concepts, exercise breakdowns, solved examples, and why RD Sharma is one of the most reliable resources for Complex Numbers.
Introduction to Class 11 RD Sharma Solutions Chapter 13 - Complex Numbers
Complex Numbers mark a significant expansion in the mathematical journey of Class 11 students. Introduced in Chapter 13 of RD Sharma Mathematics, this concept helps students go beyond the real number system to understand numbers that involve the square root of negative values.
This chapter is a gateway to advanced algebra, and its applications span fields like engineering, quantum physics, and signal processing. For CBSE Class 11 students, it lays a solid foundation for both board exams and competitive exams such as JEE, CUET, and Olympiads.
What is a Complex Number?
In mathematics, a complex number is a number that has two parts: a real part and an imaginary part. These numbers are an extension of the real number system and are essential in higher-level algebra and engineering mathematics.
z = x + iy
- x is the real part of the complex number.
- y is the imaginary part.
- i is the imaginary unit, defined as: i² = –1
For example, if we have a complex number z = 3 + 4i, then:
- Real part: 3
- Imaginary part: 4
Complex numbers are used in various fields like physics, electrical engineering, and signal processing. They help solve equations that have no real solutions, such as x² + 1 = 0.
RD Sharma Solutions Class 11 Maths Chapter 13 - Complex Numbers Question with Answers
1. Express √−49 as a complex number.
√−49 = √49 × √−1 = 7i
2. Write the real and imaginary parts of z = 4 − 7i.
Real part = 4, Imaginary part = −7
3. Find the sum of z₁ = 2 + 3i and z₂ = 4 − 5i.
z₁ + z₂ = 6 − 2i
4. Multiply z₁ = 1 + 2i and z₂ = 3 + 4i.
(1 + 2i)(3 + 4i) = −5 + 10i
5. Find the modulus of z = 6 − 8i.
|z| = √(6² + (−8)²) = 10
6. Find the conjugate of z = 3 + 4i.
ẑ = 3 − 4i
7. Find z × ẑ for z = 2 + 5i.
(2 + 5i)(2 − 5i) = 29
8. Square of z = 1 + i?
(1 + i)² = 2i
9. Express 1/(1 + i) in a + bi form.
Result = ½ − ½i
10. Simplify i¹⁰.
i¹⁰ = −1
11. If z = 3 + 4i, find |z|².
|z|² = 25
12. Polar form of z = 1 + i?
√2(cos π/4 + i sin π/4)
13. Show that z + ẑ = 2Re(z).
z = x + iy → z + ẑ = 2x = 2Re(z)
14. If z = 5cosθ + i5sinθ, find |z|.
|z| = 5
15. Find (2 − i)³.
(2 − i)³ = 2 − 11i
16. Find the multiplicative inverse of z = 3 − 4i.
Inverse = (3 + 4i)/25
17. Express (2 + 3i)/(1 − 2i) in standard form.
Answer = −4/5 + 7i/5
18. If z = 7 + 24i, verify |z| = 25.
|z| = √(49 + 576) = 25 ✔
19. Prove |z₁·z₂| = |z₁||z₂|.
Let z₁ = 2 + 3i, z₂ = 1 − 4i → |z₁·z₂| = √221 = |z₁||z₂|
20. If |z| = 1, show ẑ = 1/z.
zẑ = 1 ⇒ ẑ = 1/z
21. Argument of z = −1 + √3i?
arg(z) = 2π/3 (second quadrant)
22. Find i³⁵.
i³⁵ = −i
23. Write z = 4(cosθ + i sinθ) in rectangular form.
z = 4cosθ + i4sinθ
24. Solve z² = −36.
z = ±6i
25. If z = 2 + 3i, find 1/|z|².
|z|² = 13 ⇒ 1/|z|² = 1/13
Frequently Asked Questions
A complex number is a number that has two parts: a real part and an imaginary part. It is written in the form:
z=x+iyz = x + iyz=x+iy
Here, xxx is the real part, yyy is the imaginary part, and iii is the imaginary unit where i2=−1i^2 = -1i2=−1. This concept helps in solving equations that cannot be solved using only real numbers.
Complex numbers are represented on the Argand Plane, where:
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The x-axis represents the real part
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The y-axis represents the imaginary part
Each complex number becomes a point or vector in this 2D space, similar to plotting coordinates.
Complex numbers are used in:
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Electrical Engineering (AC circuits, impedance)
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Quantum Mechanics
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Signal Processing
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Control Systems
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Fluid Dynamics
They help in representing oscillations, wave patterns, and other scientific phenomena where real numbers alone are not enough.
Complex Numbers are covered in Chapter 13 of the RD Sharma Class 11 Maths book. The chapter includes theoretical explanations, solved examples, and multiple exercises to build conceptual clarity and problem-solving skills.
Yes. For JEE, complex numbers are a high-weightage topic in algebra. Understanding the basics in Class 11 is essential for mastering advanced concepts like De Moivre’s Theorem, Euler’s form, and applications in coordinate geometry and calculus. For NEET, it helps improve logical reasoning and mathematical confidence.