Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations – Exercise 4.2 NCERT Solutions

Introduction

Exercise 4.2 focuses on basic operations with complex numbers. You will learn how to add, subtract, multiply, and divide complex numbers, as well as how to simplify expressions using the properties of the imaginary unit $i$. This exercise strengthens your ability to handle algebraic manipulation of complex numbers.

Key Concepts

1. Addition/Subtraction:

$$(a+ib)\pm (c+id)=(a\pm c)+i(b\pm d)$$

2. Multiplication:

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

3. Division:

$$\frac{a+ib}{c+id}=\frac{(a+ib)(c-id)}{c^2+d^2}$$

4. Conjugate:

$$\bar{z}=a-ib$$

5. Modulus:

$$\mathrm{\mid }z\mathrm{\mid }=\sqrt{a^2+b^2}$$

NCERT Questions with Solutions (10)

Q1. Simplify $(3+2i)+(4+5i)$.

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

Q2. Simplify $(6+3i)-(2+7i)$.

$$=(6-2)+i(3-7)=4-4i$$

Q3. Multiply $(2+3i)(4+5i)$.

$$=(2\cdot 4-3\cdot 5)+i(2\cdot 5+3\cdot 4)=(8-15)+i(10+12)=-7+22i$$

Q4. Multiply $(1+i)(1-i)$.

$$=1^2-i^2=1-(-1)=2$$

Q5. Find conjugate of $z=5+7i$.

$$\bar{z}=5-7i$$

Q6. Find modulus of $z=3+4i$.

$$\mathrm{\mid }z\mathrm{\mid }=\sqrt{3^2+4^2}=\sqrt{9+16}=5$$

Q7. Divide $\frac{3+2i}{1-i}$.

$$=\frac{(3+2i)(1+i)}{1^2+(-1{)}^2}=\frac{3+3i+2i+2i^2}{2}=\frac{3+5i-2}{2}=\frac{1+5i}{2}=\frac{1}{2}+\frac{5}{2}i$$

Q8. Divide $\frac{4+3i}{2+i}$.

$$=\frac{(4+3i)(2-i)}{2^2+1^2}=\frac{8-4i+6i-3i^2}{5}=\frac{8+2i+3}{5}=\frac{11+2i}{5}=\frac{11}{5}+\frac{2}{5}i$$

Q9. Simplify $(2+5i)(2-5i)$.

$$=2^2-(5i{)}^2=4-(-25)=29$$

Q10. Simplify $(1+2i{)}^2$.

$$=1^2+2\cdot 1\cdot 2i+(2i{)}^2=1+4i-4=-3+4i$$

FAQs (10)

1. How do you add complex numbers? Add real parts and imaginary parts separately.

2. How do you subtract complex numbers? Subtract real parts and imaginary parts separately.

3. How do you multiply complex numbers? Use distributive law and $i^2=-1$.

4. How do you divide complex numbers? Multiply numerator and denominator by conjugate of denominator.

5. What is conjugate of complex number? $\bar{z}=a-ib$.

6. What is modulus of complex number? $\mathrm{\mid }z\mathrm{\mid }=\sqrt{a^2+b^2}$.

7. What is result of $(a+ib)(a-ib)$? $a^2+b^2$.

8. Why use conjugate in division? To remove imaginary part from denominator.

9. Is modulus always positive? Yes, modulus is non‑negative.

10. Why is this exercise important? It builds skills in algebraic manipulation of complex numbers.

Conclusion

Exercise 4.2 has 10 solved questions and 10 FAQs that strengthen your understanding of operations on complex numbers. This builds the foundation for solving quadratic equations with complex roots in Class 11 Maths.

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New Syllabus-Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations – Exercise 4.2 NCERT Solutions