Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations – Exercise 4.3 NCERT Solutions

Introduction

Exercise 4.3 focuses on quadratic equations with complex roots. You will learn how to solve quadratic equations using the quadratic formula, identify when roots are real or complex, and express complex roots in standard form. This exercise builds the foundation for advanced algebra and calculus.

Key Concepts

1. Quadratic Equation:

$$a{x}^2+bx+c=0,a\mathrm{\ne }0$$

2. Quadratic Formula:

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

3. Discriminant:

$$D=b^2-4ac$$

• If $D>0$: real and distinct roots

• If $D=0$: real and equal roots

• If $D<0$: complex conjugate roots

NCERT Questions with Solutions (10)

Q1. Solve $x^2+4x+5=0$.

$$D=4^2-4(1)(5)=16-20=-4$$

$$x=\frac{-4\pm \sqrt{-4}}{2}=\frac{-4\pm 2i}{2}=-2\pm i$$

Q2. Solve $x^2+2x+2=0$.

$$D=2^2-4(1)(2)=4-8=-4$$

$$x=\frac{-2\pm \sqrt{-4}}{2}=\frac{-2\pm 2i}{2}=-1\pm i$$

Q3. Solve $x^2+x+1=0$.

$$D=1^2-4(1)(1)=1-4=-3$$

$$x=\frac{-1\pm \sqrt{-3}}{2}=\frac{-1\pm i\sqrt{3}}{2}$$

Q4. Solve $x^2+6x+13=0$.

$$D=36-52=-16$$

$$x=\frac{-6\pm \sqrt{-16}}{2}=\frac{-6\pm 4i}{2}=-3\pm 2i$$

Q5. Solve $x^2+2x+5=0$.

$$D=4-20=-16$$

$$x=\frac{-2\pm \sqrt{-16}}{2}=\frac{-2\pm 4i}{2}=-1\pm 2i$$

Q6. Solve $x^2+3x+10=0$.

$$D=9-40=-31$$

$$x=\frac{-3\pm \sqrt{-31}}{2}=\frac{-3\pm i\sqrt{31}}{2}$$

Q7. Solve $x^2+5x+9=0$.

$$D=25-36=-11$$

$$x=\frac{-5\pm \sqrt{-11}}{2}=\frac{-5\pm i\sqrt{11}}{2}$$

Q8. Solve $x^2+4x+8=0$.

$$D=16-32=-16$$

$$x=\frac{-4\pm \sqrt{-16}}{2}=\frac{-4\pm 4i}{2}=-2\pm 2i$$

Q9. Solve $x^2+7x+20=0$.

$$D=49-80=-31$$

$$x=\frac{-7\pm \sqrt{-31}}{2}=\frac{-7\pm i\sqrt{31}}{2}$$

Q10. Solve $x^2+2x+10=0$.

$$D=4-40=-36$$

$$x=\frac{-2\pm \sqrt{-36}}{2}=\frac{-2\pm 6i}{2}=-1\pm 3i$$

FAQs (10)

1. What is discriminant? $D=b^2-4ac$.

2. When are roots real? If $D\ge 0$.

3. When are roots complex? If $D<0$.

4. What are complex conjugate roots? Roots of form $a\pm ib$.

5. What is quadratic formula? $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.

6. Can quadratic equation have imaginary roots? Yes, if discriminant is negative.

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

8. What is conjugate of complex root? If root is $a+ib$, conjugate is $a-ib$.

9. Why are complex roots important? They extend solutions beyond real numbers.

10. Why is this exercise important? It builds skills in solving quadratic equations with complex roots.

Conclusion

Exercise 4.3 has 10 solved questions and 10 FAQs that strengthen your understanding of quadratic equations with complex roots. This completes the Complex Numbers and Quadratic Equations chapter in Class 11 Maths.

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New Syllabus-Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations – Exercise 4.3 NCERT Solutions