Our ncert solutions for Miscellaneous Exercise on Chapter - 5 Class 11 maths 5. Complex Numbers and Quadratic Equations - ncert solutions - Toppers Study is the best material for English Medium students cbse board and other state boards students.

# Miscellaneous Exercise on Chapter - 5 Class 11 maths 5. Complex Numbers and Quadratic Equations - ncert solutions - Toppers Study

Topper Study classes prepares ncert solutions on practical base problems and comes out with the best result that helps the students and teachers as well as tutors and so many ecademic coaching classes that they need in practical life. Our ncert solutions for Miscellaneous Exercise on Chapter - 5 Class 11 maths 5. Complex Numbers and Quadratic Equations - ncert solutions - Toppers Study is the best material for English Medium students cbse board and other state boards students.

## Miscellaneous Exercise on Chapter - 5 Class 11 maths 5. Complex Numbers and Quadratic Equations - ncert solutions - Toppers Study

CBSE board students who preparing for **class 11 ncert solutions maths and Mathematics** solved exercise **chapter 5. Complex Numbers and Quadratic Equations** available and this helps in upcoming exams
2024-2025.

### You can Find Mathematics solution Class 11 Chapter 5. Complex Numbers and Quadratic Equations

- All Chapter review quick revision notes for chapter 5. Complex Numbers and Quadratic Equations Class 11
- NCERT Solutions And Textual questions Answers Class 11 Mathematics
- Extra NCERT Book questions Answers Class 11 Mathematics
- Importatnt key points with additional Assignment and questions bank solved.

NCERT Solutions do not only help you to cover your syllabus but also will give to textual support in exams 2024-2025 to complete **exercise -5 maths class 11 chapter 5** in english medium. So revise and practice these all cbse study materials like class 11 maths chapter -5 in english ncert book. Also ensure to repractice all syllabus within time or before board exams for ncert class 11 maths ex -5 in english.

See all solutions for class 11 maths chapter 5 exercise 5 in english medium solved questions with answers.

### Miscellaneous Exercise on Chapter - 5 class 11 Mathematics Chapter 5. Complex Numbers and Quadratic Equations

Sure! The following topics will be covered in this article

- Miscellaneous Exercise On Chapter - 5 Class 11 Maths 5. Complex Numbers And Quadratic Equations - Ncert Solutions - Toppers Study
- Class 11 Ncert Solutions
- Solution Chapter 5. Complex Numbers And Quadratic Equations Class 11
- Solutions Class 11
- Chapter 5. Complex Numbers And Quadratic Equations Miscellaneous Exercise On Chapter - 5 Class 11

**Notice**: Undefined offset: 5 in

**/home/atpeduca/public_html/toppersstudy.com/view-home.php**on line

**123**

## 5. Complex Numbers and Quadratic Equations

### | Miscellaneous Exercise on Chapter - 5 |

## Miscellaneous Exercise on Chapter - 5 Class 11 maths 5. Complex Numbers and Quadratic Equations - ncert solutions - Toppers Study

**Miscellaneous Exercise on Chapter 5**

**Q2.** **For any two complex numbers z _{1} and z_{2}, prove that **

**Re (z _{1}z_{2}) = Re z_{1} Re z_{2} – Im z_{1} Im z_{2}**

**Solution: **

*Let z _{1} = a + ib, z_{2} = c + id *

*Re z _{1} = a, Re z_{2} = c, Im z_{1} = b, Im z_{2} = d ….. (1) *

*z _{1}z_{2} = (a + ib) (c + id)*

* = ac + iad + ibc + bd i ^{2}*

* = ac + iad + ibc + bd (-1)*

* = ac + iad + ibc **-** bd *

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

*Comparing real and imaginary part we obtain,*

*Re(z _{1}z_{2}) = ac *

*-*

*bd, Im(z*

_{1}z_{2}) = ad + bc*Now we take real part *

⇒ *Re(z _{1}z_{2}) = ac *

*-*

*bd*

⇒ *Re(z _{1}z_{2}) = Re z_{1} Re z_{2} – Im z_{1} Im z_{2 } [using (1) ] *

**Hence, proved **

Multiplying numerator and denominator by 28 + 10i

On multiplying numerator and denominator by (2 – *i*), we get

On comparing real and imaginary part we obtain

**Q14. ****Find the real numbers x and y if (x – iy) (3 + 5i) is the conjugate of –6 – 24i.**

**Solution: **

Let z = (x – iy) (3 + 5i)

= 3x + 5xi - 3yi - 5yi^{2}

= 3x + 5xi - 3yi + 5y

= 3x + 5y + 5xi - 3yi

= (3x + 5y) + (5x - 3y)i

Comparing both sides and equating real and imaginary parts, we get

3x + 5y = – 6 …… (1)

5x - 3y = 24 …… (2)

Multiplying equation (i) by 3 and equation (ii) by 5 and then adding them,

* (x + iy)*^{3}* = u + iv*

⇒* x ^{3 }+ 3 . x^{2 }. iy + 3 . x . (iy)*

^{2}*+ (iy)*

^{3}=*u + iv*

⇒* x ^{3 }+ 3x^{2 }y i + 3 x y^{2} i*

^{2}*+ y*

^{3}i^{3}=*u + iv*

⇒* x ^{3 }+ 3x^{2 }y i - 3 x y^{2} *

*- i y*

^{3}=*u + iv*

⇒* x ^{3 }*

*-*

*3 x y*

^{2}+ 3x^{2 }y i*- iy*

^{3}=*u + iv*

⇒*( x ^{3 }*

*-*

*3 x y*

^{2}) +*i*

*(3x*

^{2 }y*-*

*y*

^{3}) =*u + iv*

On equating both sides real and imaginary parts, we get;

u = *x ^{3 }*

*-*

*3 x y*

^{2}, …… (1)*v = 3x ^{2 }y *

*-*

*y*

^{3}, ……(2)

**Q18. ****Find the number of non-zero integral solutions of the equation |1 ****-** *i***| ^{x} = 2^{x}**

^{}

Thus, 0 is the only integral solution of the given equation. Therefore, the number of nonzero integral solutions of the given equation is 0.

**Q19. ****If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that**

**(a ^{2} + b^{2}) (c^{2} + d^{2}) (e^{2} + f ^{2}) (g^{2} + h^{2}) = A^{2} + B^{2}**

**Solution : **

(a + ib) (c + id) (e + if) (g + ih) = A + iB ……….. (1) given

Replacing i by (-i) we have

(a - ib) (c - id) (e - if) (g - ih) = A - iB ……….. (2)

Multiplying (1) and (2)

(a + ib) (a - ib) (c + id) (c - id) (e + if) (e - if) (g + ih) (g - ih) = (A + iB) (A - iB)

(*a*^{2} + *b*^{2}) (*c*^{2} + *d*^{2}) (*e*^{2} + *f *^{2}) (*g*^{2} + *h*^{2}) = A^{2} + B^{2 } [ ∵ (x + iy) (x - iy) = x^{2} + y^{2}]

Hence, proved

Or m = 4k

Hence, least positive integer is 1.

Therefore, the least positive integral value of *m *is 4 × 1 = 4

##### Other Pages of this Chapter: 5. Complex Numbers and Quadratic Equations

### Select Your CBSE Classes

Important Study materials for classes 06, 07, 08,09,10, 11 and 12. Like CBSE Notes, Notes for Science, Notes for maths, Notes for Social Science, Notes for Accountancy, Notes for Economics, Notes for political Science, Noes for History, Notes For Bussiness Study, Physical Educations, Sample Papers, Test Papers, Mock Test Papers, Support Materials and Books.

*Mathematics Class - 11th*

NCERT Maths book for CBSE Students.

books

## Study Materials List:

##### Solutions ⇒ Class 11th ⇒ Mathematics

## Topper's Study

New Books