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**NCERT Solutions for Class 11 Maths Chapter 5 – Complex Numbers and Quadratic Equations Miscellaneous Exercise-** NCERT solutions for Class 11 Maths Chapter 5 miscellaneous exercise summarises questions from all topics of the NCERT chapter complex numbers and quadratic equations. Class 11 Maths Chapter 5 miscellaneous exercise solutions present questions from basics of complex numbers, algebra of complex numbers, modulus and argument of complex numbers and conjugate of the complex numbers. A few more topics covered in the miscellaneous exercise chapter 5 Class 11 are the polar representation of complex numbers and solutions of quadratic equations with negative discriminant. Along with the NCERT syllabus Class 11 Maths chapter 5 miscellaneous solutions, the following exercises and their solutions are also available. Students can utilise these for their preparation.

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- NCERT Solutions for Class 11 Maths Chapter 5 - Complex Numbers and Quadratic Equations Miscellaneous Exercise- Download Free PDF
- NCERT Solutions for Class 11 Maths Chapter 5 – Complex Numbers and Quadratic Equations Miscellaneous Exercise
- Question:1 Evaluate .
- More About NCERT Solutions for Class 11 Maths Chapter 5 Miscellaneous Exercise
- Topics Covered in Miscellaneous Exercise Class 11 Chapter 5
- Benefits of NCERT Solutions for Class 11 Maths Chapter 5 Miscellaneous Exercise
- Key Features of Class 11 Maths Ch 5 Miscellaneous Exercise Solutions
- NCERT Solutions of Class 11 Subject Wise
- Subject Wise NCERT Exampler Solutions

****As per the CBSE Syllabus 2023-24, this class 11 chapter 5 maths miscellaneous solutions is now designated as Chapter 4**.

**Access Complex Numbers And Quadratic Equations Class 11 Chapter 5****- ****Miscellaneous Exercise **

Answer:

The given problem is

Now, we will reduce it into

Now,

Therefore, answer is

Question:2 For any two complex numbers and , prove that

Answer:

Let two complex numbers are

Now,

Hence proved

Question:3 Reduce to the standard form.

Answer:

Given problem is

Now, we will reduce it into

Now, multiply numerator an denominator by

Therefore, answer is

Question:4 If , prove that

Answer:

the given problem is

Now, multiply the numerator and denominator by

Now, square both the sides

On comparing the real and imaginary part, we obtain

Now,

Hence proved

Question:5(i) Convert the following in the polar form:

Answer:

Let

Now, multiply the numerator and denominator by

Now,

let

On squaring both and then add

Now,

Since the value of is negative and is positive this is the case in II quadrant

Therefore,

Therefore, the required polar form is

Question:5(ii) Convert the following in the polar form:

Answer:

Let

Now, multiply the numerator and denominator by

Now,

let

On squaring both and then add

Now,

Since the value of is negative and is positive this is the case in II quadrant

Therefore,

Therefore, the required polar form is

Question:6 Solve each of the equation:

Answer:

Given equation is

Now, we know that the roots of the quadratic equation are given by the formula

In this case the value of

Therefore,

Therefore, the solutions of requires equation are

Question:7 Solve each of the equation:

Answer:

Given equation is

Now, we know that the roots of the quadratic equation are given by the formula

In this case the value of

Therefore,

Therefore, the solutions of requires equation are

Question:8 Solve each of the equation: .

Answer:

Given equation is

Now, we know that the roots of the quadratic equation are given by the formula

In this case the value of

Therefore,

Therefore, the solutions of requires equation are

Question:9 Solve each of the equation:

Answer:

Given equation is

Now, we know that the roots of the quadratic equation are given by the formula

In this case the value of

Therefore,

Therefore, the solutions of requires equation are

Question:10 If , find .

Answer:

It is given that

Then,

Now, multiply the numerator and denominator by

Now,

Therefore, the value of

is

Question:11 If , prove that .

Answer:

It is given that

Now, we will reduce it into

On comparing real and imaginary part. we will get

Now,

Hence proved

Question:12(ii) Let Find

Answer:

It is given that

Therefore,

NOw,

Now,

Therefore,

Therefore, the answer is 0

Question:13 Find the modulus and argument of the complex number .

Answer:

Let

Now, multiply the numerator and denominator by

Therefore,

Square and add both the sides

Therefore, the modulus is

Now,

Since the value of is negative and the value of is positive and we know that it is the case in II quadrant

Therefore,

Argument

Therefore, Argument and modulus are respectively

Question:14 Find the real numbers x andy if is the conjugate of .

Answer:

Let

Therefore,

Now, it is given that

Compare (i) and (ii) we will get

On comparing real and imaginary part. we will get

On solving these we will get

Therefore, the value of x and y are 3 and -3 respectively

Question:15 Find the modulus of .

Answer:

Let

Now, we will reduce it into

Now,

square and add both the sides. we will get,

Therefore, modulus of

is 2

Question:16 If , then show that

Answer:

it is given that

Now, expand the Left-hand side

On comparing real and imaginary part. we will get,

Now,

Hence proved

Question:17 If and are different complex numbers with , then find .

Answer:

Let

and

It is given that

and

Now,

Therefore, value of is 1

Question:18 Find the number of non-zero integral solutions of the equation .

Answer:

Given problem is

Now,

x = 0 is the only possible solution to the given problem

Therefore, there are 0 number of non-zero integral solutions of the equation

Question:19 If then show that

Answer:

It is given that

Now, take mod on both sides

Square both the sides. we will get

Hence proved

Question:20 If then find the least positive integral value of .

Answer:

Let

Now, multiply both numerator and denominator by

We will get,

We know that

Therefore, the least positive integral value of is 4

All the 20 questions of miscellaneous exercise chapter 5 Class 11 are important and can use these NCERT solutions for Class 11 Maths chapter 5 miscellaneous exercise as a tool for getting good scores in the Class 11 exams. The concepts covered in the Class 11 Maths chapter 5 miscellaneous solutions not only help in preparation for the Class 11 exam. These are useful for higher studies in the field of mathematics and engineering also. The applications of complex numbers are used in various fields of engineering. For example, electrical circuit analysis uses complex numbers

**Also Read| **Complex Numbers And Quadratic Equations Class 11th Notes

Miscellaneous Exercise Topics in NCERT Solutions for Class 11 Maths Chapter 5 - Complex Numbers and Quadratic Equations:

- Complex Numbers
- Algebraic Operations with Complex Numbers
- Modulus and Conjugate of Complex Numbers
- Argand Plane and Polar Representation
- Quadratic Equations

NCERT syllabus Class 11 Maths chapter 5 miscellaneous exercise solutions can be accessed easily and downloaded for future reference

Solving miscellaneous exercise chapter 5 Class 11 helps students to make sure that whether they have familiarised them with the concepts discussed in the chapter.

Class 11 Maths chapter 5 miscellaneous solutions also helps in the preparation for competitive exams like JEE Main.

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Download EBook**Comprehensive Coverage:**The miscellaneous exercise class 11 chapter 5 solutions comprehensively address all 20 questions from the Miscellaneous Exercise in Chapter 5, ensuring a thorough understanding of complex numbers and quadratic equations.**Conceptual Clarity:**The class 11 chapter 5 maths miscellaneous solutions provide clear explanations, step-by-step procedures, and valuable insights to help students grasp challenging mathematical concepts effectively.**Relevance to Exams:**Designed in alignment with the Class 11 syllabus, the Class 11 maths miscellaneous exercise chapter 5 solutions serve as an invaluable resource for exam preparation, offering a strategic approach to solving miscellaneous exercise questions.**Preparation for Higher Studies:**The covered concepts serve as a robust foundation for advanced studies in mathematics and engineering, with a focus on real-world applications of complex numbers.**Competitive Exam Readiness:**The class 11 maths ch 5 miscellaneous exercise solutions aid in preparing for competitive exams, such as JEE Main, by reinforcing key mathematical principles and enhancing problem-solving skills.**User-Friendly Access:**The class 11 chapter 5 miscellaneous exercise solutions are easily accessible, providing students with the convenience of free PDF downloads for future reference, study, and revision.**Verification of Understanding:**Students can use the class 11 chapter 5 maths miscellaneous solutions solutions to verify their understanding of Chapter 5 concepts, ensuring a strong foundation in complex numbers and quadratic equations.

- NCERT Solutions for Class 11 Maths
- NCERT Solutions for Class 11 Physics
- NCERT Solutions for Class 11 Chemistry
- NCERT Solutions for Class 11 Biology

1. Who introduced the symbol i for root(-1)?

Euler introduced symbol i for root(-1) first time

2. Who represented a+ib as ordered pair (a, b)?

W,R. Hamilton

3. Find Z1.Z2 if Z1=a+ib and Z2=c+id

Z1Z2=(ac-bd)+i(ad+bc)

4. Is the statement “all real numbers are complex numbers” true?

Yes, the statement is true. Example: 1 can be written as 1+0i

5. What is the argument of (1+i)/(1-i)

The argument of (1+i)/(1-i) is pi/2

6. How many questions are given in the miscellaneous examples of Class 11 NCERT Maths chapter 5?

5 questions are solved in the miscellaneous examples of complex numbers and quadratic equations.

7. How many questions are given in NCERT solutions for Class 11 Maths chapter 5 miscellaneous exercise?

Twenty questions of miscellaneous exercise chapter 5 Class 11 are solved in the Class 11 Maths chapter 5 miscellaneous exercise solutions

8. Give the ordered pair of (1+i)/(1-i)?

(1+i)/(1-i)=i

So the ordered pair is (0,1)

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