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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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**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:
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.
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.
Euler introduced symbol i for root(-1) first time
W,R. Hamilton
Z1Z2=(ac-bd)+i(ad+bc)
Yes, the statement is true. Example: 1 can be written as 1+0i
The argument of (1+i)/(1-i) is pi/2
5 questions are solved in the miscellaneous examples of complex numbers and quadratic equations.
Twenty questions of miscellaneous exercise chapter 5 Class 11 are solved in the Class 11 Maths chapter 5 miscellaneous exercise solutions
(1+i)/(1-i)=i
So the ordered pair is (0,1)
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