JEE Exam  >  JEE Notes  >  Mathematics (Maths) for JEE Main & Advanced  >  NCERT Solutions: Complex Numbers

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Exercise 4.1:

Q1: Express the given complex number in the form a  + ib:NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations
Ans:

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Q2: Express the given complex number in the form a + ib: i9+ i19
Ans:

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Q3: Express the given complex number in the form a  + ib: i–39
Ans:

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Q4: Express the given complex number in the form a  + ib: 3(7 +  i7) +  i(7 +  i7)
Ans:

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Q5: Express the given complex number in the form a +  ib: (1 – i) – (–1  i6)
Ans:

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Q6: Express the given complex number in the form a  + ib:NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations
Ans:

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Q7: Express the given complex number in the form a +  ib:NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations
Ans:

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Q8: Express the given complex number in the form a  + ib:  (1 – i)4
Ans:

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Q9: Express the given complex number in the form a  + ib:NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations
Ans:

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Q10: Express the given complex number in the form a +  ib:NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations
Ans:

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Q11: Find the multiplicative inverse of the complex number 4 – 3i
Ans:
Let z = 4 – 3i
Then,  NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations= 4 +3and NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations
Therefore, the multiplicative inverse of 4 – 3i is given by
 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Q12: Find the multiplicative inverse of the complex number NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations
Ans:
Let z = NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations
Therefore, the multiplicative inverse of NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations is given by

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Q13: Find the multiplicative inverse of the complex number –i
Ans: Let z = –i

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations
Therefore, the multiplicative inverse of –i is given by

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Q14: Express the following expression in the form of a  + ib.

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations
Ans:

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Exercise Miscellaneous

Question 1: Evaluate: NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Answer:

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 

Question 2: For any two complex numbers z1 and z2, prove that

Re (z1z2) = Re zRe z2 – Im z1 Im z2

Answer:

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 

Question 3: Reduce NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations to the standard form.

Answer:

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations [ On Multiplying numerator and denomunator by (14 + 5i)

 

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 

Question 4: If xiy = NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations prove that NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations.

Answer:

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations [ On Multiplying numerator and denomunator by (c + id)]

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 

Question 5: Convert the following in the polar form:

(i) NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations,

(ii) NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 

Answer:

 (i) Here, NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Let cos θ = –1 and r sin θ = 1

On squaring and adding, we obtain

r2 (cos2 θ + sin2 θ) = 1+ 1

r2 (cos2 θ + sin2 θ) = 2
r2 = 2     [cos2 θ + sin2 θ = 1]

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

z = r cos θ +  i r sin θ

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

This is the required polar form.

(ii) Here, NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Let cos θ = –1 and r sin θ = 1

On squaring and adding, we obtain

r2 (cos2 θ + sin2 θ) = 1+ 1
r2 (cos2 θ  +sin2 θ) = 2

r2 = 2                        [cos2 θ + sin2 θ = 1]

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

z = r cos θ  + i r sin θ

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

This is the required polar form.


Question 6: Solve the equation

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Answer:

The given quadratic equation is NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

This equation can also be written as NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

On comparing this equation with ax2 +  bx +  c = 0, we obtain

a = 9, b = –12, and c = 20

Therefore, the discriminant of the given equation is

D = b2 – 4ac = (–12)2 – 4 × 9 × 20 = 144 – 720 = –576

Therefore, the required solutions are

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 

Question 7: Solve the equation NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Answer:

The given quadratic equation is NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

This equation can also be written as NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

On comparing this equation with ax2 +  bx +  c = 0, we obtain

a = 2, b = –4, and c = 3

Therefore, the discriminant of the given equation is

D = b2 – 4ac = (–4)2 – 4 × 2 × 3 = 16 – 24 = –8

Therefore, the required solutions are

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 

Question 8: Solve the equation 27x2 – 10+ 1 = 0

Answer:

The given quadratic equation is 27x2 – 10x + 1 = 0

On comparing the given equation with ax2 +  bx +  c = 0, we obtain

a = 27, b = –10, and c = 1

Therefore, the discriminant of the given equation is

D = b2 – 4ac = (–10)2 – 4 × 27 × 1 = 100 – 108 = –8

Therefore, the required solutions are

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 

Question 9: Solve the equation 21x2 – 28+ 10 = 0

Answer:

The given quadratic equation is 21x2 – 28x + 10 = 0

On comparing the given equation with ax2 +  bx +  = 0, we obtain

a = 21, b = –28, and c = 10

Therefore, the discriminant of the given equation is

D = b2 – 4ac = (–28)2 – 4 × 21 × 10 = 784 – 840 = –56

Therefore, the required solutions are

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 

Question 9: If NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations  find NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations.

Answer:

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 

Question 10: If a +  ib = NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations, prove that a2 +  b2 = NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Answer:

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

On comparing real and imaginary parts, we obtain

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Hence, proved.


Question 10: Let  NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations. Find

(i) NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations,

(ii) NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 

Answer:

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

(i) NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

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

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

On comparing real parts, we obtain

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations 

(ii) NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

On comparing imaginary parts, we obtain

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Question 11: Find the modulus and argument of the complex number NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations.

Answer:

Let NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations, then

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

On squaring and adding, we obtain

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 

Therefore, the modulus and argument of the given complex number are  NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations respectively.


Question 12: Find the real numbers x and y if (xiy) (3+5i) is the conjugate of –6 – 24i.

Answer:

Let NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

It is given that, NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Equating real and imaginary parts, we obtain

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

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

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Putting the value of x in equation (i), we obtain

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Thus, the values of and y are 3 and –3 respectively.

 

Question 13: Find the modulus of NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations .

Answer:

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 

Question 14: If (x  + iy)3 = u  + iv, then show that NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations.

Answer:

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

On equating real and imaginary parts, we obtain

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Hence, proved.

 

Question 15: If α and β are different complex numbers with NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations = 1, then find NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations.

Answer:

Let α = a  + ib and β = x +  iy

It is given that, NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

  NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations  

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 

Question 16: Find the number of non-zero integral solutions of the equation NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations.

Answer:

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Thus, 0 is the only integral solution of the given equation. Therefore, the number of non-zero integral solutions of the given equation is 0.

 

Question 17: If (a  + ib) (c +  id) (e +  if) (g  + ih) = A + iB, then show that

(a2  + b2) (c2+   d2) (e2 +  f2) (g2 +  h2) = A2  +B2.

 

Answer:

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

On squaring both sides, we obtain

(a2  + b2) (c2 +  d2) (e2 +  f2) (g2  + h2) = A2 + B2

Hence, proved.

 

Question 18: If NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations, then find the least positive integral value of m.

Answer:

 NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

Therefore, the least positive integer is 1.

Thus, the least positive integral value of m is 4 (= 4 × 1).

The document NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
All you need of JEE at this link: JEE
209 videos|443 docs|143 tests

Up next

FAQs on NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

1. What are complex numbers?
Ans. Complex numbers are numbers that can be expressed in the form a + bi, where "a" and "b" are real numbers, and "i" is the imaginary unit (√-1).
2. How do you add complex numbers?
Ans. To add complex numbers, simply add the real parts together and the imaginary parts together. For example, (3 + 2i) + (1 + 4i) = 4 + 6i.
3. How do you multiply complex numbers?
Ans. To multiply complex numbers, use the distributive property and remember that i^2 = -1. For example, (3 + 2i)(1 + 4i) = 3 + 12i + 2i + 8i^2 = 3 + 14i - 8 = -5 + 14i.
4. How do you find the conjugate of a complex number?
Ans. The conjugate of a complex number a + bi is a - bi. It is found by changing the sign of the imaginary part. For example, the conjugate of 3 + 2i is 3 - 2i.
5. How do you divide complex numbers?
Ans. To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. Simplify the expression and then separate the real and imaginary parts to get the quotient.
209 videos|443 docs|143 tests
Download as PDF

Up next

Explore Courses for JEE exam
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Related Searches

mock tests for examination

,

Free

,

MCQs

,

video lectures

,

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

,

Sample Paper

,

past year papers

,

study material

,

ppt

,

Previous Year Questions with Solutions

,

pdf

,

Important questions

,

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

,

Exam

,

Semester Notes

,

Objective type Questions

,

Summary

,

Viva Questions

,

Extra Questions

,

shortcuts and tricks

,

NCERT Solutions Class 11 Maths Chapter 4 - Complex Numbers and Quadratic Equations

,

practice quizzes

;