Conjugate of Complex Number MCQ Quiz in मराठी - Objective Question with Answer for Conjugate of Complex Number - मोफत PDF डाउनलोड करा

Last updated on Apr 22, 2025

पाईये Conjugate of Complex Number उत्तरे आणि तपशीलवार उपायांसह एकाधिक निवड प्रश्न (MCQ क्विझ). हे मोफत डाउनलोड करा Conjugate of Complex Number एमसीक्यू क्विझ पीडीएफ आणि बँकिंग, एसएससी, रेल्वे, यूपीएससी, स्टेट पीएससी यासारख्या तुमच्या आगामी परीक्षांची तयारी करा.

Latest Conjugate of Complex Number MCQ Objective Questions

Top Conjugate of Complex Number MCQ Objective Questions

Conjugate of Complex Number Question 1:

If |z - 1 - i| = 1, then the locus of a point represented by the complex number 3(z - i) - 4 is ______.

  1. Circle with center (1, 0) and radius 3.
  2. Circle with centre (-1, 0) and radius 4
  3. Circle with centre (-1, 0) and radius 3
  4. Circle with centre (1, 0) and radius 4

Answer (Detailed Solution Below)

Option 3 : Circle with centre (-1, 0) and radius 3

Conjugate of Complex Number Question 1 Detailed Solution

Given:

|z - 1 - i| = 1

Let us solve the question graphically,

|z| = 1

F1 Neha Shraddha 16.07.2021 D1

for |z - 1 - i|, the centre is shifted to (1, 1)

F1 Neha Shraddha 16.07.2021 D2

3(z - i) - 4

For (z - i), the graph is shifted down and centre is now at (1, 0)

F1 Neha Shraddha 16.07.2021 D3

3(z - i) (the center is shifted to (3, 0)

F1 Neha Shraddha 16.07.2021 D4

3(z - i) - 4,

the circle is shifted left by 4 units, 

F1 Neha Shraddha 16.07.2021 D5

∴ we get a circle having a radius of 3 and centre at (-1, 0).

Conjugate of Complex Number Question 2:

Multiply 3 – 2i by its conjugate.

  1. 7
  2. 11
  3. 13
  4. 10

Answer (Detailed Solution Below)

Option 3 : 13

Conjugate of Complex Number Question 2 Detailed Solution

Concept:

Let z = x + iy be a complex number,

Where x is called the real part of the complex number or Re (z) and y is called the Imaginary part of the complex number or Im (z)

Conjugate of z = z̅ = x - iy  

Calculation:

Let z = 3 – 2i

Conjugate of z is 3 + 2i

(3 – 2i)(3 + 2i) = 32 – (2i)2

= 9 – 4i2

= 9 – 4(-1)            (∵ i2 = -1)

= 13 

∴ The required value is 13.

Conjugate of Complex Number Question 3:

The value of cos h2z - sin h2z is ___ .

  1. cos h(2z)
  2. 1
  3. sin h(2z)
  4. 0

Answer (Detailed Solution Below)

Option 2 : 1

Conjugate of Complex Number Question 3 Detailed Solution

Concept:

1. cosh z=ez+ez2      ----(1)

2. sinh z=ezez2      ----(2)

3. a2 - b2 = (a -b)(a + b)

Calculation:

cos h2z - sin h2z = (cos hz + sin hz)(cos hz - sin hz)

From equation (1) and (2)

⇒ (ez+ez2+ezez2)(ez+ez2ezez2) 

⇒ 2ez2×2ez2=ez×ez

ez+z=e0=1

Conjugate of Complex Number Question 4:

Let z = 5 + i  then zz̅ = ?

  1. 24
  2. 25
  3. 26
  4. 27

Answer (Detailed Solution Below)

Option 3 : 26

Conjugate of Complex Number Question 4 Detailed Solution

Concept:

Let z = x + iy be a complex number.

Modulus of z = |z|=x2+y2=Re(z)2+Im(z)2

Conjugate of z = z̅ = x - iy  

 

Calculation:

z = 5 + i 

Now, Conjugate of z = z̅ = 5 - i 

To Find: zz̅

zz̅ = (5 + i)(5 - i)

= 25 - i2

= 25 + 1       [∵ i2 = -1]

= 26

Hence, option 3 is correct 

Conjugate of Complex Number Question 5:

What is the conjugate of the given complex number?

z = 3i + 7

  1. z¯ = 3 + 7i
  2. z¯ = -3i + 7
  3. z¯ = 3i - 7
  4. z¯ = -3i - 7

Answer (Detailed Solution Below)

Option 2 : z¯ = -3i + 7

Conjugate of Complex Number Question 5 Detailed Solution

Concept:

If a complex number is a ± bi, then the complex conjugate will be a ∓  bi and vice-versa.

Where a & b are real numbers and i = Imaginary number =1

Calculation:

Given the complex number z = 3i +  7 

∴ The complex conjugate of this number = -3i + 7

Option 2 is correct

Conjugate of Complex Number Question 6:

Let p be the conjugate of the complex number q = x + iy . Find the value of pq. 

  1. x2
  2. x2 + y2
  3. x2 - y2
  4. y2

Answer (Detailed Solution Below)

Option 2 : x2 + y2

Conjugate of Complex Number Question 6 Detailed Solution

Concept:

Conjugate of a complex number:

For any complex number z = x + iy  the conjugate z̅ is given by  z̅ = x - iy

Calculation: 

q = x + iy

As p is conjugate to q 

∴ p = x - iy

Now, pq = (x + iy)(x - iy)

pq = x2 + ixy - ixy - i2y2

pq = x2 + y2

∴ pq = x2 + y2

Conjugate of Complex Number Question 7:

A real and positive value of a and b will satisfy the equation 2ab(ZZ¯) = a+ib, Z = (b + ia) if:

  1. 2a = b
  2. a = -b
  3. a = 2b
  4. a = b

Answer (Detailed Solution Below)

Option 4 : a = b

Conjugate of Complex Number Question 7 Detailed Solution

Concept:

A complex number (Z):  Complex number is the combination of a real number and an imaginary number. It is given by

Z = x + iy, where 'x' and 'y' are the real and imaginary part of Z and i = √-1 

Conjugate of a complex number: When the i of a complex number is replaced with - i, we get the conjugate of that complex number.

Z¯ = x  iy

Re(Z) = x

\Img(Z) = y

|Z| = x2 + y2

Formula used:

1. |Z1Z2|=|Z1||Z2|

2. ZZ¯ =|Z|2

3. (a - b)2 = a2 + b2 - 2ab

Calculation:

Given that,

2ab(ZZ¯) = a+ib     -----(1)

Z = (b + ia)     ----(2)

Therefore, a conjugate of Z

Z̅ = b - ia      ----(3)

Hence, from equation (1)

2ab(b + iab  ia) = a+ib

Taking modulus of both sides,

2ab|(b + iab  ia)| = |a+ib|

2ab|b + ia||b  ia|| = |a+ib|    

2abb2 + a2b2 + a2 =a2 + b2   

Taking square of both side

a2 + b2 - 2ab = 0

⇒ (a - b)2 = 0

⇒ a = b

Hence, option 4 is correct.

Conjugate of Complex Number Question 8:

Conjugate of 143i is

  1. 425+3i25
  2. 14+3i16
  3. 14+2i
  4. 4253i25

Answer (Detailed Solution Below)

Option 4 : 4253i25

Conjugate of Complex Number Question 8 Detailed Solution

Concept:

  • To get the conjugate of complex number change the sign of imaginary part.
  • Suppose z is a complex number z, such as z = x + yi. Then, conjugate of z is given by,

z̅ = x – iy (i2 = -1)

  • To simplify the fraction, we multiply the numerator and the denominator by the complex conjugate of the denominator
  • (a+b)(ab)=(a2b2)

 

Calculation:

Given complex number: 143i

To find: Conjugate of 143i = ?

 

First we need to simplify the given complex number.

To simplify we multiply the numerator and the denominator by the complex conjugate of the denominator

In 143i, denominator = 43i

Here real part is 4 and imaginary part = -3i

∴ Conjugate of 43i is 4 + 3i                                     (change the sign of imaginary part.)

Now, multiply and divide by 4 + 3i, we get

143i×4+3i4+3i=4+3i((4)2(3i)2)

=4+3i16(9)(1)                                        (∵ i2 = -1)

=4+3i16+(9)

=425+3i25

Now, conjugate of 425+3i25 is 4253i25                         (change the sign of imaginary part)

Hence, option (4) is correct.

Conjugate of Complex Number Question 9:

For any complex number, if |Z| = 1, then the value of  2(Z + Z¯)  2(1Z + 1Z¯) will be

  1. 1
  2. 0
  3. -1
  4. 2

Answer (Detailed Solution Below)

Option 2 : 0

Conjugate of Complex Number Question 9 Detailed Solution

Concept:

Properties of |Z|: If Z = x + iy is a complex number then the following properties are applicable for |Z|.

1. |Z| = |Z¯|

2. |z|2 = Z̅Z

3. |z1 + z2| = |Z1¯ + Z2¯|

Calculation:

Given that,

|Z| = 1 

⇒ |Z|2 = 1

⇒ Z Z̅  = 1                  (∵ |z|2 = Z̅Z)

Z=1Z¯        ----(1)

Therefore, the value of 2(Z + Z¯)  2(1Z + 1Z¯)

=  2(Z  1Z¯)  2( Z¯  1Z)

But, from equation (1) Z=1Z¯

⇒ 2(Z  1Z¯)  2( Z¯  1Z) = 0

Hence, option 2 is correct.

Conjugate of Complex Number Question 10:

Find the conjugate of 123i

  1. 213+313i
  2. 213313i
  3. 413313i
  4. More than one of the above
  5. None of the above

Answer (Detailed Solution Below)

Option 2 : 213313i

Conjugate of Complex Number Question 10 Detailed Solution

Concept:

Conjugate of a Complex Number:

Conjugate of a complex number z = x + iy is x - iy and which is denoted as z.

For example, the conjugate of the complex number z = 3 - 4i is 3 + 4i. 

Solution:

123i

123i×2+3i2+3i

=2+3i22+32

=2+3i4+9

=2+3i13

=213+313i

Conjugate of this complex number is 213313i

∴ The correct option is (2)

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