Let Z be a complex number and Z denotes the conjugate of Z- If 2Z-3 Z-27-23 i1-i- then which of the following is-are correct-

Given : nbsp;2Z 3 Z= 27+23 i1+i Assuming nbsp;Z=x+iy ⇒Z=x iy Now, nbsp; 2Z 3 Z nbsp;= 3(Z Z) Z nbsp;= 3(2iy) (x+iy) nbsp;= x+5iy And 27+23 i1 ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Conjugate of a Complex Number

Let Z be a co...

Question

Let $Z$ be a complex number and $¯ ¯¯ ¯ Z$ denotes the conjugate of $Z$ . If $2 Z - 3 ¯ ¯¯ ¯ Z = \frac{- 27 + 23 i}{1 + i}$ , then which of the following is/are correct?

R e (Z) = 2

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

I m (Z) = 5

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

R e (Z) = 5

No worries! We‘ve got your back. Try BYJU‘S free classes today!

I m (Z) = 2

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B $I m (Z) = 5$
Given : $2 Z - 3 ¯ ¯¯ ¯ Z = \frac{- 27 + 23 i}{1 + i}$
Assuming $Z = x + i y \Rightarrow ¯ ¯¯ ¯ Z = x - i y$
Now,
$2 Z - 3 ¯ ¯¯ ¯ Z = 3 (Z - ¯ ¯¯ ¯ Z) - Z = 3 (2 i y) - (x + i y) = - x + 5 i y$
And
$\frac{- 27 + 23 i}{1 + i} = \frac{- 27 + 23 i}{1 + i} \times \frac{1 - i}{1 - i} = \frac{- 27 + 27 i + 23 i + 23}{1 + 1} = - 2 + 25 i$

Therefore,
$- x + 5 i y = - 2 + 25 i \Rightarrow x = 2, y = 5$

Suggest Corrections

Similar questions

Q. Let

Z

be a complex number and

¯ ¯¯ ¯ Z

denotes the conjugate of

Z

. If

2 Z - 3 ¯ ¯¯ ¯ Z = \frac{- 27 + 23 i}{1 + i}

, then which of the following is/are correct?

Q. Let

z

is a complex number and

¯ ¯ ¯ z

is conjugate of

z

. If

(1 + i) z = (1 - i) ¯ ¯ ¯ z,

then the value of

z + i ¯ ¯ ¯ z

Q. Let

z

is a complex number and

¯ ¯ ¯ z

is conjugate of

z

. If

(1 + i) z = (1 - i) ¯ ¯ ¯ z,

then the value of

z + i ¯ ¯ ¯ z

Q. Let

z_{i}, {¯ ¯ ¯ z}_{i} (i = 1, 2, . ., 5)

are the complex roots of the equation

x^{10} + (13 x - 1)^{10} = 0,

where the bar denotes complex conjugation. Then the value of

5 \sum i = 1 \frac{1}{z_{i} {¯ ¯ ¯ z}_{i}}

Q. If

z

and

ω

are two complex numbers such that

| z ω | = 1

and

arg (z) - arg (ω) = \frac{3 π}{2},

then

arg (\frac{1 - 2 ¯ ¯ ¯ z ω}{1 + 3 ¯ ¯ ¯ z ω})

is
(Here

arg (z)

denotes the principal argument of complex number

z

)

Join BYJU'S Learning Program

Explore more

Conjugate of a Complex Number

Standard XII Mathematics

Join BYJU'S Learning Program

Let Z be a complex number and ¯¯¯¯Z denotes the conjugate of Z. If 2Z−3¯¯¯¯Z=−27+23i1+i, then which of the following is/are correct?

The correct option is B Im(Z)=5Given : 2Z−3¯¯¯¯Z=−27+23i1+i Assuming Z=x+iy⇒¯¯¯¯Z=x−iy Now, 2Z−3¯¯¯¯Z=3(Z−¯¯¯¯Z)−Z=3(2iy)−(x+iy)=−x+5iy And −27+23i1+i=−27+23i1+i×1−i1−i=−27+27i+23i+231+1=−2+25i Therefore, −x+5iy=−2+25i⇒x=2,y=5

Let $Z$ be a complex number and $¯ ¯¯ ¯ Z$ denotes the conjugate of $Z$ . If $2 Z - 3 ¯ ¯¯ ¯ Z = \frac{- 27 + 23 i}{1 + i}$ , then which of the following is/are correct?