Consider the complex numbers `z_(1)` and `z_(2)` Satisfying the relation `|z_(1)+z_(2)|^(2)=|z_(1)| + |z_(2)|^(2)` Complex number `z_(1)barz_(2)` is

Monjur Alahi

Consider the complex numbers `z_(1)` and `z_(2)` Satisfying the relation `|z_(1)+z_(2)|^(2)=|z_(1)| + |z_(2)|^(2)` Complex number `z_(1)barz_(2)` is
A. purely real
B. purely imaginary
C. zero
D. none of theses


0 like 0 dislike
1 views
Share with your friends

1 Answers

Salim Ahmed Kawsar

Call

Correct Answer - B
Given that
`|z_(1) + z_(2)|^(2) = |z_(1)|^(2) + |z_(2)|^(2)`
`rArr |z_(1)|^(2) + |z_(2)|^(2) + z_(1)barz_(1) + barz_(1)z_(2) = |z_(1)|^(2)+|z_(2)|^(2)`
`rArr z_(1)barz_(1) + barz_(1)z_(2) = 0" "(1)`
`rArr (z_(1))/(z_(2)) + (barz_(1))/(barz_(2))" "("dividinbg by " z_(2)barz_(2))`
`rArr (z_(1))/(z_(2))+bar((z_(1)/(z_(2)))) = 0" "(2)`
From (1), `z_(2) barz_(2)` is purely imaginary. From (2) `z_(1)//z_(2)` is purely imaginary. Hence,
`arg((z_(1))/(z_(2))) = pm (pi)/(2) or arg(z_(1)) - arg(z_(2)) = pm(pi)/(2)`
Also, `i(z_(1)//z_(2))` is purely real. Hence its possible arguments are 0 and `pi`.

0 like 0 dislike
1 views
Talk Doctor Online in Bissoy App
For real numbers x and y, we write xRy ⇒ x – y + √2 is an irrational number. Then the relation R is

For each value of x ∈ R, x − x + √2 that is √2 is an irrational number. It is reflexive. Let x = √2 and y =2 then x −...

2 Answers 1 views
In a senior secondary school,80 students play football or hockey.The number that play football is 5 more than twice the number that play hockey.If 15 students play both games and every student in the school plays at least one game,find (i)the number of students that play football (ii)the number of students that play football but not hockey (iii)the number of students that play hockey but not football.

I thought F or H = 80 is giving by n(F u H) = n(F) + n(H) - n(F n H)

2 Answers 8 views
In a senior secondary school,80 students play football or hockey.The number that play football is 5 more than twice the number that play hockey.If 15 students play both games and every student in the school plays at least one game,find (i)the number of students that play football (ii)the number of students that play football but not hockey (iii)the number of students that play hockey but not football. 2 Answers 3 views
The product of two numbers is -20/9 and one of the numbers is 4 then the other number is ………………… 

Correct option is (D) -5/9 Let the other numbr be x. Then \(x\times4=-\frac{20}9\) \(\Rightarrow\) \(x=\frac{-20}{9\times4}\) \(=\frac{-5}{9}.\)

2 Answers 1 views
The product of two numbers is -20/9 and one of the numbers is 4 then the other number is …………………  2 Answers 1 views
If A={a,b,c,d}, then on A . (i) write the identity relation `I_(A)` . (ii) write a reflexive relation which is not the identity relation.

Correct Answer - (i) {(a,a),(b,b),(c,c),(d,d)} `" "` {(a,a),(b,b),(c,c),(d,d),(a,c),(b,d)}

2 Answers 1 views
For real numbers x and y, define xRy if and only if x – y + √2 is an irrational number. Then the relation R is

(A) is the correct choice.

2 Answers 1 views
Consider the relation `4l^(2)-5m^(2)+6l+1=0` , where `l,m in R` The number of tangents which can be drawn from the point (2,-3) to the above fixed cir

Correct Answer - 3 Let the equation of the circle be `x^(2)+y^(2)+2gx+2fy+c=0` (1) The line `lx +my+1=0` will touch circle (1) if the length of perpendicular from the center `( -g,...

2 Answers 2 views
Consider a quadratic equaiton `az^(2) + bz + c=0`, where a,b,c are complex number. If equaiton has two purely imaginary roots, then which of the follo

Correct Answer - D Let `z_(1)` (purely imginary ) be a root of the given equation Then, `z_(1) = - barz_(1)` and `underline(az_(1)^(2) + bz_(1) + c)=0" "(1)` `rArr az_(1)^(2)...

2 Answers 1 views
Consider the complex number `z = (1 - isin theta)//(1+ icos theta)`. The value of `theta` for which z is purely imaginary are

Correct Answer - D `z=(1-i sin theta)/(1+ i cos theta) = ((1- isin theta)( 1- icos theta))/((1+ i cos theat)(1-i cos theta))` `=((1-sin theta cos theta) - icos theta + sintheta)/((1+...

2 Answers 1 views
X