Let w =( z 2-3 z +6/ z +1) and z =1+ i ( where i =√-1), then | w | and amp w respectively are

Question

Let w =( z 2-3 z +6/ z +1) and z =1+ i ( where i =√-1), then | w | and amp w respectively are (A) 2,-(π/4) (B) √2,-(π/4) (C) 2, (3 π/4) (D) √2, (3 π/4)

Tardigrade Admin · Accepted Answer

w =((1+ i )2-3(1+ i )+6/2+ i )=(3- i /2+ i )=((3- i )(2- i )/5)=(5-5 i /5)=1- i text Hence | w |=√2 text and amp. w =-(π/4)

Q. Let $w = \frac{z ^{2} - 3 z + 6}{z + 1}$ and $z = 1 + i ($ where $i = - 1)$ , then $∣ w ∣$ and amp $w$ respectively are

$2, - \frac{π}{4}$

$2, - \frac{π}{4}$

$2, \frac{3 π}{4}$

$2, \frac{3 π}{4}$

$w = \frac{( 1 + i ) ^{2} - 3 ( 1 + i ) + 6}{2 + i} = \frac{3 - i}{2 + i} = \frac{( 3 - i ) ( 2 - i )}{5} = \frac{5 - 5 i}{5} = 1 - i$
$Hence ∣ w ∣ = 2 and amp. w = - \frac{π}{4}$

Q. Let w=z+1z2−3z+6​ and z=1+i( where i=−1​), then ∣w∣ and amp w respectively are

2,−4π​

2​,−4π​

2,43π​

2​,43π​

w=2+i(1+i)2−3(1+i)+6​=2+i3−i​=5(3−i)(2−i)​=55−5i​=1−i Hence ∣w∣=2​ and amp. w=−4π​

Q. Let $w = \frac{z ^{2} - 3 z + 6}{z + 1}$ and $z = 1 + i ($ where $i = - 1)$ , then $∣ w ∣$ and amp $w$ respectively are

$2, - \frac{π}{4}$

$2, - \frac{π}{4}$

$2, \frac{3 π}{4}$

$2, \frac{3 π}{4}$

$w = \frac{( 1 + i ) ^{2} - 3 ( 1 + i ) + 6}{2 + i} = \frac{3 - i}{2 + i} = \frac{( 3 - i ) ( 2 - i )}{5} = \frac{5 - 5 i}{5} = 1 - i$
$Hence ∣ w ∣ = 2 and amp. w = - \frac{π}{4}$