Let z = ((√3/2) + (i/2))5 + ((√3/2) - (i/2))5 . If R(z) and I[z] respectively denote the real and imaginary parts of z, then :

Question

Let z = ((√3/2) + (i/2))5 + ((√3/2) - (i/2))5 . If R(z) and I[z] respectively denote the real and imaginary parts of z, then : (A) R(z) > 0 and I(z) > 0 (B) R(z) < 0 and I(z) > 0 (C) R(z) = -3 (D) I(z) = 0

Tardigrade Admin · Accepted Answer

z=((√3+i/2))5+((√3-i/2))5 z=(eiπ/6)5 +(e-i π/6)5 =ei5π/6 +e-i5π/6 = cos (5π/6) +i ( sin5π/6) + cos((-5π/6)) +i sin((-5π/6)) = 2 cos (5π/6) < 0 I(z) = 0 and Re(z) < 0

Q. Let $z = (\frac{3}{2} + \frac{i}{2})^{5} + (\frac{3}{2} - \frac{i}{2})^{5}$ . If $R (z)$ and $I [z]$ respectively denote the real and imaginary parts of $z$ , then :

$R (z) > 0 an d I (z) > 0$

$R (z) < 0 an d I (z) > 0$

$R (z) = - 3$

$I (z) = 0$

$z = (\frac{3 + i}{2})^{5} + (\frac{3 - i}{2})^{5}$
$z = (e^{iπ /6})^{5} + (e^{- iπ /6})^{5}$
$= e^{i 5 π /6} + e^{- i 5 π /6}$
$= cos \frac{5 π}{6} + i \frac{s i n 5 π}{6} + cos (\frac{- 5 π}{6}) + i sin (\frac{- 5 π}{6})$
$= 2 cos \frac{5 π}{6} < 0$
$I (z) = 0 an d R e (z) < 0$

Q. Let z=(23​​+2i​)5+(23​​−2i​)5. If R(z) and I[z] respectively denote the real and imaginary parts of z, then :

R(z)>0andI(z)>0

R(z)<0andI(z)>0

R(z)=−3

I(z)=0