If z = ((√3/2) + (i/2))5 + ((√3/2) - (i/2))5 , then

Question

If z = ((√3/2) + (i/2))5 + ((√3/2) - (i/2))5 , then (A) Re (z) = 0 (B) Im (z) = 0 (C) Re (z) > 0, Im (z) > 0 (D) Re (z) > 0, Im (z) < 0

Tardigrade Admin · Accepted Answer

On simplification, we get z = 2 [5C0 ((√3/2))2 + 5C2 ((√3/2))3 ((i/2))2 + 5C4 ((√3/2))((i/2))4] Since i2 = - 1 and i4 = 1, z will not contain any i and hence Im (z) = 0

Q. If $z = (\frac{3}{2} + \frac{i}{2})^{5} + (\frac{3}{2} - \frac{i}{2})^{5}$ , then

$R e (z) = 0$

$I_{m} (z) = 0$

$R e (z) > 0, I_{m} (z) > 0$

$R e (z) > 0, I_{m} (z) < 0$

On simplification, we get $z = 2 [^{5} C_{0} (\frac{3}{2})^{2} +^{5} C_{2} (\frac{3}{2})^{3} (\frac{i}{2})^{2} +^{5} C_{4} (\frac{3}{2}) (\frac{i}{2})^{4}]$
Since $i^{2} = - 1$ and $i^{4} = 1, z$ will not contain any i and hence $I_{m} (z) = 0$

Q. If z=(23​​+2i​)5+(23​​−2i​)5, then

Re(z)=0

Im​(z)=0

Re(z)>0,Im​(z)>0

Re(z)>0,Im​(z)<0