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Joint Entrance Examination
Mathematics
Complex Numbers
Question 278
Question 278 of 455
Medium
MCQ
If
z
=
(
3
2
+
i
2
)
5
+
(
3
2
−
i
2
)
5
z=\left(\dfrac{\sqrt{3}}2+\dfrac i2\right)^5+\left(\dfrac{\sqrt{3}}2-\dfrac i2\right)^5
z
=
(
2
3
+
2
i
)
5
+
(
2
3
−
2
i
)
5
, and
R
(
z
)
,
I
(
z
)
R(z),I(z)
R
(
z
)
,
I
(
z
)
denote the real and imaginary parts, then:
A
I
(
z
)
=
0
I(z)=0
I
(
z
)
=
0
B
R
(
z
)
>
0
R(z)>0
R
(
z
)
>
0
and
I
(
z
)
>
0
I(z)>0
I
(
z
)
>
0
C
R
(
z
)
<
0
R(z)<0
R
(
z
)
<
0
and
I
(
z
)
>
0
I(z)>0
I
(
z
)
>
0
D
R
(
z
)
=
−
I
(
z
)
R(z)=-I(z)
R
(
z
)
=
−
I
(
z
)
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Solutions · 1
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Official solution
z
=
e
i
5
π
/
6
+
e
−
i
5
π
/
6
=
2
cos
π
6
=
3
z=e^{i5\pi/6}+e^{-i5\pi/6}=2\cos\dfrac\pi6=\sqrt{3}
z
=
e
i
5
π
/6
+
e
−
i
5
π
/6
=
2
cos
6
π
=
3
(using
3
2
+
i
2
=
e
i
π
/
6
\dfrac{\sqrt{3}}2+\dfrac i2=e^{i\pi/6}
2
3
+
2
i
=
e
iπ
/6
). So
I
(
z
)
=
0
I(z)=0
I
(
z
)
=
0
,
R
(
z
)
=
3
.
R(z)=\sqrt{3}.
R
(
z
)
=
3
.
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