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AP EAPCET 23-Aug-2021 Shift 1 Paper
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© examsnet.com
Question : 9
Total: 160
If
z
2
+
z
+
1
=
0
where
z
is a complex number, then
(
z
+
1
z
)
3
+
(
z
4
+
1
z
4
)
3
is equal to
1
0
-1
-2
Validate
Solution:
Given,
z
2
+
z
+
1
=
0
(
z
+
1
z
)
3
+
(
z
4
+
1
z
4
)
3
=
(
z
2
+
1
z
)
3
+
(
z
8
+
1
z
4
)
3
=
(
−
z
z
)
3
+
(
z
8
+
1
z
4
)
3
[
∵
z
2
+
z
+
1
=
0
⇒
z
2
+
1
=
−
z
]
=
−
1
+
(
z
8
+
1
z
4
)
3
.....(1)
∵
z
2
+
1
=
−
z
, squaring both side,
z
4
+
1
+
2
z
2
=
z
2
⇒
z
4
+
1
=
−
z
2
Again, squaring it
⇒
z
8
+
1
+
2
z
4
=
z
4
⇒
z
8
+
1
=
−
z
4
Putting in Eq. (i), we get
(
z
+
1
z
)
3
+
(
z
4
+
1
z
4
)
3
=
−
1
+
(
−
z
4
z
4
)
3
=
−
1
+
(
−
1
)
=
−
2
© examsnet.com
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