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JEE Main Math Class 11 Sequences and Series Part 1 Questions
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© examsnet.com
Question : 35
Total: 100
Let
z
=
1
+
ai be a complex number,
a
>
0
,
such that
z
3
is areal number. Then the sum
1
+
z
+
z
2
+
.
.
.
.
+
z
11
is equal to:
(Online April 10, 2016)
1365
√
3
i
−
1365
√
3
i
−
1250
√
3
i
1250
√
3
i
Validate
Solution:
z
=
1
+
a
i
z
2
=
1
−
a
2
+
2
a
i
z
2
⋅
z
=
{
(
1
−
a
2
)
+
2
a
i
}
{
1
+
a
i
}
=
(
1
−
a
2
)
+
2
a
i
+
(
1
−
a
2
)
a
i
−
2
a
2
∵
z
3
is real
⇒
2
a
+
(
1
−
a
2
)
a
=
0
a
(
3
−
a
2
)
=
0
⇒
a
=
√
3
(
a
>
0
)
1
+
z
+
z
2
.
.
.
.
.
.
.
.
.
.
z
11
=
z
12
−
1
z
−
1
=
(
1
+
√
3
i
)
12
−
1
1
+
√
3
i
−
1
=
(
1
+
√
3
i
)
12
−
1
√
3
i
(
1
+
√
3
i
)
12
=
2
12
(
1
2
+
√
3
2
i
)
12
=
2
12
(
cos
π
3
+
i
sin
π
3
)
12
=
2
12
(
cos
4
π
+
i
sin
4
π
)
=
2
12
⇒
2
12
−
1
√
3
i
=
4095
√
3
i
=
−
4095
3
√
3
i
=
−
1365
√
3
i
© examsnet.com
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