If a, b, c, d are real numbers and
is a non
real cube root of unity such that
then
equals :
Answers
Answered by
3
Answer:
Since,
w
2
=
1
w
and
w
=
1
w
2
Given relation may be rewritten as
1
a
+
w
+
1
b
+
w
+
1
c
+
w
=
2
w
and
1
a
+
w
2
+
1
b
+
w
2
+
1
c
+
w
2
=
2
w
2
Clearly
w
and
w
2
are the roots of
1
a
+
x
+
1
b
+
x
+
1
c
+
x
=
2
x
⇒
(
b
+
x
)
(
c
+
x
)
+
(
a
+
x
)
(
c
+
x
)
+
(
a
+
x
)
(
b
+
x
)
(
a
+
x
)
(
b
+
x
)
(
c
+
x
)
=
2
x
⇒
x
[
3
x
2
+
2
(
a
+
b
+
c
)
x
+
b
c
+
c
a
+
a
b
]
=
2
[
a
b
c
+
(
b
c
+
c
a
+
a
b
)
x
+
(
a
+
b
+
c
)
x
2
+
x
3
]
⇒
x
3
−
(
b
c
+
c
a
+
a
b
)
x
−
2
a
b
c
=
0
If
α
is the third root of this equation then sum of the roots
α
+
w
+
w
2
=
0
⇒
α
=
1
Hence,
1
is the root of equation, we get
1
a
+
1
+
1
b
+
1
+
1
c
+
1
=
2
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