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Byju's Answer
Standard XII
Mathematics
Greatest Binomial Coefficients
If x=a+b, y...
Question
If
x
=
a
+
b
,
y
=
a
ω
+
b
ω
2
,
z
=
a
ω
2
+
b
ω
, then xyz equals to where,
ω
is the cube root of unity.
A
a
+
b
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B
a
2
+
b
2
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C
a
3
+
b
3
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D
a
4
+
b
4
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Solution
The correct option is
C
a
3
+
b
3
Given :
x
=
a
+
b
,
y
=
a
ω
+
b
ω
2
,
z
=
a
ω
2
+
b
ω
Consider,
x
y
z
=
(
a
+
b
)
(
a
ω
+
b
ω
2
)
(
a
ω
2
+
b
ω
)
=
(
a
2
ω
+
a
b
ω
2
+
a
b
ω
+
b
2
ω
2
)
(
a
ω
2
+
b
ω
)
=
a
3
ω
3
+
a
2
b
ω
2
+
a
2
b
ω
4
+
a
b
2
ω
3
+
a
2
b
ω
3
+
a
b
2
ω
2
+
a
b
2
ω
4
+
b
3
ω
3
=
a
3
+
b
3
+
a
2
b
(
ω
2
+
ω
4
+
ω
3
)
+
a
b
2
(
ω
3
+
ω
2
+
ω
4
)
.......
[
∵
ω
3
=
1
]
=
a
3
+
b
3
+
a
2
b
(
ω
2
+
ω
+
1
)
+
a
b
2
(
ω
2
+
ω
+
1
)
=
a
3
+
b
3
.....
[
∵
(
ω
2
+
ω
+
1
)
=
0
,
ω
3
=
1
]
Suggest Corrections
0
Similar questions
Q.
Let
x
=
a
+
b
,
y
=
a
ω
+
b
ω
2
,
z
=
a
ω
2
+
b
ω
where
ω
being cube root of unity. Then find the value of
x
y
z
Q.
If
x
=
a
+
b
,
y
=
a
ω
2
+
b
ω
and
z
=
a
ω
+
b
ω
2
, where
ω
is a complex cube root of unity, then
the product of
x
y
z
is equal to ____
Q.
If
x
=
a
+
b
,
y
=
a
ω
+
b
ω
2
and
z
=
a
ω
2
+
b
ω
where
w
is complex cube root of unity
then (i)
x
+
y
+
z
=
0
(ii)
x
y
z
=
a
3
+
b
3
Q.
If
x
=
a
+
b
,
y
=
a
ω
+
b
ω
2
,
z
=
a
ω
2
+
b
ω
,
ω
is cube root of unity then value of
x
3
+
y
3
+
z
3
Q.
Let
x
=
a
+
b
,
y
=
a
ω
+
b
ω
2
,
z
=
a
ω
2
+
b
ω
, where
ω
is the cube root of unity. Then prove than
x
2
+
y
2
+
z
2
=
6
a
b
.
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