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Byju's Answer
Standard XII
Mathematics
Roots of a Quadratic Equation
If 1,α 1 ,α...
Question
If
1
,
α
1
,
α
2
,
α
3
and
α
4
be the roots of
x
5
−
1
=
0
, then
ω
−
α
1
ω
2
−
α
1
.
ω
−
α
2
ω
2
−
α
2
.
ω
−
α
3
ω
2
−
α
3
.
ω
−
α
4
ω
2
−
α
4
=
A
1
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B
ω
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C
ω
2
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D
None of these
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Solution
The correct option is
B
ω
Since
1
,
α
1
,
α
2
,
α
3
,
α
4
are roots of the equation
x
5
−
1
=
0
Thus
x
5
−
1
=
(
x
−
1
)
(
x
−
α
1
)
(
x
−
α
2
)
(
x
−
α
3
)
(
x
−
α
4
)
⇒
x
5
−
1
(
x
−
1
)
=
(
x
−
α
1
)
(
x
−
α
2
)
(
x
−
α
3
)
(
x
−
α
4
)
........1
Putting
x
=
w
in 1 we get,
⇒
w
5
−
1
(
w
−
1
)
=
(
w
−
α
1
)
(
w
−
α
2
)
(
w
−
α
3
)
(
w
−
α
4
)
⇒
w
2
−
1
(
w
−
1
)
=
(
w
−
α
1
)
(
w
−
α
2
)
(
w
−
α
3
)
(
w
−
α
4
)
............2
Putting
x
=
w
2
in 1 we get
⇒
w
10
−
1
(
w
2
−
1
)
=
(
w
2
−
α
1
)
(
w
2
−
α
2
)
(
w
2
−
α
3
)
(
w
2
−
α
4
)
⇒
w
−
1
(
w
2
−
1
)
=
(
w
2
−
α
1
)
(
w
2
−
α
2
)
(
w
2
−
α
3
)
(
w
2
−
α
4
)
..........3
Dividing 2 by 3 we get,
⇒
(
w
−
α
1
)
(
w
−
α
2
)
(
w
−
α
3
)
(
w
−
α
4
)
(
w
2
−
α
1
)
(
w
2
−
α
2
)
(
w
2
−
α
3
)
(
w
2
−
α
4
)
=
(
w
2
−
1
)
2
(
w
−
1
)
2
=
w
4
+
1
−
2
w
2
w
2
+
1
−
2
w
=
w
+
1
−
2
w
2
w
2
+
1
−
2
w
=
−
w
2
−
2
w
2
−
w
−
2
w
=
w
Suggest Corrections
0
Similar questions
Q.
If
1
,
α
1
,
α
2
,
α
3
,
α
4
be the roots of
z
5
−
1
=
0
and
ω
be an imaginary cube root of unity,
then
(
ω
−
α
1
ω
2
−
α
1
)
(
ω
−
α
2
ω
2
−
α
2
)
(
ω
−
α
3
ω
2
−
α
3
)
(
ω
−
α
4
ω
2
−
α
4
)
is ?
Q.
If
α
1
,
α
2
,
α
3
,
α
4
be the roots of
x
5
−
1
=
0
then find
ω
−
α
1
ω
2
−
α
1
⋅
ω
−
α
2
ω
2
−
α
2
⋅
ω
−
α
3
ω
2
−
α
3
⋅
ω
−
α
4
ω
2
−
α
4
Q.
If
1
a
+
ω
+
1
b
+
ω
+
1
c
+
ω
+
1
d
+
ω
=
2
ω
, where
a
,
b
,
c
are real and
ω
is a non-real cube root of unity, then
Q.
If '
ω
' is a complex cube root of unity,then
ω
(
1
3
+
2
9
+
4
27
.
.
.
∞
)
+
ω
(
1
2
+
3
8
+
9
32
.
.
.
∞
)
=
Q.
If
ω
is complex (non real) cube root of
1
then show that
∣
∣ ∣ ∣
∣
1
ω
ω
2
ω
ω
2
1
ω
2
1
ω
∣
∣ ∣ ∣
∣
=
0
.
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