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Byju's Answer
Standard XII
Mathematics
Algebra of Limits
If α, β are...
Question
If
α
,
β
are roots of
a
x
2
+
b
x
+
c
=
0
then
1
α
3
+
1
β
3
=
?
A
3
a
b
c
−
b
3
a
3
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B
3
a
b
−
b
3
a
2
c
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C
3
a
b
c
−
b
3
c
3
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D
b
2
−
2
a
c
a
c
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Solution
The correct option is
C
3
a
b
c
−
b
3
c
3
⇒
α
and
β
are roots of the equation
a
x
2
+
b
x
+
c
=
0
⇒
α
β
=
c
a
------ ( 1 )
⇒
α
3
β
3
=
c
3
a
3
----- ( 2 )
⇒
α
+
β
=
−
b
a
------ ( 3 )
⇒
(
α
+
β
)
3
=
α
3
+
β
3
+
3
α
β
(
α
+
β
)
⇒
(
−
b
a
)
3
=
α
3
+
β
3
+
3
(
c
a
)
(
−
b
a
)
[ By using ( 1 ) and ( 3 ) ]
⇒
−
b
3
a
3
=
α
3
+
β
3
−
3
b
c
a
2
∴
α
3
+
β
3
=
−
b
3
a
3
+
3
b
c
a
2
∴
α
3
+
β
3
=
−
b
3
+
3
a
b
c
a
3
∴
α
3
+
β
3
=
3
a
b
c
−
b
3
a
3
----- ( 4 )
Now,
⇒
1
α
3
+
1
β
3
=
α
3
+
β
3
α
3
β
3
=
3
a
b
c
−
b
3
a
3
c
3
a
3
[ By using ( 2 ) and ( 4 ) ]
=
3
a
b
c
−
b
3
a
3
×
a
3
c
3
=
3
a
b
c
−
b
3
c
3
∴
1
α
3
+
1
β
3
=
=
3
a
b
c
−
b
3
c
3
Suggest Corrections
0
Similar questions
Q.
If
α
,
β
are the roots of
a
x
2
+
b
x
+
c
=
0
then match the elements of list I with elements of list II:
List I
List II
A)
α
β
+
β
α
=
1)
c
2
a
2
B)
α
2
+
β
2
α
−
2
+
β
−
2
=
2)
c
5
[
3
a
b
c
−
b
3
]
a
8
C)
α
3
+
β
3
3)
b
2
−
2
a
c
a
c
D)
α
5
β
8
+
α
8
β
5
=
4)
3
a
b
c
−
b
3
a
3
The correct match from List-I to List-II:
A
,
B
,
C
,
D
Q.
If
a
>
b
>
c
and
a
3
+
b
3
+
c
3
=
3
a
b
c
,
then the quadratic equations
a
x
2
+
b
x
+
c
=
0
has roots which are
Q.
If the roots of the equation
(
c
2
−
a
b
)
x
2
−
2
(
a
2
−
b
c
)
x
+
(
b
2
−
a
c
)
=
0
are real and equal, show that either
a
=
0
or
a
3
+
b
3
+
c
3
=
3
a
b
c
[
H
i
n
t
:
D
=
4
a
(
a
3
+
b
3
+
c
3
−
3
a
b
c
)
]
Q.
If
a
3
+
b
3
+
c
3
=
3
a
b
c
and
a
+
b
+
c
=
0
show that
(
b
+
c
)
2
3
b
c
+
(
c
+
a
)
2
3
a
c
+
(
a
+
b
)
2
3
a
b
=
1
Q.
If
a
3
+
b
3
+
c
3
=
3
a
b
c
and
a
+
b
+
c
=
0
, then find
(
b
+
c
)
2
3
b
c
+
(
c
+
a
)
2
3
a
c
+
(
a
+
b
)
2
3
a
b
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