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Byju's Answer
Standard XIII
Mathematics
AM,GM,HM Inequality
If a, b, c ∈ ...
Question
If
a
,
b
,
c
∈
R
+
and
a
2
b
c
+
4
a
b
c
+
4
b
c
=
81
such that (
2
a
+
b
+
c
+
4
) assumes its least value, then
A
a
is a prime number
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B
a
+
b
+
c
is a prime number
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C
b
+
c
−
a
is a prime number
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D
a
3
+
b
3
+
c
3
is a product of two prime numbers
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Solution
The correct option is
D
a
3
+
b
3
+
c
3
is a product of two prime numbers
b
c
(
a
2
+
4
a
+
4
)
=
81
⇒
b
c
(
a
+
2
)
2
=
81
AM
≥
GM
(
a
+
2
)
+
(
a
+
2
)
+
b
+
c
4
≥
[
(
a
+
2
)
2
b
c
]
1
4
2
a
+
b
+
c
+
4
≥
12
equality holds when
a
+
2
=
b
=
c
=
3
⇒
a
=
1
,
b
=
3
,
c
=
3
Suggest Corrections
0
Similar questions
Q.
If
a
,
b
,
c
∈
R
+
and
a
2
b
c
+
4
a
b
c
+
4
b
c
=
81
such that (
2
a
+
b
+
c
+
4
) assumes its least value, then
Q.
Assertion :If
a
,
b
,
c
∈
R
and
2
a
+
3
b
+
6
c
=
0
, then the equation
a
x
2
+
b
x
+
c
=
0
has at least one real root in
(
0
,
1
)
. Reason: If
f
(
x
)
is a polynomial which assumes both positive and negative values, then it has at least one real root.
Q.
Let a, b, c
ϵ
R such that a + b + c = 0 and a + b - c = 0, then the polynomial function
f
(
x
)
=
a
x
2
+
b
x
+
c
;
(
a
>
0
)
attains its least value at 'x' equal to
Q.
If
a
,
b
,
c
∈
R
+
are such that
2
a
,
b
,
4
c
are in A.P. and
c
,
a
,
and
b
are in G.P., then
Q.
If
a
,
b
,
c
,
d
are four positive real numbers such that
a
b
c
d
=
1
then least value of
(
a
+
4
)
(
b
+
4
)
(
c
+
4
)
(
d
+
4
)
is
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