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Byju's Answer
Standard XII
Mathematics
Theorems for Continuity
If fx and ...
Question
If
f
(
x
)
and
g
(
x
)
are continuous functions, the
∫
ln
1
/
λ
ln
λ
f
(
x
2
/
4
)
[
f
(
x
)
−
f
(
−
x
)
]
g
(
x
2
/
4
)
[
g
(
x
)
+
g
(
−
x
)
]
d
x
is
A
dependent on
λ
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B
a non-zero constant
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C
zero
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D
None of these
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Solution
The correct options are
A
dependent on
λ
C
zero
I
=
∫
log
1
λ
log
λ
f
(
x
2
/
4
)
[
f
(
x
)
−
f
(
−
x
)
]
g
(
x
2
/
4
)
[
g
(
x
)
+
g
(
−
x
)
]
d
x
=
∫
−
log
λ
log
λ
f
(
x
2
/
4
)
[
f
(
x
)
−
f
(
−
x
)
]
g
(
x
2
/
4
)
[
g
(
x
)
+
g
(
−
x
)
]
=
0
Let
h
(
x
)
=
f
(
x
2
/
4
)
[
f
(
x
)
−
f
(
−
x
)
]
g
(
x
2
/
4
)
[
g
(
x
)
+
g
(
−
x
)
]
∴
h
(
−
x
)
=
f
(
(
−
x
)
2
/
4
)
[
f
(
−
x
)
−
f
(
x
)
]
g
(
(
−
x
)
2
/
4
)
[
g
(
−
x
)
+
g
(
x
)
]
∴
h
(
−
x
)
=
f
(
x
2
/
4
)
[
f
(
−
x
)
−
f
(
x
)
]
g
(
x
2
/
4
)
[
g
(
x
)
+
g
(
−
x
)
]
∴
h
(
−
x
)
=
−
f
(
x
2
/
4
)
[
f
(
x
)
−
f
(
−
x
)
]
g
(
x
2
/
4
)
[
g
(
x
)
+
g
(
−
x
)
]
∴
h
(
−
x
)
=
−
h
(
x
)
Thus, the function is odd.
∴
I
=
0
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0
Similar questions
Q.
If
f
and
g
are continuous functions, then
ln
1
λ
∫
ln
λ
f
(
x
2
4
)
[
f
(
x
)
−
f
(
−
x
)
]
g
(
x
2
4
)
[
g
(
x
)
+
g
(
−
x
)
]
d
x
,
λ
>
0
,
is
Q.
If
f
(
x
)
and
g
(
x
)
are continuous functions, then the value of
ln
1
/
λ
∫
ln
λ
f
(
x
2
4
)
[
f
(
x
)
−
f
(
−
x
)
]
g
(
x
2
4
)
[
g
(
x
)
+
g
(
−
x
)
]
d
x
is
Q.
Let
g
(
x
)
be a polynomial of degree one and
f
(
x
)
be defined by
f
(
x
)
=
{
g
(
x
)
,
x
≤
0
|
x
|
s
i
n
x
,
x
>
0
If
f
(
x
)
is continuous satisfying
f
′
(
1
)
=
f
(
−
1
)
, then
g
(
x
)
is
Q.
If
f
(
x
)
,
g
(
x
)
are two differentiable functions on
[
0
,
2
]
satisfying
f
′′
(
x
)
=
g
′′
(
x
)
,
f
′
(
1
)
=
2
g
′
(
1
)
=
4
and
f
(
2
)
=
3
g
(
2
)
=
9
, then
Q.
The functions
f
(
x
)
and
g
(
x
)
are positive and continuous. If
f
(
x
)
is increasing and
g
(
x
)
is decreasing, then
1
∫
0
f
(
x
)
[
g
(
x
)
−
g
(
1
−
x
)
]
d
x
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