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Question
Mathematics
Let f be a positive function. If I1=∫ limits1-kk x f[x(1-x)] d x and I2=∫ limits1-kk f[x(1-x)] d x, where 2 k-1>0 . Then, (I1/I2) is
Q. Let
f
be a positive function.
If
I
1
=
1
−
k
∫
k
x
f
[
x
(
1
−
x
)]
d
x
and
I
2
=
1
−
k
∫
k
f
[
x
(
1
−
x
)]
d
x
, where
2
k
−
1
>
0.
Then,
I
2
I
1
is
1328
216
IIT JEE
IIT JEE 1997
Integrals
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A
2
17%
B
k
31%
C
2
1
38%
D
1
14%
Solution:
Given,
I
1
=
1
−
k
∫
k
x
f
[
x
(
1
−
x
)]
d
x
⇒
I
1
=
1
−
k
∫
k
(
1
−
x
)
f
[(
1
−
x
)
x
]
d
x
=
1
−
k
∫
k
f
[(
1
−
x
)]
d
x
]
−
1
−
k
∫
k
x
f
(
1
−
x
)
]
d
x
⇒
I
1
=
I
2
−
I
1
⇒
I
2
I
1
=
2
1