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KVPY SX SB Exam 2015 Question Paper
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© examsnet.com
Question : 12
Total: 120
Define a function f : R → R by
f(x) = max
{
|
x
|
,
|
x
−
1
|
,
.
.
.
,
|
x
−
2
n
|
}
where n is a fixed natural number. Then
2
n
∫
0
f
(
x
)
d
x
is -
n
n
2
3n
3
n
2
Validate
Solution:
2
n
∫
0
f
(
x
)
d
x
=
n
∫
0
f
(
x
)
d
x
+
2
n
∫
n
f
(
x
)
d
x
=
n
∫
0
|
x
−
2
n
|
d
x
+
2
n
∫
n
|
x
|
d
x
=
n
∫
0
|
2
n
−
x
|
d
x
+
2
n
∫
n
x
.
d
x
=
[
2
n
x
−
x
2
2
]
0
n
+
[
x
2
2
]
n
2
n
=
(
2
n
2
−
n
2
2
)
+
(
4
n
2
2
−
n
2
2
)
=
3
n
2
2
+
3
n
2
2
=
3
n
2
© examsnet.com
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