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KVPY SX SB Exam 2012 Question Paper
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© examsnet.com
Question : 88
Total: 120
Consider
L
=
3
√
2012
+
3
√
2013
+
.
.
.
+
3
√
3011
L
=
3
√
2013
+
3
√
2014
+
.
.
.
+
3
√
3012
and
I
=
3012
∫
2012
3
√
x
d
x
, then
L
+
R
<
2
I
L
+
R
=
2
I
L
+
R
>
2
I
√
L
R
=
I
Validate
Solution:
L
=
3
√
2012
+
3
√
2013
+
.
.
.
+
3
√
3011
……………(1)
L
=
3
√
2013
+
3
√
2014
+
.
.
.
+
3
√
3012
……………(2)
I
=
3012
∫
2012
x
1
∕
3
d
x
[Let
f
(
x
)
=
x
1
∕
3
]
n
=
b
−
a
h
=
3012
−
2012
1
=
1000
I
=
(
b
−
a
)
n
[
f
(
a
)
+
f
(
a
+
h
)
+
f
(
a
+
2
h
)
+
.
.
.
+
f
(
a
+
(
n
−
1
)
h
)
]
=
[
f
(
2010
)
+
f
(
2013
)
+
.
.
.
+
f
(
3011
)
]
I
=
(
2012
)
1
∕
3
+
(
2013
)
1
∕
3
+
.
.
.
+
(
3011
)
1
∕
3
2
I
=
2
(
2012
)
1
∕
3
+
2
(
2013
)
1
∕
3
+
.
.
.
+
2
(
3011
)
1
∕
3
=
(
2012
)
1
∕
3
+
(
2012
)
1
∕
3
+
2
(
2013
)
1
∕
3
+
.
.
.
.
.
+
(
2
)
(
3011
)
1
∕
3
+
(
3012
)
1
∕
3
−
(
3012
)
1
∕
3
=
(
2012
)
1
∕
3
+
L
+
R
−
(
3012
)
1
∕
3
2
I
<
L
+
R
© examsnet.com
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