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EAMCET Engineering 2010 Solved Paper
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© examsnet.com
Question : 46
Total: 160
If
I
n
=
π
∕
4
∫
0
tan
n
x
dx
, then
I
2
+
I
4
,
I
3
+
I
5
,
I
4
+
I
6
,
.
.
.
, are in
arithmetic progression
geometric progression
harmonic progression
arithmetico-geometric progression
Validate
Solution:
I
n
=
π
∕
4
∫
0
tan
n
x
dx
We have,
I
r
+
2
=
π
∕
4
∫
0
tan
r
+
2
x
dx
=
π
∕
4
∫
0
tan
r
x
⋅
tan
2
x
dx
and
I
r
=
π
∕
4
∫
0
tan
r
x
dx
Then,
I
r
+
I
r
+
2
=
π
∕
4
∫
0
tan
r
x
dx
+
π
∕
4
∫
0
tan
r
x
⋅
tan
2
x
dx
=
π
∕
4
∫
0
tan
r
x
(
1
+
tan
2
x
)
dx
=
π
∕
4
∫
0
tan
r
x
⋅
s
e
c
2
x
dx
[Put
t
=
tan
x
⇒
dt
=
s
e
c
2
x
dx
]
=
1
∫
0
t
r
dt
⇒
[
t
r
+
1
r
+
1
]
0
1
=
(
1
r
+
1
)
So,
I
r
+
I
r
+
2
=
1
r
+
1
ie,
I
2
+
I
4
=
1
3
I
3
+
I
5
=
1
4
I
4
+
I
6
=
1
5
which are clearly in HP.
© examsnet.com
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