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KVPY SX SB Exam 20141 Question Paper
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© examsnet.com
Question : 1
Total: 120
PART - I
MATHEMATICS
Let
C
0
be a circle of radius I. For n ≥ 1, let C
n
be a circle whose area equals the area of a square inscribed in C
n
−
1
.
Then
∞
∑
i
=
0
Area
(
C
i
)
equals
π
2
π
−
2
π
2
1
π
2
π
2
π
−
2
Validate
Solution:
∞
∑
i
=
0
Area
(
C
i
)
=
π
r
0
2
+
π
r
1
2
+
π
r
2
2
+
π
r
3
2
+
.
.
.
.
.
∞
Area of
C
n
=
π
r
n
2
=
(
√
2
r
n
−
1
)
2
r
n
2
=
2
π
r
2
n
−
1
so
r
1
2
=
2
π
r
0
2
,
r
2
2
=
2
π
r
1
2
=
2
π
(
2
π
r
0
2
)
r
3
2
=
2
π
(
r
2
2
)
=
2
π
(
2
π
2
π
r
0
2
)
So
∞
∑
i
=
0
Area
(
C
i
)
=
π
[
r
0
2
+
2
π
r
0
2
+
2
π
.2
π
r
0
2
+
.
.
.
∞
]
=
π
r
0
2
1
−
2
π
=
π
2
r
0
2
π
−
2
∀
r
0
=
1
=
π
2
π
−
2
© examsnet.com
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