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KVPY 202021 SB SX Exam Previous Papers
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© examsnet.com
Question : 2
Total: 120
Consider a regular 10 gon with its vertices on the unit circle. With one vertex fixed, draw straight lines to the other 9 vertices. Call them
L
1
,
L
2
,
.
.
.
.
.
,
L
9
and denote their lengths by
l
1
,
l
2
,
.
.
.
.
.
,
l
9
respectively. Then the product
l
1
l
2
,
.
.
.
.
.
.
,
l
9
is
10
10
√
3
50
√
3
20
Validate
Solution:
Let
α
=
e
(
i
2
π
10
)
=
e
i
π
5
Now,
z
10
−
1
=
(
z
−
1
)
(
z
−
α
)
.
.
.
.
.
.
(
z
−
α
9
)
....(1)
so,
l
1
l
2
.
.
.
.
.
l
9
=

1
−
α


1
−
α
2

.
.
.
.
.
.

1
−
α
9

=

(
1
−
α
)
(
1
−
α
2
)
.
.
.
.
.
.
(
1
−
α
9
)

=

lim
z
→
1
z
10
−
1
z
−
1

=
10
© examsnet.com
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