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KVPY SA Exam 2018 Question Paper
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© examsnet.com
Question : 9
Total: 80
Let a,b,c be the side-lengths of a triangle, and l, m, n be the lengths of its medians. Let
K
=
l
+
m
+
n
a
+
b
+
c
Then, as a, b, c vary, K can assume every value in the interval
(
1
4
,
2
3
)
(
1
2
,
4
5
)
(
3
4
,
1
)
(
4
5
,
5
4
)
Validate
Solution:
As AD is medium ⇒
A
D
<
A
B
+
A
C
2
⇒
l
<
b
+
c
2
Similarly
m
<
c
+
a
2
and
n
<
a
+
b
2
⇒
l
+
m
+
n
+
a
+
b
+
c
⇒
l
+
m
+
n
a
+
b
+
c
<
1
......(i)
Also in the
∆
BGC
BG + GC > BC
⇒
2
3
(
m
+
n
)
>
a
Similarly
2
3
(
n
+
l
)
>
b
and
2
3
(
l
+
m
)
>
c
Hence
4
3
(
l
+
m
+
n
)
>
a
+
b
+
c
l
+
m
+
n
a
+
b
+
c
>
3
4.
.
.
.
.
(ii)
By (i) and (ii)
l
+
m
+
n
a
+
b
+
c
∈
(
3
4
,
1
)
© examsnet.com
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