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NIMCET 2009 Question Paper with solutions
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© examsnet.com
Question : 22
Total: 120
If
a
,
b
and
c
are unit vectors, then
|
a
−
b
|
2
+
|
b
−
c
|
2
+
|
c
−
a
|
2
does not exceed
9
4
8
6
Validate
Solution:
|
a
+
b
+
c
|
2
=
|
a
2
+
|
b
|
+
|
c
|
2
+
2
(
a
⋅
b
+
b
⋅
c
+
c
⋅
a
)
⇒
1
+
1
+
1
+
2
(
a
⋅
b
+
b
⋅
c
+
c
⋅
a
)
≥
0
(
∵
∣
a
+
b
+
c
|
2
≥
0
)
⇒
2
(
a
⋅
b
+
b
⋅
c
+
c
⋅
a
)
≥
−
3
⇒
−
2
(
a
⋅
b
+
b
+
c
+
c
−
a
)
≤
3
⇒
6
−
2
(
a
⋅
b
+
b
⋅
c
+
c
⋅
a
)
≤
9
⇒
2
(
|
a
|
2
+
b
|
2
+
|
c
|
2
−
a
⋅
b
−
b
⋅
c
−
c
⋅
a
≤
9
(
|
a
|
2
+
|
b
|
2
−
2
a
⋅
b
)
+
(
|
b
|
2
+
|
c
|
2
−
2
b
⋅
c
)
+
(
|
c
|
2
+
|
a
|
−
2
c
.
a
)
≤
9
⇒
|
a
−
b
|
2
+
|
b
−
c
|
2
+
|
c
−
a
|
2
≤
9
© examsnet.com
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