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KEAM 2011 Math Paper
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© examsnet.com
Question : 25
Total: 120
A unit vector in the XOY- plane that makes an angle 30° with the vector
⋀
i
+
⋀
j
makes an angle 60° with
⋀
i
−
⋀
j
is
1
4
[
(
√
6
+
√
2
)
⋀
i
−
(
√
6
−
√
2
)
⋀
j
]
1
2
[
(
√
6
−
√
2
)
⋀
i
+
(
√
6
+
√
2
)
⋀
j
]
1
4
[
(
√
6
−
√
2
)
⋀
i
+
(
√
6
+
√
2
)
⋀
j
]
1
3
[
(
√
6
+
√
2
)
⋀
i
+
(
√
2
−
√
6
)
⋀
j
]
1
4
[
(
√
6
+
√
2
)
⋀
i
+
(
√
6
−
√
2
)
⋀
j
]
Validate
Solution:
Let a unit vector in
X
O
Y
-plane is,
∧
a
=
a
|
a
|
a
=
a
1
i
+
a
2
j
,
|
a
|
=
√
a
1
2
+
a
2
2
Given vectors let
b
=
i
+
j
and
c
=
i
−
j
From question;
∧
a
⋅
b
=
|
∧
a
|
⋅
|
b
|
cos
30
∘
=
1
⋅
√
2
⋅
√
3
2
⇒
√
6
2
⇒
a
⋅
b
|
a
|
=
√
6
2
⇒
a
1
+
a
2
=
√
6
2
|
a
|
.....(i)
and
∧
a
⋅
c
=
|
∧
a
|
|
c
|
⋅
cos
60
∘
=
1
⋅
√
2
⋅
1
2
a
⋅
c
|
a
|
=
√
2
2
⇒
a
1
−
a
2
=
√
2
2
|
a
|
.....(ii)
From Eqs. (i) and (ii),
2
a
1
=
(
√
6
+
√
2
)
|
a
|
2
a
1
=
1
4
(
√
6
+
√
2
)
|
a
|
⇒
a
1
|
a
|
=
1
4
(
√
6
+
√
2
)
and
a
2
=
1
4
(
√
6
−
√
2
)
|
a
|
⇒
a
2
|
a
|
=
1
4
(
√
6
−
√
2
)
So, the unit vector is
∧
a
=
1
|
a
|
(
a
1
i
+
a
2
j
)
∧
a
=
1
4
{
(
√
6
+
√
2
)
i
+
(
√
6
−
√
2
)
j
}
© examsnet.com
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