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UPSC CDS 2 2021 Math Paper
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© examsnet.com
Question : 37
Total: 100
If
cos
2
θ
−
3
cos
θ
+
2
sin
2
θ
=
1
Where
0
∘
<
θ
<
90
∘
then what is
sin
2
θ
+
cos
θ
equal to
5
4
3
2
7
4
2
Validate
Solution:
👈: Video Solution
Given,
cos
2
θ
−
3
cos
θ
+
2
sin
2
θ
=
1
Where,
0
∘
<
θ
<
90
∘
⇒
cos
2
θ
−
3
cos
θ
+
2
=
sin
2
θ
⇒
cos
2
θ
−
3
cos
θ
+
2
=
1
−
cos
2
θ
[
∵
sin
2
A
+
cos
2
A
=
1
]
⇒
2
cos
2
θ
−
3
cos
θ
+
1
=
0
⇒
2
cos
2
θ
−
2
cos
θ
−
cos
θ
+
1
=
0
⇒
2
cos
θ
(
cos
θ
−
1
)
−
1
(
cos
θ
−
1
)
=
0
⇒
(
cos
θ
−
1
)
(
2
cos
θ
−
1
)
=
0
⇒
cos
θ
=
1
or
cos
θ
=
1
2
∵
In the given interval
θ
≠
0
⇒
cos
θ
≠
1
∴
cos
θ
=
1
2
(only)
sin
θ
=
√
1
−
cos
2
θ
=
√
1
−
1
4
=
√
3
4
=
√
3
2
Now,
sin
2
θ
+
cos
θ
=
(
√
3
2
)
2
+
1
2
=
3
4
+
1
2
=
3
4
+
2
4
=
5
4
Hence,
sin
2
θ
+
cos
θ
=
5
4
.
© examsnet.com
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