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UPSC CDS 2 2020 Math Paper
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© examsnet.com
Question : 29
Total: 100
Consider the following statements :
1) The equation
2
sin
2
θ
−
cos
θ
+
4
=
0
is possible for all
θ
.
2)
tan
θ
+
cot
θ
cannot be less than
2
,
where
0
<
θ
<
π
2
Which of the above statements is/are correct?.
1 only
2 only
Both 1 and 2
Neither 1 nor 2
Validate
Solution:
Consider statement 1 :
2
cos
2
θ
+
cos
θ
−
6
=
0
⇒
2
cos
2
θ
+
4
cos
θ
−
3
cos
θ
−
6
=
0
2
cos
θ
(
cos
θ
+
4
)
−
3
(
cos
θ
+
4
)
=
0
(
cos
θ
+
4
)
(
2
cos
θ
−
3
)
=
0
cos
θ
=
−
4
and
cos
θ
=
3
2
We know that
cos
θ
can never take value of -4 and
3
2
Hence, there is no value of
θ
for which the above equation holds.
Consider statement 2 :
A
.
M
.
≥
G
.
M
.
tan
θ
+
cot
θ
2
≥
√
tan
θ
×
cot
θ
tan
θ
+
cot
θ
2
≥
√
tan
θ
×
1
tan
θ
tan
θ
+
cot
θ
≥
2
tan
θ
+
cot
θ
is always greater than or equal to
2
,
where
0
<
θ
<
π
2
.
Hence, only statement (2) is correct.
© examsnet.com
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