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UPSC CDS 2 2018 Math Paper
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© examsnet.com
Question : 67
Total: 100
Consider the following statements :
1.
cos
θ
+
sec
θ
can never be equal to 1.5.
2.
sec
2
θ
+
cosec
2
θ
can never be less them 4.
Which of the statements given above is/are correct ?
[2018 CDS-II]
1 only
2 only
Both 1 and 2
. Neither 1 nor 2
Validate
Solution:
Using AM–GM inequality :
cos
θ
+
sec
θ
2
≥
√
cos
θ
⋅
sec
θ
⇒
cos
θ
+
sec
θ
2
≥
√
1
⇒
cos
θ
+
sec
θ
≥
2
Hence, statement (1) is correct
sec
2
θ
+
cosec
2
θ
=
1
+
tan
2
θ
+
1
+
cot
2
θ
=
2
+
tan
2
θ
+
cot
2
θ
Using AM–GM inequality:
tan
2
θ
+
cot
2
θ
2
≥
√
tan
2
θ
⋅
cot
2
θ
⇒
tan
2
θ
+
cot
2
θ
≥
2
∴
2
+
t
a
n
2
θ
+
cot
2
θ
≥
4
Hence, statement (2) is also correct.
© examsnet.com
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