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UPSC 2014 CDS I Math Paper
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© examsnet.com
Question : 98
Total: 100
Consider the following :
1. The arithmetic mean of two unequal positive numbers is always greater than their geometric mean.
2. The geometric mean of two unequal positive numbers is always greater than their harmonic mean.
Which of the above statements is/are correct ?
[2014 CDS-I]
1 only
2 only
Both 1 and 2
Neither 1 nor 2
Validate
Solution:
Let a and b (a > b) be two unequal positive numbers.
A
M
−
G
M
=
a
+
b
2
−
√
a
b
=
a
+
b
−
2
√
a
b
2
=
1
2
(
√
a
−
√
b
)
2
>
0
⇒ AM > GM
Hence, statement 1 is correct
G
M
−
H
M
−
√
a
b
−
2
a
b
a
+
b
=
√
a
b
(
a
+
b
−
2
√
a
b
a
+
b
)
=
√
a
b
(
a
+
b
)
(
√
a
−
√
b
)
2
>
0
⇒ GM > HM
Hence, statement 2 is correct
© examsnet.com
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