Consider the following statements 1 There exists a positive real number m such that cos x 2m 1 2 mn ...........

Quantitative Aptitude (CDS) Number System

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Consider the following statements

1. There exists a positive real number m such that cos x = 2^{m + 1} .

2. mn $\geq$ m + n, for all m, n belonging to set of natural numbers.

Which of the above statements is/are correct?

Only 1

Only 2

Both 1 and 2

Neither 1 nor 2

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Explanation:

1. As, cos x lies between −1 and 1, then cos x = 2^m + 1 does not exist for positive value of m.
2. $\because$ AM $\geq$ GM
$\therefore \: \frac{m+n}{2}\geq \sqrt{mn} mn\geq (m+n) does\: not\: hold\: \: true$
$for\: m,n \: belonging\: to\: natural\: numbers.$
Let's take an example
If m = 1, then n $\geq$ 1 + n this cannot be true.
Hence, neither 1 nor 2 is correct.