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Write 'True' or 'False' and justify your answer in each of the following:
The value of $ 2 \sin \theta $ can be $ a+\frac{1}{a} $, where $ a $ is a positive number, and $ a
eq 1 $
Given:
The value of \( 2 \sin \theta \) can be \( a+\frac{1}{a} \), where \( a \) is a positive number, and \( a ≠ 1 \).
To do:
We have to find whether the given statement is true or false.
Solution:
$a$ is a positive number and $a ≠1$
This implies,
$AM>GM$
AM and GM of two numbers $a$ and $b$ are $\frac{(a+b)}{2}$ and $\sqrt{a b}$.
Therefore,
$\frac{a+\frac{1}{a}}{2}>\sqrt{a \times \frac{1}{a}}$
$(a+\frac{1}{a})>2$
$2 \sin \theta>2$ ($2 \sin \theta=a+\frac{1}{a}$)
$\sin \theta>1$ which is not possible. [Since $-1 \leq \sin \theta \leq 1$]
Hence, the value of $2 \sin \theta$ cannot be $a+\frac{1}{a}$.
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