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Question
Write 'True' or 'False' and justify your answer in the following:
The value of \[\sin \theta\] is \[x + \frac{1}{x}\], where 'x' is a positive real number.
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Solution
This statement is false.
Explanation:
\[\sin\theta = x + \frac{1}{x}\]
\[ \Rightarrow - 1 \leq x + \frac{1}{x} \leq 1\]
\[ \Rightarrow x + \frac{1}{x} \leq 1\]
\[ \Rightarrow x^2 + 1 \leq x\]
\[ \Rightarrow x^2 + 1 - x \leq 0\]
\[\text{Take } x = 1, \]
\[ \Rightarrow 1 + 1 - 1 \leq 0\]
\[ \Rightarrow 1 \leq 0\]
\[\text{Which is false, so x is not always a positive real number.}\]
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