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8. Introduction to Trigonometry
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આપેલ વિધાન સત્ય છે કે અસત્ય છે કારણ સહિત દર્શાવો :
$a \neq 1$ હોય તેવી કોઈક ધન સંખ્યા $a$ માટે $2 \sin \theta$ નું મૂલ્ય $(a + \frac{1}{a})$ હોઈ શકે.
Option A
Option B
Option C
Option D
Solution
False.
Given, $a$ is a positive number and $a \neq 1,$ then $A M>G M$
$\Rightarrow$ $\frac{a+\frac{1}{a}}{2}>\sqrt{a \cdot \frac{1}{a}} \Rightarrow\left(a+\frac{1}{a}\right)>2$
[since, $AM$ and $GM$ of two number's $a$ and $b$ are $\frac{(a+b)}{2}$ and $\sqrt{a b}$, respectively]
$2 \sin \theta>2$ $\left[\because 2 \sin \theta=a+\frac{1}{a}\right]$
$\sin \theta>1$ $[\because-1 \leq \sin \theta \leq 1]$
Which is not possible.
Hence,the value of $2 \sin \theta$ can not be $a+\frac{1}{a}$
Standard 10
Mathematics
