MCQ
${{\tan }^{-1}}x+{{\tan }^{-1}}y+{{\tan }^{-1}}z=\pi ,$ તો ............. .
- A$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$
- ✓$\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}=1$
- C$xy+yz+zx=1$
- D${{x}^{3}}+{{y}^{3}}+{{z}^{3}}=1$
ધારો કે $\tan^{-1} x = \alpha , \tan^{-1} y = \beta, \tan^{-1} z = \gamma \left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)$
જયા $\alpha , \beta, \gamma \in$
$x = \tan \alpha, y = \tan \beta, 1 = \tan \gamma$
હવે $\tan^{-1} x + \tan^{-1} y +\tan^{-1}1 = \pi$
$\therefore \alpha +\beta + \gamma = \pi$
$\therefore \alpha +\beta = \pi - \gamma$
$\therefore \tan (\alpha +\beta) = \tan (\pi - \gamma)$
$\therefore \frac{\tan \alpha+ \tan \beta}{1- \tan \alpha \times \tan \beta}= - \tan \gamma$
$\therefore \frac{x+y}{1- xy}= -z$
$\therefore x+y+z = xyz$
$\therefore \frac{1}{yz} + \frac{1}{xz} +\frac{1}{xy} = 1$ $(xyz$ વડે ભાગતા)
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$f(x)=\left\{\begin{array}{cc}2 \sin \left(-\frac{\pi x}{2}\right), & \text { if } x<-1 \\ \left|a x^{2}+x+b\right|, & \text { if }-1 \leq x \leq 1 \\ \sin (\pi x), & \text { if } x>1\end{array}\right.$
વડે વ્યાખ્યાયીત છે. જો $f(x)$ એ $R$ પર સતત હોય, તો $a+b $ ..... .