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Question

If $$\displaystyle \cos \:^{-1}x+\cos \:^{-1}y+\cos ^{-1}z=3\pi$$ then the value of $$xy+yz+zx$$ is


A
3
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B
1
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C
3
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D
none of these
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Solution

The correct option is C $$3$$
If  $$y=cos^{-1}(x)$$   $$ R\epsilon[\pi,0]$$ where$$ R$$ is the range of $$ y$$.

Therefore in the above case, which is only possible if
$$cos^{-1}(x)=cos^{-1}(y)=cos^{-1}(z)=\pi$$

Hence
$$x=y=z=-1$$

Therefore $$xy+yz+zx$$
$$=3 (-1)^2=3$$

Mathematics

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