If fx-cos log x- then fx2- fy2-1-2fx2-y2-fx2 y2 has the value

The correct option is D None of these Given: f(x) = cos (log x) ⇒ f(x^2) = cos (log (x^2)) ⇒ f(x^2) = cos (2 log (x)) Similarly, f(y^2) = cos (2 log (y)) No ...

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Byju's Answer

Standard XII

Mathematics

Composite Function

If fx=cos log...

Question

If $f (x) = c o s (l o g x)$ , then $f (x^{2}), f (y)^{2} - \frac{1}{2} {f (\frac{x^{2}}{y^{2}}) + f (x^{2} y^{2})}$ has the value

$- 2$

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$- 1$

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$\frac{1}{2}$

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None of these

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Solution

The correct option is D
None of these

Given:
$f (x) = c o s (l o g x) \Rightarrow f (x^{2}) = c o s (l o g (x^{2})) \Rightarrow f (x^{2}) = c o s (2 l o g (x))$
Similarly,
$f (y^{2}) = c o s (2 l o g (y))$
Now,
$f (\frac{x^{2}}{y^{2}}) = c o s (l o g x^{2} - l o g y^{2})$
and
$f (x^{2} y^{2}) = c o s (l o g x^{2} y^{2}) = c o s (l o g x^{2} + l o g y^{2}) \Rightarrow f (\frac{x^{2}}{y^{2}}) + f (x^{2} y^{2}) = c o s ((2 l g o x - 2 l o g y)) + c o s ((2 l o g x + 2 l o g y)) \Rightarrow f (\frac{x^{2}}{y^{2}}) + f (x^{2} y^{2}) = 2 c o s (2 l o g x) c o s (2 l o g y) \Rightarrow \frac{1}{2} [f (\frac{x^{2}}{y^{2}}) + f (x^{2} y^{2})] = c o s (2 l o g x) c o s (2 l o g y) \Rightarrow f (x^{2}) f (y^{2}) - \frac{1}{2} [f (x^{2} y^{2}) + f (\frac{x^{2}}{y^{2}})] = c o s (2 l o g x) c o s (2 l o g y) - c o s (2 l o g x) c o s (2 l o g y) = 0$

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If f(x)=cos(log x), then f(x2), f(y)2−12{f(x2y2)+f(x2y2)} has the value

If $f (x) = c o s (l o g x)$ , then $f (x^{2}), f (y)^{2} - \frac{1}{2} {f (\frac{x^{2}}{y^{2}}) + f (x^{2} y^{2})}$ has the value