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  4. If f'(x)=g(x) and g'(x)=-f(x) for all x and f(1)=5, f'(1)=4, then the value of f2(1)+g2(1) is equal to

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Q. If $ f'(x)=g(x) $ and $ g'(x)=-f(x) $ for all x and $ f(1)=5,\,\,f'(1)=4, $ then the value of $ {{f}^{2}}(1)+{{g}^{2}}(1) $ is equal to

J & K CETJ & K CET 2012

Solution:

Given, $ f(1)=5 $ and $ g(x)=f'(x) $ and $ f'(1)=4 $
$ \therefore $ $ {{f}^{2}}(1)+{{g}^{2}}(1)={{(5)}^{2}}+{{[f'(1)]}^{2}} $
$ =25+{{(4)}^{2}} $ $ =25+16 $
$ =41 $