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Question

If the functions $$ \displaystyle f\left ( x \right )=\sin \left ( x+a \right ) $$ and $$ \displaystyle g\left ( x \right )=b\sin x+c\cos x $$ satisfy $$ \displaystyle f\left ( 0 \right )=g\left ( 0 \right ) $$ and $$ \displaystyle {f}'\left ( 0 \right )={g}'\left ( 0 \right ) $$ then


A
b=π2
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B
b=cosa
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C
c=sina
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D
c=cosa
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Solution

The correct options are
B $$ \displaystyle b=\cos a $$
D $$ \displaystyle c=\sin a $$
Given, $$f(x)=\sin(x+a)\quad $$
and $$ g(x)=b\sin x+c\cos x\\ f(0)=g(0)\Rightarrow \sin a=c\\ f'(x)=\cos(x+a)\\ g'(x)=b\cos x-c\sin x\\ f'(0)=g'(0)\Rightarrow \cos a=b$$

Mathematics

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