A) 0, 2
B) 1, 1
C) 2, 0
D) 2, 1
Correct Answer: C
Solution :
Given function is continuous at all point in \[(-\,\infty ,\,\,6)\] and at \[x=1,\,\,x=3\] function is continuous. If function \[f(x)\] is continuous at \[x=1,\] then \[\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,\,f(x)=\underset{x\to {{1}^{+}}}{\mathop{\lim }}\,\,f(x)\]\[\Rightarrow \,\,\,1+\sin \frac{\pi }{2}=a+b\] \[\therefore \,\,\,a+b=2\] .....(i) If at \[x=3,\] function is continuous, then \[\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,\,f(3)=\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,\,f(x)\] \[\Rightarrow \,\,3a+b=6\tan \frac{3\pi }{12}\] \[\therefore \,\,\,3a+b=6\] .....(ii) From (i) and (ii), \[a=2,\,\,b=0\] .
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