A) \[P=5,Q=-6,R=3\]
B) \[P=-5,Q=6,R=3\]
C) \[P=-5,Q=6,R=3\]
D) \[P=3,Q=2,R=3\]
Correct Answer: A
Solution :
\[f'(x)=P{{e}^{2x}}+Q{{e}^{x}}+Rx\] \[\Rightarrow \]\[f'(x)=2P{{e}^{2x}}+Q{{e}^{x}}+R\] \[\Rightarrow \]\[31=2P{{e}^{2\log 2}}+Q{{e}^{\log 2}}+R\] \[\Rightarrow \]\[8P+2Q+R=31\] ?(i) Also, \[0=P+Q\] ?(ii) \[\And \int_{0}^{\log 4}{(f(x)-Rx)dx}=\frac{39}{2}\] \[\Rightarrow \]\[\int_{0}^{\log 4}{(P{{e}^{2x}}+Q{{e}^{x}})dx}=\frac{39}{2}\] \[\Rightarrow \]\[\left[ \frac{P}{2}{{e}^{2x}}+Q{{e}^{x}} \right]_{0}^{\log 4}=\frac{39}{2}\] \[\Rightarrow \]\[\frac{P}{2}{{e}^{2\log 4}}+Q{{e}^{\log 4}}-\frac{P}{2}-Q=\frac{39}{2}\] \[\Rightarrow \]\[15P+6Q=39\] ?(iii) Solving (i), (ii) and (iii), we get \[P=5,Q=-6,R=3\]
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