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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