A) \[3{{R}_{1}}=2{{R}_{2}}\]
B) \[2{{R}_{1}}=3{{R}_{2}}\]
C) \[{{R}_{1}}={{R}_{2}}\]
D) \[{{R}_{1}}=2{{R}_{2}}\]
Correct Answer: D
Solution :
We have \[{{R}_{2}}=\int\limits_{-2}^{3}{x}f(x)dx=\int\limits_{-2}^{3}{(1-x)f(1-x)dx}\] \[\left[ \text{Using}\,\int\limits_{a}^{b}{{}}f(x)dx=\int\limits_{a}^{b}{f(a+b-x)dx} \right]\] \[\Rightarrow \]\[{{R}_{2}}=\int\limits_{-2}^{3}{(1-x)f(x)dx}\] \[(\because f(x)=f(1-x)on[-2,3])\] \[\therefore \]\[{{R}_{2}}+{{R}_{2}}=\int\limits_{-2}^{3}{xf(x)dx}+\int\limits_{-2}^{3}{(1-x)}f(x)dx\] \[=\int\limits_{-2}^{3}{f(x)dx}={{R}_{1}}\]\[\Rightarrow \]\[2{{R}_{2}}={{R}_{1}}\]You need to login to perform this action.
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