JEE Main 12 Jan 2019 Morning Question 14
Question: Let f and g be continuous functions on [0, a] such that $ f(x)=f(a-x) $ and $ g(x)=g(a-x)=4, $ then $ \int\limits_0^{a}{f(x)g(x)dx} $ is equal to [JEE Main Online Paper Held On 12-Jan-2019 Morning]
Options:
A) $ 2\int\limits_0^{a}{f(x)dx} $
B) $ 4\int\limits_0^{a}{f(x)dx} $
C) $ -3\int\limits_0^{a}{f(x)dx} $
D) $ \int\limits_0^{a}{f(x)dx} $
Show Answer
Answer:
Correct Answer: A
Solution:
Here, $ f(x)=f(a-x) $ and $ g(x)+g(a-x)=4 $
Let $ I=\int\limits_0^{a}{f(x)g(x)dx}=\int\limits_0^{a}{f}(a-x)g(a-x)dx $
$ =\int\limits_0^{a}{f}(x)(4-g(x))dx $ $ =4\int\limits_0^{a}{f}(x)dx-\int\limits_0^{a}{f}(x)g(x)dx $
$ =4\int\limits_0^{a}{f}(x)dx-I\Rightarrow I=2\int\limits_0^{a}{f}(x)dx $
