Question

Observe the following Lists

List-I	List-II
$A\left){\int }_0^{a}f\right(x)dx$	1) ${\int }_0^{a}f(a+x)dx$
B) ${\int }_{-a}^{a}f\left(x\right)dx$	2) ${\int }_0^{a}f(a-x)dx$
C) ${\int }_0^{2a}f\left(x\right)dx$	3) 0, if $f\left(x\right)$ is odd
D) ${\int }_0^{na}f\left(x\right)dx$	4) $0$, if $f(2a-x)$ $=-f\left(x\right)$
	5) $(n-1){\int }_0^{a}f\left(x\right)dx$, if period of f(x) is $a$.
	6) ${\int }_0^{a}f\left(x\right)dx$, if period of $f\left(x\right)$ is $a$.

• A. $A-1,B-2,C-4,D-6$
• B. $A-2,B-3,C-4,D-6$
• C. $A-2,B-3,C-5,D-6$
• D. $A-2,B-1,C-4,D-5$

Answer 1

Answer: (B)

$A-2,B-3,C-4,D-6$

From property ${\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f(a-x)dx$
Gives A-2
From property ${\int }_{-a}^{a}f\left(x\right)dx=0$ if $f\left(x\right)$ is odd
Gives B-3
From property ${\int }_0^{2a}f\left(x\right)dx={\int }_0^{2a}f(2a-x)dx$
If $f(2a-x)=-f\left(x\right)$ $\Rightarrow {\int }_0^{2a}f\left(x\right)dx=-{\int }_0^{2a}f\left(x\right)dx\to 2{\int }_0^{2a}f\left(x\right)dx=0\Rightarrow {\int }_0^{2a}f\left(x\right)dx=0$
Gives C-4
From property ${\int }_0^{na}f\left(x\right)dx={\int }_0^{a}f\left(x\right)dx$, if period of $f\left(x\right)$ is $a$.
Gives D-6
Hence, option 'B' is correct.