Question

Statement-1: if satisfies $f(x+y)=f\left(x\right)+f\left(y\right)\forall x,yϵR$ then ${\int }_{-5}^{5}f\left(x\right)dx=0$

Statement -2: if $f$ is an odd function then ${\int }_{-a}^{a}f\left(x\right)dx=0$

• A. Statement I is True, Statement 2 is True: Statement 2 is a correct explanation for Statement 1
• B. Statement 1 is True, Statement 2 is True; Statement 2 Not a correct explanation for Statement 1
• C. Statement 1 is True, Statement 2 is False
• D. Statement 1 is False, Statement 2 is True

Answer 1

The correct option is A Statement I is True, Statement 2 is True: Statement 2 is a correct explanation for Statement 1

Statement-1: $f(x+y)=f\left(x\right)+f\left(y\right)\forall x,yϵR$
Let y be constant
$f^{\prime }(x+y)=f^{\prime }\left(x\right)$
$\to$ and
$f\left(2x\right)=2f\left(x\right)$
$\Rightarrow f\left(x\right)=ax$ $f\left(0\right)=2f\left(0\right)\Rightarrow f\left(0\right)=0$
$f\left(x\right)=ax\to {\int }_{-5}^{5}ax=0$
Since $f\left(x\right)=ax$ is odd function
Statement-2: true and explain statement 1
So, A.