Question

Assertion :If $f,g$ and $h$ be continuous function on $[0,a]$ such that $f\left(x\right)=f(a-x),$ $g\left(x\right)=-g(a-x)$ and $3h\left(x\right)-4h(a-x)=5,$ then ${\int }_0^{a}f\left(x\right)g\left(x\right)h\left(x\right)dx=0$ Reason: ${\int }_0^{a}f\left(x\right)g\left(x\right)dx=0$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

${\int }_0^{a}f\left(x\right)g\left(x\right)h\left(x\right)dx={\int }_0^{a}f(a-x)g(x-a)h(a-x)dx$

$=-{\int }_0^{a}f\left(x\right)g\left(x\right)\left(\frac{3h\left(x\right)-5}{4}\right)dx$

$=-\frac{3}{4}{\int }_0^{a}f\left(x\right)g\left(x\right)h\left(x\right)dx+\frac{5}{4}{\int }_0^{a}f\left(x\right)g\left(x\right)dx$

$\therefore \frac{7}{4}{\int }_0^{a}f\left(x\right)g\left(x\right)h\left(x\right)dx=\frac{5}{4}{\int }_0^{a}f\left(x\right)g\left(x\right)dx=0$

$\because {\int }_0^{a}f\left(x\right)g\left(x\right)dx={\int }_0^{a}f(a-x)g(a-x)dx={\int }_0^{a}f\left(x\right)g(-x)dx$

$\Rightarrow {\int }_0^{a}f\left(x\right)g\left(x\right)dx=0$