Mathematics - Definite Integration Question with Solution | TestHub

Statement-1: Let $\mathrm{a},\mathrm{b},\mathrm{c}$ be non zero real numbers and $\mathrm{f}(\mathrm{x})={\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}$ satisfying ${\int }_0^1\left(1+{\cos }^{8}x\right)f(x)dx={\int }_0^2\left(1+{\cos }^{8}x\right)f(x)dx$ then the equation $f(x)=0$ has at least one root in $(0,2)$. Statement-2: If ${\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{g}(\mathrm{x})\mathrm{dx}$ vanishes and $\mathrm{g}(\mathrm{x})$ is continuous then the equation $\mathrm{g}(\mathrm{x})=0$ has at least one real root in ( $a,b$ ).