NDA 2026 — Mathematics PYQ

To solve this problem, we use the property of definite integrals: ${\int }_a^{b}h(x)dx={\int }_a^{b}h(a+b-x)dx$.

Step 1: Simplify the expression

Given $g(x)-f(x)=f(4-x)$, we can write $g(x)=f(x)+f(4-x)$.

The integral becomes:

$$I={\int }_0^4\frac{f(x)}{f(x)+f(4-x)}dx\dots \text{(Equation 1)}$$

Step 2: Apply the integral property

Applying the property ${\int }_0^{4}h(x)dx={\int }_0^{4}h(4-x)dx$:

$$I={\int }_0^4\frac{f(4-x)}{f(4-x)+f(4-(4-x))}dx$$

$$I={\int }_0^4\frac{f(4-x)}{f(4-x)+f(x)}dx\dots \text{(Equation 2)}$$

Step 3: Combine and Solve

Adding Equation 1 and Equation 2:

$$2I={\int }_0^4\frac{f(x)}{f(x)+f(4-x)}dx+{\int }_0^4\frac{f(4-x)}{f(4-x)+f(x)}dx$$

$$2I={\int }_0^4\frac{f(x)+f(4-x)}{f(x)+f(4-x)}dx$$

$$2I={\int }_0^{4}1dx$$

$$2I=[x{]}_0^4=4-0=4$$

$$I=\frac{4}{2}=2$$

Therefore, the value of the integral is $2$, and the correct option is (c).