JEE 2022 — Mathematics PYQ

To solve for $f(x)$, we first simplify the integral equation by separating the terms involving $x$:

Since the integrals are definite integrals with constant limits, they result in constant values. Let:

• $A={\int }_0^{1}f(t)dt$

• $B={\int }_0^{1}tf(t)dt$

Now, the equation becomes:

Step 1: Solve for constants $A$ and $B$

Substitute $f(t)=(1+A)t-B$ back into the definitions of $A$ and $B$.

For A:

For B:

Step 2: Solve the system of equations

From Eq. 1: $A=1-2B$. Substitute this into Eq. 2:

Now find $A$:

Step 3: Determine the function $f(x)$

Substitute $A$ and $B$ back into our linear form:

Step 4: Check the given points

• (A) $(2,4)$: $f(2)=\frac{18(2)-4}{13}=\frac{32}{13}\ne 4$

• (B) $(1,2)$: $f(1)=\frac{18(1)-4}{13}=\frac{14}{13}\ne 2$

• (C) $(4,17)$: $f(4)=\frac{18(4)-4}{13}=\frac{72-4}{13}=\frac{68}{13}\ne 17$

• (D) $(6,8)$: $f(6)=\frac{18(6)-4}{13}=\frac{108-4}{13}=\frac{104}{13}=8$

The point $(6,8)$ satisfies the equation.

Correct Option: (D)