JEE 2023 — Mathematics PYQ

Step 1: Simplify the equation

The term on the right-hand side, ${\int }_0^{2}f(t)dt$, is a definite integral, which means it is a constant. Let:

Now the equation becomes a first-order linear differential equation:

Step 2: Solve the differential equation

Using the integrating factor $I.F.=e^{\int 1dx}=e^x$:

Integrating both sides with respect to $x$:

Step 3: Use the initial condition

Given $f(0)=e^{-2}$:

So, the function is:

Step 4: Find the value of C

Substitute $f(x)$ back into our assumption $C={\int }_0^{2}f(t)dt$:

Dividing by $e^{-2}$:

Step 5: Calculate $f(2)$

Substitute $C$ back into $K$:

Thus, $f(x)=(e^{-2}-1)+e^{-x}$.

Step 6: Final Calculation

We need to find $2f(0)-f(2)$:

Correct Option: (A) 1