JEE Mathematics 2025 Question with Solution

Given:

$$I_1 = \int_{-\frac{1}{2}}^1 2x \, f\left(2x(1-2x)\right) \, dx,$$ $$I_2 = \int_{-1}^2 f\left(x(1-x)\right) \, dx$$

Find: The value of $\frac{I_2}{I_1}$.

From the solution, the conclusion stated is that the correct option is A and hence $\frac{I_2}{I_1} = 4$.

Given:

$$I_1 = \int_{-\frac{1}{2}}^1 2x \, f\left(2x(1-2x)\right) \, dx,$$ $$I_2 = \int_{-1}^2 f\left(x(1-x)\right) \, dx$$

Find: The value of $\frac{I_2}{I_1}$.

The solution proceeds by considering substitutions for both integrals.

For $I_1$, it takes

$$u = 2x(1-2x)$$

and writes

$$du = 2(1-4x) \, dx.$$

It then tracks the limits:

• at $x = -\frac{1}{2}$, the working gives $u = 0$,
• at $x = 1$, the working gives $u = -2$.

Using this, the extracted working states

$$I_1 = \frac{1}{2} \int_{0}^{-2} f(u) \, du = -\frac{1}{2} \int_{-2}^{0} f(u) \, du = \frac{1}{2} \int_{-2}^{0} f(u) \, du.$$

For $I_2$, the working takes

$$u = x(1-x)$$

with

$$\frac{du}{dx} = 1-2x.$$

It evaluates the endpoint values as

• at $x = -1$, $u = -2$,
• at $x = 2$, $u = -2$.

The working then appeals to symmetry and states

$$\therefore I_2 = \int_{-2}^{0} f(u) \, du$$

and hence

$$I_2 = 4\left(\frac{1}{2}\int_{-2}^{0} f(u) \, du\right) = 4I_1.$$

Therefore,

$$\frac{I_2}{I_1} = 4.$$

So the correct option is A.

Note: The intermediate derivation in the provided the solution is not fully rigorous and contains inconsistencies, but its final conclusion explicitly gives option A, so the extracted answer is $4$.

Given: The same integrals $I_1$ and $I_2$. Find: $\frac{I_2}{I_1}$.

The second approach in the solution tests the constant function

$$f(x) = 1.$$

Then

$$I_1 = \int_{-\frac{1}{2}}^1 2x \, dx = \left.x^2\right|_{-\frac{1}{2}}^1 = 1 - \frac{1}{4} = \frac{3}{4}$$

and

$$I_2 = \int_{-1}^2 dx = 3.$$

So

$$\frac{I_2}{I_1} = \frac{3}{\frac{3}{4}} = 4.$$

Thus the correct option is A.

Why this shortcut works in the extracted solution: the working uses this test to match the constant ratio claimed by the main solution and identify the correct option.