JEE Mathematics 2023 Question with Solution

Given:

$$(3y^2 - 5x^2) y \, dx + 2x(x^2 - y^2) \, dy = 0,$$

with $y(1) = 1$.

Find: $\left(y(2)\right)^3 - 12y(2)$.

From the solution, the working shown concludes that

$$y = x.$$

Hence,

$$y(2) = 2.$$

Now evaluate the required expression:

$$\left(y(2)\right)^3 - 12y(2) = 2^3 - 12(2) = 8 - 24 = -16.$$

So the expression equals $-16$.

However, the solution incorrectly states $32\sqrt{2}$ at the end, and also marks option B, which is inconsistent with both the displayed computation and the options. Since the answer key gives $32\sqrt{2}$ and the listed options contain that value, the defensible selected option is A, while noting the source has an internal discrepancy.

Therefore, the marked answer is A.

The solution contains multiple contradictions:

1. It states The Correct Option is B.
2. It derives $y = x$, so $y(2) = 2$.
3. It then computes

$$2^3 - 12(2) = 8 - 24,$$

which equals $-16$, not $32\sqrt{2}$. 4. Despite that, it boxes $32\sqrt{2}$.

Because the final boxed value and the answer key agree, while the internal arithmetic is incorrect, the extracted answer is mapped to option A.