JEE Mathematics 2025 Question with Solution

Given:

• $M$ is the set of all real matrices of order $3 \times 3$.
• $S = \{-3,-2,-1,1,2\}$.
• the solution states the final result as Correct Answer: $125$.

Find: $\alpha$.

From the solution:

$$|S_1| = 5^6$$ $$|S_2| = 0$$

and the union count is written as

$$n(S_1 \cup S_2 \cup S_3) = |S_1| + |S_2| + |S_3| - (\text{intersections}) = 125$$

Therefore, according to the provided the solution, the required value is $125$.

Conclude: $\alpha = 125$.

Given: The solution is incomplete and internally inconsistent as a full derivation, but it explicitly displays Correct Answer: $125$.

Identify principle: The page discusses counting matrices in $S_1$, $S_2$, and $S_3$ using symmetry conditions and the trace condition, then applying inclusion-exclusion.

Observed working from HTML:

1. For symmetric matrices, the page states

$$|S_1| = 5^6$$

2. For skew-symmetric matrices, the page states diagonal entries would have to be $0$, which is not in $S$, so

$$|S_2| = 0$$

3. It then mentions using inclusion-exclusion for

$$n(S_1 \cup S_2 \cup S_3)$$

and concludes the value is $125$.

Because no valid, complete derivation for $\alpha$ is present in the supplied HTML beyond that explicit conclusion, the answer must be taken from the solution's stated result.

Conclude: The correct numerical answer extracted from the source is $125$.