JEE Mathematics 2023 Question with Solution

Given:

• The system is

Find: Which statement is NOT correct.

Using the determinant of the coefficient matrix,

So the system has a unique solution when $\Delta \neq 0$, that is, when $\alpha \neq -5$.

When $\alpha = -5$, compute the sub-determinants:

Now analyze the cases:

• If $\alpha = -5$ and $\beta = 8$, then $\Delta = 0$ but at least one of $\Delta_1, \Delta_2, \Delta_3$ is non-zero, so the system is inconsistent. Hence option A is correct.
• If $\alpha = -5$ and $\beta = 9$, then $\Delta = \Delta_1 = \Delta_2 = \Delta_3 = 0$, so the system has infinitely many solutions. Hence option D is correct.
• If $\alpha \neq -5$, then $\Delta \neq 0$, so the system has a unique solution. Hence option C is correct.

For option B, the given values are $\alpha = -6$ and $\beta = 9$. Then

so the system has a unique solution, not infinitely many solutions.

Therefore, the statement that is NOT correct is option C according to the solution. There is a discrepancy because the working shows option B is false, but the solution explicitly states The Correct Option is C. the correct option is C.

Given:

Find: Which option is not correct.

From the first two equations, eliminate variables to reduce the system. The extracted second approach in the solution states that the system reduces to

Hence:

• unique solution if $\alpha - 2\beta + 7 \neq 0$
• no solution if $\alpha - 2\beta + 7 = 0$ and $2\alpha - 8\beta + 3 \neq 0$
• infinitely many solutions if both are zero.

Substitute $\alpha = -6$ and $\beta = 9$:

So the system has a unique solution. This again shows that option B is false by the working.

However, the solution headline explicitly says The Correct Option is C. Therefore, the extracted answer must follow the solution conclusion, while noting the inconsistency between the heading and the algebraic working.