Determinant: For the system of linear equations 2x-y+3z=5 3x+2y-z=7 4x+5y+alpha… | JEE Main 2023

JEE Main 2023 (Online) 10th April Morning ShiftQuestion 1

For the system of linear equations $2x-y+3z=5$ $3x+2y-z=7$ $4x+5y+\alpha z=\beta$, which of the following is NOT correct?

Determinant

The system is inconsistent for $\alpha=-5$ and $\beta=8$The system has a unique solution for $\alpha\neq-5$ and $\beta=8$The system has infinitely many solutions for $\alpha=-5$ and $\beta=9$The system has infinitely many solutions for $\alpha=-6$ and $\beta=9$✓ Correct answer

Solution

Determinant $=7(\alpha+5)$. $\alpha=-5$: rank $A=2$; rank $[A|b]=3$ for $\beta=8$ (inconsistent, A true) and $2$ for $\beta=9$ (infinite, C true). $\alpha\ne-5$: unique solution (B true). $\alpha=-6$ gives determinant $-7\neq0$, so a unique solution, not infinitely many. NOT correct: (D).

Determinant - JEE Mathematics question (JEE Main 2023 (Online) 10th April Morning Shift)

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