JEE 2023 — Mathematics PYQ

To solve this, we evaluate the determinant of the coefficient matrix, $D$.

1. Find the Main Determinant ($D$)

Expanding along the first row:

• $D=1(3\beta -14)-1(3\alpha -7)+1(2\alpha -\beta )$

• $D=3\beta -14-3\alpha +7+2\alpha -\beta$

• $D=2\beta -\alpha -7$

2. Evaluate the Options

• Option (1): If $\alpha =\beta$ and $\alpha \ne 7$.

  • Substituting $\beta =\alpha$ into $D$: $D=2\alpha -\alpha -7=\alpha -7$.

  • Since $\alpha \ne 7$, then $D\ne 0$.

  • When $D\ne 0$, the system has a unique solution. This statement is TRUE.

• Option (2): If $\alpha =\beta =7$.

  • Substituting these into $D$: $D=2(7)-7-7=0$.

  • Now check $D_x$ (the determinant where the $x$-column is replaced by constants):

    $$D_x=\left|\begin{array}{ccc}6& amp;1& amp;1\\ 3& amp;7& amp;7\\ 14& amp;2& amp;3\end{array}\right|=6(21-14)-1(9-98)+1(6-98)=42+89-92=39$$

  • Since $D=0$ and $D_x\ne 0$, the system has no solution. This statement is TRUE.

• Option (3): For points on the line $x-2y+7=0$.

  • The condition for $D=0$ is $2\beta -\alpha -7=0$, which is equivalent to $x-2y+7=0$ where $x=\alpha$ and $y=\beta$.

  • For infinitely many solutions, we need $D=0$ AND $D_x=D_y=D_z=0$.

  • We already found in Option (2) that at $(7,7)$, $D_x\ne 0$. In fact, $D_x=0$ only for a specific point on that line, not for every point. This statement is FALSE.

• Option (4): Unique point on $x+2y+18=0$.

  • Infinite solutions require $D=0$ and $D_x=0$. This gives us a specific intersection point for $(\alpha ,\beta )$.

  • Since the line $x+2y+18=0$ is not parallel to the line $D=0$, they will intersect at exactly one point. This statement is TRUE.

Correct Answer (The statement that is NOT true): (3)