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3 and 4 .Determinants and Matrices
hard
For the system of equations
$x+y+z=6$
$x+2 y+\alpha z=10$
$x+3 y+5 z=\beta$, which one of the following is NOT true?
A
System has a unique solution for $\alpha=3, \beta \neq 14$.
B
System has no solution for $\alpha=3, \beta=24$.
C
System has a unique solution for $\alpha=-3, \beta=14$.
D
System has infinitely many solutions for $\alpha=3$, $\beta=14$.
(JEE MAIN-2023)
Solution
$x+y+z$
$x+2 y+\alpha z=10$
$x+3 y+5 z=\beta$
$D=\left|\begin{array}{lll}1 & 1 & 1 \\ 1 & 2 & \alpha \\ 1 & 3 & 5\end{array}\right|=1(10-3 \alpha)-1(5-\alpha)+1(3-z)$
$=10-3 \alpha-5+\alpha+1$
$=6-2 \alpha$
For unique solution $6-2 \alpha \neq 0 \Rightarrow \alpha \neq 3$
Standard 12
Mathematics
