- Home
- Standard 12
- Mathematics
3 and 4 .Determinants and Matrices
medium
If the system of equations $\alpha x+y+z=5, x+2 y+$ $3 z=4, x+3 y+5 z=\beta$ has infinitely many solutions, then the ordered pair $(\alpha, \beta)$ is equal to:
A
$(1,-3)$
B
$(-1,3)$
C
$(1,3)$
D
$(-1,-3)$
(JEE MAIN-2022)
Solution
For infinitely many solutions,
$\Delta=0=\Delta_{ x }=\Delta_{ y }=\Delta_{ z }$
$\Delta=\left|\begin{array}{lll}\alpha & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 5\end{array}\right|=0$
$\Rightarrow \alpha(10-9)-1(5-3)+1(3-2)=0$
$\Rightarrow \alpha-2+1=0$
$\Rightarrow \alpha=1$
$\Delta_{x}=\left|\begin{array}{lll}5 & 2 & 3 \\ \beta & 3 & 5\end{array}\right|=0$
$\Rightarrow 5(10-9)-1(20-3 \beta)+1(12-2 \beta)$
$\Rightarrow 5-20+3 \beta+12-2 \beta$
$\Rightarrow-3+\beta=0$
$\Rightarrow \beta=3$
Standard 12
Mathematics
