- Home
- Standard 12
- Mathematics
3 and 4 .Determinants and Matrices
hard
Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations
$x+y+z=1$ ; $2 x+N y+2 z=2$ ; $3 x+3 y+N z=3$
has unique solution is $\frac{k}{6}$, then the sum of value of $k$ and all possible values of $N$ is
A
$18$
B
$19$
C
$20$
D
$21$
(JEE MAIN-2023)
Solution
$x+y+z=1$
$2 x+N y+2 z=2$
$3 x+3 y+N z=3$
$\Delta=\left|\begin{array}{ccc}1 & 1 & 1 \\2 & N & 2 \\3 & 3 & N\end{array}\right| =( N -2)( N -3)$For unique solution $\Delta \neq 0$
So $N \neq 2,3$
$\Rightarrow P ($ system has unique solution $)=\frac{4}{6}$
So $k =4$
Therefore sum $=4+1+4+5+6=20$
Standard 12
Mathematics
