Question

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 + Ny +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

Answer 1

Answer: (C)

$ x+y+z=1$
$ 2 x+N y+2 z=2$
$ 3 x+3 y+N z=3$
$\Delta=\begin{vmatrix}1 & 1 & 1 \\2 & N & 2 \\3 & 3 & N\end{vmatrix}$
$ =( 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 $=4+1+4+5+6=20$