Question

Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations
$x+y+z=1$
$2x+Ny+2z=2$
$3x+3y+Nz=3$
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)

$x+y+z=1$
$2x+Ny+2z=2$
$3x+3y+Nz=3$
$\Delta =\left|\begin{array}{ccc}1& 1& 1\\ 2& N& 2\\ 3& 3& N\end{array}\right|$
$=(N-2)(N-3)$
For unique solution $\Delta \ne 0$
So $N\ne 2,3$
$\Rightarrow P($ system has unique solution $)=\frac{4}{6}$
So $k=4$
Therefore sum $=4+1+4+5+6=20$