Let N denote the number that turns up when a fair die is rolled.

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 a unique solution is $\dfrac{k}{6}$, then the sum of $k$ and all possible values of $N$ is

Accepted Answer

For unique solution $\Delta 
eq 0$

i.e. $\left|\begin{array}{ccc} 1 & 1 & 1 \ 2 & N & 2 \ 3 & 3 & N \end{array}ight| 
eq 0$

$\Rightarrow\left(N^{2}-6ight)-(2 N-6)+(6-3 N) 
eq 0$

$\Rightarrow N^{2}-5 N+6 
eq 0$

$	herefore N 
eq 2$ and $N 
eq 3$

$	herefore $ Probability of not getting 2 or 3 in a throw of dice $=\frac{2}{3}$

As given $\frac{2}{3}=\frac{k}{6} \Rightarrow k=4$

$	herefore$ Required value $=1+4+5+6+4=20$

Solutions · 1