Probability Question 1
Question 1 - 24 January - Shift 1
Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations
$x+y+z=1$
$2 x+Ny+2 z=2$
$3 x+3 y+Nz=3$
has unique solution is $\frac{k}{6}$, then the sum of value of k and all possible values of N is
(1) 18
(2) 19
(3) 20
(4) 21
Show Answer
Answer: (3)
Solution:
Formula: Consistency of a system of equations, Probability of occurrence of an event
$x+y+z=1$
$2 x+Ny+2 z=2$
$3 x+3 y+Nz=3$
$\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 $=4+1+4+5+6=20$
