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  • Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations <img alt="\begin{aligned} & x+y+z=1 \\ & 2 x+\mathrm{Ny}+2

Let N denote the number that turns up when a fair die is rolled. If the probability that the system of
equations

\begin{aligned} & x+y+z=1 \\ & 2 x+\mathrm{Ny}+2 z=2 \\ & 3 x+3 y+\mathrm{N} z=3 \end{aligned}

has unique solution is \frac{k}{6}   then the sum of value of k and all possible values of N is

 

Option: 1

21


Option: 2

18


Option: 3

20


Option: 4

19


Answers (1)

best_answer

for unique solu.

\begin{aligned} & \Delta \neq 0 \\ & \left|\begin{array}{ccc} 1 & 1 & 1 \\ 2 & \mathrm{~N} & 2 \\ 3 & 3 & \mathrm{~N} \end{array}\right| \neq 0 \\ & \Rightarrow\left(\mathrm{N}^2-6\right)-(2 \mathrm{~N}-6)+(6-3 \mathrm{~N}) \neq 0 \\ & \Rightarrow \mathrm{N}^2-5 \mathrm{~N}+6 \neq 0 \end{aligned}

\Rightarrow \mathrm{N} \neq 3 \quad \& \quad \mathrm{~N} \neq 2

\text { Hence } N \text { can be }\{1,4,5,6\} \text { Fav case : } \frac{4}{6}=\frac{K}{6} \Rightarrow k=4

\operatorname{sum}=20

 

Posted by

shivangi.shekhar

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