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  • Two pair dice are thrown. The number on them are taken as and a system of linear equations <img alt="x+y+z= 5" src

Two pair dice are thrown. The number on them are taken as \lambda \, and\, \mu , and a system of linear equations
x+y+z= 5
x+2y+3z= \mu
x+3y+\lambda z= 1
is constructed, If p is the probability that the system has a unique solution and q is the probability that the system has no solution, then :
Option: 1 p= \frac{1}{6}\, and\, q= \frac{5}{36}
Option: 2 p= \frac{5}{6}\, and\, q= \frac{1}{36}
Option: 3 p= \frac{1}{6}\, and\, q= \frac{1}{36}
Option: 4 p= \frac{5}{6}\, and\, q= \frac{5}{36}

Answers (1)

best_answer

For unique solution \Delta \neq 0

\Rightarrow \begin{vmatrix} 1& 1 &1 \\ 1 &2 &3 \\ 1& 3& \lambda \end{vmatrix}\neq0
\Rightarrow \lambda\neq 5
\therefore p= \frac{5}{6}

For no solution

\Delta = 0 \Rightarrow \lambda= 5 \: \: and
\Delta_{1} \neq 0 \Rightarrow \begin{vmatrix} 1 & 1 &5 \\ 1& 2 &\mu \\ 1& 3 & 1 \end{vmatrix} \neq 0\Rightarrow \mu\neq3
\therefore q= \frac{1}{6}\cdot \frac{5}{6}= \frac{5}{36}

Posted by

Kuldeep Maurya

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