JEE MAIN - Mathematics (2021 - 26th August Evening Shift - No. 9)
Two fair dice are thrown. The numbers 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 :
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 :
$$p = {1 \over 6}$$ and $$q = {1 \over 36}$$
$$p = {5 \over 6}$$ and $$q = {5 \over 36}$$
$$p = {5 \over 6}$$ and $$q = {1 \over 36}$$
$$p = {1 \over 6}$$ and $$q = {5 \over 36}$$
Explanation
$$D \ne 0 \Rightarrow \left| {\matrix{
1 & 1 & 1 \cr
1 & 2 & 3 \cr
1 & 3 & \lambda \cr
} } \right| \ne 0 \Rightarrow \lambda \ne 5$$
For no solution D = 0 $$\Rightarrow$$ $$\lambda$$ = 5
$${D_1} = \left| {\matrix{ 1 & 1 & 5 \cr 1 & 2 & \mu \cr 1 & 3 & 1 \cr } } \right| \ne 0 \Rightarrow \mu \ne 3$$
$$p = {5 \over 6}$$
$$q = {1 \over 6} \times {5 \over 6} = {5 \over {36}}$$
Option (b).
For no solution D = 0 $$\Rightarrow$$ $$\lambda$$ = 5
$${D_1} = \left| {\matrix{ 1 & 1 & 5 \cr 1 & 2 & \mu \cr 1 & 3 & 1 \cr } } \right| \ne 0 \Rightarrow \mu \ne 3$$
$$p = {5 \over 6}$$
$$q = {1 \over 6} \times {5 \over 6} = {5 \over {36}}$$
Option (b).
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