Probability: Two fair dice are thrown. The numbers on them are taken as lambda… | JEE Main 2021

JEE Main 2021 (Online) 26th August Evening ShiftQuestion 1

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 :

Probability

$p=\frac{1}{6}$ and $q=\frac{1}{36}$$p=\frac{5}{6}$ and $q=\frac{1}{36}$$p=\frac{1}{6}$ and $q=\frac{5}{36}$$p=\frac{5}{6}$ and $q=\frac{5}{36}$✓ Correct answer

Solution

$\Delta=\begin{vmatrix}1&1&1\\1&2&3\\1&3&\lambda\end{vmatrix}=\lambda-5$. Unique solution iff $\lambda\ne5$: $p=\frac56$. If $\lambda=5$: $R_2-R_1$: $y+2z=\mu-5$; $R_3-R_1$: $2y+4z=-4\Rightarrow y+2z=-2$. No solution iff $\mu\ne3$, i.e. 5 values of $\mu$: $q=\frac5{36}$. Option D.

Probability - JEE Mathematics question (JEE Main 2021 (Online) 26th August Evening Shift)

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