JEE Main 2021 — Mathematics Algebra

$$|\begin{matrix}1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & \lambda \end{matrix}|=\lambda -5$$∣111​123​13λ​∣=λ−5

$${\Delta }_{1}=|\begin{matrix}1 & 1 & 5 \\ 2 & 3 & \mu \\ 3 & \lambda & 1\end{matrix}|=10\lambda +3\mu -\lambda \mu -44$$Δ1​=∣123​13λ​5μ1​∣=10λ+3μ−λμ−44

$${\Delta }_{2}=|\begin{matrix}1 & 1 & 5 \\ 1 & 3 & \mu \\ 1 & \lambda & 1\end{matrix}|=5\lambda +\mu -\lambda \mu -13$$Δ2​=∣111​13λ​5μ1​∣=5λ+μ−λμ−13

$${\Delta }_{3}=|\begin{matrix}1 & 1 & 5 \\ 1 & 2 & \mu \\ 1 & 3 & 1\end{matrix}|=-2\mu +6$$Δ3​=∣111​123​5μ1​∣=−2μ+6

For unique solution

$$\lambda \neq 5,\mu \in {1,2,3,4,5,6}$$λ=5,μ∈1,2,3,4,5,6

$$\Rightarrow p=\frac{5}{6}\times 1=\frac{5}{6}$$⇒p=65​×1=65​

For No Solution

$$\lambda =5,\mu \neq 3$$λ=5,μ=3

$$\Rightarrow q=\frac{1}{6}\times \frac{5}{6}=\frac{5}{36}$$⇒q=61​×65​=365​