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CBSE Class 12 Math 2023 All Sets Solved Paper

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Question : 17 of 20
Marks: +1, -0
If (1212313a1)\begin{pmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & a & 1 \end{pmatrix}is a non-singular matrix and a ∈\in A , then the set A is:
Solution:  
Let P = (1212313a1)\begin{pmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & a & 1 \end{pmatrix}
| P | = ∣1212313a1∣=∣1211102a−20∣\begin{vmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & a & 1 \end{vmatrix} = \begin{vmatrix} 1 & 2 & 1 \\ 1 & 1 & 0 \\ 2 & a-2 & 0 \end{vmatrix} (R2→R2→R1R3→R3−R1)\begin{pmatrix} R_2 \rightarrow R_2 \rightarrow R_1 \\ R_3 \rightarrow R_3 - R_1 \end{pmatrix}
⇒∣P∣=a−4\Rightarrow |P| = a - 4
But | P | ≠\neq 0 ⇒\Rightarrow a ≠\neq 4
∴a∈R−{4}\therefore a \in \mathbb{R} - \{4\}
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