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  • Let  and <img alt="\mathrm{S}_{2}" src="https://entrancecorner.oncodecogs.com/gi

Let \mathrm{S}_{1} and \mathrm{S}_{2} be respectively the sets of all a \in \mathbb{R}-\{0\} for which the system of linear equations 
\begin{aligned} & a x+2 a y-3 a z=1 \\ & (2 a+1) x+(2 a+3) y+(a+1) z=2 \\ & (3 a+5) x+(a+5) y+(a+2) z=3\end{aligned}
has unique solution and infinitely many solutions. Then

Option: 1

S_{1} is an infinite set and n\left(S_{2}\right)=2


Option: 2

\mathrm{S}_{1}=\Phi and \mathrm{S}_{2}=\mathbb{R}-\{0\}


Option: 3

n\left(S_{1}\right)=2 and S_{2} is an infinite set 


Option: 4

S_{1}=\mathbb{R}-\{0\} and S_{2}=\Phi


Answers (1)

best_answer

\Delta=\left|\begin{array}{ccc} a & 2 a & -3 a \\ 2 a+1 & 2 a+3 & a+1 \\ 3 a+5 & a+5 & a+2 \end{array}\right|

\Delta=a\left(15 a^{2}+31 a+36\right)=0
\quad a=0
\Delta \neq 0 \text { for all } a \in \mathbb{R}-\{0\}
\therefore S_{1}=\mathbb{R}-\{0\}, \mathrm{S}_{2}=\Phi

Posted by

Divya Prakash Singh

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