Let S 1 and S 2 be respectively the sets of all a R - { 0 } for which the system of linear equations 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 has a unique solution and

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Solution

Q.
Let $S_{1}$ and $S_{2}$ be respectively the sets of all $a \in R - {0}$ for which the system of linear equations

$a x + 2 a y - 3 a z = 1$

$(2 a + 1) x + (2 a + 3) y + (a + 1) z = 2$

$(3 a + 5) x + (a + 5) y + (a + 2) z = 3$

has a unique solution and infinitely many solutions. Then [2023]

1 $S_{1} = Φ$ and $S_{2} = R - {0}$

2 $S_{1}$ is an infinite set and $n (S_{2}) = 2$

3 $S_{1} = R - {0}$ and $S_{2} = Φ$

4 $n (S_{1}) = 2$ and $S_{2}$ is an infinite set.

Ans.
(3)

$| A |$ $=$ $| \begin{matrix} a & 2 a & - 3 a \\ 2 a + 1 & 2 a + 3 & a + 1 \\ 3 a + 5 & a + 5 & a + 2 \end{matrix} |$

$= a \times$ $| \begin{matrix} 1 & 2 & - 3 \\ 2 a + 1 & 2 a + 3 & a + 1 \\ 3 a + 5 & a + 5 & a + 2 \end{matrix} |$

Applying $(C_{2} \to C_{2} - C_{1})$ , we get

$| A |$ $= a \times$ $| \begin{matrix} 1 & 1 & - 3 \\ 2 a + 1 & 2 & a + 1 \\ 3 a + 5 & - 2 a & a + 2 \end{matrix} |$

$= a \times$ $| \begin{matrix} 1 & 1 & - 3 \\ - a - 4 & 2 + 2 a & - 1 \\ 3 a + 5 & - 2 a & a + 2 \end{matrix} |$ Applying $(R_{2} \to R_{2} - R_{3})$

$= a \times [2 a^{2} + 6 a + 4 - 2 a - 3 a - 5 + (a + 4) (a + 2) - 3 [(a + 4) 2 a - (2 + 2 a) (3 a + 5)]]$

$= a \times (2 a^{2} + 6 a + 4 - 2 a - 3 a - 5 + a^{2} + 6 a + 8) - 3 (2 a^{2} + 8 a - 6 a^{2} - 16 a - 10)$

$| A |$ $= a (15 a^{2} + 31 a + 37)$

$| A |$ $\neq 0 \forall a \in R as 15 a^{2} + 31 a + 37 \neq 0 \forall a \in R & a \neq 0$

$S_{1} = R - {0}$

Since $| A |$ can never be 0.

There cannot be infinitely many solutions for any value of $a$ .

$∴$ $S_{2} = ϕ$

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