For the system of linear equations 2 x + 4 y + 2 a z = b , x + 2 y + 3 z = 4 , 2 x - 5 y + 2 z = 8 which of the following is NOT correct? [2023]

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Solution

Q.
For the system of linear equations

$2 x + 4 y + 2 a z = b,$

$x + 2 y + 3 z = 4,$

$2 x - 5 y + 2 z = 8$

which of the following is NOT correct [2023]

1 It has infinitely many solutions if $a = 3, b = 8$

2 It has infinitely many solutions if $a = 3, b = 6$

3 It has unique solution if $a = b = 8$

4 It has unique solution if $a = b = 6$

Ans.
(2)

We have, system of linear equations

$2 x + 4 y + 2 a z = b$

$x + 2 y + 3 z = 4$

$2 x - 5 y + 2 z = 8$

It can be written as $A X = B$ ,

where, $A = [\begin{matrix} 2 & 4 & 2 a \\ 1 & 2 & 3 \\ 2 & - 5 & 2 \end{matrix}], B = [\begin{matrix} b \\ 4 \\ 8 \end{matrix}], X = [\begin{matrix} x \\ y \\ z \end{matrix}]$

$| A |$ $= 2 (4 + 15) - 4 (2 - 6) + 2 a (- 5 - 4) = 54 - 18 a$

If $| A |$ $= 0$ , then $a = 3$ and the system has infinitely many solutions.

If $| A |$ $\neq 0$ , then $a \neq 3$ and the system has a unique solution.

$A_{11} = 19, A_{12} = 4, A_{13} = - 9, A_{21} = - 38, A_{22} = - 8, A_{23} = 18, A_{31} = 0, A_{32} = 0, A_{33} = 0$

$adj A = [\begin{matrix} A_{11} & A_{21} & A_{31} \\ A_{12} & A_{22} & A_{32} \\ A_{13} & A_{23} & A_{33} \end{matrix}] = [\begin{matrix} 19 & - 38 & 0 \\ 4 & - 8 & 0 \\ - 9 & 18 & 0 \end{matrix}]$

For infinitely many solutions:

$(adj A) B = O \Rightarrow [\begin{matrix} 19 & - 38 & 0 \\ 4 & - 8 & 0 \\ - 9 & 18 & 0 \end{matrix}] [\begin{matrix} b \\ 4 \\ 8 \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$

$\Rightarrow 19 b - 152 = 0 \Rightarrow b = 8$

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