If the system of equations 2 x + 3 y - z = 5 , x + y + 3 z = - 4 , 3 x - y + z = 7 has infinitely many solutions, then 13 is equal to _____ . [2024]

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Solution

Q.
If the system of equations $2 x + 3 y - z = 5,$ $x + α y + 3 z = - 4,$ $3 x - y + β z = 7$ has infinitely many solutions, then $13 α β$ is equal to _____ . [2024]

1 1210

2 1110

3 1220

4 1120

Ans.
(4)

Given, system of equations can be written as $A X = B,$

where $A = [\begin{matrix} 2 & 3 & - 1 \\ 1 & α & 3 \\ 3 & - 1 & β \end{matrix}], B = [\begin{matrix} 5 \\ - 4 \\ 7 \end{matrix}]$ and $X = [\begin{matrix} x \\ y \\ z \end{matrix}]$

Using Cramer's rule for infinite solutions,

$D = D_{1} = D_{2} = D_{3} = 0$

$D_{3} =$ $| \begin{matrix} 2 & 3 & 5 \\ 1 & α & - 4 \\ 3 & - 1 & 7 \end{matrix} |$ $= 2 (7 α - 4) - 3 (7 + 12) + 5 (- 1 - 3 α) = 0$

$\Rightarrow α = - 70$

Similarly,

$D_{2} =$ $| \begin{matrix} 2 & 5 & - 1 \\ 1 & - 4 & 3 \\ 3 & 7 & β \end{matrix} |$ $= 2 (- 4 β - 21) - 5 (β - 9) - 1 (7 + 12) = 0$

$\Rightarrow β = - \frac{16}{13}$

$∴$ $13 α β = 13 \times (- 70) \times (\frac{- 16}{13}) = 1120$

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