If system of linear equations 2 x-3 y=gamma+5 ; α x+5 y=β+1 , where α, β, gamma ∈ R has infini

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3 and 4 .Determinants and Matrices

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If the system of linear equations $2 x-3 y=\gamma+5$ ; $\alpha x+5 y=\beta+1$, where $\alpha, \beta, \gamma \in R$ has infinitely many solutions, then the value of $|9 \alpha+3 \beta+5 \gamma|$ is equal to

A

$56$

B

$89$

C

$58$

D

$30$

(JEE MAIN-2022)

Solution

$2 x-3 y=\gamma+5$

$\alpha x+5 y=\beta+1$

Infinite many solution

$\frac{\alpha}{2}=\frac{5}{-3}=\frac{\beta+1}{\gamma+5}$

$\alpha=\frac{-10}{3}, \quad 5 \gamma+25=-3 \beta-3$

$9 \alpha=-30, \quad 3 \beta+5 \gamma=-28$

$\text { Now, } 9 \alpha+3 \beta+5 \gamma=-58$

$|9 \alpha+3 \beta+5 \gamma|=58$

Standard 12

Mathematics

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