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  • If the system of equations <img alt="\mathrm{\begin{aligned} &x+y+z=6 \\ &2 x+5 y+\alpha z=\beta \\ &x+2 y+3 z=14 \end{aligned} }" src="/latex-image/?%5Cmathrm%7B%5Cbegin%7Balign

If the system of equations
\mathrm{\begin{aligned} &x+y+z=6 \\ &2 x+5 y+\alpha z=\beta \\ &x+2 y+3 z=14 \end{aligned} }
has infinitely many solutions, then \alpha+\beta  is equal to

Option: 1

8


Option: 2

36


Option: 3

44


Option: 4

48


Answers (1)

best_answer

\text{For infinite solutions} \\ \begin{aligned} &\Delta=0 \\ \Rightarrow &\left|\begin{array}{lll} 1 & 1 & 1 \\ 2 & 5 & \alpha \\ 1 & 2 & 3 \end{array}\right|=0 \\ \Rightarrow & (15-2 \alpha)-1(6-\alpha)+(4-5)=0 \\ \Rightarrow & 15-2^{\alpha}-6+\alpha-1=0 \\ \Rightarrow & \alpha=8 \end{aligned} \\ \text{Also } \Delta \mathrm{z=0}\\ \begin{aligned} &\Rightarrow\left|\begin{array}{ccc} 1 & 1 & 6 \\ 2 & 5 & \beta \\ 1 & 2 & 14 \end{array}\right|=0 \\ &\Rightarrow(70-2 \beta)-1(28-\beta)+6(4-5)=0 \\ &\Rightarrow 36-\beta=0 \\ &\Rightarrow \beta=36 \\ &=\alpha+\beta=44 \\ &\therefore \text { option (c) } \end{aligned}

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shivangi.bhatnagar

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