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  • Let S be the set of values of  , for which the system of equations <img alt="\begin{aligned} &amp

Let S be the set of values of \lambda , for which the system of equations

\begin{aligned} & 6 \lambda x-3 y+3 z=4 \lambda^2 ,\\ & 2 x+6 \lambda y+4 z=1, \end{aligned}
3 x+2 y+3 \lambda z=\lambda has no solution. Then 12 \sum_{1 \in S}|\lambda| is equal to _________.
 

 

Option: 1

24


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

\begin{aligned} & \Delta=\left|\begin{array}{ccc} 6 \lambda & -3 & 3 \\ 2 & 6 \lambda & 4 \\ 3 & 2 & 3 \lambda \end{array}\right|=0 \\ & 2 \lambda\left(9 \lambda^2-4\right)+(3 \lambda-6)+(2-9 \lambda)=0 \\ & 18 \lambda^3-14 \lambda-4=0 \\ & (\lambda-1)(3 \lambda+1)(3 \lambda+2)=0 \\ & \Rightarrow \lambda=1,-1 / 3,-2 / 3 \end{aligned}

For each values of   \lambda, \Delta_1=\left|\begin{array}{ccc} 6 \lambda & -3 & 4 \lambda^2 \\ 2 & 6 \lambda & 1 \\ 3 & 2 & \lambda \end{array}\right| \neq 0

12\left(1+\frac{1}{3}+\frac{2}{3}\right)=24

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vinayak

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