A) \[1\]
B) \[2\]
C) \[-1\]
D) \[3\]
Correct Answer: D
Solution :
Key Idea: The system of equations\[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z={{d}_{1}},\]\[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z={{d}_{2}}\]\[{{a}_{3}}x+{{b}_{3}}y+{{c}_{3}}z={{d}_{3}}\]are consistent, if \[\left| \begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\ \end{matrix} \right|=0\] The given system of equations\[x+y+z=6,\,\,x+2y+3z=10\],\[x+2y+\lambda z=10\]are consistent, if \[\left| \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 2 & \lambda \\ \end{matrix} \right|=0\] \[\Rightarrow \] \[1(2\lambda -6)-1(\lambda -3)+1(2-2)=0\] \[\Rightarrow \] \[\lambda -3=0\] \[\Rightarrow \] \[\lambda =3\]
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