Question

Investigate for what values of $\lambda ,\mu$ the simultaneous equation $x+y+z=6;x+2y+3z=10$ & $x+2y+\lambda z=\mu$ have an infinite number of solutions

• A. $\lambda =4,\mu =11$
• B. $\lambda =3,\mu =10$
• C. $\lambda =2,\mu =8$
• D. $\lambda =1,\mu =11$

Answer 1

Answer: (B)

$\lambda =3,\mu =10$

$\Delta =\left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 2& \lambda \end{array}\right|=1(2\lambda -6)-1(\lambda -3)+1(2-2)=2\lambda -6-\lambda +3+0=\lambda -3{\Delta }_1=\left|\begin{array}{ccc}6& 1& 1\\ 10& 2& 3\\ \mu & 2& \lambda \end{array}\right|=6(2\lambda -6)-1(10\lambda -3\mu )+1(20-2\mu )=12\lambda -36-10\lambda +3\mu +20-2\mu =2\lambda +\mu -16$

For infinite solution $\Delta =0,{\Delta }_1\Rightarrow \lambda =3\Rightarrow \mu =10$