Question

Match the following for system of linear equations
2x -3y + 5z =12
$3x+y+\lambda z=\mu$
x -7y + 8z =17
$\begin{array}{cccc}& \text{Column - I}& & \text{Column - I}\\ \left(P\right)& \text{Unique solution}& \left(1\right)& \lambda =2,\mu =7\\ \left(Q\right)& \text{Infinite solution}& \left(2\right)& \lambda \ne 2,\mu =7\\ \left(R\right)& \text{No solution}& \left(3\right)& \lambda \ne 2,\mu \ne 7\\ \left(S\right)& \text{Consistent system}& \left(4\right)& \lambda \in R,\mu \ne 7\\ & \text{equation}& & \\ & & \left(5\right)& \lambda =2,\mu \ne 7\end{array}$

• A. $P\to 2;Q\to 1;R\to 5;S\to 1,2$
• B. $P\to 2,3;Q\to 1;R\to 5;S\to 1,2,3$
• C. $P\to 2,3;Q\to 1;R\to 4,5;S\to 1,2,3$
• D. $P\to 3;Q\to 1;R\to 4,5;S\to 2,3$

Answer 1

The correct option is B $P\to 2,3;Q\to 1;R\to 5;S\to 1,2,3$

$D=\left|\begin{array}{ccc}2& -3& 5\\ 3& 1& \lambda \\ 1& -7& 8\end{array}\right|=88-110+11\lambda =(11\lambda -22)$
$D_1=\left|\begin{array}{ccc}12& -3& 5\\ \mu & 1& \lambda \\ 17& -7& 9\end{array}\right|=33\lambda =11\mu +1=11(3\lambda -\mu +1)$
$D_2=\left|\begin{array}{ccc}2& 12& 5\\ 3& \mu & \lambda \\ 1& 17& 8\end{array}\right|=11\mu -22\lambda -33=11(\mu -2\lambda -3)$
$D_3=\left|\begin{array}{ccc}2& -3& 12\\ 3& 1& \mu \\ 1& -7& 17\end{array}\right|=11(\mu -7)$
For unique solve $\to D\ne 0$
$\lambda \ne 2,\mu \in R$
For infinte solution, $D=0,D_1=0,D_2=0,D_3=0$
$\lambda =2,3\lambda -\mu +1=0,\mu -2\lambda -3=0$
$\mu =7$
So $\lambda =2\&\mu =7$
For no solution $\to \lambda =2,\mu \ne 7$
Unique solution and infinite solution make consistent system.