Phrases Previous Year Questions (PYQs) – Determinants – Page 1 of 2

Given System of Equations:

• $x+2y+2z=5$
• $x+2y+3z=6$
• $x+2y+\lambda z=\mu$

Goal: Find values of $\lambda$ and $\mu$ such that the system has infinitely many solutions

Step 1: Write Augmented Matrix

$[A|B]=\left[\begin{array}{cccc}1& 2& 2& 5\\ 1& 2& 3& 6\\ 1& 2& \lambda & \mu \end{array}\right]$

Step 2: Row operations: Subtract $R_1$ from $R_2$ and $R_3$

$\Rightarrow \left[\begin{array}{cccc}1& 2& 2& 5\\ 0& 0& 1& 1\\ 0& 0& \lambda -2& \mu -5\end{array}\right]$

Step 3: For infinitely many solutions, rank of coefficient matrix = rank of augmented matrix < number of variables (3)

This happens when the third row becomes all zeros:

$\lambda -2=0\text{and}\mu -5=0$

$\Rightarrow \lambda =2,\mu =5$

✅ Final Answer: $\boxed{{\displaystyle \lambda =2,\mu =5}}$