Matrices & Determinants
System of Equations — Infinite Solutions (Cramer's Rule)
nta_pyq_2024_jan
Grade 12
Question:
Let the system of equations $x+2y+3z=5$, $2x+3y+z=9$, $4x+3y+\lambda z=\mu$ have infinite number of solutions. Then $\lambda+2\mu$ is equal to:
Step-by-Step Solution
Key Concept: For infinite solutions, $\Delta=0$ and $\Delta_1=\Delta_2=\Delta_3=0$. Set up and solve the determinant conditions to find $\lambda$ and $\mu$.
$\Delta=0\Rightarrow\lambda=-13$. $\Delta_1=\begin{vmatrix}5&2&3\\9&3&1\\\mu&3&-13\end{vmatrix}=0\Rightarrow\mu=15$. Check $\Delta_2,\Delta_3=0$ — satisfied. $\lambda+2\mu=-13+30=17$.
Correct Answer: 2