Matrices & Determinants
System of Equations — Infinitely Many Solutions
nta_pyq_2024_jan
Grade 12
Question:
If the system of linear equations $x-2y+z=-4$, $2x+\alpha y+3z=5$, $3x-y+\beta z=3$ has infinitely many solutions, then $12\alpha+13\beta$ is equal to
Step-by-Step Solution
Key Concept: For infinite solutions, $D=D_1=D_2=D_3=0$. Solve for $\alpha$ and $\beta$ from two of the determinant conditions, then compute $12\alpha+13\beta$.
$D=0\Rightarrow\alpha\beta-3\alpha+4\beta=17$. $D_2=\begin{vmatrix}1&-4&1\\2&5&3\\3&3&\beta\end{vmatrix}=0\Rightarrow13\beta-9-36-9=0\Rightarrow\beta=\frac{54}{13}$. Substituting: $\frac{54\alpha}{13}-3\alpha+\frac{216}{13}=17\Rightarrow15\alpha=5\Rightarrow\alpha=\frac{1}{3}$. $12\alpha+13\beta=4+54=58$.
Correct Answer: 4