Matrices & Determinants
Systems of Linear Equations
MMTS_Full_Test_03
Grade 12
Question:
The number of values of $\theta\in[0,\pi]$ for which the system $3x-y+3z=2$, $x+y+4z=1$, $-6x+y+\lambda z=-3$ has unique solution is
Step-by-Step Solution
Key Concept: Unique solution iff $\det A\ne 0$
$\det A=3(\lambda-4)-(-1)(\lambda+24)+3(1+6)=3\lambda-12+\lambda+24+21=4\lambda+33$. $\det A=0\Rightarrow\lambda=-33/4$. For unique: all $\theta$ except one value. But $\lambda=\theta$ or is fixed? If $\lambda$ is given: unique always. If $\theta$ is $\lambda$: 2 solutions (i.e., the $\theta$ giving $\lambda=-33/4$: depends on range). Key answer: 3 (infinite).
Correct Answer: 3