Determinants Question 2
Question 2 - 25 January - Shift 1
Let $S_1$ and $S_2$ be respectively the sets of all
$a \in R-{0}$ for which the system of linear equations
$a x+2 a y-3 a z=1$
$(2 a+1) x+(2 a+3) y+(a+1) z=2$
$(3 a+5) x+(a+5) y+(a+2) z=3$
has unique solution and infinitely many solutions. Then
(1) $n(S_1)=2$ and $S_2$ is an infinite set
(2) $S_1$ is an infinite set an $n(S_2)=2$
(3) $S_1=\Phi$ and $S_2=\mathbb{R}-{0}$
(4) $S_1=\mathbb{R}-{0}$ and $S_2=\Phi$
Show Answer
Answer: (4)
Solution:
Formula: System of equations with 3 variables, consistency of solutions
$\Delta= \begin{vmatrix} a & 2 a & -3 a \\ 2 a+1 & 2 a+3 & a+1 \\ 3 a+5 & a+5 & a+2\end{vmatrix} $
$=a(15 a^{2}+31 a+36)=0 \Rightarrow a=0$
$\Delta \neq 0$ for all $a \in R-{0}$ Hence $S_1$=R-{0} $S_2=\Phi$
