### 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} $S_2=\Phi$