Let S1 and S2 be respectively the sets of all a ∈ 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

Question

Let S1 and S2 be respectively the sets of all a ∈ 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 (A) S1=Φ and S2=R- 0  (B) S1 is an infinite set and n(S2)=2 (C) S1=R- 0 and S2=Φ (D) n ( S 1)=2 and S 2 is an infinite set

Tardigrade Admin · Accepted Answer

Δ=|a 2 a -3 a 2 a+1 2 a+3 a+1 3 a+5 a+5 a+2| =a(15 a2+31 a+36)=0 ⇒ a=0 Δ ≠ 0 text for all a ∈ R- 0 Hence S1=R- 0 S2=Φ

Q. 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

$S_{1} = Φ$ and $S_{2} = R - {0}$

$S_{1}$ is an infinite set and $n (S_{2}) = 2$

$S_{1} = R - {0}$ and $S_{2} = Φ$

$n (S_{1}) = 2$ and $S_{2}$ is an infinite set

$Δ = ∣ ∣ a 2 a + 1 3 a + 5 2 a 2 a + 3 a + 5 - 3 a a + 1 a + 2 ∣ ∣$
$= a (15 a^{2} + 31 a + 36) = 0 \Rightarrow a = 0$
$Δ \neq = 0 for all a \in R - {0}$
Hence $S_{1} = R - {0} S_{2} = Φ$

Q. Let S1​ and S2​ be respectively the sets of all a∈R−{0} for which the system of linear equations ax+2ay−3az=1 (2a+1)x+(2a+3)y+(a+1)z=2 (3a+5)x+(a+5)y+(a+2)z=3 has unique solution and infinitely many solutions. Then

S1​=Φ and S2​=R−{0}

S1​ is an infinite set and n(S2​)=2

S1​=R−{0} and S2​=Φ

n(S1​)=2 and S2​ is an infinite set