Find the condition for which system of
ax + 4y + z = 0
2y + 3z – 1 = 0
3x – bz + 2 = 0 has
(i) Unique sol. (ii) Infinite sol (iii) No sol.
Text Solution
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Sol. Given system
ax + 4y + z = 0
0x + 2y + 3z = 1
3x + 0y – bz = –2
Δ = 
Δ = a(–2b) –4(–9) + 1(–6)
Δ = –2ab + 30
(i) For unique sol.
Δ ≠ 0
⇒ –2ab + 30 ≠ 0
ab – 15 ≠ 0 ⇒ ab ≠ 15
(ii) For Infinite Sol.
Δ = Δ x = Δ y = Δ z = 0
Δ = 0 ⇒ ab = 15 ....(i)
Δ x = 
= 0
⇒ 0 – 4(–b + 6) + 1(+4) = 0
–b + 6 – 1 = 0 ⇒ b = 5
& Δ y = 
= 0
⇒ a(–b + 6) + 1(–3) = 0 ⇒ –ab + 6a – 3 = 0 &
b = +5 ⇒ –5a + 6a – 3 = 0 ⇒ a = 3
& Δ z = 
= 0 ⇒ a(–4) –4(–3) = 0
⇒ –4a + 12 = 0 ⇒ a = 3
(iii) For no. Sol. ab = 15 but a ≠ 3 or b ≠ 5
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