If system of linear equations (a-1) x+z=α, x+(b-1) y=β and y+(c-1) z=γ where a, b, c ∈ I does not have a unique solution, then maximum possible value of |a+b+c| is

Question

If system of linear equations (a-1) x+z=α, x+(b-1) y=β and y+(c-1) z=&gamma; where a, b, c ∈ I does not have a unique solution, then maximum possible value of |a+b+c| is (A) 0 (B) 1 (C) 3 (D) 4

Tardigrade Admin · Accepted Answer

(a-1) x+0 y+z=α ....(1) x+(b-1) y+0 z=β ....(2) 0 x+y+(c-1) z=&gamma;....(3) For no unique solution D =0 |(a-1) 0 1 1 (b-1) 0 0 1 (c-1)|=0 (a-1)(b-1)(c-1)+1=0 ∴ a=2 ; b=2 ; c=0 Hence, | a + b + c |=4.

Q. If system of linear equations $(a - 1) x + z = α, x + (b - 1) y = β$ and $y + (c - 1) z = γ$ where $a, b, c \in I$ does not have a unique solution, then maximum possible value of $∣ a + b + c ∣$ is

0

1

3

4

$(a - 1) x + 0 y + z = α$ ....(1)
$x + (b - 1) y + 0 z = β$ ....(2)
$0 x + y + (c - 1) z = γ$ ....(3)
For no unique solution $D = 0$
$∣ ∣ (a - 1) 10 0 (b - 1) 1 10 (c - 1) ∣ ∣ = 0$
$(a - 1) (b - 1) (c - 1) + 1 = 0$
$∴ a = 2; b = 2; c = 0$
Hence, $∣ a + b + c ∣ = 4$ .

Q. If system of linear equations (a−1)x+z=α,x+(b−1)y=β and y+(c−1)z=γ where a,b,c∈I does not have a unique solution, then maximum possible value of ∣a+b+c∣ is

0

1

3

4

(a−1)x+0y+z=α ....(1) x+(b−1)y+0z=β....(2) 0x+y+(c−1)z=γ....(3) For no unique solution D=0 ∣∣​(a−1)10​0(b−1)1​10(c−1)​∣∣​=0 (a−1)(b−1)(c−1)+1=0 ∴a=2;b=2;c=0 Hence, ∣a+b+c∣=4.

Q. If system of linear equations $(a - 1) x + z = α, x + (b - 1) y = β$ and $y + (c - 1) z = γ$ where $a, b, c \in I$ does not have a unique solution, then maximum possible value of $∣ a + b + c ∣$ is

$(a - 1) x + 0 y + z = α$ ....(1)
$x + (b - 1) y + 0 z = β$ ....(2)
$0 x + y + (c - 1) z = γ$ ....(3)
For no unique solution $D = 0$
$∣ ∣ (a - 1) 10 0 (b - 1) 1 10 (c - 1) ∣ ∣ = 0$
$(a - 1) (b - 1) (c - 1) + 1 = 0$
$∴ a = 2; b = 2; c = 0$
Hence, $∣ a + b + c ∣ = 4$ .