The values of λ and μ for which the system of linear equations x+y+z=2 x+2 y+3 z=5 x+3 y+λ z=μ has infinitely many solutions are, respectively

Question

The values of λ and μ for which the system of linear equations x+y+z=2 x+2 y+3 z=5 x+3 y+λ z=μ has infinitely many solutions are, respectively (A) 5 and 7 (B) 6 and 8 (C) 4 and 9 (D) 5 and 8

Tardigrade Admin · Accepted Answer

For infinite many solutions D = D 1= D 2= D 3=0 Now D=|1&1&1 1&2&3 1&3&λ|=0 1 .(2 λ-9)-1 .(λ-3)+1 .(3-2)=0 ∴ λ=5 Now D1=|2&1&1 5&2&3 μ&3&5|=0 2(10-9)-1(25-3 μ)+1(15-2 μ)=0 μ=8

Q. The values of $λ$ and $μ$ for which the system of linear equations
$x + y + z = 2$
$x + 2 y + 3 z = 5$
$x + 3 y + λ z = μ$
has infinitely many solutions are, respectively

5 and 7

6 and 8

4 and 9

5 and 8

For infinite many solutions
$D = D_{1} = D_{2} = D_{3} = 0$
Now $D = ∣ ∣ 111123 13 λ ∣ ∣ = 0$
$1. (2 λ - 9) - 1. (λ - 3) + 1. (3 - 2) = 0$
$∴ λ = 5$
Now $D_{1} = ∣ ∣ 25 μ 123135 ∣ ∣ = 0$
$2 (10 - 9) - 1 (25 - 3 μ) + 1 (15 - 2 μ) = 0$
$μ = 8$

Q. The values of λ and μ for which the system of linear equations x+y+z=2 x+2y+3z=5 x+3y+λz=μ has infinitely many solutions are, respectively