Given:
System of linear equations,
x+y+z=4μ,
x+2y+2λz=10μ,
x+3y+4λz2=μ2+15.
⇒Δ=∣11112312λ4λ2∣
⇒Δ=(8λ2−6λ)−(4λ2−2λ)+(3−2)
⇒Δ=8λ2−6λ−4λ2+2λ+1
⇒Δ=4λ2−4λ+1
⇒Δ=(2λ−1)2
For unique solution, Δ=0.
⇒2λ−1=0
⇒λ=21
Let, Δ=0,λ=21
We know that, for infinite solutions, Δx=Δy=Δz=0
⇒Δx=∣4μ10μμ2+15123111∣
⇒Δx=4μ(2−3)−(10μ−μ2−15)+(30μ−2μ2−30)
⇒Δx=−4μ−10μ+μ2+15+30μ−2μ2−30
⇒Δx=−μ2+16μ−15
⇒(μ−15)(μ−1)=0
So, for infinite solution λ=21,μ=1 or 15.