For which value(s) of λ do the pair of linear equations λx + y = λ² and x + λy = 1 have

(i) no solution?

(ii) infinitely many solutions?

(iv) a unique solution?

The given pair of linear equations -

x + y - ² = 0and x + y - 1 = 0

Comparing with ax + by + c = 0;

Here, a₁ = , b₁ = 1, c₁ = - ²;

And a₂ = 1, b₂ = , c₂ = - 1;

a₁ /a₂ = λ/1

b₁ /b₂ = 1/λ

c₁ /c₂ = λ²

(i) For no solution,

a₁/a₂ = b₁/b₂ c₁/c₂

i.e. = 1/²

so, ² = 1;

and ²

Here, we take only = - 1 because at = 1, the system of linear equations has infinitely many solutions.

(ii) For infinitely many solutions,

a₁/a₂ = b₁/b₂ = c₁/c₂

i.e. = 1/ = ²

so = 1/ gives = 1;

= ² gives = 1,0;

Hence satisfying both the equations

= 1 is the answer.

(iii) For a unique solution,

a₁/a₂ b₁/b₂

so 1/

hence, ² 1;

1;

So, all real values of except