For which value(s) of λ do the pair of linear equations λx + y = λ^{2} 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, a_{1} = , b_{1} = 1, c_{1} = - ^{2};

And a_{2} = 1, b_{2} = , c_{2} = - 1;

a_{1} /a_{2} = λ/1

b_{1} /b_{2} = 1/λ

c_{1} /c_{2} = λ^{2}

(i) For no solution,

a_{1}/a_{2} = b_{1}/b_{2} c_{1}/c_{2}

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,

a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2}

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,

a_{1}/a_{2} b_{1}/b_{2}

so 1/

hence, ^{2} 1;

1;

So, all real values of except

1