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


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