Question

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?
(iii) a unique solution?
NCERT Class X
Mathematics - Exemplar Problems
Chapter _PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

Answer 1

Solution :

If the a1x+b1Y+ c1 = 0,a2x+b2y+c2 = 0 system has unique solution then

a1/a2 ≠ b1/b2

Given linear equations λx + y = λ2 and x + λy = 1

λ/ 1 ≠ 1 / λ   ⇒ λ2 ≠ 1  ⇒  λ ≠ ±1

∴ λ ≠ ±1 then it has unique solution.

If the a1x+b1Y+ c1 = 0,a2x+b2y+c2 = 0 system has infinitely many solutions then

a1/a2 = b1/b2 = c1/c2.

λ/ 1 = 1 / λ  = λ2 / 1

∴ λ = ±1 then it has infinitely many solutions.

If the a1x+b1Y+ c1 = 0,a2x+b2y+c2 = 0 system has no solution then

a1/a2 = b1/b2 ≠ c1/c2.

λ/ 1 = 1 / λ  ≠  λ2 / 1

 λ3 ≠ 1 ⇒ λ ≠ 1

∴ λ = ±1 then it has no solution.

Hope u understand