Find the value(s) of k for which the pair of linear equations kx + y = k² and x + ky = 1 have infinitely many solutions.

We have the linear equations,

kx + y = k² …(i)

x + ky = 1 …(ii)

We can rewrite equations (i) & (ii) as,

kx + y – k² = 0 …(iii)

x + ky – 1 = 0 …(iv)

Comparing equation (iii) from a₁x + b₁y + c₁ = 0, we get

a₁ = k, b₁ = 1 and c₁ = -k²

Comparing equation (iv) from a₂x + b₂y + c₂ = 0, we get

a₂ = 1, b₂ = k and c₂ = -1

Also, given that kx + y = k²and x + ky = 1 have infinitely many solutions. So, we can write as

Putting in values,

Solving ,

⇒ k² = 1

⇒ k = √1

⇒ k = 1 …(v)

Also solving ,

⇒ k³ = 1

Or k³ = 1³

⇒ k = 1 …(vi)

Hence, k = 1 because it satisfies both equations (v) and (vi).