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Find the value(s) of $k$ for which the pair of linear equations $kx + y = k^2$; $x + ky = 1$ have infinitely many solutions.
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For infintiely many solutions
$\frac{k}{1} = \frac{1}{k} = \frac{k^2}{1}$
$\Rightarrow k^2 = 1$ and $k^3 = 1$
$\Rightarrow k = \pm 1$ and $k = 1$
$\therefore k = 1$
$\frac{k}{1} = \frac{1}{k} = \frac{k^2}{1}$
$\Rightarrow k^2 = 1$ and $k^3 = 1$
$\Rightarrow k = \pm 1$ and $k = 1$
$\therefore k = 1$