Find the set of values of $k$ for which the equation $kx^2 + 2x + 1 = 0$ has distinct real roots.
The standard equation $\;$ $ax^2 + bx + c = 0$ $\;$ has distinct real roots when $\;$ $b^2 - 4ac > 0$
Given equation: $\;$ $kx^2 + 2x + 1 = 0$
Comparing the given quadratic equation with the standard quadratic equation, we have,
$a = k$, $\;$ $b = 2$, $\;$ $c = 1$
$\therefore \;$ For real distinct roots, $\;$ $2^2 - 4 \times k \times 1 > 0$
i.e. $\;$ $4 - 4k > 0$
i.e. $\;$ $1 - k > 0$
i.e. $\;$ $1 > k$
$\implies$ $k < 1$
$\therefore \;$ The given quadratic equation has distinct real roots for $\;$ $\left\{k \mid k < 1, \; k \in R \right\}$