If a random variable $X$ has the following probability distribution of $X$…

The sum of probabilities in the distribution must be 1: $P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6) + P(X=7) = 1$.

Substituting the given expressions: $0 + k + 2k + 2k + 3k + k^2 + 2k^2 + (7k^2 + k) = 1$.

Combining like terms yields $10k^2 + 9k = 1$, which rearranges to the quadratic equation $10k^2 + 9k - 1 = 0$.

Factoring gives $(10k - 1)(k + 1) = 0$, with solutions $k = \frac{1}{10}$ or $k = -1$. Since probability must be non-negative, $k = \frac{1}{10}$.

To find $P(X \geq 6)$, sum $P(X=6) + P(X=7)$:

$P(X=6) = 2k^2 = 2(\frac{1}{10})^2 = \frac{2}{100}$

$P(X=7) = 7k^2 + k = 7(\frac{1}{10})^2 + \frac{1}{10} = \frac{17}{100}$

$P(X \geq 6) = \frac{2}{100} + \frac{17}{100} = \frac{19}{100}$

This matches option A.