Find the value of $k$ so that the equations $x ^ { 2 } - x - 12 = 0$ and $k x ^ { 2 } + 10 x + 3 = 0$ may have one root in common.

Let the common root of the equations x^2 - x - 12 = 0 and kx^2 + 10x + 3 = 0 be alpha. From the first equation, we factor to find the roots: (x - 4)(x + 3) = 0, meaning alpha must be either 4 or -3. If alpha is 4, we substitute it into the second equation to get k(16) + 10(4) + 3 = 0. Solving 16k + 43 = 0 gives k = -43/16, which is not an option. If alpha is -3, substituting it into the second equation gives k(9) + 10(-3) + 3 = 0, which simplifies to 9k - 27 = 0 and yields k = 3. Therefore, the value of k is 3.

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