Given two lines 2x + ky + 7 = 0 and 27x - 18y + 25 = 0.
We need to find the value of k for which these two lines are perpendicular to each other.

Perpendicular Lines:
Two lines are said to be perpendicular, if the product of their slopes is equal to -1.
Mathematically, it can be written as:
m1*m2 = -1
Where m1 and m2 are the slopes of the two lines.

Slopes of the given lines:
To find the slopes of the given lines, we need to calculate their coefficients of x and y.
For the first line, 2x + ky + 7 = 0
Coefficient of x = 2
Coefficient of y = k
Therefore, the slope of the first line = m1 = -2/k

For the second line, 27x - 18y + 25 = 0
Coefficient of x = 27
Coefficient of y = -18
Therefore, the slope of the second line = m2 = 18/27

Product of Slopes:
Now, we need to calculate the product of the slopes of the two lines.
m1*m2 = -2/k * 18/27
m1*m2 = -2/3k
To make the product of slopes equal to -1, we need to have -2/3k = -1
Therefore, we get,
k = 3

Hence, the value of k for which the lines 2x + ky + 7 = 0 and 27x - 18y + 25 = 0 are perpendicular to each other, is k = 3.
Therefore, Option C. k = 3 is the correct answer.

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