Consider the system of two linear equations as follows: $$3x + 21y + p = 0$$; and $$qx + ry - 7 = 0$$, where p, q, and r are real numbers.
Which of the following statements DEFINITELY CONTRADICTS the fact that the lines represented by the two equations are coinciding?

In order for the line two be coincident, the ratio of the coefficients of the variables and the constants from both the equations must be the same. 

That is, $$\frac{3}{q}=\frac{21}{r}=\frac{p}{-7}$$

Let's consider each option individually.
Checking through the easier options first.

Option A: p and q must have opposite signs
In order for  $$\frac{3}{q}=\frac{p}{-7}$$ to be true, p and q must be of opposite signs as pq = -21
Hence, this option does not contradict the lines coinciding. 

Option D: r and q must have the same signs

Similar to option A, for $$\frac{3}{q}=\frac{21}{r}$$ to hold true, q and r must be of the same signs. 
Hence, this option too does not contradict the lines coinciding. 

Option E: p cannot be zero
In order for q and r to have real values, the fractions $$\frac{3}{q}=\frac{21}{r}=\frac{p}{-7}$$ must not be zero. 
Putting p=0 would give the value of p/(-7) as 0, which would then not give real values of q and r
Hence, p can never be zero. This statement, too, does not contradict the two lines being coincident. 

Now that A, D, and E are not our answers, we need to consider the more complex options. 

Option B: r is the smallest amongst p, q, r
We would want $$\frac{3}{q}=\frac{21}{r}$$ to hold true.
If we take q and r to be positive, we must take p to be negative, in that case p would be negative and the smallest. So, in order to consider the possibility of r being the smallest, we must take q and r as negative values.
Now, with q and r negative, a smaller number would have a bigger magnitude or a bigger absolute value.
We can try putting in values at this point to see if this would contradict the lines coinciding.
taking r as -21 and q as -3, we can take p as 7
These values would make the lines coincident, with r being the smallest value.
Hence, this options too does not necessarily contradcit the lines coinciding.

Option C: q is the largest amongst p, q, r
Follwoing the same logic as Optoin B, we cannot take q and r to be negative, as that would make p positive, making p the largest value.
Therefore, in order to consider the possibility of q being the largest, we must take positive values of q and r
Now, comparing $$\frac{3}{q}=\frac{21}{r}$$, if q is larger than r, then the numerator of the first term would be smaller than the numerator of the second term, and the denominator of the first term would be larger than the denominator of the second term
Making the fraction $$\frac{3}{q}$$ strictly smaller than $$\frac{21}{r}$$
If this option is true, the lines can never coincide, as the ratios of the coefficient can never be equal.

Therefore, Option C would be the correct answer.