Solution

For the line: \(px+qy+r=0\)

solve for \(y\):

\(y=-\frac pqx-\frac rq\)

Slope \(=-\frac pq\)

x-intercept: set \(y=0:\)

\(x=-\frac rp\)

Now check each statement:

I. If p and q have the same sign, slope is negative.

\(-\frac pq<0\)

because \(\frac{p}{q}>0\). True.

II. If p and r have the same sign, x-intercept is negative.

\(x=-\frac rp<0\)

because \(\frac{r}{p}>0.\) True.

III. If slope is negative and x-intercept is positive, q and r have the same sign.

Slope negative means: \(-\frac pq<0 \Rightarrow \frac pq>0\)

Thus p and q have the same sign. Positive x-intercept means:

\(-\frac rp>0 \Rightarrow \frac rp<0\)

Thus r and p have opposite signs.

Since q has the same sign as p, q and r must have opposite signs, not the same sign.

Therefore, the statements that must be true are I and II only.

Answer: C