Line A passes through point (r, s) on the coordinate plane. Is the slope of A > 0?
(1) Line A passes through point (0, 0).
(2) Line A passes through point (u, t), where r > u and t > s.

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(B) The slope of a line is determined by the formula:
Slope = m = (change in y)/(change in x).

To establish the sufficiency of the statements, substitute the given values into the formula.

Statement (1): m = (s – 0)/(r – 0) tells us that the slope of A is s/r, but, because it doesn't tell us whether s and r are positive or negative, the slope cannot be determined. Thus, Statement (1) is insufficient.
Note: Be careful that you do not assume, because s and r do not have a negative sign before them, that they are positive. The both variables can be positive or negative.

Statement (2): m = (s – t)/(r – u) tells us that the numerator in this equation is negative, because t > s. The denominator (r – u) is positive because r > u. So the slope of line A is negative, and thus Statement (2) is sufficient.

Since Statement (1) is insufficient and Statement (2) is sufficient, the correct answer is B.
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In the question it is mentioned that line A passes through (r, s) in the coordinate plane. In option 1 it is clearly stated that line A passes through (0, 0). So, the slope becomes (s – 0)/(r – 0) = s/r.
When it is mentioned that in the coordinate plane there is a point (r, s) so it can be fathomed that r is in +ve x quadrant and s is also in +ve Y quadrant. If it is mentioned that (r, s) an arbitrary point lies in coordinate plane and in any of the 4 quadrants ,then option 1 will be insufficient to state about slope of A (=ve or -ve)?

That is correct.

No. We can not assume that. It is not stated that r and s are positive numbers.
Furthermore, nothing specific is stated about (r, s), except that it is a point on the coordinate plane. Therefore (r, s) can be anywhere on the coordinate plane.