Question

If the product of j and k does not equal zero, is j $<$ 0 and k $>$ 0?
(1) (-j, k) lies above the x-axis and to the right of the y-axis.
(2) (j, -k) lies below the x-axis and to the left of the y-axis.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient.

Hint:

Quickly translate coordinate plane locations into inequalities: \[\begin{array}{rl} \bullet & \text{Right of y-axis: x-coordinate \textgreater 0} \\ \bullet & \text{Left of y-axis: x-coordinate \textless 0} \\ \bullet & \text{Above x-axis: y-coordinate \textgreater 0} \\ \bullet & \text{Below x-axis: y-coordinate \textless 0} \\ \end{array}\] This allows for rapid evaluation of such statements.

Answer 1

The Correct Option is D

Step 1: Understanding the Question
This is a Yes/No question about the signs of the variables j and k. The condition \(jk \neq 0\) means that neither j nor k is zero. The question asks if j is negative AND k is positive. This corresponds to the second quadrant in a j-k plane.

Step 2: Analysis of Statement (1)
Statement (1) describes the location of the point (-j, k) in the Cartesian coordinate system. \[\begin{array}{rl} \bullet & \text{"lies above the x-axis" means the y-coordinate is positive. So, \(k > 0\).} \\ \bullet & \text{"to the right of the y-axis" means the x-coordinate is positive. So, \(-j > 0\).} \\ \end{array}\] If we have the inequality \(-j > 0\), we can multiply both sides by -1. Remember to flip the inequality sign when multiplying by a negative number. \[ j < 0 \] So, from statement (1), we have concluded that \(j < 0\) and \(k > 0\). This provides a definitive "Yes" to the question.
Therefore, Statement (1) ALONE is sufficient.

Step 3: Analysis of Statement (2)
Statement (2) describes the location of the point (j, -k). \[\begin{array}{rl} \bullet & \text{"lies below the x-axis" means the y-coordinate is negative. So, \(-k < 0\).} \\ \bullet & \text{"to the left of the y-axis" means the x-coordinate is negative. So, \(j < 0\).} \\ \end{array}\] From statement (2), we have also concluded that \(j < 0\) and \(k > 0\). This also provides a definitive "Yes" to the question.
Therefore, Statement (2) ALONE is sufficient.
Step 4: Final Answer
Since each statement alone is sufficient to answer the question, the correct answer is (D).