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থিম নির্বাচন

আপনার পছন্দের থিম বেছে নিন।

Instruction: Each problem consists of a question followed by two statements. Decide whether the data in the statements are sufficient to answer the question. If \(xy < yz < 0\), is \(y > 0\)? (1) \(x < z\). (2) \(x < 0\).

A. statement (1) alone is sufficient, but statement (2) alone is not sufficient to answer the question
B. statement (2) alone is sufficient, but statement (1) alone is not sufficient to answer the question
C. both statements taken together are sufficient to answer the question, but neither statement alone is sufficient
D. each statement alone is sufficient
E. statements (1) and (2) together are not sufficient, and additional data is needed to answer the question.
Explanation: (A) Given: \(xy < yz < 0\). The condition \(yz < 0\) means y and z have opposite signs. The condition \(xy < yz\) means \(xy - yz < 0 \implies y(x - z) < 0\). (1) If \(x < z\), then \(x - z < 0\). For \(y(x - z) < 0\) to be true, if \(x-z\) is negative, \(y\) must be positive. This guarantees \(y > 0\). So (1) is sufficient. (2) If \(x < 0\). If y is positive, \(y > 0\). Example: \(x = -5, y = 2, z = -1 \implies -10 < -2 < 0\). Yes, \(y > 0\). If y is negative, \(y < 0\). Example: \(x = -5, y = -2, z = 1 \implies 10 < -2 < 0\). No. Since the answer can be yes or no, (2) is not sufficient. Thus, (1) alone is sufficient.