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.
Explanation: Given the circle, DE is a chord. We need to know if \(DE < 6\). Statement 1: \(r=5\). The maximum length of a chord is the diameter, \(2r = 10\). Since \(DE\) can be any length up to 10 (e.g., 3, 6, 9), we cannot definitively say if \(DE < 6\). Not sufficient. Statement 2: Let \(d\) be the distance from the center to \(DE\). \(d < 4\). The chord length \(L\) relates to \(r\) and \(d\) by the Pythagorean theorem: \((L/2)^2 + d^2 = r^2\). Since \(r\) is unknown, we cannot determine \(L\). Not sufficient. Together: \(r=5\) and \(d < 4\). From the relation, \((DE/2)^2 + d^2 = 5^2 = 25\). \((DE/2)^2 = 25 - d^2\). Since \(d < 4\), \(d^2 < 16\). So, \(25 - d^2 > 25 - 16 = 9\). \((DE/2)^2 > 9\). \(DE/2 > 3\). \(DE > 6\). Since \(DE\) **must** be greater than \(6\), the answer to the question "is DE < 6?" is definitively "No". Therefore, the statements TOGETHER ARE sufficient. The provided answer/explanation is incorrect as it states: "Statement -2: যদি ধরেনি, center হবে Chord এর distance 4 তখন, Chord এর length হয় 6 কিন্তু যেহেতু distance 4 থেকে ছোট, length অবশ্যই 6 থেকে বেশি।" The problem is data sufficiency, not solving. Together is sufficient to answer a definitive NO.
