answer:First, let's understand the definition of a purely imaginary number. A complex number z = a + bi is considered purely imaginary if a = 0 and b neq 0. Now let's analyze the given condition "a=0": 1. Necessary condition: Indeed, for a complex number to be purely imaginary, its real part a must be zero. So, the condition is necessary. 2. Sufficient condition: However, having a=0 alone does not guarantee that the complex number is purely imaginary, as the imaginary part b could also be zero, resulting in a zero complex number. So, the condition is not sufficient. Therefore, the condition "a=0" is a necessary but not sufficient condition for "the complex number z = a + bi being a purely imaginary number". Hence, the correct answer is: boxed{text{B: Necessary but not sufficient condition}}

answer:To determine how many times the digit 4 will be written when listing the integers from 1 to 1000, we can break it down into different place values: units, tens, and hundreds. 1. Units place: The digit 4 will appear in the units place once every 10 numbers (4, 14, 24, ..., 994). Since there are 1000 numbers, and the pattern repeats every 10 numbers, the digit 4 will appear in the units place 1000/10 = 100 times. 2. Tens place: Similarly, the digit 4 will appear in the tens place once every 100 numbers (40-49, 140-149, ..., 940-949). There are 10 sets of 100 in 1000, so the digit 4 will appear in the tens place 10 * 10 = 100 times. 3. Hundreds place: The digit 4 will appear in the hundreds place for 100 consecutive numbers (400-499). This happens only once in the range from 1 to 1000, so the digit 4 will appear in the hundreds place 100 times. Adding these up, the digit 4 will be written: 100 (units place) + 100 (tens place) + 100 (hundreds place) = boxed{300} times.

answer:In option A, since M={(3,2)} and N={(2,3)} represent different points, the sets M and N do not represent the same set. In option B, since the elements in a set are unordered, M={4,5} and N={5,4}, thus, the sets M and N represent the same set. In option C, since M={1,2} is a number set composed of two elements 1 and 2, and N={(1,2)} is a point set composed of the point (1,2), the sets M and N do not represent the same set. In option D, since M={(x,y)|x+y=1} represents a set of points, and N={y|x+y=1} represents a set of numbers, the sets M and N do not represent the same set. Therefore, the correct answer is boxed{text{B}}.

answer:To address the problem of determining the mutual arrangement of three circles that have exactly two common intersection points, we need to consider the following scenarios: 1. **Case 1: All three circles intersect each other at exactly two common points.** In this configuration, let (C_1), (C_2), and (C_3) be the three circles. Assume that: - (C_1) and (C_2) intersect at points (P) and (Q). - (C_2) and (C_3) intersect at the same points (P) and (Q). - (C_1) and (C_3) also intersect at points (P) and (Q). The intersection points (P) and (Q) are thus the only points at which any pair of these circles intersect. Therefore, the three circles (C_1), (C_2), and (C_3) all pass through points (P) and (Q), forming a common intersection. 2. **Case 2: Two circles intersect at two points, and the third circle intersects each of them at one of the two points.** In this configuration, consider: - Circles (C_1) and (C_2) that intersect at points (P) and (Q). Now introduce the third circle (C_3). For (C_3) to intersect each of (C_1) and (C_2) at one of the common points, there should be: - (C_3) intersects (C_1) at point (P). - (C_3) intersects (C_2) at point (Q). This setup ensures that we again have exactly two points of intersection among all three circles, (P) and (Q). # Conclusion: There are the following two possible mutual arrangements of three circles having exactly two common intersection points: 1. **All three circles intersect at the same two points.** 2. **One circle intersects each of the other two circles at two distinct points.** Thus, from the given analysis, the mutual arrangement of the circles must corroborate with these two cases. [ boxed{text{Thus, the mutual arrangement of the circles can be either as described in Case 1 or Case 2.}} ]