answer:To analyze the relationship between events A and B, we need to consider the definition of each type of event mentioned. 1. **Complementary Events**: Two events are complementary if the occurrence of one event means the other cannot occur. Since event A (the first coin is heads up) and event B (the second coin is tails up) can both occur simultaneously (for example, if the outcome is heads for the first coin and tails for the second), they are not complementary. 2. **Mutually Exclusive Events**: Two events are mutually exclusive if they cannot happen at the same time. Given that both A and B can occur together (as illustrated in the scenario where the first coin shows heads and the second shows tails), they are not mutually exclusive. 3. **Independent Events**: Two events are independent if the occurrence of one does not affect the occurrence of the other. In this case, the result of the first coin flip does not influence the result of the second coin flip, making A and B independent events. This aligns with option C. 4. **Calculation of P(AB)**: To further confirm our understanding, we calculate the probability of both A and B occurring. Since A and B are independent, P(AB) = P(A) cdot P(B). Given that the probability of heads or tails for a fair coin is frac{1}{2}, [P(AB) = P(A) cdot P(B) = frac{1}{2} cdot frac{1}{2} = frac{1}{4}] This calculation disproves option D, which incorrectly states P(AB) = frac{1}{3}. Therefore, the correct answer based on our analysis is C, indicating that events A and B are independent events. [boxed{C}]

answer:To solve the problem, we start by defining the variables for the number of floors in each building. Let's denote the number of floors in the Leeward Center as L. According to the problem, the Chrysler Building has 11 more floors than the Leeward Center, which can be expressed as: [ text{Chrysler} = L + 11 ] The total number of floors in both buildings is given as 35. This can be written as: [ L + (L + 11) = 35 ] Simplifying the equation, we combine like terms: [ 2L + 11 = 35 ] Subtracting 11 from both sides to isolate the terms with L: [ 2L = 35 - 11 ] [ 2L = 24 ] Dividing both sides by 2 to solve for L: [ L = frac{24}{2} ] [ L = 12 ] Now that we know the Leeward Center has 12 floors, we can find the number of floors in the Chrysler Building by adding 11 to L: [ text{Chrysler} = L + 11 ] [ text{Chrysler} = 12 + 11 ] [ text{Chrysler} = 23 ] Therefore, the Chrysler Building has boxed{23} floors.

answer:Solution: Let's take a = -frac {1}{2}, b = -1 to verify, and we know that ③ is incorrect. ① Proof: Since frac {1}{a} < frac {1}{b} < 0, it follows that a < 0, b < 0, thus ab > 0, a+b < 0, therefore a+b < ab, so ① is correct. ② From the given, we can deduce b < a < 0, then |a| < |b|, so ② is correct. ④ Proof: Since frac {b}{a} > 0, frac {a}{b} > 0, and a neq b, by the arithmetic mean inequality, we have frac {b}{a} + frac {a}{b} > 2, so ④ is correct. Therefore, the answer is boxed{text{①②④}}. By using the substitution method, we first eliminate the incorrect option ③, then use the properties of inequalities to prove ①②④, thereby determining the correct answer. This is a basic problem that directly tests the fundamental properties of inequalities. Note that the flexible application of the substitution method can effectively simplify the problem-solving process.

answer:Consider an initial monthly grade ( x ). The punishment imposes a discount of ( x % ) on this grade. 1. **Discount Calculation**: [ x% text{ of } x = frac{x}{100} times x = frac{x^2}{100} ] Therefore, the grade after punishment is given by: [ x - frac{x^2}{100} ] 2. **Function Definition**: Define the function representing the grade after punishment as: [ f(x) = x - frac{x^2}{100} ] 3. **Range of Grades**: Since the initial grade ( x ) ranges from 0 to 100, we consider ( f(x) ) on the interval ([0, 100]). 4. **Vertex of the Parabolic Function**: The function ( f(x) = x - frac{x^2}{100} ) is a downward-opening parabola. The vertex of this parabola occurs where: [ x = -frac{b}{2a} ] Here, ( a = -frac{1}{100} ) and ( b = 1 ). Thus: [ x = -frac{1}{2 cdot -frac{1}{100}} = 50 ] 5. **Maximum Grade After Punishment**: The maximum value of ( f(x) ) occurs at ( x = 50 ): [ f(50) = 50 - frac{50^2}{100} = 50 - 25 = 25 ] 6. **Minimum Grade After Punishment**: The minimum value of ( f(x) ) occurs at the endpoints ( x = 0 ) and ( x = 100 ): [ f(0) = 0 - frac{0^2}{100} = 0 ] [ f(100) = 100 - frac{100^2}{100} = 100 - 100 = 0 ] # Conclusion: (a) The highest grade after punishment is [ boxed{25} ] (b) The lowest grade after punishment is [ boxed{0} ] (c) Students who scored either very high (near 100) or very low (near 0) initially will end up with very low grades, specifically 0 after the punishment. Hence, the students who scored high are indeed justified in complaining since they end up with the same grade as those who scored low. [ boxed{text{Correct}} ]