💡Okay, so I'm trying to figure out this probability problem related to the new college entrance examination rules in Zhejiang Province. Let me read through the problem again to make sure I understand it.Each student needs to choose three elective subjects from seven. They've already chosen physics and chemistry, so they need to pick one more subject from the science subjects (biology and technology) and one subject from the humanities subjects (political science, history, and geography). They've defined some events:- E = "choose biology"- F = "choose a science subject"- G = "choose political science"- H = "choose a humanities subject"And there are four conclusions given, numbered ① to ④. I need to determine which of these conclusions are correct.Let me list out the conclusions again for clarity:1. G and H are mutually exclusive events but not complementary events.2. F and H are mutually exclusive and complementary events.3. P(F) + P(G) = 14. P(E ∪ H) = P(E) + P(H)Alright, let's tackle each conclusion one by one.**Conclusion ①: G and H are mutually exclusive events but not complementary events.**First, what does it mean for two events to be mutually exclusive? It means they can't happen at the same time. So, if G is "choose political science" and H is "choose a humanities subject," is it possible to choose both? Well, political science is a humanities subject, right? So if you choose political science (G), you are also choosing a humanities subject (H). That means G and H can occur together. Therefore, they are not mutually exclusive. Since they can occur together, they are not mutually exclusive, which makes this conclusion incorrect. Also, they are not complementary because complementary events cover all possibilities, and there are other humanities subjects besides political science (like history and geography). So, H includes more than just G, meaning G and H are not complementary. But since they aren't mutually exclusive either, conclusion ① is wrong.**Conclusion ②: F and H are mutually exclusive and complementary events.**F is "choose a science subject," and H is "choose a humanities subject." Since a student needs to choose one science subject and one humanities subject, they can't choose both F and H at the same time. So, they are mutually exclusive. Are they complementary? That means every possible outcome is either F or H, but not both. Since the student must choose one science and one humanities subject, every choice will fall into either F or H, but not both. Therefore, F and H are complementary as well. So, conclusion ② is correct.**Conclusion ③: P(F) + P(G) = 1**Let's think about this. F is choosing a science subject, which can be either biology or technology. G is choosing political science, which is a humanities subject. The probability of choosing a science subject (F) plus the probability of choosing political science (G) should be equal to 1? Hmm, not necessarily. Because choosing a science subject and choosing political science are not mutually exclusive or complementary. A student could choose a science subject and a humanities subject that's not political science, like history or geography. So, the probabilities don't add up to 1. Therefore, conclusion ③ is incorrect.**Conclusion ④: P(E ∪ H) = P(E) + P(H)**E is choosing biology, and H is choosing a humanities subject. Are these events mutually exclusive? Well, choosing biology (a science subject) and choosing a humanities subject are separate choices. You can choose biology and also choose a humanities subject. So, they are not mutually exclusive. Wait, but the formula P(E ∪ H) = P(E) + P(H) is only true if E and H are mutually exclusive. Since they are not, this formula doesn't hold. Therefore, conclusion ④ is incorrect.Wait, hold on. Let me double-check. If E and H are independent events, does that affect the formula? No, the formula P(E ∪ H) = P(E) + P(H) - P(E ∩ H) is the general formula. Since E and H are not mutually exclusive, we need to subtract the probability of their intersection. Therefore, the given formula is missing that term, making it incorrect. So, yes, conclusion ④ is incorrect.Wait again, maybe I misread. The problem says the student needs to choose one more subject from biology and technology (science) and one subject from political science, history, and geography (humanities). So, E is choosing biology, which is one of the science subjects, and H is choosing a humanities subject. So, in this context, choosing biology (E) and choosing a humanities subject (H) are separate choices. They don't overlap because biology is a science subject and H is a humanities subject. Therefore, E and H are mutually exclusive? Wait, no, because the student is choosing both a science subject and a humanities subject. So, E is part of the science choice, and H is part of the humanities choice. Wait, I'm getting confused. Let me clarify. The student has to choose one science subject (either biology or technology) and one humanities subject (political science, history, or geography). So, E is "choose biology," which is one specific science subject. H is "choose a humanities subject," which is a broader category. Therefore, choosing biology (E) doesn't prevent you from choosing a humanities subject (H). In fact, you have to choose both. So, E and H are not mutually exclusive. Therefore, P(E ∪ H) would be the probability of choosing biology or a humanities subject. But since you have to choose both, the union would actually be the entire sample space, which is 1. But that's not what the formula is saying. The formula is P(E) + P(H). Wait, let's calculate P(E ∪ H). Since E and H are not mutually exclusive, P(E ∪ H) = P(E) + P(H) - P(E ∩ H). But in this case, since choosing biology (E) and choosing a humanities subject (H) are separate choices, the intersection P(E ∩ H) is actually the probability of choosing biology and a humanities subject, which is the total probability, because every student chooses one science and one humanities subject. Wait, no. The student chooses one science subject (either biology or technology) and one humanities subject. So, the total number of possible combinations is 2 (science) * 3 (humanities) = 6.E is choosing biology, which is 3 out of 6 possibilities (since for each humanities subject, there's a combination with biology). H is choosing a humanities subject, which is all 6 possibilities because every choice includes a humanities subject. Wait, no. H is "choose a humanities subject," which is certain because every student has to choose one. So, P(H) = 1. E is "choose biology," which is 3 out of 6, so P(E) = 0.5.Therefore, P(E ∪ H) is P(E) + P(H) - P(E ∩ H). But since H is certain, P(E ∪ H) = P(E) + 1 - P(E ∩ H). But P(E ∩ H) is the probability of choosing biology and a humanities subject, which is the same as P(E), because choosing biology already implies choosing a humanities subject. So, P(E ∩ H) = P(E) = 0.5.Therefore, P(E ∪ H) = 0.5 + 1 - 0.5 = 1. But the formula given is P(E) + P(H) = 0.5 + 1 = 1.5, which is greater than 1, which is impossible for a probability. Therefore, the formula P(E ∪ H) = P(E) + P(H) is incorrect because it doesn't account for the overlap. Wait, but in reality, since H is certain, P(E ∪ H) is just P(H) = 1. So, the formula P(E) + P(H) would be 0.5 + 1 = 1.5, which is wrong. Therefore, conclusion ④ is incorrect.Hmm, I think I made a mistake earlier. Let me try again.Since the student must choose one science subject and one humanities subject, the events E and H are not mutually exclusive. However, H is certain because every student chooses a humanities subject. Therefore, P(H) = 1. So, P(E ∪ H) is the probability of choosing biology or a humanities subject. But since every student chooses a humanities subject, P(E ∪ H) = P(H) = 1. On the other hand, P(E) + P(H) = 0.5 + 1 = 1.5, which is greater than 1, which is impossible. Therefore, the formula P(E ∪ H) = P(E) + P(H) is incorrect because it doesn't subtract the overlap. But wait, in reality, since H is certain, P(E ∪ H) = P(H) = 1. So, the formula given is incorrect because it doesn't account for the fact that H is certain and E is a subset of H in a way. Wait, no, E is choosing biology, which is a science subject, and H is choosing a humanities subject. They are separate choices. So, E and H are independent events. Therefore, P(E ∪ H) = P(E) + P(H) - P(E)P(H). But since H is certain, P(H) = 1, so P(E ∪ H) = P(E) + 1 - P(E)*1 = 1. Which matches the reality. But the formula given is P(E) + P(H), which would be 0.5 + 1 = 1.5, which is wrong. Therefore, conclusion ④ is incorrect.Wait, but in reality, since H is certain, P(E ∪ H) is just 1, and P(E) + P(H) is 1.5, which is impossible. Therefore, the formula is wrong. So, conclusion ④ is incorrect.Wait, but maybe I'm overcomplicating it. Let's think about it differently. The student has to choose one science subject (biology or technology) and one humanities subject (political science, history, or geography). E is "choose biology," which is one of the two science subjects. So, P(E) = 1/2.H is "choose a humanities subject," which is certain because every student has to choose one. So, P(H) = 1.Now, E and H are not mutually exclusive because choosing biology doesn't prevent you from choosing a humanities subject. In fact, you have to choose both. So, P(E ∪ H) is the probability of choosing biology or a humanities subject. But since you have to choose both, the union is actually the entire sample space, which is 1. But the formula given is P(E) + P(H) = 1/2 + 1 = 3/2, which is greater than 1, which is impossible. Therefore, the formula is incorrect. So, conclusion ④ is incorrect.Wait, but maybe I'm misunderstanding the events. Let me clarify:- E is "choose biology" (a specific science subject)- H is "choose a humanities subject" (any of the three)So, E is a subset of the science subjects, and H is a subset of the humanities subjects. Since the student chooses one science and one humanities, E and H are independent events. Therefore, P(E ∪ H) = P(E) + P(H) - P(E)P(H). Given that P(E) = 1/2 and P(H) = 1 (since every student chooses a humanities subject), then P(E ∪ H) = 1/2 + 1 - (1/2 * 1) = 1/2 + 1 - 1/2 = 1. But the formula given is P(E) + P(H) = 1/2 + 1 = 3/2, which is incorrect. Therefore, conclusion ④ is incorrect.Wait, but in reality, since H is certain, P(E ∪ H) = P(H) = 1, which is correct. But the formula given doesn't account for the overlap, so it's wrong. Therefore, conclusion ④ is incorrect.Okay, so after going through all four conclusions, I think only conclusion ② is correct. Conclusions ①, ③, and ④ are incorrect.

💡Okay, so I have this problem here where I need to figure out which of the given equations doesn't describe y as either directly or inversely proportional to x. Let me try to understand what direct and inverse proportionality mean first.From what I remember, direct proportionality means that y increases as x increases, and it can be written as y = kx, where k is a constant. On the other hand, inverse proportionality means that y decreases as x increases, and it can be written as y = k/x or xy = k.Alright, so I need to check each equation and see if it fits either of these forms. If it doesn't, then that's the one I'm looking for.Let's go through each equation one by one.1. **Equation 1: x² + y² = 16** - Hmm, this looks like the equation of a circle with radius 4. I don't think this is directly or inversely proportional because it's not linear, and it doesn't fit the form y = kx or y = k/x. If I try to solve for y, I get y = sqrt(16 - x²), which isn't a linear relationship. So, this might be the one that doesn't describe y as proportional to x.2. **Equation 2: 2x + 3y = 6** - Let me solve for y to see what it looks like. Subtract 2x from both sides: 3y = 6 - 2x. Then divide by 3: y = (6 - 2x)/3, which simplifies to y = 2 - (2/3)x. This is a linear equation, but it's in the form y = mx + b, where m is the slope and b is the y-intercept. Since it has a y-intercept, it's not a direct proportionality because direct proportionality doesn't have a constant term. So, this might also not be directly proportional. But wait, is it inversely proportional? Let me check. Inverse proportionality would require that y = k/x, but this is y = 2 - (2/3)x, which isn't the same. So, this equation doesn't show y as inversely proportional either. Hmm, so this might be another candidate.3. **Equation 3: xy = 5** - This looks familiar. If I solve for y, I get y = 5/x. That's clearly an inverse proportionality because y is equal to a constant divided by x. So, this one does describe y as inversely proportional to x.4. **Equation 4: x = 7y** - Let's solve for y. Divide both sides by 7: y = x/7. This is a direct proportionality because y is equal to a constant times x. So, this one describes y as directly proportional to x.5. **Equation 5: x/y = 2** - Let me solve for y. Multiply both sides by y: x = 2y. Then divide both sides by 2: y = x/2. Again, this is a direct proportionality because y is equal to a constant times x. So, this one also describes y as directly proportional to x.Okay, so after going through all the equations, I see that equations 3, 4, and 5 clearly show y as either directly or inversely proportional to x. Equations 1 and 2 don't fit into these categories. Equation 1 is a circle, which isn't a linear relationship, and equation 2 is a linear equation with a y-intercept, which isn't a direct or inverse proportionality.But the question asks for the equation that does not describe y as either directly or inversely proportional to x. So, both equations 1 and 2 might seem like candidates. However, equation 2 is a linear equation, but it's not a direct proportionality because it has a y-intercept. But wait, is it possible that it's still considered inversely proportional? Let me think.Inverse proportionality is specifically y = k/x or xy = k. Equation 2 is y = 2 - (2/3)x, which is a linear equation with a negative slope and a y-intercept. It doesn't fit the form of inverse proportionality because it's not y = k/x or xy = k. So, equation 2 doesn't describe y as inversely proportional either.But equation 1 is a circle, which isn't even a function in terms of y, meaning that for some x values, there are two y values. This definitely doesn't describe y as directly or inversely proportional to x because proportionality implies a one-to-one relationship.So, between equations 1 and 2, equation 1 is more clearly not describing y as proportional to x, while equation 2 is a linear equation that doesn't fit the proportionality forms but is still a function. However, the question is asking for the equation that does not describe y as either directly or inversely proportional. Since equation 2 is a linear equation with a y-intercept, it's not a direct proportionality, but it's still a linear relationship. However, it's not an inverse proportionality either. So, both equations 1 and 2 don't describe y as proportional to x.But looking back at the options, the choices are A to E, and equation 1 is option A, while equation 2 is option B. The question is asking for the equation that does not describe y as either directly or inversely proportional. So, both A and B might seem like possible answers, but I need to choose one.Wait, let me double-check equation 2. If I rearrange it, it's y = 2 - (2/3)x. This is a linear equation, but it's not passing through the origin because of the y-intercept. Direct proportionality requires that y = kx, which passes through the origin. So, equation 2 doesn't describe y as directly proportional. It's not inversely proportional either because it's not y = k/x or xy = k.Equation 1, on the other hand, is a circle, which isn't even a function, so it doesn't describe y as proportional to x in any way.So, both equations 1 and 2 don't describe y as proportional to x, but the question is asking for which one does not describe y as either directly or inversely proportional. Since equation 1 is a circle and not a function, it's more clearly not describing y as proportional to x. Equation 2 is a linear equation with a y-intercept, which is still a function but not a proportional relationship.Therefore, the correct answer should be equation 1, which is option A.

💡First, I need to determine the total number of beads in the container initially. There is 1 green bead, 2 brown beads, and 3 red beads.Adding these together, the total number of beads is 1 + 2 + 3 = 6 beads.Tom took some beads out and left 4 beads in the container. To find out how many beads Tom took out, I subtract the number of beads left from the total number of beads.So, 6 beads (total) - 4 beads (left) = 2 beads taken out.Therefore, Tom took out 2 beads.

💡Okay, so I need to find the arc length of this parametric curve given by the equations:x = (t² - 2) sin t + 2t cos ty = (2 - t²) cos t + 2t sin twith t ranging from 0 to π/3.Hmm, I remember that the formula for the arc length of a parametric curve is the integral from t1 to t2 of the square root of (dx/dt)² + (dy/dt)² dt. So, first, I need to find the derivatives dx/dt and dy/dt.Let me start with dx/dt. So, x is given by (t² - 2) sin t + 2t cos t. To find dx/dt, I need to differentiate this with respect to t.Breaking it down, the first part is (t² - 2) sin t. Using the product rule, the derivative of that would be (2t) sin t + (t² - 2) cos t.Then, the second part is 2t cos t. Again, using the product rule, the derivative is 2 cos t - 2t sin t.So, putting it all together:dx/dt = 2t sin t + (t² - 2) cos t + 2 cos t - 2t sin t.Wait, let me simplify that. The 2t sin t and -2t sin t cancel each other out. Then, (t² - 2) cos t + 2 cos t simplifies to t² cos t - 2 cos t + 2 cos t, which further simplifies to t² cos t. Okay, that's nice and simple.Now, moving on to dy/dt. y is given by (2 - t²) cos t + 2t sin t. Let's differentiate this term by term.First, (2 - t²) cos t. Using the product rule, the derivative is (-2t) cos t + (2 - t²)(-sin t).Then, the second part is 2t sin t. Differentiating that gives 2 sin t + 2t cos t.So, putting it all together:dy/dt = -2t cos t - (2 - t²) sin t + 2 sin t + 2t cos t.Simplifying this, the -2t cos t and +2t cos t cancel each other out. Then, we have -(2 - t²) sin t + 2 sin t, which is -2 sin t + t² sin t + 2 sin t. The -2 sin t and +2 sin t cancel, leaving us with t² sin t.Alright, so now we have both derivatives:dx/dt = t² cos tdy/dt = t² sin tNow, to find the arc length, we need to compute the integral from 0 to π/3 of the square root of (dx/dt)² + (dy/dt)² dt.Let's compute (dx/dt)² + (dy/dt)²:(t² cos t)² + (t² sin t)² = t⁴ cos² t + t⁴ sin² t = t⁴ (cos² t + sin² t) = t⁴ * 1 = t⁴.So, the integrand simplifies to sqrt(t⁴) = t².Therefore, the arc length L is the integral from 0 to π/3 of t² dt.Integrating t² with respect to t gives (1/3) t³. Evaluating this from 0 to π/3:L = (1/3)( (π/3)³ - 0 ) = (1/3)( π³ / 27 ) = π³ / 81.So, the arc length of the curve is π³ divided by 81.

💡Alright, so I have this problem to prove that the second central moment, which is the variance, is less than the ordinary second moment when you take it around any point that's not the mean. Hmm, okay. Let me try to unpack this.First, I need to recall what these terms mean. The second central moment is the variance, right? It's like the average of the squared deviations from the mean. So, if I have a random variable X, and its mean is M(X), then the variance is E[(X - M(X))²]. Got that.Now, the ordinary second moment around a point C is E[(X - C)²]. So, instead of centering it at the mean, I'm centering it at some other point C. The problem says that if C is not equal to M(X), then this ordinary second moment is larger than the variance.Okay, so I need to show that E[(X - M(X))²] < E[(X - C)²] when C ≠ M(X). Hmm, how do I approach this?Maybe I can expand E[(X - C)²] and see how it relates to the variance. Let's try that. Expanding (X - C)² gives X² - 2CX + C². So, E[(X - C)²] = E[X²] - 2CE[X] + C².Wait, E[X] is just M(X), right? So, substituting that in, we get E[X²] - 2C M(X) + C². Hmm, interesting.Now, the variance is E[(X - M(X))²] which is E[X²] - (M(X))². So, variance is E[X²] minus the square of the mean.So, if I compare E[(X - C)²] and the variance, I have:E[(X - C)²] = E[X²] - 2C M(X) + C²andVar(X) = E[X²] - (M(X))².So, if I subtract Var(X) from E[(X - C)²], I get:E[(X - C)²] - Var(X) = [E[X²] - 2C M(X) + C²] - [E[X²] - (M(X))²]Simplifying this, the E[X²] terms cancel out:= -2C M(X) + C² - (- (M(X))²)= -2C M(X) + C² + (M(X))²Hmm, that's a quadratic in terms of C. Let me write it as:= C² - 2C M(X) + (M(X))²Wait a minute, that looks familiar. That's (C - M(X))²!So, E[(X - C)²] - Var(X) = (C - M(X))².Since squares are always non-negative, (C - M(X))² ≥ 0. And if C ≠ M(X), then (C - M(X))² > 0.Therefore, E[(X - C)²] - Var(X) > 0 when C ≠ M(X), which implies that E[(X - C)²] > Var(X).So, that's the proof! The variance is always less than or equal to the second moment around any other point, and strictly less if the point isn't the mean.Wait, let me double-check my steps to make sure I didn't make a mistake.1. Expanded E[(X - C)²] correctly: E[X²] - 2C E[X] + C².2. Recognized that E[X] is M(X), so substituted that in.3. Subtracted Var(X) from E[(X - C)²], which gave me C² - 2C M(X) + (M(X))².4. Noticed that this expression is a perfect square: (C - M(X))².5. Concluded that since squares are non-negative, and positive if C ≠ M(X), then E[(X - C)²] > Var(X).Yeah, that seems solid. I don't see any errors in the logic or calculations. So, I think this proves the statement.**Final Answer**The second central moment is less than the ordinary second moment for any ( C neq M(X) ), so (boxed{mu_{2} < mu_{2}^{prime}}).

💡First, I need to determine the total marks of all 25 students by multiplying the number of students by the average marks: 25 students * 70 marks/student = 1750 marks.Next, since each student's marks are doubled, I will calculate the new total marks: 1750 marks * 2 = 3500 marks.Finally, to find the new average, I will divide the new total marks by the number of students: 3500 marks / 25 students = 140 marks/student.Therefore, the new average after doubling each student's marks is 140.