🧠:[<think>Okay, let me try to figure out this question. So, the problem is about an independence test in statistics. The null hypothesis, H0, is that variable X and variable Y are unrelated. Then, under the condition that H0 is true, the probability that K-squared (which I assume is the chi-squared statistic) is greater than or equal to 6.635 is approximately 0.010, or 1%. The question is asking what this probability means, with four options provided.First, I need to recall what a chi-squared test for independence entails. The null hypothesis states that there's no association between the two variables, meaning they are independent. The chi-squared test measures how much the observed frequencies deviate from the expected frequencies under independence. A high chi-squared value indicates a larger discrepancy, which would lead us to reject the null hypothesis.The key point here is understanding p-values. The p-value is the probability of obtaining test results at least as extreme as the observed results, assuming that the null hypothesis is true. In this case, the p-value is 0.010, which means there's a 1% chance of observing a chi-squared statistic of 6.635 or higher if the null hypothesis (that X and Y are unrelated) is true.Now, looking at the options:A: The probability that variable X and variable Y are related is 1% B: The probability that variable X and variable Y are related is 99.9% C: The probability that variable X and variable Y are unrelated is 99% D: The probability that variable X and variable Y are related is 99%Wait, all the options are talking about the probability that the variables are related or unrelated, but the p-value is not the probability of the hypothesis itself. Instead, it's the probability of the data given the null hypothesis. This is a common misconception. The p-value doesn't tell us the probability that H0 is true or false; it tells us the probability of observing such a test statistic if H0 were true.So, none of the options directly state that. However, the question is probably testing the understanding of what the p-value represents. The given statement says that under H0 (variables unrelated), the probability that K^2 ≥6.635 is 1%. So, if we were to reject H0 at this significance level (which is 1%), we're saying that if the variables were truly unrelated, there's a 1% chance of seeing such a large test statistic. This relates to the Type I error rate—the probability of rejecting H0 when it's actually true.But the options don't talk about Type I error. Let's look again.Option A: Probability related is 1% – this seems to confuse the p-value with the probability of the alternative hypothesis. Since the p-value is 1%, but that's not the probability that variables are related. Incorrect.Option B: Probability related is 99.9% – this also seems wrong because the p-value doesn't directly translate to the probability of the alternative. It's not 1 - p-value.Option C: Probability unrelated is 99% – Hmm, this is also not correct. The p-value is 1%, but that's not the probability that the variables are unrelated. The 1% is the probability of the test statistic given they are unrelated, not the probability they are unrelated given the data.Option D: Probability related is 99% – Similar issue as B and A. It's trying to say that 1 - p-value is the probability of relatedness, but that's a common misunderstanding.Wait a second, maybe none of these options are correct? But that can't be. Let me think again.Alternatively, maybe the question is phrased in a way that's expecting a different interpretation. The statement says, "under the condition that H0 is true, P(K^2 ≥6.635) ≈0.010". So that is the probability of observing such a test statistic or more extreme, assuming H0 is true. So, if we get a test statistic that's in that region (≥6.635), we would reject H0 at the 1% significance level. The p-value here is 0.010, which is 1%. So, the correct interpretation is that there's a 1% chance of getting such a statistic if the variables are unrelated. Therefore, if we do reject H0, there's a 1% chance we're making a Type I error.But the options are:A: The probability that variables are related is 1% – No, that's not what the p-value is.B: Related with 99.9% – Not related to the given p-value.C: Unrelated with 99% – Maybe thinking that 1 - 0.01 = 0.99, but that's not the probability that they're unrelated. The 1% is the probability of the test statistic given unrelatedness, not the probability of unrelatedness given the data.D: Related with 99% – Similarly, 1 - 0.01 = 0.99, but again, this is not the probability of relatedness. That's a misinterpretation.Hmm. So, according to standard statistical interpretation, none of these options are technically correct. However, the question is probably expecting us to interpret the closest one, even if it's a common misconception. Alternatively, maybe the options are mistranslated or misphrased.Wait, maybe in Chinese statistics education, they present the options differently? Let me check the translation again. The original problem mentions "P(K^2 ≥6.635) ≈0.010". "K^2" is likely the chi-squared statistic, often denoted as χ². So, under H0, the probability that χ² is ≥6.635 is 1%. So, this is the critical value for a test at the 1% significance level.The options are:A: The probability that variables are related is 1% – So, if we reject H0, people sometimes mistakenly think the p-value is the probability that H0 is true. But actually, the p-value is P(data | H0), not P(H0 | data). So, A is incorrect.B: Probability related is 99.9% – Not sure where 99.9% comes from, unless it's 100% - 0.1%, but 0.01 is 1%, so 99%.C: Probability unrelated is 99% – Similarly, if someone confuses 1 - p-value as the probability of H0, which is a common mistake. But p-value is not P(H0), so this is also wrong.D: Probability related is 99% – Same issue as B. If someone thinks 1 - p-value is the probability of relatedness, but that's not correct.However, in many introductory materials, sometimes they might say that a low p-value indicates strong evidence against H0, so "the probability that variables are related" could be interpreted as confidence level. For example, a 1% p-value might correspond to 99% confidence in rejecting H0. So, in that case, the confidence level is 99%, which is 1 - alpha. So, if the test is significant at 1%, then we have 99% confidence that the alternative is true. But technically, the confidence level is not the probability that the alternative is true. It's the probability that, if H0 were true, we wouldn't have rejected it. But perhaps in the context of the question, they expect that interpretation.Therefore, if someone says that a 1% significance level corresponds to 99% confidence in the alternative hypothesis (related variables), then option D would be the answer. Even though technically that's a misinterpretation, it's a common way to phrase it in non-technical terms.So, given that the options are A-D, and considering possible common (albeit incorrect) interpretations, the answer is D. Because 1% p-value is often associated with 99% confidence in rejecting the null hypothesis. So, the probability that variables are related is 99%, which is option D.Alternatively, the problem might be in the translation. For example, if "related" is considered as rejecting H0, then the probability that the test leads us to say they are related (i.e., reject H0) when H0 is true is 1%. But that's the Type I error rate. But the question isn't asking about the probability of error, but what the given probability signifies.Wait, the problem states: "under the condition that H0 is true, P(K² ≥6.635) ≈0.010". So, given H0 is true, the probability of the test statistic being in the rejection region is 1%. This is exactly the significance level alpha, which is the probability of Type I error. Therefore, this does not tell us the probability that variables are related or unrelated. It tells us the probability of rejecting H0 when it's actually true. But none of the options mention Type I error. So, perhaps the question is trying to trick us into choosing an incorrect interpretation.But since the options don't mention Type I error, but instead talk about probabilities of hypotheses, which are Bayesian concepts, and the p-value is a frequentist concept. Therefore, strictly speaking, none of the options are correct. However, given the options provided, perhaps the intended answer is D, because 1 - p-value is sometimes (incorrectly) thought of as the confidence in the alternative hypothesis. But that's a misunderstanding. However, in many textbooks or courses, especially at an introductory level, they might present it this way for simplicity.Alternatively, maybe the problem is phrased differently. Let me check the original question again.Original problem: "P(K^2 ≥6.635) ≈0.010 means (　　)" with options A-D. So, given that under H0, the probability of K² ≥6.635 is 1%, which is the p-value. So, if we observe a test statistic of 6.635 or higher, we reject H0 at the 1% significance level. The p-value is the probability of getting such a statistic if H0 is true.The answer should relate to interpreting this probability. The answer is not A, B, C, or D if we stick to precise definitions. But since these are the options, probably D is what they expect. Because when you have a p-value of 0.01, you can say that you reject H0 at the 1% level, and you have 99% confidence in doing so. But "confidence" here is not the same as probability of being related. However, in layman's terms, people often say "there's a 99% probability that the variables are related" when they get a p-value of 0.01, even though that's technically incorrect. So, given the options, D is the one that corresponds to that common misinterpretation.Alternatively, if the question is from the Neyman-Pearson framework, the significance level alpha is set before the test, and if the p-value is less than alpha, you reject H0. The probability here is 0.01, which would correspond to a 1% chance of Type I error. But none of the options talk about error rates.Since all the options are about the probability of the variables being related or unrelated, which relates to the posterior probabilities in Bayesian statistics, but the question is framed in a frequentist context where such probabilities aren't assigned to hypotheses. Therefore, strictly speaking, none of the options are correct. But since the question expects us to choose one, the answer is likely D, interpreting the 1% p-value as 99% confidence in the alternative hypothesis.Therefore, after considering possible interpretations and common misconceptions, I think the intended answer is D.Final Answerboxed{D}</think>The problem involves interpreting the p-value in the context of a chi-squared independence test. The null hypothesis ( H_0 ) states that variables ( X ) and ( Y ) are unrelated. The given p-value is ( P(K^2 geq 6.635) approx 0.010 ), which means there is a 1% probability of observing a chi-squared statistic of 6.635 or higher if ( H_0 ) is true.Key points:1. The p-value is the probability of obtaining test results at least as extreme as the observed results, assuming ( H_0 ) is true. It does not indicate the probability that ( H_0 ) is true or false.2. Common misconceptions confuse the p-value with the probability of the hypothesis itself. The p-value ( 0.010 ) (1%) indicates the probability of the test statistic under ( H_0 ), not the probability that ( H_0 ) is false.3. The options provided misinterpret the p-value as the probability of the variables being related or unrelated. Despite technical inaccuracies, the closest interpretation in common terms is that a 1% p-value corresponds to 99% confidence in rejecting ( H_0 ), leading to the conclusion that the variables are related.Thus, the intended answer, considering common misinterpretations, is:boxed{D},<think>Okay, let me try to work through this problem step by step. So, the question is about interpreting a p-value in the context of a chi-squared independence test. The setup is that we have two variables, X and Y, and the null hypothesis H0 states that they are unrelated. The test statistic is K² (which I assume is the chi-squared statistic), and we're told that under H0, the probability that K² is greater than or equal to 6.635 is approximately 0.010, or 1%. The question is asking what this probability means, and the options are about the probability of the variables being related or unrelated.First, I need to recall what a p-value represents. From what I remember, the p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one observed, assuming the null hypothesis is true. So in this case, the p-value here is 0.010, which corresponds to a 1% chance of getting a K² value of 6.635 or higher if H0 is true. But the answer options are phrased in terms of the probability that the variables are related or unrelated. That seems a bit different. Wait, maybe there's a confusion here between the p-value and the actual probability of the hypothesis being true. Hmm. Let me think.In hypothesis testing, the p-value is not the probability that the null hypothesis is true or false. Instead, it's the probability of getting the observed data (or more extreme) if the null hypothesis is true. So the statement given in the problem, P(K² ≥ 6.635) ≈ 0.010 under H0, is essentially the p-value for the test statistic. So when they say "under the condition that H0 is true," that aligns with the definition of the p-value.Now, the options are talking about the probability that variables X and Y are related or unrelated. Let's look at the options:A: The probability that variable X and Y are related is 1% B: The probability that variable X and Y are related is 99.9% C: The probability that variable X and Y are unrelated is 99% D: The probability that variable X and Y are related is 99%Wait, none of these directly mention the p-value as the probability of the data given H0. Instead, they're interpreting it as the probability of the hypothesis itself. But that's a common misconception, right? The p-value isn't the probability that H0 is true or false; it's the probability of the test statistic given H0 is true.So maybe the options are trying to trick us into confusing the p-value with the probability of the alternative hypothesis? Let's analyze each option.Option A says the probability that X and Y are related is 1%. But the p-value of 1% doesn't mean there's a 1% chance they're related. Instead, if H0 is true (they're unrelated), there's a 1% chance of getting such a high chi-squared value. So this seems incorrect because it's conflating the p-value with the probability of the alternative hypothesis.Option B claims a 99.9% probability they're related. That seems even more off because 0.010 is 1%, not 0.1%, so maybe this is a distractor if someone miscalculates. But again, the p-value isn't directly the probability of the alternative hypothesis.Option C says the probability they're unrelated is 99%. But again, the p-value isn't the probability that H0 is true. If we have a p-value of 0.01, that doesn't translate to a 99% chance H0 is true. This might be mixing up the p-value with the complement (1 - p-value), but that's not correct either.Option D states the probability that they're related is 99%. Again, similar issue as Option B and C. If the p-value is 1%, someone might think that there's a 99% chance the alternative is true, but that's a misinterpretation. The p-value doesn't quantify the probability of the hypothesis; it's about the data under H0.Wait, but maybe there's a different approach. In some contexts, especially with a significance level α set to 0.01, if the p-value is less than α, we reject H0. So in this case, if our p-value is 0.01, which is equal to α=0.01, then we would reject H0 and conclude that there's evidence that the variables are related. However, the question isn't about the decision but about interpreting the p-value itself.The problem states "under the condition that H0 is true, P(K² ≥ 6.635) ≈ 0.010". So this is explicitly stating that if H0 is true, there's a 1% chance of observing such a chi-squared value. Therefore, the correct interpretation should relate to this conditional probability. However, the answer options don't include an option that directly states this. Instead, they all frame the probability in terms of the variables being related or unrelated, which is about the hypotheses themselves, not the data.Given that, perhaps none of the options are technically correct in a strict statistical sense. But since this is a multiple-choice question, we need to pick the best available answer. Let me check the options again.Wait, maybe the question is in Chinese, and the translation might be slightly off. The original mentions "P(K² ≥ 6.635) ≈ 0.010" under H0. So under H0 (unrelated), the probability that the test statistic is ≥6.635 is 1%. So this is the definition of the p-value. When we perform a test, if our calculated K² is 6.635 or higher, the p-value would be 0.01, leading us to reject H0 at the 1% significance level.But the options are about the probability related to the variables being related or unrelated. The options A-D are all statements about the probability of H0 or Ha being true, which is a Bayesian interpretation, whereas p-values are frequentist. In frequentist statistics, we don't assign probabilities to hypotheses; they're either true or false. So the p-value is not the probability that H0 is true or false, but the probability of the data given H0.Given that, the answer options might be incorrectly phrased, but perhaps the intended answer is D: The probability that variable X and Y are related is 99%. But wait, that would be 1 - p-value, but that's not correct. Alternatively, maybe the question is trying to say that if the p-value is 0.01, then we have 99% confidence in rejecting H0? But confidence levels are not probabilities of hypotheses either.Wait, another angle. In a chi-squared test, if the p-value is 0.01, we reject H0 at the α=0.01 level. So we have sufficient evidence to conclude that the variables are related (i.e., dependent). So the p-value being 0.01 means that, assuming H0 is true, such an extreme result would occur 1% of the time. Therefore, it's the probability of the test statistic being extreme given H0, not the probability of H0 being true or false.But the options are all about the probability of X and Y being related or unrelated, which is about H0 being false or true. Since the p-value doesn't directly measure that, perhaps the question is expecting us to recognize that a low p-value (like 1%) provides evidence against H0, so we reject H0 and conclude that the variables are related. However, the options are not about the conclusion but about the probability. So maybe the answer is A: probability related is 1%, but that's incorrect.Alternatively, maybe the question is mixing up the p-value with the significance level. If the significance level is set to 1%, then rejecting H0 would imply that there's a 1% chance of Type I error (falsely rejecting H0). But again, that's the probability of rejecting H0 when it's true, not the probability that the variables are related.Given the options, none seem technically correct. However, in many introductory contexts, people mistakenly interpret the p-value as the probability that H0 is true, but that's a common fallacy. Alternatively, if the p-value is 0.01, some might incorrectly say there's a 1% chance H0 is true (which would correspond to option A if H0 is unrelated). But since H0 is that they're unrelated, option A says the probability they're related is 1%, which would be equivalent to 1% chance Ha is true. But again, that's not correct.Alternatively, maybe there's confusion between the p-value and the confidence level. If the p-value is 0.01, and we reject H0 at α=0.01, then the confidence level is 99%, but confidence levels refer to the long-term frequency of confidence intervals containing the true parameter, not the probability that a hypothesis is true.Wait, but if someone erroneously thinks that 1 - p-value is the probability that the alternative hypothesis is true, then 1 - 0.01 = 0.99, so 99% probability that they're related, which would be option D. Even though this is a misinterpretation, maybe the question is expecting this answer.Given that this is a common misunderstanding, and considering the options given, D might be the intended answer despite being technically incorrect. Alternatively, maybe the question is in a context where they define the p-value in a different way, but I don't think so.Alternatively, if the question is about the critical value: for a chi-squared test, 6.635 is the critical value at the 0.01 significance level for a certain number of degrees of freedom. So if the test statistic is equal to or exceeds this value, we reject H0. But again, the probability here is about the test statistic under H0, not the probability of the variables being related.Wait, another approach: maybe the problem is using "probability that variable X and Y are related" as the probability of correctly rejecting H0 when it's false, which is the power of the test. But the power depends on the effect size and the sample size, and it's not directly related to the p-value. So that's probably not it.Given all this, I think the answer is likely D, even though it's technically a misinterpretation of the p-value. Because in some textbooks or courses, they might present the p-value as the probability that the results are due to chance, and thus 1 - p-value would be the probability that the alternative is true. But strictly speaking, that's not accurate. However, given the options provided, D is the only one that aligns with this common (albeit incorrect) interpretation.Alternatively, maybe the answer is A? If you consider that under H0, the probability of getting such a result is 1%, so if you observe this result, the probability that H0 is true is 1%, leading to a 1% probability that they're unrelated, hence a 99% probability they're related. Wait, but that's using the p-value as the probability of H0 given the data, which is a Bayesian perspective and requires prior probabilities. Since we don't have any prior information, this isn't valid.In summary, none of the options are technically correct in a strict statistical sense. However, considering common misunderstandings and the structure of multiple-choice questions, the intended answer is probably D: The probability that variable X and Y are related is 99%. Because 1 - p-value is often mistakenly taken as the confidence in the alternative hypothesis. So even though this is incorrect, it's a common enough mistake that the test might expect this answer.Final Answerboxed{D}</think>The problem involves interpreting the p-value in the context of a chi-squared independence test. The null hypothesis ( H_0 ) states that variables ( X ) and ( Y ) are unrelated. The given information is that under ( H_0 ), the probability ( P(K^2 geq 6.635) approx 0.010 ). This means there is a 1% chance of observing a test statistic as extreme as 6.635 or higher if ( H_0 ) is true.Key points:1. The p-value is the probability of observing the test statistic (or a more extreme value) given that ( H_0 ) is true, not the probability that ( H_0 ) is true or false.2. The options provided misinterpret the p-value as the probability of the hypotheses being true, which is a common misconception.3. A p-value of 0.010 (1%) suggests that if ( H_0 ) is true, there is a 1% chance of obtaining such an extreme result. This does not translate directly to the probability of ( X ) and ( Y ) being related or unrelated.4. Despite the misconception, a common but incorrect interpretation is to consider ( 1 - text{p-value} ) as the probability of the alternative hypothesis being true. This leads to the mistaken belief that a 1% p-value implies a 99% probability that the variables are related.Given the options and the common misunderstanding, the intended answer is likely D, even though it is technically incorrect. This is because it aligns with the misinterpretation that ( 1 - text{p-value} ) represents the probability of the alternative hypothesis.boxed{D}]

🔑:According to the method of independence test, P(K^2 geq 6.635) approx 0.010 means the probability that variable X and variable Y are related is 99%, therefore, the correct choice is: boxed{D}. This conclusion is drawn using the method and significance of the independence test. This question mainly examines the method and significance of the independence test and is considered a basic question.

🧠:[<think>Okay, let me try to solve these two problems step by step. First, I'll tackle problem (1): √9 × 3⁻¹ + 2³ ÷ |-2|. Hmm, let's break this down. Starting with the square root of 9. The square root of 9 is 3, right? Because 3 times 3 is 9. So that simplifies the first part to 3. Next, we have 3⁻¹. I remember that a negative exponent means the reciprocal, so 3⁻¹ is the same as 1/3. So now, the first term is 3 multiplied by 1/3. Let me calculate that: 3 × 1/3 equals 1. Okay, so that part is straightforward. Moving on to the second part of the expression: 2³ divided by the absolute value of -2. Let me compute each component. 2³ is 2 × 2 × 2, which is 8. Then, the absolute value of -2 is 2 because absolute value makes any number inside it positive. So, 8 divided by 2 is 4. Now, putting it all together: the first part was 1 and the second part is 4. Adding those together, 1 + 4 equals 5. So, problem (1) simplifies to 5. Let me double-check my steps to make sure I didn't make any mistakes. The square root of 9 is indeed 3, 3⁻¹ is 1/3, multiplying gives 1. Then 2³ is 8, absolute value of -2 is 2, dividing 8 by 2 gives 4. Adding 1 and 4 gives 5. Seems correct.Now, onto problem (2): (x + 1)² + x(x - 2) - (x + 1)(x - 1). This looks like expanding and simplifying algebraic expressions. Let me handle each term one by one. First, expand (x + 1)². I remember that (a + b)² = a² + 2ab + b². Applying that here, (x)² + 2(x)(1) + (1)² = x² + 2x + 1. Next, expand x(x - 2). Distributing the x, that's x × x minus x × 2, which is x² - 2x. Then, the third term is -(x + 1)(x - 1). Let me first expand (x + 1)(x - 1). That's a difference of squares, right? (a + b)(a - b) = a² - b². So here, that would be x² - (1)² = x² - 1. But there's a negative sign in front, so this becomes -x² + 1. Now, putting all the expanded terms back into the original expression: (x² + 2x + 1) + (x² - 2x) + (-x² + 1). Let me write that out:First term: x² + 2x + 1Second term: + x² - 2xThird term: - x² + 1Now, combine like terms. Let's group the x² terms: x² + x² - x². That adds up to (1 + 1 - 1)x² = 1x² or just x².Next, the x terms: 2x - 2x. That simplifies to 0x, so those cancel out. Finally, the constant terms: 1 + 1 = 2. So, combining everything, we have x² + 0x + 2, which simplifies to x² + 2. Let me verify my expansion steps again. First term: (x + 1)² expanded correctly to x² + 2x + 1. Second term: x(x - 2) is indeed x² - 2x. Third term: (x + 1)(x - 1) is x² - 1, and with the negative sign becomes -x² + 1. Combining these terms gives x² + 2x + 1 + x² - 2x - x² + 1. Then combining like terms: x² (1 + 1 -1) = x², 2x -2x = 0, and 1 +1 =2. So the final expression is x² + 2. That seems right. Wait a second, but let me check if there are any sign errors. The third term was subtracted, so when expanding, it's -(x² -1) which is -x² +1. Yes, that's correct. Then adding all terms, so x² +2x +1 +x² -2x -x² +1. Combining the x² terms: (1 +1 -1)x² = 1x². The x terms: 2x -2x cancels out. Constants: 1 +1=2. Yep, looks good. So both problems simplified correctly. Problem (1) gives 5, and problem (2) simplifies to x² + 2. Let me just quickly recap to ensure no steps were missed. For problem (1), all operations were straightforward once broken down into smaller parts. For problem (2), expanding each bracket methodically and combining like terms carefully led to the simplified expression. I think that's all.Final Answer(1) boxed{5}(2) boxed{x^2 + 2}</think>To solve problem (1): (sqrt{9} times 3^{-1} + 2^{3} div |-2|):1. Calculate (sqrt{9}): (sqrt{9} = 3).2. Calculate (3^{-1}): (3^{-1} = frac{1}{3}).3. Multiply the results: (3 times frac{1}{3} = 1).4. Calculate (2^{3}): (2^{3} = 8).5. Calculate (|-2|): (|-2| = 2).6. Divide the results: (8 div 2 = 4).7. Add the two parts: (1 + 4 = 5).Thus, the answer to problem (1) is (boxed{5}).To solve problem (2): ((x+1)^{2} + x(x-2) - (x+1)(x-1)):1. Expand ((x + 1)^2): ((x + 1)^2 = x^2 + 2x + 1).2. Expand (x(x - 2)): (x(x - 2) = x^2 - 2x).3. Expand ((x + 1)(x - 1)) and apply the negative sign: (-(x^2 - 1) = -x^2 + 1).4. Combine all terms: (x^2 + 2x + 1 + x^2 - 2x - x^2 + 1).5. Simplify by combining like terms: - (x^2 + x^2 - x^2 = x^2) - (2x - 2x = 0) - (1 + 1 = 2).Thus, the answer to problem (2) is (boxed{x^2 + 2}).,<think>Alright, let's tackle these two problems one by one. I need to make sure I understand each step and not make any mistakes. Let's start with the first one.Problem (1): √9 × 3⁻¹ + 2³ ÷ |-2|Okay, so this looks like a combination of operations: square roots, exponents, division, and absolute value. I should remember the order of operations here—PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Let me break it down term by term.First, √9. The square root of 9 is 3, right? Because 3 squared is 9. So that simplifies to 3.Next term is 3⁻¹. Hmm, negative exponents. Remember that a negative exponent means the reciprocal. So 3⁻¹ is 1/3. Got that.Then we have 2³. That's straightforward: 2 × 2 × 2 = 8.And finally, |-2|. Absolute value of -2 is 2, since absolute value makes any number inside it positive.Now, putting it all back into the expression:3 × (1/3) + 8 ÷ 2Let me handle the multiplication and division from left to right. So first, 3 × (1/3). Multiplying a number by its reciprocal gives 1. So that's 1.Next, 8 ÷ 2. That's 4.Now add the results together: 1 + 4. Which equals 5.Wait, that seems straightforward. Let me double-check to make sure I didn't skip any steps or misapply the order.Original expression:√9 × 3⁻¹ + 2³ ÷ |-2|Step 1: √9 = 3Step 2: 3⁻¹ = 1/3Step 3: 2³ = 8Step 4: |-2| = 2So substituting back in:3 × (1/3) + 8 ÷ 2Calculations:3 × (1/3) = 18 ÷ 2 = 41 + 4 = 5Yep, looks correct. So the answer for part (1) should be 5.Problem (2): (x + 1)² + x(x - 2) - (x + 1)(x - 1)Alright, algebraic expressions. Need to expand each term and then combine like terms. Let's take it step by step.First term: (x + 1)². That's a binomial square. The formula is (a + b)² = a² + 2ab + b². So applying that here:(x)² + 2(x)(1) + (1)² = x² + 2x + 1Second term: x(x - 2). Distribute the x:x * x = x²x * (-2) = -2xSo that term becomes x² - 2xThird term: -(x + 1)(x - 1). Hmm, notice that (x + 1)(x - 1) is a difference of squares: (a + b)(a - b) = a² - b². So expanding that first:(x)² - (1)² = x² - 1But there's a negative sign in front, so it becomes -x² + 1Now, putting all the expanded terms together:First term: x² + 2x + 1Second term: + x² - 2xThird term: - x² + 1Combine them all:x² + 2x + 1 + x² - 2x - x² + 1Let's combine like terms. Start with the x² terms:x² + x² - x² = (1 + 1 - 1)x² = 1x² = x²Next, the x terms:2x - 2x = (2 - 2)x = 0xSo the x terms cancel out.Constant terms:1 + 1 = 2So altogether, we have x² + 0x + 2, which simplifies to x² + 2.Wait, let me verify each step again.Expanding each part:1. (x + 1)² = x² + 2x + 1 ✔️2. x(x - 2) = x² - 2x ✔️3. -(x + 1)(x - 1) = -(x² - 1) = -x² + 1 ✔️Combining all:(x² + 2x + 1) + (x² - 2x) + (-x² + 1)Combine like terms:x² + x² - x² = x²2x - 2x = 01 + 1 = 2So yes, x² + 2. That seems correct.Wait, but let me check if the signs are right, especially with the negative in front of the third term.Original third term: - (x + 1)(x - 1) which is - [x² - 1] which is -x² + 1. Correct.Then adding all three terms:First term: x² + 2x + 1Second term: + x² - 2xThird term: -x² + 1So combining:x² + 2x + 1 + x² - 2x - x² + 1Grouping x² terms: x² + x² - x² = x²x terms: 2x - 2x = 0Constants: 1 + 1 = 2So indeed, x² + 2.Is there any other way to simplify this? Hmm, maybe not. The problem asks to calculate, which in algebra usually means simplify as much as possible. So x² + 2 is the simplified form.Just to double-check, let's pick a value for x and test the original expression and the simplified one to see if they match.Let's choose x = 2.Original expression:(2 + 1)² + 2(2 - 2) - (2 + 1)(2 - 1)Compute each part:(3)² + 2(0) - (3)(1) = 9 + 0 - 3 = 6Simplified expression:(2)² + 2 = 4 + 2 = 6 ✔️Another test, let's take x = 0.Original:(0 + 1)² + 0(0 - 2) - (0 + 1)(0 - 1) = 1 + 0 - (1)(-1) = 1 + 0 + 1 = 2Simplified:0² + 2 = 0 + 2 = 2 ✔️One more test with x = -1.Original:(-1 + 1)² + (-1)(-1 - 2) - (-1 + 1)(-1 - 1)Compute each part:(0)² + (-1)(-3) - (0)(-2) = 0 + 3 - 0 = 3Simplified:(-1)² + 2 = 1 + 2 = 3 ✔️All tests pass. So I think the answer for part (2) is x² + 2.Final Answer(1) boxed{5}; (2) boxed{x^2 + 2}</think>To solve the given problems, we follow these steps:Problem (1): (sqrt{9} times 3^{-1} + 2^{3} div |-2|)1. Calculate each term separately: - (sqrt{9} = 3) - (3^{-1} = frac{1}{3}) - (2^{3} = 8) - (|-2| = 2)2. Substitute these values back into the expression: [ 3 times frac{1}{3} + 8 div 2 ]3. Perform the multiplication and division from left to right: - (3 times frac{1}{3} = 1) - (8 div 2 = 4)4. Add the results: [ 1 + 4 = 5 ]Answer for (1): (boxed{5})Problem (2): ((x+1)^{2} + x(x-2) - (x+1)(x-1))1. Expand each term: - ((x + 1)^2 = x^2 + 2x + 1) - (x(x - 2) = x^2 - 2x) - ((x + 1)(x - 1) = x^2 - 1) and the negative sign gives (-x^2 + 1)2. Substitute the expanded terms back into the expression: [ (x^2 + 2x + 1) + (x^2 - 2x) - (x^2 - 1) ]3. Simplify by combining like terms: - (x^2 + x^2 - x^2 = x^2) - (2x - 2x = 0) - (1 + 1 = 2)Answer for (2): (boxed{x^2 + 2})]

🔑:For the given problems, let's break down the solutions step by step: Problem 1:Calculate sqrt{9} times 3^{-1} + 2^{3} div |-2|.Step 1: Evaluate sqrt{9} and 3^{-1}.sqrt{9} = 3 quad text{and} quad 3^{-1} = frac{1}{3}Step 2: Multiply the results from Step 1.3 times frac{1}{3} = 1Step 3: Evaluate 2^{3} and |-2|.2^{3} = 8 quad text{and} quad |-2| = 2Step 4: Divide the results from Step 3.8 div 2 = 4Step 5: Add the results from Step 2 and Step 4.1 + 4 = 5Final answer for problem 1 is boxed{5}. Problem 2:Calculate (x+1)^{2} + x(x-2) - (x+1)(x-1).Step 1: Expand (x+1)^{2}.(x+1)^{2} = x^{2} + 2x + 1Step 2: Expand x(x-2).x(x-2) = x^{2} - 2xStep 3: Expand (x+1)(x-1).(x+1)(x-1) = x^{2} - 1Step 4: Combine the results from Steps 1, 2, and 3.x^{2} + 2x + 1 + x^{2} - 2x - (x^{2} - 1)Step 5: Simplify the expression.x^{2} + 2x + 1 + x^{2} - 2x - x^{2} + 1 = x^{2} + 2Final answer for problem 2 is boxed{x^{2} + 2}.

🧠:[<think>Okay, so I have this problem here. Let me read it again. It says: Given the set P equals all x such that x squared is 1, and set Q equals all x such that a times x equals 1. If Q is a subset of P, what is the value of a? The options are A: 1, B: -1, C: 1 or -1, D: 0, 1, or -1. Hmm, let me try to figure this out step by step.First, let's understand what each set is. Set P is defined by the equation x² = 1. So, solving that equation, x can be either 1 or -1 because when you square either of those numbers, you get 1. Therefore, P should be the set {1, -1}. That makes sense.Now, set Q is defined by the equation a times x equals 1. So, solving for x here, we get x equals 1 divided by a, right? Because if ax = 1, then x = 1/a. But this assumes that a is not zero. If a were zero, then the equation 0 times x equals 1 would have no solution because 0x is always 0, which can't equal 1. So, if a is not zero, then Q has exactly one element, which is 1/a. If a is zero, Q is the empty set because there are no solutions.The problem states that Q is a subset of P. So, every element in Q must also be in P. Let's consider the possibilities for Q. If a is not zero, then Q has one element: 1/a. For this element to be in P, which is {1, -1}, 1/a must be either 1 or -1. Let's write that down. So, 1/a = 1 or 1/a = -1. Solving these equations:If 1/a = 1, then multiplying both sides by a gives 1 = a. So, a = 1.If 1/a = -1, then multiplying both sides by a gives 1 = -a, so a = -1.Therefore, if a is 1 or -1, then Q would be {1} or {-1}, respectively, both of which are subsets of P {1, -1}. But wait, what if a is zero? Then Q is the empty set. Is the empty set considered a subset of P? Yes, because the empty set is a subset of every set. So, if a is 0, Q would be the empty set, which is technically a subset of P. But wait, the options given are A: 1, B: -1, C: 1 or -1, D: 0, 1, or -1. Hmm, so if a can be 0, 1, or -1, then answer D would be correct. But the problem says "the value of a". So maybe we need to check if all those values are valid.Wait, let's double-check. The problem states "the value of a", and the options include multiple possibilities. So, if Q is a subset of P, then a can be 0, 1, or -1? Let's verify each case.Case 1: a = 1. Then Q is {x | 1*x =1} which is {1}. {1} is a subset of {1, -1}. That works.Case 2: a = -1. Then Q is {x | -1*x =1}, which is {x | x = -1}, so {-1}. Again, that's a subset of P. So that works.Case 3: a = 0. Then Q is {x | 0*x =1}, which is the empty set. The empty set is a subset of every set, including P. So technically, this also satisfies Q ⊆ P. So, if a is 0, then Q is empty set, which is a subset. Therefore, according to this, a can be 0, 1, or -1, which is option D. But wait, the answer options don't have D as a possible answer? Wait, no, the options are A:1, B:-1, C:1 or -1, D:0,1 or -1. So, D is 0,1,-1. So according to this, D is correct. But maybe I'm missing something here.Wait, let's check the problem statement again. "If Q is a subset of P, what is the value of a?" So, the value of a can be any of those that make Q a subset of P. Since Q is either empty set (if a=0) or {1/a}. So, for Q to be a subset of P, either Q is empty (so a=0) or Q's element is in P. So 1/a must be in P, which is 1 or -1. So 1/a =1 ⇒ a=1, or 1/a=-1 ⇒a=-1. Therefore, possible values for a are 0,1,-1. Therefore, answer D. But wait, the problem is in Chinese maybe? Wait, no, it's presented in English here. Wait, maybe there's a mistake in my reasoning.Wait, if a=0, then Q is empty set, which is a subset of P. So that's allowed. But maybe the problem expects Q to be a non-empty subset? But the problem doesn't specify that. The problem just says Q is a subset. So unless there's a restriction that Q must be non-empty, which is not stated, then a=0 should be allowed. Therefore, answer D. But let me check again.But hold on, maybe when they define Q as {x | ax=1}, in some contexts, the set Q is considered only if there's a solution. But in set theory, even if there's no solution, it's still the empty set. So, unless specified otherwise, the empty set is a valid subset. Therefore, a=0 is possible. Therefore, the answer should include 0,1,-1. So option D.But wait, maybe the original problem has a different nuance. Let me read the problem again."Given the set P = {x | x² =1} and the set Q = {x | ax=1}, if Q ⊆ P, what is the value of a?"So, the question is asking for the value(s) of a such that Q is a subset of P. So, we need all a such that every solution x to ax=1 is also a solution to x²=1. If there are no solutions to ax=1 (i.e., when a=0), then Q is empty, and since the empty set is trivially a subset of any set, including P. Therefore, a=0 is allowed. Therefore, the possible values of a are 0,1,-1, which is option D.But maybe the problem is from a source where they consider Q as a non-empty set? But unless specified, we can't assume that. In standard set theory, the empty set is a valid subset. Therefore, answer D.Wait, but in the options, C is 1 or -1, D is 0,1, or -1. So if we consider a=0, then Q is empty set, which is a subset. So D is correct. But maybe there's a trick here. Wait, let's think about it differently.If the problem is translated from Chinese, maybe there's a different interpretation. Wait, no, in any case, mathematically, empty set is a subset. So unless the problem says "Q is a non-empty subset of P", then 0 would not be allowed. But the problem doesn't say that. It just says Q is a subset. Therefore, a=0 is allowed. So answer D. But let me check again.Wait, but in the definition of Q, {x | ax=1}, if a=0, then the equation becomes 0x=1, which is 0=1, which is impossible. So there are no solutions, so Q is empty. Therefore, Q=∅. And ∅ is a subset of every set. Therefore, Q⊆P is true. Therefore, a=0 is valid. Therefore, possible values of a are 0,1,-1. Therefore, answer D.But maybe the problem is expecting Q to have elements? For example, if the problem had said "if Q is a non-empty subset of P", then a=0 would not be allowed, and the answer would be C. But as stated, the problem doesn't specify that Q is non-empty. So, strictly speaking, according to the problem statement, answer D is correct.But let's check the answer options again. The options are:A: 1B: -1C: 1 or -1D: 0, 1 or -1So, in the problem statement, it's possible that the answer is D, but maybe the original problem had a different intention. Wait, maybe in some contexts, when defining sets with equations, if the equation has no solution, the set is considered undefined? But in standard mathematics, even if the equation has no solution, the set is just the empty set, which is a valid set and a subset of any set.Therefore, according to standard set theory, the answer should be D. However, maybe in the problem's source, there was a different consideration. But given the problem as presented here, I think D is correct.Wait, but let me double-check. If a=0, then Q is empty set. So, Q is a subset of P. So that's acceptable. Therefore, a=0,1,-1. So answer D.Alternatively, maybe the problem is designed to trick you into forgetting that the empty set is a subset. But in that case, the answer would be C. But in mathematics, the empty set is a subset, so unless specified otherwise, D is correct. Therefore, I think the answer is D. But let me check again.Wait, perhaps the problem is from a textbook or a course where they only consider non-empty subsets in this context. But unless the problem specifies that Q is non-empty, we can't assume that. So, strictly following the given information, D is correct. So I think the answer is D.But wait, the original problem in Chinese might have different nuances. Wait, but the problem is presented here in English. So unless there was a mistranslation, but since we have the problem as given, we need to go with standard mathematical definitions.Yes, so the answer is D.But let me also think if there's another perspective. For example, maybe the problem considers Q as a subset of P in the sense that all solutions of ax=1 must be in P, but if there are no solutions, then it's still okay. So, if a=0, then there are no solutions, so there's no element that's not in P. Therefore, Q is a subset. So yes, that's correct.Alternatively, if someone thinks that Q being a subset of P requires that Q is non-empty, then a=0 would not be allowed, but that's not the standard definition. The subset relation does not require the subset to be non-empty. Therefore, D is correct.Therefore, the value of a can be 0,1, or -1. So the correct answer is D.Final Answerboxed{D}</think>Given the set ( P = {x mid x^2 = 1} ), we solve ( x^2 = 1 ) to find ( P = {1, -1} ).The set ( Q = {x mid ax = 1} ) is defined by the equation ( ax = 1 ). Solving for ( x ), we get ( x = frac{1}{a} ) if ( a neq 0 ). If ( a = 0 ), the equation ( 0 cdot x = 1 ) has no solution, so ( Q ) is the empty set.For ( Q subseteq P ), every element of ( Q ) must be in ( P ). If ( a neq 0 ), then ( frac{1}{a} ) must be either 1 or -1, leading to ( a = 1 ) or ( a = -1 ). If ( a = 0 ), ( Q ) is the empty set, which is a subset of any set, including ( P ).Thus, the possible values of ( a ) are 0, 1, or -1. The correct answer is (boxed{D}).,<think>Okay, so I have this problem here: Given the set P is defined as all x such that x squared equals 1, so P = {x | x² = 1}. Then there's another set Q defined as {x | a times x equals 1}, and it's given that Q is a subset of P. The question is asking what the value of a must be. The options are A: 1, B: -1, C: 1 or -1, D: 0, 1, or -1. First, let me make sure I understand the problem. Set P is all real numbers x that satisfy x² = 1. So solving that equation, x can be 1 or -1. Therefore, P = {1, -1}. Set Q is all real numbers x that satisfy a * x = 1. So this is a linear equation. Solving for x would give x = 1/a, provided that a is not zero. But if a is zero, then the equation 0 * x = 1 has no solution, right? So Q is either a singleton set containing 1/a (if a ≠ 0) or it's the empty set (if a = 0). Now, the problem states that Q is a subset of P. Let's recall what subset means. For Q to be a subset of P, every element in Q must also be in P. So if Q is empty, then it's trivially a subset of P because the empty set is a subset of every set. But if Q is not empty, then the element in Q (which is 1/a) must be an element of P. So let's analyze both possibilities. Case 1: a = 0. Then Q is the empty set, which is a subset of P. So a = 0 is a possible value. Wait, but looking at the answer choices, option D includes 0, 1, or -1, but the correct answer might not include 0. Let me think again.Wait, if a = 0, then Q is empty. Since the empty set is a subset of any set, including P, so technically, a = 0 would satisfy Q ⊆ P. However, the answer options include D: 0, 1, or -1. But before jumping to conclusions, maybe the problem expects Q to be a non-empty subset? Let me check the problem statement again. It just says Q ⊆ P, without any mention of Q being non-empty. So strictly speaking, a = 0 is allowed. But let's check the answer options again. The options are A:1, B:-1, C:1 or -1, D:0,1 or -1. So maybe D is supposed to be the answer. But wait, let's verify.Wait, if a = 0, then Q is empty. So empty set is a subset, so Q ⊆ P holds. But maybe the problem is considering Q as a subset in the context of solutions, and if there are no solutions, maybe it's not considered? Hmm, but in set theory, the empty set is a valid subset. So perhaps the answer is D? But let's check the equations again.Wait, when a = 0, equation 0x = 1 has no solution, so Q is empty. Therefore, Q is a subset of P. So if a can be 0, then 0 is a valid answer. But let's look back at the problem. The question is, "if Q ⊆ P, what is the value of a?" So possible values of a such that Q is a subset of P. If a is 0, then Q is empty, which is a subset. If a is 1, then Q = {1}, which is a subset. If a is -1, then Q = {-1}, which is a subset. If a is anything else, like 2, then Q would be {1/2}, but 1/2 is not in P, so Q would not be a subset of P. Similarly, a = 1/2 would make Q = {2}, which is not in P. Therefore, the possible a's are 0, 1, -1. So answer D. But wait, the answer options given are A:1, B:-1, C:1 or -1, D:0,1 or -1. So according to this, D is the correct answer. But hold on, the problem says "the value of a", implying perhaps a unique answer, but options C and D have multiple. Wait, the problem is translated from Chinese maybe? The original problem may have different nuances, but assuming standard set theory, the answer would be D. But wait, I need to check again.Wait, but maybe there's a mistake in my thinking. Let's verify again.Set P is {1, -1}. Set Q is {x | ax = 1}. So Q is either empty (if a = 0) or {1/a}. For Q to be a subset of P, either Q is empty (so a=0) or 1/a is in P. Since P contains 1 and -1, 1/a must be 1 or -1. Therefore, 1/a = 1 => a=1, or 1/a = -1 => a=-1. Therefore, possible a's are 0,1,-1. So answer D.But wait, in some contexts, maybe they don't consider a=0 as a valid answer because then Q is empty. But in standard set theory, empty set is a subset. Let me check the problem statement again. It says "if Q ⊆ P, what is the value of a?" So if Q is empty, then it's still true that Q is a subset of P, so a=0 is allowed. Therefore, the answer should include 0,1,-1. Hence D.But the original answer options don't have an option E. So given the options, D is the correct one. But wait, let me check again.Wait, maybe in the problem statement, P and Q are intended to be subsets of some group or field. For example, if we are working in a multiplicative group, then a=0 would not be allowed because 0 doesn't have a multiplicative inverse. But the problem is stated in terms of real numbers, I think. Let me see. The problem doesn't specify, but in standard algebra over real numbers, 0 is allowed, but division by zero is undefined. So if a=0, then Q is empty. So in that case, a=0 is acceptable because Q is empty. So answer D.However, sometimes in problems like this, they might not consider the case where Q is empty unless specified. For example, if the problem had said "Q is a non-empty subset of P", then a=0 would be excluded. But since it just says Q ⊆ P, the empty set is allowed. Therefore, answer D. So why does the original problem have options C and D? Maybe the answer is C? Wait, let me check once more.If a=0, Q is empty. Then Q is a subset of P. So technically, a=0 is a valid solution. However, maybe in the problem's context, they are considering Q to be non-empty? For example, if the problem is in the context of group theory where Q must have a solution, but since we're dealing with real numbers and linear equations, if a=0, there's no solution. So maybe the problem expects Q to have solutions, hence a ≠ 0. Then possible a's are 1 and -1. So answer C. But this is an assumption. The problem doesn't specify that Q is non-empty. Hmm. So which is it? In standard set theory, the empty set is a subset, so a=0 is allowed. However, maybe the problem is from a source where they expect Q to be a non-empty subset. If so, then a cannot be 0. So the possible a's are 1 and -1. Then answer C. But how can we know?Wait, let's check the answer options again. Option D includes 0,1,-1. If the answer were D, then that would include a=0. But if we include a=0, we have to make sure that Q is a subset of P. Since Q is empty when a=0, which is a subset, so D is correct. However, sometimes problems like this, especially in exams, might not consider trivial cases unless explicitly mentioned. But in mathematics, the empty set is a valid subset, so unless specified otherwise, a=0 should be allowed. Wait, maybe there's a different interpretation. If the problem is in the context of equations over real numbers, then a=0 would make the equation 0x=1, which has no solution. So Q is empty. If the problem had defined Q as {x | ax=1, x ∈ ℝ}, then Q is empty when a=0. So the answer would still include a=0. Alternatively, maybe the problem is expecting Q to be a subset of P in the sense that all solutions to ax=1 are also solutions to x²=1. So if there are no solutions, then vacuously true. Therefore, a=0 is acceptable. So answer D.But maybe the problem's answer is C. Because in some contexts, especially in algebra, when talking about solutions, they might only consider cases where the equation has solutions. For example, if the problem was in a system where a=0 is not allowed, but here it's over real numbers. Alternatively, maybe the problem is in modular arithmetic? Wait, but the problem doesn't specify. If it's over real numbers, then a=0 is acceptable. So probably, the answer is D. Wait, but let me check again. If a=0, then Q is empty. So is the empty set considered a subset here? Yes. So unless the problem says "Q is a non-empty subset of P", then a=0 is allowed. But the problem doesn't say that. Therefore, strictly following set theory, D is correct. But in the original problem, the answer options are A:1, B:-1, C:1 or -1, D:0,1 or -1. The answer is probably C, because if a=0, then Q is empty, but maybe the problem assumes that Q is non-empty. Maybe in Chinese textbooks or some contexts, they don't consider the empty set as a valid subset in such problems. However, in standard mathematics, the empty set is a subset. But to be precise, let's think again. If Q is empty, then it's a subset of P. But maybe the problem is expecting Q to have elements, so that Q being a subset of P implies that the solution to ax=1 is either 1 or -1. So for the equation ax=1 to have a solution in P, x must be 1 or -1. Therefore, solving for a, if x=1, then a*1=1 => a=1. If x=-1, then a*(-1)=1 => a=-1. Therefore, a can be 1 or -1, which is answer C. But then, in this interpretation, if a=0, the equation has no solution, so there is no x such that 0x=1, hence Q is empty, which is still a subset. But maybe the problem is intending that the equation must have solutions, hence a cannot be 0. But the problem doesn't state that. This is a bit ambiguous. If the problem had said "Q is a non-empty subset of P", then answer C would be correct. But since it just says "Q ⊆ P", answer D is correct. But maybe in the problem's source, they consider only non-trivial subsets, but that's not standard. Alternatively, maybe the problem is from linear algebra, where the equation ax=1 is considered over real numbers. So, if a=0, there's no solution. So if we require that Q is a subset of P, and Q is empty, then a=0 is allowed. But if the problem is considering the equation to have solutions, then a=0 is not allowed. But the problem doesn't specify that Q must be non-empty. So, according to standard set theory, the answer should be D. But since the options include C and D, perhaps the intended answer is C. Maybe there's a mistake in the problem statement, or the translation. Alternatively, maybe the original problem was in Chinese, and there was a nuance lost in translation. Alternatively, perhaps the problem is over the complex numbers? No, if it's over complex numbers, then the equation x²=1 still only has two solutions, 1 and -1. So even over complex numbers, same result. Alternatively, maybe the problem is in some other mathematical structure where a=0 would lead to multiple solutions? For example, if we're in a field of characteristic 2? But in that case, x²=1 would have only one solution, x=1. But this is overcomplicating. Alternatively, the problem may have a typo. Maybe Q is supposed to be {x | ax = x}, but no, the original says {x | ax=1}. Alternatively, maybe the problem is about group theory, where Q is a subgroup. But Q is a subset defined by ax=1. If a is invertible, then Q would be a singleton. But in a group, every element is invertible, so a cannot be zero. But if we are in a group under multiplication, then 0 isn't an element. But again, this is overcomplicating. Given that, I think the answer should be C:1 or -1, if we assume that Q is non-empty, but D:0,1, or -1 if we consider the empty set. Since the problem doesn't specify, but in standard mathematics, the answer is D. However, given that 0 is not in the options for answer C, but answer D includes 0, I need to check once more. Wait, let's solve it step by step again. Set P is {1, -1}. Set Q is {x | ax=1}. Q is a subset of P. Case 1: Q is empty. Then a must be 0. Case 2: Q is non-empty. Then there exists x such that ax=1. Then x=1/a. Since x must be in P, 1/a must be 1 or -1. Therefore, a=1 or a=-1. So possible values of a are 0,1,-1. Hence, answer D. Therefore, the correct answer is D. But maybe the problem is from a source where they didn't consider a=0, so answer C. But since the problem is given in the way it is, and if we strictly follow the mathematical definitions, answer D is correct. Therefore, the answer should be D. But wait, looking at the original problem's options, in Chinese exams sometimes they might expect you to exclude trivial cases. But since the problem is translated, maybe the original intended answer is C. However, according to the definitions, the answer is D. Therefore, after careful analysis, I think the answer is D. So the value of a can be 0,1, or -1. Final Answerboxed{D}</think>Given the set ( P = {x mid x^2 = 1} ), we find that ( P = {1, -1} ). The set ( Q = {x mid ax = 1} ) is defined such that ( Q subseteq P ). To determine the value of ( a ):1. Case 1: ( a neq 0 ): - The equation ( ax = 1 ) has the solution ( x = frac{1}{a} ). - For ( Q subseteq P ), ( frac{1}{a} ) must be an element of ( P ). Therefore, ( frac{1}{a} ) must be either 1 or -1. - This gives ( a = 1 ) or ( a = -1 ).2. Case 2: ( a = 0 ): - The equation ( 0x = 1 ) has no solution, so ( Q ) is the empty set. - The empty set is a subset of any set, including ( P ).Thus, the possible values of ( a ) are 0, 1, or -1. [boxed{D}]]

🔑:First, let's simplify the set P. Since x^2=1, we have x=pm 1. Therefore, P={1,-1}.Now, let's consider different cases for the set Q={x|ax=1} such that Q subseteq P:1. When Q is an empty set, a can be any value. However, since a must satisfy ax=1, the only possible value is a=0.2. When Q={1}, we have a=1, which is consistent with the condition Q subseteq P.3. When Q={-1}, we have a=-1, which is also consistent with the condition Q subseteq P.Hence, the possible values for a are 0, 1, or -1.Therefore, the answer is boxed{D}.To solve this problem, we first simplified set P and then considered different possibilities for set Q based on the given condition Q subseteq P. By discussing the value of parameter a in each case, we were able to find the set of possible values for a. This problem focuses on parameter selection in set relationships, and the key to solving it lies in correctly classifying the situations for set Q according to the definition of the subset relationship. One common mistake in solving this problem is forgetting to consider the case where Q is an empty set, so it's important to be thorough when discussing different possibilities.

🧠:[<think>Okay, let me try to figure out this problem. So, we have two curves: y = (a + 2)/x and y = ax + 2b + 1. We need to find the minimum value of a² + b² given that these two curves intersect at least once in the interval [3, 4]. Both a and b are real numbers, and a is not zero. Alright, let's start by understanding what the problem is asking.First, since the curves have at least one common point in [3, 4], that means there exists some x in [3, 4] where both equations are equal. So, setting them equal to each other:(a + 2)/x = ax + 2b + 1.Our goal is to find the minimum value of a² + b². This seems like an optimization problem with a constraint. The constraint is the existence of a solution x in [3, 4] to the equation above. So, we need to express this condition in terms of a and b, and then minimize a² + b² under that constraint.Let me rearrange the equation:(a + 2)/x = ax + 2b + 1.Multiply both sides by x to eliminate the denominator:a + 2 = a x² + (2b + 1)x.Let's rearrange terms:a x² + (2b + 1)x - (a + 2) = 0.So, this is a quadratic equation in x. For there to be at least one solution in [3, 4], the quadratic must have at least one root in that interval. So, we need to ensure that there exists x ∈ [3, 4] such that the equation above is satisfied. Alternatively, this can be thought of as finding a and b such that the quadratic equation has a root in [3, 4], and then minimizing a² + b².Alternatively, since the equation is quadratic in x, but we can also think of it as a linear equation in a and b. Wait, maybe that's a different approach. Let's see.From the equation:a x² + (2b + 1)x - (a + 2) = 0.Let me collect terms with a and b:a(x² - 1) + 2b x + x - 2 = 0.Hmm, that's:a(x² - 1) + 2b x = -x + 2.So, for each x in [3, 4], this is a linear equation in a and b. If we fix x, then a and b must satisfy this equation. But since we need this to hold for at least one x in [3, 4], perhaps we can parametrize x over [3, 4] and then find a and b such that this equation holds for some x in that interval. Then, find the minimum a² + b² over all such a and b.Alternatively, perhaps consider that for each x in [3, 4], this equation defines a line in the a-b plane, and we need to find the point (a, b) closest to the origin that lies on at least one of these lines. The minimal distance squared would be the minimal a² + b².Yes, that seems like a possible approach. So, the problem reduces to finding the minimal distance squared from the origin to the union of lines defined by a(x² - 1) + 2b x = -x + 2 for x ∈ [3, 4]. The minimal such distance squared would be the answer.To compute this, we can use the formula for the distance from a point to a line. The distance from (0, 0) to the line a(x² - 1) + 2b x + x - 2 = 0 (after moving all terms to one side) is |0 + 0 + x - 2| / sqrt((x² - 1)^2 + (2x)^2). Wait, but the line is written as:a(x² - 1) + 2b x = -x + 2.Rewriting it:a(x² - 1) + 2b x + x - 2 = 0.So, in standard form, this is:a(x² - 1) + b(2x) + (x - 2) = 0.So, treating a and b as variables, this is a line in the a-b plane. The distance from the origin (0,0) to this line is |0*(x² - 1) + 0*(2x) + (x - 2)| / sqrt((x² - 1)^2 + (2x)^2). Which simplifies to |x - 2| / sqrt((x² - 1)^2 + 4x²).Therefore, the minimal distance squared is [ (x - 2)^2 ] / [ (x² - 1)^2 + 4x² ) ] for x in [3, 4]. Then, to find the minimal a² + b², we need to find the minimum of (x - 2)^2 / [ (x² - 1)^2 + 4x² ) ] over x ∈ [3, 4], and then the minimal a² + b² would be the square of the minimal distance, which is the same as that expression. Wait, actually, the distance squared is [ (x - 2)^2 ] / [ (x² - 1)^2 + 4x² ) ]. So, the minimal a² + b² is the minimal value of this expression over x ∈ [3, 4].Therefore, if we can compute f(x) = (x - 2)^2 / [ (x² - 1)^2 + 4x² ) ] for x ∈ [3, 4], and find its minimum, that would give the minimal a² + b². Let me verify that logic.Given that for each x in [3,4], the line a(x² -1) + 2b x + (x - 2) = 0 has a distance from the origin of |x - 2| / sqrt( (x² -1)^2 + (2x)^2 ). The minimal a² + b² is the square of the minimal distance, because the closest point on the line to the origin is the point where the distance is minimized, and the square of the distance is exactly a² + b² for that point. So yes, that seems correct.Therefore, we can write:a² + b² ≥ [ (x - 2)^2 ] / [ (x² -1)^2 + (2x)^2 ]for each x in [3, 4], and equality holds when (a, b) is the closest point on the line corresponding to x. Therefore, the minimal a² + b² over all possible x in [3,4] is the minimum of the right-hand side over x ∈ [3,4].Therefore, we need to compute f(x) = (x - 2)^2 / [ (x² - 1)^2 + 4x² ] for x in [3,4], and find its minimum.Let me compute this function f(x). Let's first simplify the denominator:(x² -1)^2 + 4x².Let me expand (x² -1)^2: x⁴ - 2x² +1. Then adding 4x² gives x⁴ - 2x² +1 +4x² = x⁴ + 2x² +1 = (x² +1)^2.Ah, that's a nice simplification! So, denominator becomes (x² +1)^2. Therefore, f(x) = (x -2)^2 / (x² +1)^2.So, now our task is to find the minimum of f(x) = (x -2)^2 / (x² +1)^2 for x in [3,4].That's a simpler expression. So, let's define f(x) = (x -2)^2 / (x² +1)^2, and find its minimum on [3,4].To find the minimum, we can take the derivative of f(x) with respect to x, set it to zero, and solve for x. Also, check the endpoints x=3 and x=4.First, let's compute f'(x). Let me use the quotient rule.f(x) = numerator / denominator, where numerator = (x -2)^2, denominator = (x² +1)^2.The derivative f'(x) = [num’ * den - num * den’] / den².Compute num’ = 2(x -2).Den’ = 2(x² +1)(2x) = 4x(x² +1).Therefore,f’(x) = [2(x -2)(x² +1)^2 - (x -2)^2 * 4x(x² +1)] / (x² +1)^4.Factor out common terms in numerator:2(x -2)(x² +1)[ (x² +1) - 2x(x -2) ] / (x² +1)^4.Simplify:2(x -2)[ (x² +1) - 2x(x -2) ] / (x² +1)^3.Now, let's compute the expression in the brackets:(x² +1) - 2x(x -2) = x² +1 -2x² +4x = -x² +4x +1.Therefore, f’(x) = 2(x -2)(-x² +4x +1) / (x² +1)^3.Set f’(x) = 0:The denominator is always positive, so critical points occur when numerator is zero:2(x -2)(-x² +4x +1) = 0.Set each factor to zero:x -2 = 0 => x = 2, but x is in [3,4], so not in the interval.Then, -x² +4x +1 = 0.Multiply both sides by -1:x² -4x -1 = 0.Solve quadratic equation:x = [4 ± sqrt(16 +4)] / 2 = [4 ± sqrt(20)] / 2 = [4 ± 2*sqrt(5)] / 2 = 2 ± sqrt(5).Compute numerical values:sqrt(5) ≈ 2.236, so 2 + sqrt(5) ≈ 4.236, which is greater than 4, and 2 - sqrt(5) ≈ -0.236, which is negative. Thus, neither of these roots are in [3,4]. Therefore, in the interval [3,4], the derivative f’(x) does not have any zeros. Therefore, the function f(x) is either increasing or decreasing on [3,4].We can check the sign of f’(x) on [3,4]. Let's pick a test point in [3,4], say x=3.Compute f’(3):First, x=3:(x -2) = 1, positive.(-x² +4x +1) = -9 +12 +1 = 4, positive.Therefore, numerator: 2*1*4 = 8, positive. Denominator is positive, so f’(3) >0.Similarly, at x=4:(x -2)=2, positive.(-x² +4x +1)= -16 +16 +1=1, positive.Thus, f’(4)=2*2*1 / something positive= positive. Therefore, f’(x) >0 throughout [3,4]. Therefore, f(x) is increasing on [3,4]. Therefore, the minimum of f(x) occurs at x=3, and the maximum at x=4.Therefore, the minimal value of f(x) on [3,4] is f(3).Compute f(3):(3 -2)^2 / (3² +1)^2 = 1 / (10)^2 = 1/100 = 0.01.Therefore, the minimal value of a² + b² is 1/100, which is 0.01.Wait, but let me confirm that. Since f(x) is increasing on [3,4], then its minimum is at x=3, which is 1/100, and maximum at x=4, which is (4-2)^2/(4² +1)^2 = 4/17² = 4/289 ≈ 0.0138. Therefore, the minimum is indeed 1/100.But wait, hold on. So the minimal a² + b² is 1/100? Let me check again.Wait, but this seems too straightforward. Let me verify by actually calculating a and b. If x=3 is the point where the curves intersect, then substituting x=3 into the original equation:(a +2)/3 = a*3 + 2b +1.Multiply both sides by 3:a +2 = 9a + 6b +3.Bring all terms to left:a +2 -9a -6b -3 =0 => -8a -6b -1=0 => 8a +6b = -1.So, 8a +6b = -1. Then, the minimal a² + b² subject to 8a +6b = -1.This is a standard problem: minimize a² + b² with a linear constraint. The minimal distance is |C| / sqrt(A² + B²) where the line is Ax + By = C. Here, the line is 8a +6b = -1, so A=8, B=6, C= -1. The distance from origin is | -1 | / sqrt(8² +6²) = 1 / sqrt(64 +36) = 1/sqrt(100)=1/10. Therefore, the minimal a² + b² is (1/10)^2=1/100. So, yes, that's correct. Therefore, the minimal value is 1/100, achieved when (a, b) is the closest point on the line 8a +6b = -1 to the origin.Therefore, the minimal a² + b² is 1/100.Hence, the answer should be boxed{dfrac{1}{100}}.But let me check another point to ensure that there isn't a lower value. For instance, if we consider x=4, but since the function is increasing, the minimal is at x=3, so x=4 gives a larger value. So, this seems correct.Alternatively, suppose we take another x in [3,4], say x=3.5. Compute f(3.5):(3.5 -2)^2 / (3.5² +1)^2 = (1.5)^2 / (12.25 +1)^2 = 2.25 / (13.25)^2 ≈ 2.25 / 175.56 ≈ 0.0128, which is larger than 0.01. So, indeed, f(x) is increasing.Therefore, the minimal value occurs at x=3, giving a² + b²=1/100.Thus, the answer is 1/100.Final Answerboxed{dfrac{1}{100}}</think>Given the curves ( y = frac{a + 2}{x} ) and ( y = ax + 2b + 1 ), we need to find the minimum value of ( a^2 + b^2 ) given that the curves intersect at least once in the interval ([3, 4]).First, we set the equations equal to each other for some ( x in [3, 4] ):[frac{a + 2}{x} = ax + 2b + 1]Multiplying both sides by ( x ) and rearranging terms, we get:[a x^2 + (2b + 1)x - (a + 2) = 0]Rewriting this equation, we have:[a(x^2 - 1) + 2bx + x - 2 = 0]This is a linear equation in ( a ) and ( b ) for each ( x in [3, 4] ). The distance from the origin ((0, 0)) to the line ( a(x^2 - 1) + 2bx + (x - 2) = 0 ) is given by:[frac{|x - 2|}{sqrt{(x^2 - 1)^2 + (2x)^2}}]Simplifying the denominator, we get:[sqrt{(x^2 + 1)^2} = x^2 + 1]Thus, the distance squared is:[frac{(x - 2)^2}{(x^2 + 1)^2}]We need to find the minimum value of this expression for ( x in [3, 4] ). The function ( f(x) = frac{(x - 2)^2}{(x^2 + 1)^2} ) is increasing on ([3, 4]) because its derivative ( f'(x) ) is positive in this interval. Therefore, the minimum value occurs at ( x = 3 ).Evaluating ( f(x) ) at ( x = 3 ):[f(3) = frac{(3 - 2)^2}{(3^2 + 1)^2} = frac{1}{10^2} = frac{1}{100}]Thus, the minimum value of ( a^2 + b^2 ) is (boxed{dfrac{1}{100}}).,<think>Okay, so I need to find the minimum value of ( a^2 + b^2 ) given that the curves ( y = frac{a + 2}{x ) and ( y = ax + 2b + 1 ) intersect at least once in the interval [3, 4]. Both ( a ) and ( b ) are real numbers, and ( a ) is not zero. Alright, let's start by understanding what the problem is asking.First, since the curves must intersect at some x-value between 3 and 4, that means there exists some ( x in [3, 4] ) such that ( frac{a + 2}{x} = ax + 2b + 1 ). So, maybe I can set those two equations equal to each other and solve for ( b ) in terms of ( a ) and ( x ), then substitute back into the expression ( a^2 + b^2 ). That might help me find the minimum value.Let me try that. Setting the two equations equal:[frac{a + 2}{x} = ax + 2b + 1]I can solve for ( b ):Subtract 1 from both sides:[frac{a + 2}{x} - 1 = ax + 2b]Then subtract ( ax ):[frac{a + 2}{x} - 1 - ax = 2b]So, dividing both sides by 2:[b = frac{1}{2} left( frac{a + 2}{x} - 1 - ax right )]Okay, so ( b ) is expressed in terms of ( a ) and ( x ). Now, substituting this back into ( a^2 + b^2 ), we get:[a^2 + left( frac{1}{2} left( frac{a + 2}{x} - 1 - ax right ) right )^2]Hmm. That seems a bit complicated. Maybe I can simplify this expression. Let me compute the expression inside the square first:[frac{a + 2}{x} - 1 - ax = frac{a + 2}{x} - ax - 1]Hmm, perhaps combine terms? Let me see:Let me write all terms over x, maybe? Let's try:First, the term ( frac{a + 2}{x} ) is already over x. The term ( -ax ) can be written as ( -frac{a x^2}{x} ), and -1 can be written as ( -frac{x}{x} ). So:[frac{a + 2}{x} - frac{a x^2}{x} - frac{x}{x} = frac{a + 2 - a x^2 - x}{x}]So the entire expression becomes:[frac{a + 2 - a x^2 - x}{x} = frac{ -a x^2 + a - x + 2 }{x }]So then, ( b = frac{1}{2} cdot frac{ -a x^2 + a - x + 2 }{x } )Therefore, ( b = frac{ -a x^2 + a - x + 2 }{2x } )So, substituting back into ( a^2 + b^2 ), we get:[a^2 + left( frac{ -a x^2 + a - x + 2 }{2x } right )^2]Hmm. So, if I can write this as a function of ( a ) and ( x ), then for each ( x in [3, 4] ), I can find the minimum value of ( a^2 + b^2 ), and then take the minimum over all ( x ) in [3,4]. But maybe there's a better way.Alternatively, perhaps treating ( x ) as a variable in [3,4], and for each ( x ), the equation ( frac{a + 2}{x} = ax + 2b + 1 ) must hold for some ( a ) and ( b ). Then, the problem reduces to minimizing ( a^2 + b^2 ) over all ( a neq 0 ) and ( b ) such that this equation holds for some ( x in [3,4] ).But perhaps another approach: For each ( x in [3,4] ), we can consider the equation ( frac{a + 2}{x} = ax + 2b + 1 ), solve for ( b ) in terms of ( a ) (as done above), then express ( a^2 + b^2 ) as a function of ( a ), and find the minimum with respect to ( a ). Then, among all these minima for ( x in [3,4] ), find the smallest one. That might work.So, let's formalize that. For a fixed ( x in [3,4] ), express ( b ) in terms of ( a ):[b = frac{1}{2} left( frac{a + 2}{x} - 1 - ax right )]Then, ( a^2 + b^2 = a^2 + left( frac{1}{2} left( frac{a + 2}{x} - 1 - ax right ) right )^2 )Let me denote this function as ( f(a) ), for fixed ( x ). Then, we can find the minimum of ( f(a) ) with respect to ( a ), and then find the minimum over ( x in [3,4] ).To minimize ( f(a) ), take the derivative with respect to ( a ), set it to zero.So first, let's compute ( f(a) ):[f(a) = a^2 + left( frac{ -a x^2 + a - x + 2 }{2x } right )^2]Wait, perhaps simplifying the expression for ( b ):Earlier, we had:[b = frac{ -a x^2 + a - x + 2 }{2x }]So, ( b = frac{ a(-x^2 + 1) + (-x + 2) }{2x } )So, the expression for ( a^2 + b^2 ) is:[a^2 + left( frac{ a(-x^2 + 1) + (-x + 2) }{2x } right )^2]Let me denote ( c = -x^2 + 1 ), ( d = -x + 2 ), so then:[b = frac{ a c + d }{ 2x }]Thus, ( a^2 + b^2 = a^2 + left( frac{ a c + d }{ 2x } right )^2 )Which can be written as:[f(a) = a^2 + frac{ (a c + d )^2 }{ 4x^2 }]To find the minimum of this function with respect to ( a ), take the derivative:[f'(a) = 2a + frac{ 2( a c + d ) c }{ 4x^2 } = 2a + frac{ (a c + d ) c }{ 2x^2 }]Set derivative to zero:[2a + frac{ (a c + d ) c }{ 2x^2 } = 0]Multiply both sides by ( 2x^2 ):[4x^2 a + c(a c + d ) = 0]Expand:[4x^2 a + a c^2 + c d = 0]Factor out a:[a(4x^2 + c^2 ) + c d = 0]Therefore, solving for a:[a = - frac{ c d }{4x^2 + c^2 }]Recall that ( c = -x^2 + 1 ), so let's substitute back:[a = - frac{ (-x^2 + 1)(-x + 2) }{4x^2 + (-x^2 + 1)^2 }]Simplify numerator and denominator.First, numerator:[(-x^2 + 1)(-x + 2) = (1 - x^2)(-x + 2) = -(1 - x^2)(x - 2) = -( -x^3 + 2x^2 + x - 2 ) = x^3 - 2x^2 - x + 2]Wait, actually:Wait, let's compute (1 - x^2)(-x + 2):Multiply term by term:1*(-x) = -x1*2 = 2(-x^2)*(-x) = x^3(-x^2)*2 = -2x^2So overall:- x + 2 + x^3 - 2x^2Combine like terms:x^3 - 2x^2 - x + 2So, numerator is ( x^3 - 2x^2 - x + 2 )Denominator:4x^2 + (-x^2 + 1)^2First compute (-x^2 +1)^2 = x^4 - 2x^2 + 1Thus, denominator is 4x^2 + x^4 - 2x^2 +1 = x^4 + 2x^2 +1Notice that x^4 + 2x^2 +1 is (x^2 +1)^2So denominator is (x^2 +1)^2Therefore, a is:[a = - frac{ x^3 - 2x^2 - x + 2 }{ (x^2 +1)^2 }]So that's the value of a that minimizes ( a^2 + b^2 ) for a given x in [3,4]. Then, once we have a in terms of x, we can compute b using the earlier expression:[b = frac{ -a x^2 + a - x + 2 }{2x }]Then, substitute a into this equation to find b in terms of x, then compute ( a^2 + b^2 ) as a function of x, and find its minimum over x in [3,4].So, let's proceed step by step.First, compute a:[a = - frac{ x^3 - 2x^2 - x + 2 }{ (x^2 +1)^2 } = frac{ -x^3 + 2x^2 + x -2 }{ (x^2 +1)^2 }]Now, let's compute the numerator of a:Numerator: -x³ + 2x² + x -2Let me write it as: (-x³ + 2x²) + (x -2) = -x²(x -2) +1*(x -2) = (-x² +1)(x -2) = -(x² -1)(x -2) = -(x -1)(x +1)(x -2)Hmm, that's an interesting factorization, but maybe not necessary here. Anyway, proceed.Now, compute b:[b = frac{ -a x^2 + a - x + 2 }{2x }]Let's substitute the expression for a:First, compute -a x² + a:Factor a:a(-x² +1) = a(1 -x²)Recall that a is expressed as:a = [ -x³ + 2x² + x -2 ] / (x² +1)^2Thus,a(1 -x² ) = [ -x³ + 2x² + x -2 ](1 -x² ) / (x² +1)^2Compute the numerator:Multiply out (-x³ + 2x² + x -2)(1 -x² )Let me do term by term:- x³ *1 = -x³- x³*(-x² ) = +x^52x²*1 = 2x²2x²*(-x² ) = -2x^4x*1 = xx*(-x² ) = -x³-2*1 = -2-2*(-x² ) = +2x²Combine all terms:x^5 -2x^4 -x³ -x³ +2x² +2x² +x -2Simplify:x^5 -2x^4 -2x³ +4x² +x -2So, numerator is x^5 -2x^4 -2x³ +4x² +x -2Then, subtract x and add 2:Wait, the entire expression for -a x² +a -x +2 is:[a(1 -x² ) ] -x +2Which is [ (-x³ + 2x² +x -2)(1 -x² ) / (x² +1)^2 ] -x +2But we just computed the numerator of a(1 -x² ) as x^5 -2x^4 -2x³ +4x² +x -2Thus,a(1 -x² ) = (x^5 -2x^4 -2x³ +4x² +x -2 ) / (x² +1)^2Therefore,-a x² +a -x +2 = [ (x^5 -2x^4 -2x³ +4x² +x -2 ) / (x² +1)^2 ] -x +2To combine these terms, let's write -x +2 as [ -x +2 ]*(x² +1)^2 / (x² +1)^2 )So:[ x^5 -2x^4 -2x³ +4x² +x -2 -x(x² +1)^2 +2(x² +1)^2 ] / (x² +1)^2Let me compute the numerator:First, expand -x(x² +1)^2:First compute (x² +1)^2 = x^4 +2x² +1Then, multiply by -x: -x^5 -2x³ -xSimilarly, expand 2(x² +1)^2 = 2x^4 +4x² +2So, adding these to the existing terms:Original numerator: x^5 -2x^4 -2x³ +4x² +x -2Add (-x^5 -2x³ -x ) and (2x^4 +4x² +2 )Combine term by term:x^5 - x^5 = 0-2x^4 +2x^4 = 0-2x³ -2x³ = -4x³4x² +4x² =8x²x -x =0-2 +2 =0So total numerator: 0x^5 +0x^4 -4x³ +8x² +0x +0 = -4x³ +8x²Thus, numerator is -4x³ +8x² = -4x²(x -2 )Therefore, the entire expression for -a x² +a -x +2 is [ -4x²(x -2 ) ] / (x² +1)^2Therefore, b is:b = [ -4x²(x -2 ) / (x² +1)^2 ] / (2x ) = [ -4x²(x -2 ) ] / [ 2x (x² +1)^2 ] = [ -2x(x -2) ] / (x² +1)^2Simplify:b = [ -2x(x -2) ] / (x² +1)^2 = [ -2x² +4x ] / (x² +1)^2So, b = ( -2x² +4x ) / (x² +1)^2So now, we have expressions for a and b in terms of x:a = [ -x³ +2x² +x -2 ] / (x² +1)^2b = [ -2x² +4x ] / (x² +1)^2Now, we need to compute ( a^2 + b^2 ). Let's compute each term:First, compute a^2:[ (-x³ +2x² +x -2 ) / (x² +1)^2 ]^2Similarly, b^2:[ (-2x² +4x ) / (x² +1)^2 ]^2So, ( a^2 + b^2 = [ (-x³ +2x² +x -2 )^2 + (-2x² +4x )^2 ] / (x² +1)^4 )Let me compute the numerator:First, expand (-x³ +2x² +x -2 )^2:Let me denote this as (A)^2 where A = -x³ +2x² +x -2Expanding A^2:(-x³)^2 + (2x²)^2 + (x)^2 + (-2)^2 + 2*(-x³)*(2x²) + 2*(-x³)*x + 2*(-x³)*(-2) + 2*(2x²)*x + 2*(2x²)*(-2) + 2*x*(-2)Compute term by term:x^6 +4x^4 +x^2 +4 + (-4x^5) + (-2x^4) + (4x³) + (4x³) + (-8x²) + (-4x )Combine like terms:x^6 -4x^5 + (4x^4 -2x^4 ) + (4x³ +4x³ ) + (x^2 -8x² ) + (-4x ) +4Simplify:x^6 -4x^5 +2x^4 +8x³ -7x² -4x +4Now, compute (-2x² +4x )^2:= ( -2x² )^2 + (4x )^2 + 2*(-2x²)(4x )=4x^4 +16x² + (-16x³ )So, 4x^4 -16x³ +16x²Adding both squared terms:A^2 + B^2 = [x^6 -4x^5 +2x^4 +8x³ -7x² -4x +4] + [4x^4 -16x³ +16x² ]Combine like terms:x^6 -4x^5 + (2x^4 +4x^4 ) + (8x³ -16x³ ) + (-7x² +16x² ) + (-4x ) +4Simplify:x^6 -4x^5 +6x^4 -8x³ +9x² -4x +4Therefore, the numerator of ( a^2 + b^2 ) is ( x^6 -4x^5 +6x^4 -8x³ +9x² -4x +4 ), and the denominator is ( (x² +1)^4 )So, ( a^2 + b^2 = frac{ x^6 -4x^5 +6x^4 -8x³ +9x² -4x +4 }{ (x² +1)^4 } )Hmm. That seems quite a complicated expression. Maybe we can factor the numerator or find a way to simplify this?Let me try to factor the numerator. Let's call the numerator N(x) = x^6 -4x^5 +6x^4 -8x³ +9x² -4x +4Is there a way to factor this? Let me check for possible rational roots using Rational Root Theorem. Possible roots are ±1, ±2, ±4. Let me plug in x=1:N(1) =1 -4 +6 -8 +9 -4 +4 =1-4= -3; -3+6=3; 3-8=-5; -5+9=4; 4-4=0; 0+4=4 ≠0x=2: 64 - 128 + 96 -64 +36 -8 +4= 64-128=-64; -64+96=32; 32-64=-32; -32+36=4; 4-8=-4; -4+4=0. So x=2 is a root.Therefore, (x -2) is a factor. Let's perform polynomial division.Divide N(x) by (x -2). Let's use synthetic division:Coefficients of N(x): 1 (x^6), -4 (x^5), 6 (x^4), -8 (x³), 9 (x²), -4 (x), 4 (constant)Using x=2:Bring down 1Multiply by 2: 2Add to next coefficient: -4 +2 = -2Multiply by 2: -4Add to next coefficient:6 +(-4)=2Multiply by 2:4Add to next coefficient:-8 +4=-4Multiply by2:-8Add to next coefficient:9 +(-8)=1Multiply by2:2Add to next coefficient:-4 +2=-2Multiply by2:-4Add to last coefficient:4 +(-4)=0So the quotient polynomial is x^5 -2x^4 +2x³ -4x² +x -2Thus, N(x) = (x -2)(x^5 -2x^4 +2x³ -4x² +x -2 )Now, let's factor the quintic. Try x=2 again:Coefficients:1, -2, 2, -4, 1, -2Using x=2:Bring down 1Multiply by2:2, add to -2:0Multiply by2:0, add to 2:2Multiply by2:4, add to -4:0Multiply by2:0, add to1:1Multiply by2:2, add to -2:0Thus, (x -2) is a factor again. So quotient is x^4 +0x^3 +2x^2 +0x +1Wait, let me check:After division by (x -2) again:Using synthetic division with x=2:Coefficients:1, -2, 2, -4,1,-2Bring down 1Multiply by2:2, add to -2:0Multiply by2:0, add to2:2Multiply by2:4, add to -4:0Multiply by2:0, add to1:1Multiply by2:2, add to -2:0So quotient polynomial is x^4 +0x^3 +2x^2 +0x +1, which is x^4 +2x² +1Thus, N(x) = (x -2)^2 (x^4 +2x² +1 )But x^4 +2x² +1 is (x² +1)^2. So,N(x) = (x -2)^2 (x² +1)^2Therefore, numerator simplifies to (x -2)^2 (x² +1)^2Denominator is (x² +1)^4Therefore, ( a^2 + b^2 = frac{ (x -2)^2 (x² +1)^2 }{ (x² +1)^4 } = frac{ (x -2)^2 }{ (x² +1)^2 } )Wow! That simplifies everything. So, after all that, ( a^2 + b^2 = frac{(x -2)^2}{(x^2 +1)^2} ). That's a much simpler expression. So, for each x in [3,4], the minimal value of ( a^2 + b^2 ) is ( frac{(x -2)^2}{(x^2 +1)^2} ). Therefore, to find the minimum value over x in [3,4], we just need to find the minimum of this function on [3,4].So, now the problem reduces to minimizing ( f(x) = frac{(x -2)^2}{(x^2 +1)^2} ) for x ∈ [3,4].Let's analyze this function. Since x is in [3,4], (x -2) is positive, so (x -2)^2 is increasing in x. The denominator is (x² +1)^2, which is also increasing in x for x >0. So, the function is the square of (x -2)/(x² +1). Let's see whether the function is increasing or decreasing.Compute derivative of f(x):Let me write f(x) = [(x -2)/(x² +1)]². Let’s compute the derivative using chain rule:f’(x) = 2[(x -2)/(x² +1)] * [ (1)(x² +1) - (x -2)(2x) ] / (x² +1)^2Simplify the derivative:First, compute the inner derivative:d/dx [ (x -2)/(x² +1) ] = [1*(x² +1) - (x -2)(2x)] / (x² +1)^2Compute numerator:x² +1 -2x(x -2) = x² +1 -2x² +4x = -x² +4x +1Therefore, derivative:f’(x) = 2*(x -2)/(x² +1) * (-x² +4x +1)/(x² +1)^2 )= 2*(x -2)*(-x² +4x +1) / (x² +1)^3So, f’(x) = [ -2(x -2)(x² -4x -1) ] / (x² +1)^3Note that x² -4x -1 is equivalent to -( -x² +4x +1 )Wait, but let's keep it as is.So, critical points occur where numerator is zero:Either x =2, which is not in [3,4], or x² -4x -1 =0Solve x² -4x -1=0:x = [4 ± sqrt(16 +4)]/2 = [4 ± sqrt(20)]/2 = [4 ± 2*sqrt(5)]/2 = 2 ± sqrt(5)Compute 2 + sqrt(5) ≈ 2 + 2.236 ≈ 4.236, which is outside [3,4]2 - sqrt(5) ≈ -0.236, also outside [3,4]Therefore, the derivative f’(x) in [3,4] has no critical points. Therefore, f(x) is monotonic on [3,4]. Let's check the sign of f’(x):Given that x ∈ [3,4], so x -2 ≥1 >0, and denominator (x² +1)^3 >0 always.The numerator is -2(x -2)(x² -4x -1 )We need to check the sign of (x² -4x -1 ) in [3,4].Compute at x=3: 9 -12 -1 = -4At x=4: 16 -16 -1= -1Thus, x² -4x -1 is negative in [3,4], so (x² -4x -1 ) <0Therefore, the numerator is -2*(positive)*(negative) = -2*(negative) = positiveTherefore, f’(x) >0 in [3,4]. So the function f(x) is increasing on [3,4]. Therefore, its minimum occurs at x=3, and maximum at x=4.Therefore, the minimal value of ( a^2 + b^2 ) is f(3) = (3-2)^2 / (9 +1)^2 = 1 / 100 = 0.01Wait, let's compute it:At x=3:( f(3) = frac{(3 -2)^2}{(3² +1)^2} = frac{1}{(9 +1)^2} = frac{1}{10^2} = frac{1}{100} = 0.01 )At x=4:( f(4) = frac{(4 -2)^2}{(16 +1)^2} = frac{4}{17²} = frac{4}{289} ≈0.0138 )Since the function is increasing, the minimum is 1/100 at x=3.Therefore, the minimal value of ( a^2 + b^2 ) is 1/100.But let me verify once again to be sure. Wait, the problem states that the curves have at least one common point in [3,4]. Therefore, there exists some x in [3,4] where the curves intersect, so the minimal value is the minimum of f(x) over x ∈ [3,4], which is 1/100. Therefore, the answer should be 1/100, which is 0.01, boxed as boxed{dfrac{1}{100}}.But just to confirm, let me check with x=3, compute a and b:At x=3:a = [ -27 + 18 +3 -2 ] / (9 +1)^2 = (-27 +18= -9; -9+3=-6; -6-2=-8 ) /100 = -8/100 = -2/25b = [ -2*9 +12 ] / (10)^2 = ( -18 +12 = -6 ) /100 = -6/100 = -3/50Compute ( a^2 + b^2 = (4/625) + (9/2500 ) = (16/2500 ) + (9/2500 ) =25/2500 =1/100 ). Correct.Similarly, at x=4:a = [ -64 +32 +4 -2 ] / (16 +1)^2 = (-64 +32= -32; -32+4= -28; -28 -2= -30 ) / 289 = -30/289b= [ -32 +16 ] /289= (-16)/289Compute ( a^2 + b^2 = (900 + 256 ) / (289)^2 = 1156 / 83521 = 4/289 ≈0.0138 ). Correct.Thus, the minimal value is indeed 1/100.Final AnswerThe minimum value of (a^2 + b^2) is boxed{dfrac{1}{100}}.</think>Given that (a, b in mathbb{R}), (a neq 0), and the curves (y = frac{a + 2}{x}) and (y = ax + 2b + 1) have at least one common point in the interval ([3, 4]), we need to find the minimum value of (a^2 + b^2).First, we set the equations equal to each other for some (x in [3, 4]):[frac{a + 2}{x} = ax + 2b + 1]Solving for (b), we get:[b = frac{1}{2} left( frac{a + 2}{x} - 1 - ax right)]Substituting this expression for (b) into (a^2 + b^2), we simplify and find that:[a^2 + b^2 = frac{(x - 2)^2}{(x^2 + 1)^2}]We then need to minimize this expression over the interval ([3, 4]).The function (f(x) = frac{(x - 2)^2}{(x^2 + 1)^2}) is analyzed. By examining the derivative, we find that (f(x)) is increasing on ([3, 4]). Therefore, the minimum value occurs at (x = 3):[f(3) = frac{(3 - 2)^2}{(3^2 + 1)^2} = frac{1}{10^2} = frac{1}{100}]Thus, the minimum value of (a^2 + b^2) is boxed{dfrac{1}{100}}.]

🔑:Since the curves y = frac{a + 2}{x} and y = ax + 2b + 1 share a common point, we can equate the two expressions:frac{a + 2}{x} = ax + 2b + 1Multiply both sides by x and rearrange the terms to get:a + 2 = ax^2 + 2bx + xRearrange the terms to the standard form of a linear equation in a and b:(x^2 - 1)a + 2bx + (x - 2) = 0Now, consider (a, b) as a point on this line. The distance d from the origin to the line is given by:d = frac{|x - 2|}{sqrt{(x^2 - 1)^2 + (2x)^2}}Squaring both sides and simplifying, we see that a^2 + b^2 is equal to the square of this distance:a^2 + b^2 = d^2 = left( frac{x - 2}{x^2 + 1} right)^2Let t = x - 2, so x in [3, 4] implies t in [1, 2], and x = t + 2. Substitute this into the equation above:a^2 + b^2 = left( frac{t}{(t + 2)^2 + 1} right)^2 = left( frac{t}{t^2 + 4t + 5} right)^2 = left( frac{1}{t + frac{5}{t} + 4} right)^2Now, consider the function f(t) = t + frac{5}{t} + 4 for t in [1,2]. Its derivative is f'(t) = 1 - frac{5}{t^2}, which is negative for all t in [1,2]. Thus, f(t) is decreasing on [1,2].The maximum value of f(t) on [1,2] occurs at t = 1, which is f(1) = 1 + 5 + 4 = 10. Therefore, the minimum value of a^2 + b^2 occurs when t = 1, which is:a^2 + b^2 = left( frac{1}{10} right)^2 = boxed{frac{1}{100}}

🧠:[<think>Okay, let's see. I need to find the sum of a₁ + a₂ + ... + a₂₀₁₄, where each aₙ is defined as f(n) + f(n+1). The function f(n) has different expressions depending on whether n is odd or even. Let me parse this step by step.First, let me understand the function f(n). It says:f(n) = n if n is odd,f(n) = -n if n is even.So, for any integer n, if it's odd, f(n) is just n, and if it's even, f(n) is negative n. Got it. Then, aₙ is defined as the sum of f(n) and f(n+1). So, for each term aₙ, I need to evaluate f(n) and f(n+1), then add them together. Then sum all these aₙ from n=1 to n=2014.Let me start by analyzing aₙ for different n. Since n can be either odd or even, and n+1 would be the opposite parity. So, maybe I can break it down into two cases: when n is odd and when n is even.Case 1: n is odd.Then f(n) = n (since n is odd), and f(n+1) = -(n+1) (since n+1 is even). So, aₙ = n - (n+1) = n - n -1 = -1.Case 2: n is even.Then f(n) = -n (since n is even), and f(n+1) = n+1 (since n+1 is odd). So, aₙ = -n + (n+1) = -n + n +1 = +1.Wait, so depending on whether n is odd or even, aₙ is either -1 or +1. That simplifies things a lot. Let me check that again.If n is odd:aₙ = f(n) + f(n+1) = n + (-(n+1)) = n - n -1 = -1. Correct.If n is even:aₙ = f(n) + f(n+1) = (-n) + (n+1) = -n + n +1 = +1. Correct.So, each term aₙ alternates between -1 and +1 depending on the parity of n. So, starting from n=1 (which is odd), the first term a₁ is -1, then n=2 (even), a₂ is +1, n=3 (odd), a₃ is -1, and so on. Therefore, the sequence aₙ is alternating between -1 and +1.Therefore, the sum S = a₁ + a₂ + ... + a₂₀₁₄ would be the sum of 2014 terms alternating between -1 and +1. Let's see how that works.Since the sequence starts with n=1 (odd), the first term is -1, the second term (n=2) is +1, the third term (n=3) is -1, etc. So, the sequence of aₙ is: -1, +1, -1, +1, ..., up to the 2014th term.Now, if we pair the terms, maybe we can see a pattern. For example, each pair of terms (-1 +1) would sum to 0. Let me check:a₁ + a₂ = -1 +1 = 0,a₃ + a₄ = -1 +1 = 0,and so on. So, every two terms sum to zero. If that's the case, then the total sum would depend on how many such pairs there are.But wait, 2014 is an even number. Since 2014 divided by 2 is 1007, there would be 1007 pairs, each summing to 0. Therefore, the total sum S would be 0. But wait, let me check again.Wait, if the first term is -1, then each pair (n odd and n even) sums to 0. So, if the total number of terms is even, then all pairs cancel out. So, for 2014 terms, which is 1007 pairs, each pair sums to 0. Hence, the total sum is 0. So, answer is C: 0.But wait, hold on. Let me verify with a small example. Let's compute the sum for a smaller number of terms, say n=1 to 4.Compute a₁ + a₂ + a₃ + a₄.a₁ = f(1) + f(2) = 1 + (-2) = -1,a₂ = f(2) + f(3) = -2 + 3 = +1,a₃ = f(3) + f(4) = 3 + (-4) = -1,a₄ = f(4) + f(5) = -4 + 5 = +1.So, sum is -1 +1 -1 +1 = 0. Correct. So, 4 terms sum to 0.Similarly, 2 terms sum to 0, 6 terms sum to 0, etc. So, for any even number of terms, the sum is 0. Therefore, 2014 terms, which is even, would sum to 0. Therefore, the answer is C: 0.But wait, wait. Let me check with an odd number of terms. For example, 1 term: a₁ = -1, sum is -1.3 terms: a₁ + a₂ + a₃ = -1 +1 -1 = -1.So, for odd number of terms, the sum is -1, and even number of terms, sum is 0.But in the problem, we have 2014 terms, which is even, so sum is 0. Therefore, answer is C.But let me check once again to avoid any mistake.Wait, let's compute aₙ for n=1 to n=4 again:n=1: odd, so a₁ = f(1) + f(2) = 1 + (-2) = -1.n=2: even, so a₂ = f(2) + f(3) = -2 + 3 = +1.n=3: odd, a₃ = f(3) + f(4) = 3 + (-4) = -1.n=4: even, a₄ = f(4) + f(5) = -4 +5 = +1.So, indeed, the terms alternate between -1 and +1. So, the sum over an even number of terms is zero. Therefore, for 2014 terms, the answer is 0, which is option C.Wait, but let me check n=5 to n=8:a₅ = f(5) + f(6) =5 + (-6) = -1,a₆ = f(6) + f(7) =-6 +7= +1,a₇= f(7)+f(8)=7 +(-8)=-1,a₈= f(8)+f(9)= -8 +9=+1.Same pattern. So, yes, every pair of terms cancels. Therefore, the sum is 0.Alternatively, perhaps the problem is designed such that each aₙ alternates between -1 and +1, so over even terms, the sum is zero, over odd terms, the sum is -1.Therefore, given that 2014 is even, the answer is 0, so option C.But wait, the answer options are A: -1, B:2012, C:0, D:-2012.So, the correct answer is C:0.But wait, let me check again. Wait, if each pair of terms (like a₁+a₂, a₃+a₄, etc.) sums to zero, then the total sum for 2014 terms is 2014/2 = 1007 pairs, each summing to zero, so total sum is 0. That's straightforward.But let me check for n=2014. Since n goes up to 2014, which is even. The last term is a_{2014} = f(2014) + f(2015). Since 2014 is even, f(2014) = -2014, and f(2015) = 2015 (since 2015 is odd). Therefore, a_{2014} = -2014 +2015 = 1. So, yes, the last term is +1, which is consistent with the pattern.Thus, the sequence starts with -1, +1, -1, +1, ..., +1. Since 2014 is even, the last term is +1. So, all pairs cancel. Hence, sum is zero.Therefore, the answer is C:0.But wait, just to make sure I didn't make any mistake in the function definitions. Let me confirm:Given f(n) is n if odd, -n if even.Therefore, for even n, f(n) is negative.Therefore, for aₙ = f(n) + f(n+1). So, when n is odd, f(n) =n, f(n+1)= - (n+1). Therefore, aₙ =n - (n+1)= -1. When n is even, f(n)= -n, f(n+1)=n+1, so aₙ= -n + (n+1)=1. Correct.Therefore, the pattern is indeed -1,1,-1,1,..., and since the number of terms is even, the total sum is zero.Therefore, the answer is C:0.I think that's solid. So, I would choose option C.Final Answerboxed{C}</think>Given the function ( f(n) ) defined as:[f(n) = begin{cases} n & text{if } n text{ is odd} -n & text{if } n text{ is even}end{cases}]we need to find the value of ( a_1 + a_2 + cdots + a_{2014} ) where ( a_n = f(n) + f(n+1) ).First, we analyze ( a_n ) for different values of ( n ):- If ( n ) is odd, then ( f(n) = n ) and ( f(n+1) = -(n+1) ). Therefore, ( a_n = n - (n+1) = -1 ).- If ( n ) is even, then ( f(n) = -n ) and ( f(n+1) = n+1 ). Therefore, ( a_n = -n + (n+1) = 1 ).Thus, the sequence ( a_n ) alternates between -1 and 1 starting with -1 for ( n = 1 ). To find the sum ( a_1 + a_2 + cdots + a_{2014} ), we note that each pair of consecutive terms ( (a_1 + a_2), (a_3 + a_4), ldots ) sums to 0. Since there are 2014 terms, which is an even number, we can form 1007 pairs each summing to 0. Therefore, the total sum is:[a_1 + a_2 + cdots + a_{2014} = 1007 times 0 = 0]Thus, the value is (boxed{C}).,<think>Okay, let me try to figure out this problem. So, we have a function f(n) defined piecewise: if n is odd, then f(n) = n; if n is even, f(n) = -n. Then there's another sequence a_n which is defined as f(n) + f(n+1). The question is asking for the sum of a_1 + a_2 + ... + a_{2014}, and the options are given as A: -1, B: 2012, C: 0, D: -2012.First, let me make sure I understand the definitions. The function f(n) is straightforward: for odd n, it's just n, and for even n, it's negative n. Then, each term a_n is the sum of f(n) and f(n+1). So, depending on whether n is odd or even, and n+1 is odd or even, the expression for a_n will change.Since n and n+1 are consecutive integers, one of them must be even and the other odd. That might be important. So, for each term a_n, we have one odd and one even number. Let's test this with some small values to see if we can spot a pattern.Let's compute a_1 first. n=1. Since 1 is odd, f(1) = 1. n+1 = 2, which is even, so f(2) = -2. Therefore, a_1 = 1 + (-2) = -1.Next, a_2. n=2, which is even, so f(2) = -2. n+1 = 3, which is odd, so f(3) = 3. Therefore, a_2 = -2 + 3 = 1.a_3: n=3 (odd), f(3)=3; n+1=4 (even), f(4)=-4. So, a_3=3 + (-4) = -1.a_4: n=4 (even), f(4)=-4; n+1=5 (odd), f(5)=5. So, a_4 = -4 + 5 = 1.Hmm, so the pattern here is alternating between -1 and 1 for a_n. Let's verify with a_5 and a_6:a_5: n=5 (odd), f(5)=5; n+1=6 (even), f(6)=-6. So, a_5=5 -6 = -1.a_6: n=6 (even), f(6)=-6; n+1=7 (odd), f(7)=7. So, a_6= -6 +7 = 1.Yep, so it's alternating -1, 1, -1, 1, etc. So, each pair of consecutive terms (a_1, a_2), (a_3, a_4), etc., adds up to (-1 + 1) = 0. Wait, but if we look at the sum of two consecutive terms, they cancel each other out. Let's check:a_1 + a_2 = -1 + 1 = 0a_3 + a_4 = -1 + 1 = 0a_5 + a_6 = -1 + 1 = 0So, every two terms sum to zero. Therefore, if we have an even number of terms, the total sum would be zero. But wait, the problem is asking for the sum up to a_{2014}. Let's see how many terms that is.The sum is from n=1 to n=2014, so there are 2014 terms. If every two terms sum to zero, then the total sum would be (2014 / 2) * 0 = 0. Therefore, the answer might be C: 0. But wait, let me check if this pattern holds for all terms. Wait, but maybe there's a catch here.Wait, let's think again. The terms are a_1, a_2, ..., a_{2014}. Since n starts at 1 and ends at 2014, so the terms are indeed 2014 terms. If the terms alternate between -1 and 1, starting with -1 for a_1, then every pair of terms cancels. So, 2014 is an even number, so 2014 divided by 2 is 1007 pairs, each pair summing to 0. So, total sum is 0. Therefore, the answer is C: 0. But wait, let me check this again with another approach to confirm.Alternatively, maybe we can express a_n in terms of n. Let's consider two cases: when n is odd and when n is even.Case 1: n is odd. Then, since n is odd, f(n) = n. Then n+1 is even, so f(n+1) = -(n+1). Therefore, a_n = n - (n+1) = -1.Case 2: n is even. Then, f(n) = -n. n+1 is odd, so f(n+1) = n+1. Therefore, a_n = -n + (n+1) = 1.Therefore, regardless of the value of n, a_n is either -1 (if n is odd) or 1 (if n is even). Therefore, the sequence a_n alternates between -1 and 1 starting with a_1 = -1. Therefore, the sum from a_1 to a_{2014} is equal to the number of odd n's times (-1) plus the number of even n's times 1. But since in the sequence from n=1 to n=2014, there are equal numbers of odd and even n's. Wait, 2014 is even, so there are 1007 odd numbers (since starting at 1, odd, even, odd, even... up to 2014, which is even) so the number of odd n's is 2014 / 2 = 1007, and the same number of even n's. Therefore, the total sum would be 1007*(-1) + 1007*(1) = 1007*(-1 + 1) = 1007*0 = 0. So, yes, that confirms it. Therefore, the sum is 0. So, answer C.Wait, but let me check with a small example. Let's compute the sum up to a_4. That's a_1 + a_2 + a_3 + a_4 = -1 +1 -1 +1 = 0. Similarly, up to a_6: same thing. So, every even number of terms cancels out. Since 2014 is even, it cancels out. Therefore, the answer is 0. So, the correct answer is C.But let me check if the initial term a_1 is -1. Yes, because n=1 is odd, so f(1)=1, n+1=2 is even, f(2)=-2, so a_1=1 + (-2)=-1. Then a_2: n=2 even, f(2)=-2, n+1=3 odd, f(3)=3, so a_2=-2 +3=1. Then the pattern continues. So, yes, each odd n gives a_n=-1, each even n gives a_n=1. So, the terms alternate starting with -1. Therefore, since there are equal numbers of odd and even terms up to 2014, they cancel out. Therefore, answer C:0.Wait, but maybe I made a mistake here. Let me verify with n=2014. The last term is a_{2014}. Since n=2014 is even, a_{2014}=1. So, the last term is 1. Since 2014 is even, the number of even terms is 1007 (from n=2 to n=2014, stepping by 2), and the number of odd terms is 1007 (from n=1 to n=2013, stepping by 2). Wait, but 2014 divided by 2 is 1007. So, 1007 odd terms and 1007 even terms. Therefore, 1007*(-1) + 1007*(1) = 0. Therefore, the answer is 0. So, the answer is C. Therefore, I think the correct answer is C:0.Wait, but let me check one more time. Suppose we compute the sum S = a_1 + a_2 + ... + a_{2014}. Each a_n = f(n) + f(n+1). Therefore, S = sum_{n=1}^{2014} [f(n) + f(n+1)].But this can be rewritten as sum_{n=1}^{2014} f(n) + sum_{n=1}^{2014} f(n+1). Which is equal to sum_{n=1}^{2014} f(n) + sum_{n=2}^{2015} f(n). Because when we shift the index from n to n+1, the lower limit becomes 2 and upper limit becomes 2015. So, if we write this out:First sum: f(1) + f(2) + f(3) + ... + f(2014)Second sum: f(2) + f(3) + ... + f(2014) + f(2015)Therefore, adding them together, we get:f(1) + 2*f(2) + 2*f(3) + ... + 2*f(2014) + f(2015)Therefore, S = f(1) + f(2015) + 2*sum_{k=2}^{2014} f(k)But maybe this is overcomplicating. Alternatively, we can notice that the sum telescopes. Let's see:S = [f(1) + f(2) + f(3) + ... + f(2014)] + [f(2) + f(3) + ... + f(2015)]So, grouping terms:= f(1) + [f(2) + f(2)] + [f(3) + f(3)] + ... + [f(2014) + f(2014)] + f(2015)= f(1) + 2*f(2) + 2*f(3) + ... + 2*f(2014) + f(2015)But if we think of this sum, maybe we can relate it to another expression. Alternatively, perhaps there's a telescoping effect. Wait, but maybe not. Alternatively, let's compute S directly using this telescoping approach.Wait, another way: since S = sum_{n=1}^{2014} [f(n) + f(n+1)] = sum_{n=1}^{2014} f(n) + sum_{n=1}^{2014} f(n+1) = sum_{n=1}^{2014} f(n) + sum_{n=2}^{2015} f(n) = [sum_{n=1}^{2015} f(n) - f(2015)] + [sum_{n=1}^{2015} f(n) - f(1)] ].Wait, no. Wait, the first sum is from 1 to 2014, and the second sum is from 2 to 2015, so:sum_{n=1}^{2014} f(n) + sum_{n=2}^{2015} f(n) = sum_{n=1}^{2014} f(n) + sum_{n=2}^{2015} f(n)= f(1) + sum_{n=2}^{2014} f(n) + sum_{n=2}^{2014} f(n) + f(2015)= f(1) + 2*sum_{n=2}^{2014} f(n) + f(2015)But this might not be helpful. Alternatively, if we subtract the two sums:Wait, maybe this approach isn't the best. Let me go back. Since each a_n is either -1 or 1, and alternates, then the sum of 2014 terms with equal numbers of -1 and 1 is zero. Therefore, the answer is C:0. Alternatively, if we consider the telescoping sum.Wait, if we expand S:S = [f(1) + f(2)] + [f(2) + f(3)] + [f(3) + f(4)] + ... + [f(2014) + f(2015)]Wait, no, that's not correct. Wait, no, each term a_n is f(n) + f(n+1). So, S is the sum from n=1 to n=2014 of [f(n) + f(n+1)]. So expanding this, we have:= [f(1) + f(2)] + [f(2) + f(3)] + [f(3) + f(4)] + ... + [f(2014) + f(2015)]So, when we expand all the terms, we get:f(1) + 2*f(2) + 2*f(3) + ... + 2*f(2014) + f(2015)So, S = f(1) + f(2015) + 2*(f(2) + f(3) + ... + f(2014))But perhaps we can compute this expression. Let's compute each part.First, compute f(1): since 1 is odd, f(1)=1.Compute f(2015): 2015 is odd (since 2014 is even, 2014 +1=2015 is odd), so f(2015)=2015.Then, the sum S = 1 + 2015 + 2*(f(2) + f(3) + ... + f(2014))Now, compute the sum from n=2 to n=2014 of f(n). Let's denote this sum as T.T = sum_{k=2}^{2014} f(k)Since f(k) is -k when k is even, and k when k is odd. Therefore, we can split T into sums over even and odd k.Let's separate T into two parts: sum over even k from 2 to 2014, and sum over odd k from 3 to 2013 (since 2014 is even, the last odd number before 2014 is 2013).First, sum over even k: E = sum_{m=1}^{1007} (-2m) because the even numbers from 2 to 2014 are 2,4,...,2014. There are 2014/2=1007 terms.Similarly, sum over odd k: O = sum_{m=1}^{1006} (2m +1) + 2013? Wait, no. Wait, odd numbers from 3 to 2013. Let's check:From k=2 to k=2014, the odd numbers are 3,5,...,2013. The number of terms here is (2013 -3)/2 +1 = (2010)/2 +1=1005 +1=1006 terms. Wait, but let me check.Alternatively, since the first odd number greater than or equal to 2 is 3 (if starting at k=2, the next odd is 3). The last odd number less than or equal to 2014 is 2013. So, the number of odd numbers from 3 to 2013 inclusive is ((2013 -3)/2) +1 = (2010/2) +1=1005 +1=1006.Therefore, the sum over odd k from 3 to 2013 is O = sum_{m=1}^{1006} (2m +1). Wait, let's check:If m starts at 1, then 2m+1 = 3,5,..., which is correct. The last term when m=1006 is 2*1006 +1=2013. So yes, that works.Therefore, O = sum_{m=1}^{1006} (2m +1) = 2*sum_{m=1}^{1006} m + sum_{m=1}^{1006}1 = 2*(1006*1007)/2 + 1006*1 = 1006*1007 + 1006 = 1006*(1007 +1)=1006*1008.Wait, hold on:sum_{m=1}^{n} (2m +1) = 2*sum(m) + sum(1) = 2*(n(n+1)/2) + n = n(n+1) +n = n(n+2)Wait, let's test n=1: 2*1 +1=3. Using the formula, 1*(1+2)=3, correct.So, for n=1006, the sum O = 1006*(1006 +2)=1006*1008.Therefore, O = 1006*1008.Similarly, the sum over even k from 2 to 2014 is E = sum_{m=1}^{1007} (-2m) = -2*sum_{m=1}^{1007} m = -2*(1007*1008)/2 = -1007*1008.Therefore, total sum T = E + O = (-1007*1008) + (1006*1008) = [ -1007 +1006 ] *1008 = (-1)*1008= -1008.Therefore, going back to the expression for S:S = f(1) + f(2015) + 2*T = 1 + 2015 + 2*(-1008) = (1 + 2015) - 2016 = 2016 -2016=0.So, indeed, S=0. Therefore, the answer is C:0.So, through two different approaches, we arrived at the same answer. First, by noticing the alternating pattern and that the number of terms is even, leading to cancellation. Second, by expanding the sum and computing the telescoping effect, which also results in 0. Therefore, the correct answer is C:0.Final Answerboxed{C}</think>Given the function ( f(n) ) defined as:[f(n) = begin{cases} n & text{if } n text{ is odd} -n & text{if } n text{ is even}end{cases}]and the sequence ( a_n = f(n) + f(n+1) ), we need to find the value of ( a_1 + a_2 + ldots + a_{2014} ).First, we analyze the sequence ( a_n ):- If ( n ) is odd, then ( f(n) = n ) and ( f(n+1) = -(n+1) ). Thus, ( a_n = n - (n+1) = -1 ).- If ( n ) is even, then ( f(n) = -n ) and ( f(n+1) = n+1 ). Thus, ( a_n = -n + (n+1) = 1 ).Therefore, the sequence ( a_n ) alternates between -1 and 1 starting with ( a_1 = -1 ). Next, we consider the sum ( a_1 + a_2 + ldots + a_{2014} ). Since there are 2014 terms, which is an even number, the terms can be paired as follows:- Each pair ( (a_1, a_2) ), ( (a_3, a_4) ), ..., ( (a_{2013}, a_{2014}) ) sums to ( -1 + 1 = 0 ).There are ( frac{2014}{2} = 1007 ) such pairs, each summing to 0. Therefore, the total sum is:[1007 times 0 = 0]To confirm, we use another approach by expanding the sum:[S = sum_{n=1}^{2014} a_n = sum_{n=1}^{2014} [f(n) + f(n+1)]]This can be rewritten as:[sum_{n=1}^{2014} f(n) + sum_{n=1}^{2014} f(n+1) = sum_{n=1}^{2014} f(n) + sum_{n=2}^{2015} f(n)]Combining these sums, we get:[f(1) + 2 sum_{n=2}^{2014} f(n) + f(2015)]Calculating ( f(1) = 1 ) and ( f(2015) = 2015 ), and splitting the sum into even and odd terms, we find that the intermediate terms cancel out, leading to a total sum of 0.Thus, the value of ( a_1 + a_2 + ldots + a_{2014} ) is (boxed{C}).]

🔑:Since f(n)= begin{cases}n,ntext{ is odd} -n,ntext{ is even}end{cases},When n=2k-1 (kinmathbb{N}^*), a_n=f(n)+f(n+1)=n-(n+1)=-1,When n=2k (kinmathbb{N}^*), a_n=f(n)+f(n+1)=-n+(n+1)=1,Thus, a_1+a_2+...+a_{2014}=0.Hence, the answer is boxed{text{C}}.Given f(n)= begin{cases}n,ntext{ is odd} -n,ntext{ is even}end{cases}, when n=2k-1 (kinmathbb{N}^*), a_n=n-(n+1)=-1, and when n=2k (kinmathbb{N}^*), a_n=-n+(n+1)=1, which can be derived.This problem tests the properties of piecewise sequences and reasoning ability, which is a basic question.

🧠:[<think>Alright, let's tackle this problem step by step. The expression we need to calculate is:[frac{3.2 - 2.95}{0.25 times 2 + frac{1}{4}} + frac{2 times 0.3}{2.3 - 1 frac{2}{5}}]Hmm, okay. Let's break this down into smaller parts. Maybe start with the numerator and denominators of each fraction separately, then combine them at the end. Let me write down the two main fractions first:First fraction: (frac{3.2 - 2.95}{0.25 times 2 + frac{1}{4}})Second fraction: (frac{2 times 0.3}{2.3 - 1 frac{2}{5}})Alright, let's handle the first fraction. The numerator is 3.2 minus 2.95. Let me compute that:3.2 minus 2.95. Let's see. 3.2 is the same as 3.20, right? So subtracting 2.95 from 3.20.3.20 - 2.95 = ?Hmm, let's do that carefully. 3.20 minus 2.95. Starting from the rightmost digits:0 - 5. Can't do that, so borrow 1 from the tenths place. The tenths place is 2, so borrow 1, making it 1, and the hundredths place becomes 10. So 10 - 5 = 5.Now the tenths place is 1 - 9. Again, can't do that, so borrow 1 from the units place. The units place is 3, so it becomes 2, and the tenths place becomes 11. 11 - 9 = 2.Now the units place is 2 - 2 = 0.So putting it all together: 0.25.Wait, let me verify that. 3.20 - 2.95. Alternatively, maybe converting both numbers to fractions?3.2 is 32/10 or 16/5, and 2.95 is 295/100 or 59/20. Let's subtract them:16/5 - 59/20. To subtract, common denominator is 20.16/5 is 64/20. 64/20 - 59/20 = 5/20 = 1/4 = 0.25. Okay, that's correct. So the numerator of the first fraction is 0.25.Now the denominator of the first fraction is 0.25 × 2 + 1/4.Let's compute each term:0.25 × 2 = 0.5.1/4 is 0.25.So adding them together: 0.5 + 0.25 = 0.75.Alternatively, in fractions: 0.25 × 2 is 1/4 × 2 = 1/2, and 1/4. So 1/2 + 1/4 = 3/4. Either way, 0.75 or 3/4.Therefore, the first fraction is 0.25 / 0.75, or (1/4) / (3/4). Dividing fractions, multiply by reciprocal: (1/4) × (4/3) = 1/3. So the first fraction simplifies to 1/3.Okay, that's the first part. Now moving on to the second fraction: (frac{2 times 0.3}{2.3 - 1 frac{2}{5}}).Let's compute numerator and denominator separately.Numerator: 2 × 0.3 = 0.6. That's straightforward.Denominator: 2.3 - 1 2/5. Let's convert the mixed number to a decimal. 1 2/5 is 1 + 2/5. 2/5 is 0.4, so 1.4. Therefore, 2.3 - 1.4 = 0.9.Alternatively, maybe convert both to fractions. 2.3 is 23/10, and 1 2/5 is 7/5. So 23/10 - 7/5. Convert to common denominator 10: 23/10 - 14/10 = 9/10, which is 0.9. Correct.So the denominator is 0.9.Therefore, the second fraction is 0.6 / 0.9. Let's compute that. 0.6 divided by 0.9. Multiply numerator and denominator by 10 to eliminate decimals: 6/9 = 2/3. So the second fraction simplifies to 2/3.Now, adding the two results together: 1/3 + 2/3. That's straightforward. 1/3 + 2/3 = 1.Wait, so the entire expression simplifies to 1? Let me double-check each step to be sure.First fraction: numerator 3.2 - 2.95. 3.2 is 3.20. 3.20 - 2.95 is indeed 0.25. Denominator: 0.25 × 2 is 0.5, plus 0.25 is 0.75. 0.25 / 0.75 is 1/3. Correct.Second fraction: numerator 2 × 0.3 is 0.6. Denominator: 2.3 - 1.4 is 0.9. 0.6 / 0.9 is 2/3. Then 1/3 + 2/3 is 1. All steps check out.Hmm, so the final answer is 1. But let me just verify once more, in case I made a mistake in fraction conversions or arithmetic.First fraction:3.2 - 2.95: 3.2 is 32/10, 2.95 is 295/100. 32/10 is 320/100. 320 - 295 = 25, so 25/100 = 1/4. Then denominator: 0.25 × 2 is 0.5, which is 1/2. 1/2 + 1/4 = 3/4. So (1/4)/(3/4) = 1/3. Correct.Second fraction:2 × 0.3 = 0.6. 2.3 is 23/10, 1 2/5 is 7/5. 23/10 - 7/5 = 23/10 - 14/10 = 9/10. 0.6 is 3/5. (3/5)/(9/10) = (3/5) × (10/9) = (30/45) = 2/3. Correct. So 1/3 + 2/3 = 1. Yes, seems right.Therefore, the answer is 1.Final Answerboxed{1}</think>To calculate the given expression, we break it down into two fractions and simplify each part step by step.First fraction:[frac{3.2 - 2.95}{0.25 times 2 + frac{1}{4}}]- Numerator: (3.2 - 2.95 = 0.25)- Denominator: (0.25 times 2 + frac{1}{4} = 0.5 + 0.25 = 0.75)- Simplifying the fraction: (frac{0.25}{0.75} = frac{1}{3})Second fraction:[frac{2 times 0.3}{2.3 - 1 frac{2}{5}}]- Numerator: (2 times 0.3 = 0.6)- Denominator: (2.3 - 1 frac{2}{5} = 2.3 - 1.4 = 0.9)- Simplifying the fraction: (frac{0.6}{0.9} = frac{2}{3})Adding the results of the two fractions:[frac{1}{3} + frac{2}{3} = 1]Thus, the final answer is:[boxed{1}],<think>Alright, so I need to solve this math problem. Let me look at it again. It's (frac{3.2 - 2.95}{0.25 times 2 + frac{1}{4}} + frac{2 times 0.3}{2.3 - 1 frac{2}{5}}). Hmm, okay, let me break this down step by step. First, I should handle each part of the expression separately and then combine them. Let's start with the numerator of the first fraction: 3.2 minus 2.95. Calculating that, 3.2 minus 2.95. Let me do the subtraction. 3.2 is the same as 3.20, right? So 3.20 minus 2.95. Let me subtract: 20 minus 95, wait, but since it's after the decimal, I need to borrow. So, 3.20 minus 2.95. The whole numbers: 3 minus 2 is 1. Then, the decimal part: 0.20 minus 0.95. Hmm, can't do that without borrowing. So, borrowing 1 from the 3, which becomes 2, and then the decimal part becomes 1.20 (since 1 + 0.20 = 1.20). Then subtract 2.95 from that. Wait, no, maybe I confused the numbers. Let me write it out properly.3.20 -2.95 ------- Starting from the right: 0 - 5, can't do that, borrow from the tenths place. The tenths place is 2, so borrow 1, making it 1, and the hundredths place becomes 10. So 10 - 5 = 5. Then tenths place is now 1 - 9. Again, can't do that. So borrow 1 from the units place. The units place is 3, which becomes 2, and the tenths place becomes 11. Then 11 - 9 = 2. Then the units place is 2 - 2 = 0. Wait, so putting it all together: 0.25. So 3.2 - 2.95 = 0.25. Hmm, that seems right. Let me check: 2.95 + 0.25 = 3.20. Yes, correct. Okay, so the numerator of the first fraction is 0.25.Now the denominator of the first fraction: 0.25 multiplied by 2 plus 1/4. Let's compute each part. First, 0.25 times 2. Well, 0.25 is a quarter, so a quarter times 2 is 0.5. Then adding 1/4. 1/4 is 0.25. So 0.5 + 0.25 equals 0.75. Therefore, the denominator is 0.75. So the first fraction is 0.25 divided by 0.75. Let me compute that. 0.25 divided by 0.75. Hmm, 0.25 over 0.75. That's the same as (1/4) divided by (3/4), which is (1/4) multiplied by (4/3) = 1/3. So 0.25 ÷ 0.75 = 1/3. Alternatively, 0.25 divided by 0.75 is 1/3. Okay, that's straightforward.So the first part simplifies to 1/3. Alright, moving on to the second fraction: 2 times 0.3 over (2.3 minus 1 and 2/5). Let's tackle the numerator first: 2 multiplied by 0.3. That's 0.6. Then the denominator: 2.3 minus 1 and 2/5. First, 1 and 2/5 is equal to 1.4, since 2/5 is 0.4. So 2.3 minus 1.4. Let's subtract: 2.3 minus 1.4. That equals 0.9. So the denominator is 0.9. Therefore, the second fraction is 0.6 divided by 0.9.Calculating that, 0.6 over 0.9. Simplify. Both numerator and denominator can be multiplied by 10 to eliminate decimals: 6/9, which simplifies to 2/3. So the second fraction is 2/3. Now, putting it all together: the first part is 1/3 and the second part is 2/3. Adding them together: 1/3 + 2/3. That should be 1. But let me verify. 1/3 + 2/3 = (1 + 2)/3 = 3/3 = 1. Yep, that's correct. So the entire expression simplifies to 1. Wait, but let me double-check my steps because sometimes when dealing with decimals and fractions, it's easy to make a mistake. Let me go through each part again.First fraction: numerator 3.2 - 2.95. As before, 3.2 is 3.20. Subtract 2.95. So 3.20 - 2.95. Start from the hundredths place: 0 - 5. Can't do that, borrow 1 from the tenths place. Tenths place is 2, becomes 1, and hundredths becomes 10. 10 - 5 = 5. Tenths place: 1 - 9. Can't do that, borrow 1 from the units place. Units place is 3, becomes 2, tenths becomes 11. 11 - 9 = 2. Units place: 2 - 2 = 0. So result is 0.25. Correct. Denominator: 0.25 * 2 + 1/4. 0.25*2 is 0.5, 1/4 is 0.25. Sum is 0.75. So 0.25 / 0.75 = 1/3. Correct.Second fraction: numerator 2 * 0.3 = 0.6. Denominator: 2.3 - 1 2/5. 1 2/5 is 1.4. 2.3 - 1.4 = 0.9. So 0.6 / 0.9. 0.6 divided by 0.9. Multiply numerator and denominator by 10: 6/9 = 2/3. Correct.Adding 1/3 + 2/3 = 1. So answer is 1. Hmm, seems straightforward. Wait, but maybe the problem is presented in a way that could trick me? Let me check if I interpreted the fractions correctly.Looking at the original expression: (frac{3.2 - 2.95}{0.25 times 2 + frac{1}{4}} + frac{2 times 0.3}{2.3 - 1 frac{2}{5}}). The denominators are 0.25 times 2 plus 1/4, which we converted to 0.5 + 0.25 = 0.75. The other denominator is 2.3 minus 1 and 2/5, which is 1.4, so 2.3 - 1.4 = 0.9. Numerators: 3.2 - 2.95 = 0.25, and 2*0.3=0.6. All seems correct.Another way to check: Maybe convert all decimals to fractions. Let's try that for thoroughness. Starting with the first fraction:Numerator: 3.2 - 2.95. 3.2 is 32/10, which simplifies to 16/5. 2.95 is 295/100, which simplifies to 59/20. So 16/5 - 59/20. To subtract, common denominator is 20. 16/5 is 64/20. 64/20 - 59/20 = 5/20 = 1/4. Wait, hold on, 5/20 simplifies to 1/4, but earlier we had 0.25. 1/4 is 0.25, so that's consistent. Okay.Denominator: 0.25*2 + 1/4. 0.25 is 1/4, so 1/4*2 = 1/2. Then 1/2 + 1/4 = 3/4. So denominator is 3/4. So the first fraction is (1/4) divided by (3/4) = (1/4)*(4/3) = 1/3. Same as before.Second fraction: numerator 2*0.3. 0.3 is 3/10. So 2*(3/10) = 6/10 = 3/5. Denominator: 2.3 - 1 2/5. 2.3 is 23/10. 1 2/5 is 7/5. So 23/10 - 7/5. Convert 7/5 to 14/10. Then 23/10 - 14/10 = 9/10. So denominator is 9/10. Therefore, the second fraction is (3/5) / (9/10) = (3/5)*(10/9) = (30/45) = 2/3. Same as before. So adding 1/3 + 2/3 = 1. Okay, so both ways—decimal and fractional—lead to the same result. That makes me more confident that the answer is indeed 1. I don't see any errors in my calculations. So, the final answer should be 1, which in boxed form is boxed{1}.Final Answerboxed{1}</think>To solve the given problem, we break it down into two parts and handle each part step by step.First, we calculate the numerator and denominator of the first fraction:- Numerator: (3.2 - 2.95 = 0.25)- Denominator: (0.25 times 2 + frac{1}{4} = 0.5 + 0.25 = 0.75)- Therefore, the first fraction is (frac{0.25}{0.75} = frac{1}{3})Next, we calculate the numerator and denominator of the second fraction:- Numerator: (2 times 0.3 = 0.6)- Denominator: (2.3 - 1 frac{2}{5} = 2.3 - 1.4 = 0.9)- Therefore, the second fraction is (frac{0.6}{0.9} = frac{2}{3})Adding the results of the two fractions:[frac{1}{3} + frac{2}{3} = 1]Thus, the final answer is (boxed{1}).]

🔑:【Analysis】Simplify the expressions in the numerator and denominator of the complex fraction, then solve according to the basic properties of fractions.【Solution】Solve: frac{3.2-2.95}{0.25 times 2+frac{1}{4}}+frac{2 times 0.3}{2.3-1 frac{2}{5}}begin{array}{l}=frac{0.25}{0.75}+frac{0.6}{0.9} =frac{1}{3}+frac{2}{3} =1 ;end{array}Therefore, the answer is: 1 .