🧠:[<think>Okay, let me try to work through this problem step by step. There are three parts here, so I need to tackle each one carefully. Let's start with part (1).Problem (1): Prove that the point (-2, 0) lies on the graph of the function y = k(x + 2) where k ≠ 0.Hmm, so to prove that a point lies on the graph of a function, I just need to plug in the x-coordinate into the function and check if the resulting y-coordinate matches the given point. The point is (-2, 0), so the x-value is -2 and the y-value is 0.Let's substitute x = -2 into the function:y = k(-2 + 2) = k(0) = 0.So, when x is -2, y is 0, which matches the point (-2, 0). Therefore, the point (-2, 0) lies on the graph. That seems straightforward. I think that's all for part (1). Maybe they wanted me to show the substitution, which I did. So part (1) is done.Problem (2): If the graph of this function is shifted 2 units upwards and passes through the point (1, -2), find the value of k.Alright, shifting a graph upwards involves adding a constant to the original function. The original function is y = k(x + 2). If we shift it 2 units upwards, the new function becomes y = k(x + 2) + 2.Now, this shifted graph passes through the point (1, -2). That means when x = 1, y should be -2. Let's substitute these values into the shifted function:-2 = k(1 + 2) + 2Simplify the equation:-2 = k(3) + 2Subtract 2 from both sides:-2 - 2 = 3k-4 = 3kDivide both sides by 3:k = -4/3Wait, so k is -4/3? Let me check that again. Substitute k = -4/3 into the shifted function:y = (-4/3)(x + 2) + 2Let's plug in x = 1:y = (-4/3)(1 + 2) + 2 = (-4/3)(3) + 2 = -4 + 2 = -2Yes, that works. So k = -4/3 is correct. Alright, part (2) seems done.Problem (3): If the graph of this function intersects the y-axis between the x-axis and the line y = -2, find the range of values for k.Okay, so we need to find the values of k such that the y-intercept of the function y = k(x + 2) is between 0 and -2. That is, the y-intercept should satisfy -2 < y-intercept < 0.First, let's find the y-intercept of the original function. The y-intercept occurs where x = 0. So substitute x = 0 into the function:y = k(0 + 2) = 2kTherefore, the y-intercept is (0, 2k). The problem states that the y-intercept is between the x-axis (which is y = 0) and the line y = -2. So, the y-coordinate 2k must satisfy:-2 < 2k < 0Let me write that inequality down:-2 < 2k < 0To solve for k, divide all parts of the inequality by 2:-2/2 < k < 0/2Which simplifies to:-1 < k < 0So, the range of values for k is -1 < k < 0. Let me verify that.If k is between -1 and 0, then 2k is between -2 and 0. For example, if k = -0.5, then 2k = -1, which is between -2 and 0. If k approaches 0 from the negative side, 2k approaches 0 from below. If k approaches -1 from the positive side, 2k approaches -2 from above. That seems correct.Wait, but hold on, the problem says "between the x-axis and the line y = -2". So the y-intercept should be between y = 0 and y = -2. So that's between -2 and 0. Therefore, 2k must be between -2 and 0, so indeed k is between -1 and 0. So the answer is -1 < k < 0.I think that's right. Let me check with k = -1. Then the y-intercept is 2*(-1) = -2, which is exactly on the line y = -2, but the problem says "between the x-axis and the line y = -2", so it should not include k = -1. Similarly, k = 0 is excluded because the original function is defined with k ≠ 0, and at k = 0, it wouldn't be a linear function anymore. So the open interval (-1, 0) is correct.Therefore, the range is -1 < k < 0.Summary of Answers:1. Shown by substituting x = -2 into the function, resulting in y = 0.2. After shifting the function up 2 units and substituting the point (1, -2), solving gives k = -4/3.3. The y-intercept must be between -2 and 0, leading to the inequality -1 < k < 0.I think that covers all parts. Let me just confirm once again for part (3). If k is -1, y-intercept is -2, which is on the line y = -2, so it's excluded. Similarly, k = 0 is not allowed. So yes, open interval.Final Answer(1) boxed{(-2, 0)} is on the graph. (2) boxed{-dfrac{4}{3}} (3) boxed{(-1, 0)}Wait, the first part just says to prove that the point (-2, 0) lies on the graph. The question didn't ask for the answer to be boxed for part (1), just to prove it. But the user instructions say "put your final answer within boxed{}". Maybe they want each part's answer boxed separately. Let me check the original problem statement.Looking back: The user says "put your final answer within boxed{}". But in Chinese questions, sometimes each part is to be answered with a box. The original instruction says: "You will be given a problem. Please reason step by step, and put your final answer within boxed{}". Since there are three parts, maybe three separate boxed answers. Let me check the numbering.The user wrote:(1) Prove that the point (−2,0) lies on the graph...(2) If the graph...find the value of k.(3) Find the range...So probably, each part requires a boxed answer. But part (1) is a proof, but they might just accept stating that the point is on the graph, which we showed by substitution. However, since the user instruction says "put your final answer within boxed{}", maybe they expect something. Wait, the original problem is in Chinese, but translated here. Maybe in the original, part (1) is just to state that (-2,0) is on the graph, which is already proven, but in the answer, they might just write "The point (-2, 0) lies on the graph" but as per instruction here, since the user wants answers boxed. But boxed answers are usually for numerical answers or expressions. For a proof, maybe just stating it's proven, but in Chinese tests, sometimes they want Q1 to have a specific answer. Wait, maybe part (1) doesn't require a box, but the user instructions say "put your final answer within boxed{}". Hmm.Wait, looking at the initial problem statement:"Please reason step by step, and put your final answer within boxed{}"It's possible that each of the three parts requires a boxed answer. For part (1), since it's a proof, but maybe they accept the coordinates as the answer? Wait, no, part (1) is to prove that the point lies on the graph. The answer is that it does, which we have shown. But how to present that in a box? Maybe just write "True" or something, but that's not standard. Alternatively, since the coordinates are given, maybe restate the point in a box? Hmm.Looking at similar problems, often when a proof is asked, but the answer is to be boxed, sometimes the conclusion is boxed. For example, for part (1), after proving, we can box the coordinates as confirmation. Alternatively, maybe the problem expects just the answers for parts (2) and (3) boxed, and part (1) is just the proof. However, the user instruction says "put your final answer within boxed{}", which might imply all three answers need to be boxed.Wait, in the original problem statement in Chinese, if part (1) is a proof, then the answer is just shown through substitution. But perhaps the user expects, since it's three separate questions, each part to have a box. However, part (1) is a proof, so maybe the answer is simply stating that the point lies on the graph, which we did, and perhaps no box is needed. But the user's instruction is to put the final answer within boxes.This is a bit confusing. Let me check the initial problem again:"Given a linear function y = k(x + 2) (k ≠ 0).(1) Prove that the point (-2, 0) lies on the graph of this function.(2) If the graph of this function is shifted 2 units upwards and passes through the point (1, -2), find the value of k.(3) If the graph of this function intersects the y-axis between the x-axis and the line y = -2, find the range of values for k."So part (1) is a proof. The standard way is to show substitution. Since the user asks to "put your final answer within boxed{}", maybe parts (2) and (3) require boxed answers, while part (1) is just the proof. However, in Chinese tests, sometimes all answers are boxed, even if they are proofs. But proofs are usually written out. Maybe in this case, since parts (2) and (3) have numerical answers, they need boxes, and part (1) is just a sentence. However, the user instruction says "put your final answer within boxed{}", which might mean all three answers should be boxed. But part (1) is a statement, so maybe box the coordinates?Alternatively, in some contexts, the first part might not require a box, but parts (2) and (3) do. The user might have intended that. Since I can't be sure, but given the initial problem, part (1) is a proof, which we did, and parts (2) and (3) have specific answers. Therefore, the final answers to parts (2) and (3) should be boxed, while part (1) is just the proof. However, the user might expect all answers boxed. To comply, perhaps:(1) After proving, we can box (-2, 0) as the point, but actually, the question was to prove that the point lies on the graph, not to find the point. So perhaps part (1) doesn't have a boxed answer, and parts (2) and (3) do. Alternatively, if part (1) is considered as a question where the answer is the coordinates, then box it. But in any case, in the original problem, part (1) says "prove", so maybe just the proof is needed, and the answer is the process. But the user instruction says "put your final answer within boxed{}", so maybe they expect all three parts boxed. But part (1) is a point, which is (-2, 0). Maybe box that. Parts (2) and (3) are k = -4/3 and -1 < k < 0.So perhaps:(1) boxed{(-2, 0)}(2) boxed{-dfrac{4}{3}}(3) boxed{(-1, 0)}But in part (1), they didn't ask for the point, but to prove that the point lies on the graph. But since substituting x=-2 gives y=0, so the point is indeed (-2,0). Maybe the answer is just confirming that, so boxing the point. Alternatively, maybe the answer is a statement, but since instructions say boxed, probably box the coordinates.Therefore, I think the user expects three boxed answers:1. The coordinates (-2, 0)2. The value of k3. The interval for kHence, I'll present them as such.Final Answer(1) boxed{(-2, 0)} (2) boxed{-dfrac{4}{3}} (3) boxed{(-1, 0)}</think>(1) To prove that the point ((-2, 0)) lies on the graph of the function (y = k(x + 2)), we substitute (x = -2) into the function:[y = k(-2 + 2) = k cdot 0 = 0]Thus, the point ((-2, 0)) lies on the graph.(2) When the graph is shifted 2 units upwards, the new function becomes (y = k(x + 2) + 2). This shifted graph passes through the point ((1, -2)). Substituting (x = 1) and (y = -2) into the equation:[-2 = k(1 + 2) + 2]Simplifying, we get:[-2 = 3k + 2 implies -4 = 3k implies k = -frac{4}{3}](3) The y-intercept of the function (y = k(x + 2)) is found by setting (x = 0):[y = k(0 + 2) = 2k]We need the y-intercept to be between 0 and -2, which gives the inequality:[-2 < 2k < 0]Dividing by 2, we find:[-1 < k < 0] Final Answer(1) boxed{(-2, 0)} (2) boxed{-dfrac{4}{3}} (3) boxed{(-1, 0)},<think>Alright, let's tackle this problem step by step. So, first, we have the linear function y = k(x + 2), where k is not zero. There are three parts to the problem. Let's start with part (1): proving that the point (-2, 0) lies on the graph of this function. Hmm, okay. To show that a point (a, b) is on the graph of a function, we can plug x = a into the function and see if the result is y = b. So here, the point is (-2, 0). Let me substitute x = -2 into the equation. Plugging in x = -2: y = k(-2 + 2) = k(0) = 0. So regardless of the value of k (as long as k is defined, which it is, since k ≠ 0), y will be 0 when x is -2. Therefore, the point (-2, 0) is indeed on the graph. That seems straightforward. I don't think I made any mistakes there. So part (1) is done.Moving on to part (2): If the graph of this function is shifted 2 units upwards and passes through the point (1, -2), find the value of k. Okay, shifting a function upwards 2 units would mean adding 2 to the entire function. The original function is y = k(x + 2). Shifting it up 2 units gives us y = k(x + 2) + 2. Now, this shifted graph passes through the point (1, -2). So we can substitute x = 1 and y = -2 into the equation and solve for k.So let's do that. Substitute x = 1, y = -2:-2 = k(1 + 2) + 2 -2 = 3k + 2 Now, subtract 2 from both sides: -4 = 3k Divide both sides by 3: k = -4/3Wait, let me check that again. Starting with -2 = 3k + 2. Subtract 2: -2 - 2 = 3k → -4 = 3k. Then k = -4/3. Yeah, that seems right. Let me just verify by plugging back into the shifted equation. If k is -4/3, then the shifted function is y = (-4/3)(x + 2) + 2. Let's plug in x = 1:y = (-4/3)(1 + 2) + 2 = (-4/3)(3) + 2 = -4 + 2 = -2. Which matches the point (1, -2). So that checks out. Therefore, the value of k is -4/3. Okay, part (2) done.Now part (3): If the graph of this function intersects the y-axis between the x-axis and the line y = -2, find the range of values for k. Hmm. Let me parse that. The y-intercept of the function is between the x-axis (which is y = 0) and the line y = -2. So the y-intercept is between y = -2 and y = 0. Since "between" might be a bit ambiguous, but given that it's specified as "between the x-axis and the line y = -2", which is below the x-axis, so the y-intercept should be greater than -2 and less than 0. So -2 < y-intercept < 0.First, let's find the y-intercept of the original function. The y-intercept occurs where x = 0. So plug x = 0 into y = k(x + 2):y = k(0 + 2) = 2k. So the y-intercept is (0, 2k). So we need 2k to be between -2 and 0. So:-2 < 2k < 0Divide all parts by 2:-1 < k < 0So k must be between -1 and 0. Wait, but let me make sure. The problem says "between the x-axis and the line y = -2". Since the x-axis is y = 0, and the line y = -2 is below the x-axis. So the y-intercept is between y = -2 and y = 0. So 2k must satisfy -2 < 2k < 0. Then dividing by 2: -1 < k < 0. That seems right.But let me double-check. If k is between -1 and 0, then 2k is between -2 and 0. For example, if k = -0.5, then 2k = -1, which is between -2 and 0. If k approaches -1, 2k approaches -2, but stays greater than -2. If k approaches 0 from the negative side, 2k approaches 0 from below. So yes, that makes sense. The y-intercept is 2k, so if we want it between -2 and 0, k must be between -1 and 0. Therefore, the range is -1 < k < 0.Is there any chance that the inequalities could be non-strict? The problem says "between the x-axis and the line y = -2". If "between" is exclusive, then it's strict inequalities. If it were inclusive, they might mention "including" or use ≤. Since it just says "between", I think strict inequalities are correct. So the range of k is -1 < k < 0. Alright.So summarizing the three parts:1. Proved that (-2, 0) is on the graph by substitution.2. Found k = -4/3 after shifting the graph up 2 units and substituting the point (1, -2).3. Determined that k must be between -1 and 0 to have the y-intercept between -2 and 0.I think that's all. Let me just check if there's any possible mistake here. For part (3), another way to think is that the y-intercept is 2k. So if we want it between the x-axis (y=0) and y=-2, since it's below the x-axis, 2k must be between -2 and 0. So yes, 2k ∈ (-2, 0) implies k ∈ (-1, 0). Correct. Another check for part (2): If the original function is y = k(x + 2), shifting up 2 gives y = k(x + 2) + 2. Plugging (1, -2): -2 = 3k + 2 → 3k = -4 → k = -4/3. Correct. Part (1) is straightforward. So I think all three parts are answered correctly.Final Answer(1) boxed{(-2, 0)} lies on the graph. (2) The value of ( k ) is boxed{-dfrac{4}{3}}. (3) The range of values for ( k ) is boxed{(-1, 0)}.</think>(1) To prove that the point ((-2, 0)) lies on the graph of the function (y = k(x + 2)), substitute (x = -2) into the function:[y = k(-2 + 2) = k cdot 0 = 0]Thus, the point ((-2, 0)) is on the graph.(2) When the graph is shifted 2 units upwards, the new function is (y = k(x + 2) + 2). This shifted graph passes through the point ((1, -2)). Substituting (x = 1) and (y = -2) into the equation:[-2 = k(1 + 2) + 2][-2 = 3k + 2][-4 = 3k][k = -dfrac{4}{3}](3) The y-intercept of the function (y = k(x + 2)) occurs at (x = 0):[y = k(0 + 2) = 2k]We need the y-intercept to be between the x-axis ((y = 0)) and the line (y = -2). Therefore:[-2 < 2k < 0]Dividing by 2:[-1 < k < 0] Final Answer(1) boxed{(-2, 0)} lies on the graph. (2) The value of ( k ) is boxed{-dfrac{4}{3}}. (3) The range of values for ( k ) is boxed{(-1, 0)}.]

🔑: Solution:# Part (1): Proving the point left(-2,0right) lies on the graphGiven the function y=kleft(x+2right), to prove that the point left(-2,0right) lies on the graph, we substitute x=-2 into the equation:- Start with y=kleft(x+2right)- Substitute x=-2: y=kleft(-2+2right)- Simplify: y=kcdot 0- Therefore, y=0Thus, when x=-2, y=0, proving that the point left(-2,0right) lies on the graph of the function. # Part (2): Finding the value of k after shifting the graphThe function y=kleft(x+2right), when shifted 2 units upwards, becomes y=kleft(x+2right)+2. Given that this new graph passes through the point left(1,-2right), we substitute x=1 and y=-2 into the equation:- Start with y=kleft(x+2right)+2- Substitute left(1,-2right): -2=kleft(1+2right)+2- Simplify: -2=kcdot 3 + 2- Solve for k: k=frac{-2-2}{3}=frac{-4}{3}Therefore, the value of k is boxed{-frac{4}{3}}.# Part (3): Finding the range of values for kGiven that the graph intersects the y-axis between the x-axis and the line y=-2, and knowing that y=kleft(x+2right)=kx+2k when x=0, we analyze the intersection point:- The intersection with the y-axis occurs at y=2k- The condition given is -2 < 2k < 0- Divide the inequality by 2: -1 < k < 0Therefore, the range of values for k is boxed{-1 < k < 0}.

🧠:[<think>Okay, let's try to tackle this problem step by step. The question is about a robot coloring natural numbers either red or yellow based on specific rules. The key points are:1. Red numbers: Those that can be expressed as the sum of two composite numbers.2. Yellow numbers: Those that cannot be expressed as such a sum.3. We need to find the 1992nd red number, which is given as 2001. But we need to verify this.First, let's recall what composite numbers are. A composite number is a positive integer that has at least one positive divisor other than 1 and itself. So, composite numbers start from 4, 6, 8, 9, 10, 12, etc.The problem is essentially asking which numbers can't be written as the sum of two composite numbers and then figuring out how many numbers can be written that way up to a certain point.Wait, actually, it's the other way around. The red numbers are those that can be expressed as the sum of two composites. So we need to count all numbers starting from some point that are red and find the 1992nd one.But to do this, we need to know which numbers can't be expressed as the sum of two composites (yellow numbers) and then subtract their count from the total numbers to find the red ones. Alternatively, maybe directly counting red numbers is feasible, but probably requires some pattern.First, let's note that all even numbers greater than or equal to 8 can be expressed as the sum of two even composite numbers. For example, 8 = 4 + 4, 10 = 4 + 6, 12 = 6 + 6, etc. So even numbers starting from 8 should be red.For odd numbers, a number can be written as the sum of two composites only if one is even and the other is odd. Since the sum of an even and an odd number is odd. Let's see. So, an odd number N can be written as 9 + (N - 9). If N - 9 is an even composite number, then N is red. For example, 17 = 9 + 8 (both composite). So for odd numbers, starting from N where N - 9 is even composite. Let's check N=17: 17-9=8, which is composite. So 17 is red. N=19: 19-9=10, which is composite, so 19 is red. Similarly, N=21: 21-9=12, composite. So seems like starting from 17, odd numbers can be red if N - 9 is composite. Wait, but what about lower odd numbers?Let's check odd numbers starting from 1:1: Yellow (can't be written as sum of two numbers less than it)3: Same5: Same7: Same9: Can 9 be written as sum of two composites? 9 = 4 + 5, but 5 is prime. 9 = 6 + 3, 3 is prime. 9 = 8 + 1, 1 is not composite. So 9 is yellow.11: Let's check. 11 = 4 + 7 (7 prime), 6 + 5 (5 prime), 8 + 3 (3 prime), 9 + 2 (2 prime). So 11 is yellow.13: Similarly, 13 = 4 + 9 (both composite? 4 is composite, 9 is composite. Wait, 4 + 9 = 13. So 13 can be written as sum of two composites. Wait, 13 is prime, but 4 + 9 = 13. So 13 is red? Wait, 4 is composite, 9 is composite. So yes, 13 is red. Wait, this contradicts my earlier thought.Wait, perhaps the initial approach was wrong. Let me check.Wait, 13 = 4 + 9. 4 is composite, 9 is composite. Therefore, 13 should be red. So maybe my previous assumption that odd numbers start from 17 being red is incorrect. Let's check:9: 4 + 5 (5 prime), 6 + 3 (3 prime), 8 + 1 (1 not composite). So 9 is yellow.11: 4 + 7 (prime), 6 + 5 (prime), 8 + 3 (prime), 9 + 2 (prime). So 11 is yellow.13: 4 + 9 (composite + composite). So 13 is red.15: Let's see. 15 = 6 + 9 (both composite). So 15 is red.17: 8 + 9 (both composite). So 17 is red.19: 10 + 9 (both composite). So 19 is red.21: 12 + 9, both composite.So starting from 13, odd numbers can be expressed as sum of two composites? But wait, 13 is red, but 15 is red, 17, etc. Wait, but 13 is an odd number. So why did I think earlier that starting from 17?Wait, perhaps even some lower odd numbers can be expressed as sum of two composites. Let's check 7: 7 = 4 + 3 (3 is prime). 6 +1 (1 is not composite). So 7 is yellow. 5 is yellow, 3 is yellow, 1 is yellow.So 9 is yellow, 11 is yellow, 13 is red, 15 is red, 17 is red, etc.So odd numbers from 13 onwards (if they can be expressed as 4 + composite or 6 + composite, etc.)Wait, but 13 = 4 + 9. 9 is composite. So if we can write N as 4 + composite, then N is red. Similarly, N = 6 + composite, 8 + composite, etc.Similarly, for odd numbers: Let's consider that odd numbers can be written as 9 + even composite. Wait, 9 is composite. So if N -9 is even composite, then N is red.For example:N = 13: 13 - 9 = 4, which is composite. So 13 is red.N = 15: 15 - 9 = 6, composite. So 15 is red.Similarly, 17 - 9 = 8, composite. So yes, starting from N=13, which is 9 + 4, 4 is composite.But 4 is the first composite number, so N=13 is the first odd number that can be written as 9 + 4. Then N=15 is 9 + 6, etc. So perhaps all odd numbers >= 13 can be written as 9 + an even composite. So if N -9 is an even composite, then N is red.But N -9 must be even composite. Let's check: For N >=13, N is odd. So N -9 is even. So N -9 is even and >=4 (since N >=13 => N -9 >=4). So even numbers >=4: 4,6,8,10,... So N -9 is composite if N -9 is even and >=4, but 4 is composite, 6 is composite, 8 is composite, etc. Wait, but 4 is composite, so starting at N=13 (13-9=4), N=15 (15-9=6), N=17 (17-9=8), etc. So all odd numbers >=13 can be written as 9 + (even composite). Since even numbers >=4 are composite except for 2, but N -9 is even and >=4, so 4,6,8,... All even numbers >=4 are composite except for 2, but since we are starting from 4, all N -9 (even numbers starting from 4) are composite. Hence, all odd numbers >=13 can be written as 9 + even composite, hence they are red.But wait, 4 is composite, 6 is composite, 8 is composite, 10 is composite, etc. So starting from N=13, all odd numbers can be written as 9 + even composite, so they are red. So odd numbers from 13 onwards are red.Similarly, for even numbers, starting from 8, as 4 + 4, 4 +6, etc., so they are red.Therefore, the numbers that are yellow are:1, 2, 3, 4, 5, 6, 7 (since they can't be expressed as sum of two composites)Wait, but 4 is composite. Can 4 be written as sum of two composites? 4 = composite + composite. The composites start at 4. So 4 = 4 + 0, but 0 is not natural. So 4 can't be written as sum of two composites. Similarly, 5: 4 +1 (1 not composite). 6: 4 +2 (2 prime). 6 can't be written as sum of two composites. 7: 4 +3 (prime). 8: 4 +4, which are composite. So 8 is red.So yellow numbers are 1,2,3,4,5,6,7,9,11.Wait:Check each number from 1 upwards:1: Can't be written as sum of two numbers (since 1 is the smallest). So yellow.2: Same, only 1+1, both not composite. Yellow.3: 1+2, neither composite. Yellow.4: 2+2 (prime), 1+3 (neither composite). So 4 is yellow.5: 2+3 (primes), 4+1 (1 not composite). Yellow.6: 3+3 (prime), 4+2 (prime), 5+1 (non-composite). Yellow.7: 3+4 (3 prime, 4 composite). But sum must be two composites. So 7 cannot be written as sum of two composites. Yellow.8: 4+4. Both composite. So red.9: 4+5 (5 prime), 6+3 (prime). So yellow.10: 4+6 (both composite). Red.11: 4+7 (prime), 6+5 (prime), etc. Yellow.12: 6+6. Red.13: 4+9. Both composite. Red.14: 8+6. Both composite. Red.15: 6+9. Both composite. Red.16: 8+8. Both composite. Red.17: 8+9. Both composite. Red.18: 9+9. Both composite. Red.So the yellow numbers are 1,2,3,4,5,6,7,9,11. That's 9 numbers.Then starting from 8, all numbers except 9 and 11 are red? Wait, 12 is red, 13 is red, etc. Wait, starting from 8:8: red9: yellow10: red11: yellow12: red13: red14: red15: red16: red17: redSo after 11, all numbers are red. So starting from 12 onwards, every number is red. So the only yellow numbers are 1,2,3,4,5,6,7,9,11. Total 9 numbers.Therefore, all numbers from 1 to 11, except 8,10,12? Wait, 8,10,12 are red. Wait:Wait, starting from 8:8 is red, 9 yellow, 10 red, 11 yellow, 12 red, and from 12 onwards all red. Wait, let's confirm.Numbers from 12 upwards:12 is red (6+6).13: 4 +9 (both composite).14: 8 +6 (both composite).15: 9 +6.16: 8 +8.17: 9 +8.18: 9 +9.19: 10 +9.20: 10 +10.And so on. So every number from 12 onwards is red because:- Even numbers >=12: 12,14,16,... can be written as 4 + (even composite). Since 12 = 4 +8, 14=4 +10, etc. All even numbers >=8 can be written as 4 + even composite (since even numbers starting from 8-4=4, which is composite, and 8-4=4, 10-4=6, etc.)Wait, but even numbers >=8 can be written as 4 + (even number >=4), which is composite, so they are red.For odd numbers >=13: As established before, they can be written as 9 + even composite (since N -9 is even and >=4, hence composite). Therefore, odd numbers >=13 are red.Therefore, starting from 12, all numbers are red. So the yellow numbers are only up to 11, which are 1,2,3,4,5,6,7,9,11. Total of 9 numbers.Therefore, all numbers starting from 8:8,10,12,13,14,15,... are red. So starting from 8, except 9 and 11, which are yellow.So total numbers: For numbers starting from 1, up to any N, the number of red numbers would be (N - 9) if N >=12. Wait, no. Let's see:From 1 to 7: All yellow (1-7).8: red.9: yellow.10: red.11: yellow.12 onwards: red.So total yellow numbers: 1-7 (7 numbers), plus 9 and 11: total 9.So the red numbers are:8,10,12,13,14,15,... etc.So starting from 8, every number except 9 and 11 are red. Therefore, to find the 1992nd red number, we need to count starting from 8, and every number after that is red except 9 and 11.But wait, starting from 8:Numbers are 8(red),9(yellow),10(red),11(yellow),12(red),13(red),14(red),15(red),..., so after 12, all numbers are red. Therefore, up to 11, there are two yellow numbers (9 and 11) and three red numbers (8,10,12). Wait, no:Wait, 8 is red, 9 yellow, 10 red, 11 yellow, 12 red. So up to 12, there are three red numbers:8,10,12.Then starting from 13 onwards, all numbers are red. So the total red numbers up to N (where N >=12) is (N - 7) minus the number of yellow numbers. Wait:Total numbers from 1 to N: N.Subtract yellow numbers: 1,2,3,4,5,6,7,9,11. So 9 numbers.Therefore, red numbers up to N: N - 9.But wait, let's check N=12:Red numbers up to 12: 8,10,12. That's 3.N -9 = 12 -9 =3. Correct.N=13: 8,10,12,13. 4. 13 -9=4. Correct.N=14:5. 14 -9=5. Correct. So yes, formula holds.Therefore, for N >=12, the number of red numbers up to N is N -9.So we need to find the 1992nd red number. Let R(n) be the nth red number. We need R(1992)=?If the number of red numbers up to N is N -9, then solving N -9 =1992 gives N=1992 +9=2001. Therefore, the 1992nd red number is 2001.But let's verify this.Wait, according to the formula, up to N=2001, there are 2001 -9=1992 red numbers. Therefore, the 1992nd red number is 2001.But to be thorough, let's check if there are any yellow numbers after 11. Since we concluded that starting from 12 onwards, all numbers are red. Therefore, yellow numbers are only 1,2,3,4,5,6,7,9,11. So yes, after 11, no more yellow numbers. Therefore, starting from 12, every number is red. So the number of red numbers up to N is (N - number of yellow numbers). Number of yellow numbers is 9. So red numbers up to N is N -9.Therefore, the 1992nd red number is 1992 +9=2001. Therefore, the answer is 2001.Hence, the final answer is boxed{2001}.Final Answerboxed{2001}</think>To solve the problem, we need to determine which natural numbers can be expressed as the sum of two composite numbers (colored red) and which cannot (colored yellow). We then need to find the 1992nd red number.1. Composite Numbers: These start from 4, 6, 8, 9, 10, 12, etc.2. Red Numbers: These are numbers that can be expressed as the sum of two composite numbers.3. Yellow Numbers: These are numbers that cannot be expressed as the sum of two composite numbers.Key Steps:- Even Numbers: All even numbers greater than or equal to 8 can be written as the sum of two even composite numbers (e.g., 8 = 4 + 4, 10 = 4 + 6).- Odd Numbers: Odd numbers greater than or equal to 13 can be written as the sum of 9 (a composite number) and an even composite number (e.g., 13 = 4 + 9, 15 = 6 + 9).Yellow Numbers Identified:- The numbers 1, 2, 3, 4, 5, 6, 7, 9, and 11 cannot be expressed as the sum of two composite numbers. These are the only yellow numbers.Red Numbers:- Starting from 8, all numbers except the yellow numbers (9 and 11) are red. From 12 onwards, all numbers are red.Counting Red Numbers:- The number of red numbers up to any natural number ( N ) (where ( N geq 12 )) is ( N - 9 ). This is because there are 9 yellow numbers (1, 2, 3, 4, 5, 6, 7, 9, 11).Finding the 1992nd Red Number:- To find the 1992nd red number, we solve ( N - 9 = 1992 ). This gives ( N = 1992 + 9 = 2001 ).Thus, the 1992nd red number is (boxed{2001}).,<think>Okay, so there's this problem where a robot is coloring natural numbers starting from 1. The rule is: any number that can be expressed as the sum of two composite numbers is colored red, and the rest are colored yellow. Then, if you count the red-colored numbers in ascending order, the 1992nd one is supposed to be 2001. Hmm, let me see if I can figure out why that is.First, I need to recall what composite numbers are. Composite numbers are natural numbers greater than 1 that aren't prime; so they have factors other than 1 and themselves. The smallest composite number is 4, then 6, 8, 9, etc. So primes are not composite, and neither is 1, which is a unit. So, the composite numbers start at 4.The robot colors a number red if it can be written as the sum of two composite numbers. So, numbers that can't be written that way are yellow. The problem is asking for the 1992nd red number, which is given as 2001. So I need to verify that or find out the steps to get there.First, maybe I need to figure out which numbers can be expressed as the sum of two composite numbers. Let's start with small numbers and see if there's a pattern.Starting from 1:1: Can't be expressed as sum of two composites (there are no composites less than 1). So yellow.2: Same reason. Yellow.3: Also yellow.4: 4 can't be expressed as sum of two composites. The only possible pair is 4 and 0, but 0 isn't a natural number. So 4 is yellow.5: Let's see. Possible pairs: 4 + 1, but 1 isn't composite. So 5 is yellow.6: Let's try 4 + 2. 2 is prime. 6 + 0? No. So 6 can't be expressed as sum of two composites? Wait, 6 is composite, but 6 itself. Wait, but the question is about two composite numbers. So 6 = 4 + 2, but 2 is prime. 6 = 6 + 0, which isn't valid. So 6 can't be written as sum of two composites. So 6 is yellow? Wait, but 6 is composite. Wait, but does the problem allow using the same composite number twice? For example, 4 + 4 = 8. So 8 can be written as sum of two composites. But for 6, 3 + 3 is prime + prime. 2 + 4 is prime + composite. So 6 is yellow. Hmm.7: 4 + 3 (prime). 6 + 1 (non-composite). So 7 is yellow.8: 4 + 4 = 8. Both composites. So 8 is red.9: Let's see. 4 + 5 (prime). 6 + 3 (prime). 9 + 0. Wait, 9 is composite. Wait, 9 can be written as 6 + 3 (prime). 4 + 5 (prime). 8 + 1 (non-composite). Hmm. So can't 9 be written as sum of two composites? Wait, 9 itself is composite. Wait, but the problem says "sum of two composite numbers". So if I can write 9 as sum of two composites. Let me check:Is there a composite number a and another composite number b such that a + b = 9?Possible composites less than 9: 4, 6, 8, 9. Wait, 9 is composite but if we take 9 + 0, which is invalid. So 4 + 5 (5 is prime). 6 + 3 (3 is prime). 8 + 1 (1 is not composite). So no. So 9 is yellow.Wait, but 9 is composite. The color is based on being expressible as sum of two composite numbers. So even if the number itself is composite, it's only red if it can be written as sum of two composite numbers. So 9 is composite but can't be written as sum of two composites, so it's yellow.10: Let's check. 4 + 6 = 10. Both composites. So 10 is red.11: Let's see. 4 + 7 (prime). 6 + 5 (prime). 8 + 3 (prime). 9 + 2 (prime). So no. 11 is yellow.12: 4 + 8 = 12. Both composites. So red.13: 4 + 9 = 13. 4 is composite, 9 is composite. Wait, 4 + 9 = 13? 4 + 9 is 13? No, 4 + 9 is 13? Wait, 4 + 9 is 13? Wait, 4 + 9 is 13. Wait, 13 is prime. So 13 cannot be written as sum of two composites. Wait, 13 is the sum of 4 and 9, which are both composite. But 4 + 9 is 13? Wait, 4 + 9 is 13. Wait, 4 + 9 is 13. That is correct. So 13 is red? Wait, 13 is a prime number, but 13 = 4 + 9. Both 4 and 9 are composite. Therefore, 13 is red. Hmm, that's interesting.Wait, so even though 13 is prime, it can be written as sum of two composites. So it's red. So primes can be red if they can be expressed as sum of two composites. So 13 is red.Similarly, let's check 11 again. 11: 4 + 7 (prime), 6 + 5 (prime), 8 + 3 (prime), 9 + 2 (prime). So no. So 11 is yellow.So 11 is yellow, 12 is red, 13 is red.14: Let's see. 4 + 10 (both composite). 6 + 8. Both composite. So 14 is red.15: 6 + 9. Both composite. So red.16: 8 + 8. Both composite. So red.17: 8 + 9 = 17. 8 and 9 are composite. So 17 is red.18: 9 + 9. Both composite. Red.19: Let's see. 4 + 15 = 19. 15 is composite. So 4 + 15 = 19. 4 is composite, 15 is composite. So 19 is red.20: 10 + 10. Composite. So red.21: 12 + 9. Composite. So red.22: 12 + 10. Composite. Red.23: 8 + 15. Both composite. 23 is red.Wait, 8 + 15 = 23. Both composite. So yes.Wait, 24: 12 + 12. Red.So starting from 8, most numbers are red. But some are yellow. Let's see, starting from 8:8: red9: yellow10: red11: yellow12: red13: red14: red15: red16: red17: red18: red19: red20: red21: red22: red23: red24: redSo from 8 onwards, except for 9, 11, and then... Hmm. Let's check 25: 16 + 9. Both composite. So red.Wait, but maybe there's a pattern here. So starting from 8, numbers can be written as sum of two composites except for some exceptions. Let's see. Let's list the numbers and their colors:Numbers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20...Colors: Y Y Y Y Y Y Y R Y R Y R R R R R R R R R R R...So from 8 onwards, some numbers are red, some are yellow. Wait, but 9 is yellow, 10 is red, 11 is yellow, 12 is red, then starting from 12, 13 and above are all red? Wait, 13 is red, 14 red, etc. Wait, 11 is yellow, then 12 red, 13 red. So 11 is the last yellow before 12.Wait, maybe after a certain point, all numbers can be written as sum of two composites. So perhaps there's a maximum number that is yellow, and beyond that, all numbers are red. Let's check.In the problem statement, the 1992nd red number is 2001. So if we can find that after a certain number, all numbers are red, then the count of red numbers up to 2001 is 1992. Therefore, we need to figure out how many numbers are red starting from 8 up to 2001, minus the yellows in that range.But first, let's check when all numbers become red. It's similar to the Goldbach conjecture, but for composites. Wait, in Goldbach, even numbers greater than 2 can be expressed as sum of two primes. But here, it's different. We need to express numbers as sum of two composites.I remember that after a certain point, all numbers can be expressed as sum of two composite numbers. Let's try to see when that happens.First, let's consider even and odd numbers.For even numbers:An even number n can be written as 4 + (n - 4). If n - 4 is composite, then n can be written as sum of 4 and (n - 4). Similarly, for odd numbers:An odd number n can be written as 9 + (n - 9). If n - 9 is composite, then it's a sum of two composites. So, if n is even and n >= 8, then n - 4 >= 4. But n - 4 needs to be composite. So for even numbers, n >= 8, n - 4 is composite? Let's check:n = 8: 8 - 4 = 4 (composite) -> 8 = 4 + 4. Good.n = 10: 10 - 4 = 6 (composite). 10 = 4 + 6.n = 12: 12 - 4 = 8 (composite). 12 = 4 + 8.Similarly, n = 14: 14 - 4 = 10 (composite). So yes.So for even numbers >= 8, since n - 4 is even and >= 4, which is composite (except 4 itself, which is composite). Wait, 4 is composite. So even numbers >= 8 can be written as 4 + (even number >=4), which is composite. So all even numbers >=8 can be expressed as sum of two composites. Therefore, all even numbers from 8 onwards are red.What about odd numbers?An odd number can be written as 9 + (n - 9). If n - 9 is composite, then since 9 is composite, the sum is two composites. So for n >= 13 (since n -9 >= 4), let's check:n = 13: 13 - 9 = 4 (composite). 13 = 9 + 4. So red.n = 15: 15 - 9 = 6 (composite). 15 = 9 + 6.n = 17: 17 - 9 = 8 (composite). 17 = 9 + 8.n = 19: 19 - 9 = 10 (composite). 19 = 9 + 10.Similarly, n = 21: 21 - 9 = 12 (composite). So yes.So for odd numbers >= 13, n -9 is even and >=4, so composite. Wait, n -9 is even? If n is odd, then n -9 is even. Because odd minus odd is even. So n -9 is even and >=4 when n >=13. Therefore, n -9 is even and >=4, hence composite (since all even numbers >=4 are composite except 2, but since n -9 >=4, 4,6,8,...). So n -9 is composite. Therefore, odd numbers >=13 can be written as 9 + (n -9), which is composite + composite.But wait, 4 is composite, but n -9 could be 4. For example, n =13, 13 -9 =4, which is composite. So yes. So all odd numbers >=13 can be written as sum of two composites. Therefore, starting at 13, all odd numbers are red.Therefore, combining the two:Even numbers >=8 are red.Odd numbers >=13 are red.So the yellow numbers are:All numbers from 1-7: 1,2,3,4,5,6,7: all yellow.Then even numbers: 8 is red, 10 red, 12 red, etc.Odd numbers: 9, 11 are yellow, then 13 onwards red.So the yellow numbers are:1,2,3,4,5,6,7,9,11.That's 9 numbers.Wait, let me count:1,2,3,4,5,6,7 (7 numbers), then 9,11 (2 numbers). Total 9 yellow numbers.All other numbers >=8 and even, or >=13 and odd, are red.Therefore, the red numbers start from 8,10,12,13,14,15,...Therefore, to find the 1992nd red number, we need to see how many numbers are red up to a certain number. But the problem is that given the answer is 2001, so we need to verify that 2001 is the 1992nd red number.First, let's calculate how many numbers are red from 8 to 2001. Then subtract the yellow numbers in that range.But first, let's note that all numbers >=8 are red except 9 and 11.Wait, no. Wait, numbers from 8 onwards: even numbers are red starting at 8, odd numbers are red starting at 13. So between 8 and 12:8 (red), 9 (yellow), 10 (red), 11 (yellow), 12 (red). Then from 13 onwards, all odd numbers are red.So in total, the yellow numbers after 7 are 9 and 11. So total yellow numbers are 1-7 (7 numbers), 9, 11 (2 numbers). So 9 yellow numbers in total. All others are red.Wait, but 4,5,6,7 are yellow. Wait, 4 is composite but can't be written as sum of two composites. Wait, 4 is yellow. So 4 is yellow even though it's composite. So yellow numbers are all numbers that cannot be written as sum of two composites, regardless of whether they are composite or not.So total natural numbers are infinite. But up to 2001, how many yellow numbers are there?From 1 to 2001:Total numbers: 2001.Yellow numbers:From 1-7: 7 numbers.Then 9, 11: 2 numbers.So total 9 yellow numbers. Wait, but that can't be. Wait, let's check.Wait, if all even numbers from 8 onwards are red, and all odd numbers from 13 onwards are red, then the only yellow numbers are:1,2,3,4,5,6,7 (7 numbers), and odd numbers 9,11 (2 numbers). So total 9 yellow numbers.Therefore, the rest are red. So total red numbers up to 2001 would be 2001 - 9 = 1992. Therefore, the 1992nd red number is 2001.Wait, that seems to make sense. Because if there are 9 yellow numbers in total, then the red numbers are all numbers except those 9. So if you list all numbers from 1 to 2001, exclude the 9 yellow numbers, then the total red numbers are 2001 - 9 = 1992. Therefore, the 1992nd red number is 2001.But wait, let's verify with a smaller number. Let's say up to 12.Numbers 1-12. Yellow numbers:1,2,3,4,5,6,7,9,11. Total 9. Red numbers:8,10,12. So 3 red numbers. 12 is the 3rd red number. But according to the formula, total numbers 12 - 9 yellow = 3 red, so 12 is the 3rd red number. Which matches.Similarly, up to 13:Numbers 1-13. Red numbers:8,10,12,13. So 4 red numbers. According to 13 -9=4. So 13 is the 4th red number.Similarly, up to 14: red numbers would be 8,10,12,13,14. 5 red numbers. 14-9=5. So 14 is the 5th. Correct.So this formula seems to hold. Therefore, for any number n >=13, the number of red numbers up to n is n - 9. Therefore, if we set n -9 =1992, then n=1992 +9=2001. Hence, the 1992nd red number is 2001.Therefore, the answer is 2001. So the problem was checking whether you realize that after a certain point, all numbers can be written as sum of two composites, except the 9 yellow numbers, and hence the count is straightforward.But let me just make sure there are no more yellow numbers beyond 11. For example, is 17 a red number? Yes, 9 +8=17. 23 is 9 +14=23. 15 is 6 +9. So yes. All odd numbers from 13 upwards can be written as 9 + composite. Even numbers from 8 upwards as 4 + composite. Therefore, there are no other yellow numbers. Hence, total yellow numbers are only 9.Therefore, the reasoning holds. Therefore, the answer is 2001.Final Answerboxed{2001}</think>To determine the 1992nd number colored red by the robot, we need to identify which numbers can be expressed as the sum of two composite numbers. Composite numbers are natural numbers greater than 1 that are not prime. The key steps are as follows:1. Identify Composite Numbers: Composite numbers start from 4 (e.g., 4, 6, 8, 9, 10, etc.).2. Check Sum of Two Composites: - Even numbers starting from 8 can be expressed as (4 + (n - 4)), where (n - 4) is composite. - Odd numbers starting from 13 can be expressed as (9 + (n - 9)), where (n - 9) is composite.3. Identify Yellow Numbers: Numbers that cannot be expressed as the sum of two composite numbers are 1, 2, 3, 4, 5, 6, 7, 9, and 11. These are the only yellow numbers.4. Count Red Numbers: All numbers from 1 to 2001, excluding the 9 yellow numbers, are red. Therefore, the number of red numbers up to 2001 is (2001 - 9 = 1992).Thus, the 1992nd red number is (boxed{2001}).]

🔑:Let's list the first few numbers, with the numbers in red shown in parentheses: 1, 2, 3, 4, 5, 6, 7, (8=4+4), 9, (10=4+6), 11, (12=4+8), (13=4+9), (14=4+10), (15=6+9)… It can be observed that, starting from 8, every even number can definitely be divided into the sum of two larger even numbers, making it a red number. For larger odd numbers, they can definitely be divided into the sum of an even number greater than 2 and 9. Thus, the yellow numbers are only: 1, 2, 3, 4, 5, 6, 7, 9, 11, totaling 9 numbers. Therefore, the 1992nd number is: 1992 + 9 = boxed{2001}. Hence, the answer is boxed{2001}.

🧠:[<think>Okay, so I need to solve this problem about the tangent line to the curve y = x³. Let me read it again carefully.The tangent line l at point B on the curve y = x³ intersects the x-axis at point A, and triangle OAB (where O is the origin) is an isosceles triangle with A as the apex. The slope angle of the tangent line l is one of the given options: 30°, 45°, 60°, or 120°. I need to find which one.Alright, let's break this down step by step.First, let's recall that the derivative of y = x³ is y' = 3x², which gives the slope of the tangent line at any point x. So, if point B has coordinates (a, a³), then the slope of the tangent line at B is 3a². Therefore, the equation of the tangent line at B can be written using the point-slope form: y - a³ = 3a²(x - a).Next, this tangent line intersects the x-axis at point A. To find the coordinates of point A, we can set y = 0 in the equation of the tangent line and solve for x.So, substituting y = 0 into the tangent equation:0 - a³ = 3a²(x - a)Simplify:- a³ = 3a²x - 3a³Bring all terms to one side:- a³ + 3a³ = 3a²x2a³ = 3a²xDivide both sides by 3a² (assuming a ≠ 0, which makes sense because if a were 0, the tangent line would be horizontal, and the triangle would collapse):x = (2a³)/(3a²) = (2/3)aSo, point A is at ( (2/3)a, 0 )Now, triangle OAB is isosceles with A as the apex. Apex here means that the two equal sides meet at A. So, the two equal sides would be AO and AA? Wait, that doesn't make sense. Wait, apex is the vertex where the two equal sides meet. So, in triangle OAB, vertex A is the apex, meaning sides OA and AA are equal? Wait, no. Wait, triangle OAB has vertices O, A, B. If A is the apex of the isosceles triangle, then the two equal sides must be the ones meeting at A. So, the legs OA and AB must be equal? Wait, OA is from O to A, and AB is from A to B. Hmm, maybe. Or maybe OA and AA? No, that's not possible. Wait, no, in a triangle, each vertex is connected to two sides. So, if A is the apex, then the two sides meeting at A must be equal. So, sides OA and BA. Wait, OA is from O to A, BA is from B to A. So, OA and BA need to be equal in length.So, OA = BA.Let me confirm that. If the apex is A, then the two sides adjacent to A are OA and BA. Therefore, OA = BA. So, the lengths of OA and BA must be equal.So, let's compute OA and BA.First, coordinates:- O is (0, 0)- A is ( (2/3)a, 0 )- B is (a, a³ )So, OA is the distance from O to A, which is just the x-coordinate of A since it's on the x-axis. So, OA = | (2/3)a - 0 | = (2/3)|a|.BA is the distance from B to A. Let's compute that.Coordinates of A: ( (2/3)a, 0 )Coordinates of B: (a, a³ )So, BA distance is sqrt[ (a - (2/3)a )² + (a³ - 0 )² ] = sqrt[ ( (1/3)a )² + (a³ )² ] = sqrt[ (a²/9) + a⁶ ]So, set OA = BA:(2/3)|a| = sqrt[ (a²/9) + a⁶ ]Since both sides are non-negative, we can square both sides:( (2/3)a )² = (a²/9) + a⁶Compute left side:(4/9)a² = (a²/9) + a⁶Subtract (a²/9) from both sides:(4/9)a² - (1/9)a² = a⁶(3/9)a² = a⁶Simplify:(1/3)a² = a⁶Divide both sides by a² (assuming a ≠ 0, since if a = 0, point B is the origin, and the tangent line is horizontal, leading to point A being at (0,0), so triangle OAB would collapse to a single point, which isn't a triangle):1/3 = a⁴Therefore, a⁴ = 1/3Take the fourth root:a = ± (1/3)^(1/4) = ± (3)^(-1/4) = ± 1/(3^{1/4}) ) = ± 1/∜3Hmm, okay, so a is ± 1/∜3. Let's note that 3^(1/4) is the fourth root of 3, which is the same as sqrt(sqrt(3)).But let's keep it as a = ± 1/∜3 for now.Now, the slope of the tangent line at point B is 3a². Let's compute that.First, compute a²:a² = (1/∜3)^2 = 1/(3^(1/2)) = 1/√3Therefore, slope = 3 * (1/√3) = 3/√3 = √3Alternatively, if a was negative, a² is still positive, so slope is still √3.So, the slope of the tangent line is √3.Now, the question asks for the slope angle. The slope angle is the angle that the tangent line makes with the positive x-axis. The slope is equal to the tangent of that angle.So, if the slope m = tanθ, then θ = arctan(m).Here, m = √3, so θ = arctan(√3). We know that tan(60°) = √3, so θ = 60°. Therefore, the slope angle is 60°, which is option C.Wait a second, but let me double-check everything to make sure.First, let's verify that OA = BA.Given a = 1/∜3, then OA is (2/3)a = (2/3)(1/∜3). The distance OA is (2/3)/∜3.BA is sqrt[ (a - (2/3)a )² + (a³ )² ] = sqrt[ ( (1/3)a )² + (a³ )² ]Compute (1/3)a = (1/3)(1/∜3) = 1/(3∜3). Squared is 1/(9 * 3^(1/2)) = 1/(9√3)a³ = (1/∜3)^3 = 1/(3^(3/4)). Squared is 1/(3^(3/2)) = 1/(3√3)So BA squared is 1/(9√3) + 1/(3√3) = [1 + 3]/(9√3) = 4/(9√3)Thus BA is sqrt(4/(9√3)) = 2/(3 * (3^(1/4)) )Wait, maybe there's a better way to compute.Wait, OA is (2/3)a. If a = 1/∜3, OA = (2/3)/∜3. BA squared is (a²/9 + a⁶). Let's compute that:a² = 1/√3a⁶ = (a²)^3 = (1/√3)^3 = 1/(3^(3/2)) = 1/(3√3)So, BA squared is (1/√3)/9 + 1/(3√3) = (1)/(9√3) + (1)/(3√3) = (1 + 3)/9√3 = 4/(9√3)Then BA is sqrt(4/(9√3)) = 2/(3 * (3^(1/4)) )Wait, but OA is (2/3)/∜3. Let's see:(2/3)/∜3 = 2/(3 * 3^(1/4)) = 2/(3^(5/4))BA is 2/(3 * 3^(1/4)) = 2/(3^(5/4)) same as OA. Therefore, OA = BA. So that checks out.Therefore, the calculation seems correct. Therefore, the slope is √3, so the angle is 60°, which is option C.Wait, but let me check if the angle could be 120°. Because sometimes when dealing with arctangent, angles can be in different quadrants. But since the slope is positive √3, the angle is in the first quadrant, between 0° and 90°, right? Because the tangent line at point B on the curve y = x³. If a is positive, point B is in the first quadrant; if a is negative, point B is in the third quadrant. However, the tangent line slope is 3a², which is always positive, regardless of a. So, the slope is positive, meaning the angle is measured from the positive x-axis upwards, so between 0° and 90°, so 60° is correct. 120° would be in the second quadrant, which would correspond to a negative slope, but the slope here is positive. Therefore, the answer must be 60°, option C.Therefore, I think the correct answer is C: 60°.Final Answerboxed{C}</think>Given the curve ( y = x^3 ), we need to find the slope angle of the tangent line at point ( B ) such that the tangent line intersects the x-axis at point ( A ) and forms an isosceles triangle ( triangle OAB ) with ( O ) as the origin and ( A ) as the apex.1. Coordinates and Tangent Line Equation: - Let ( B ) be the point ( (a, a^3) ). - The derivative of ( y = x^3 ) is ( y' = 3x^2 ), so the slope of the tangent line at ( B ) is ( 3a^2 ). - The equation of the tangent line at ( B ) is ( y - a^3 = 3a^2(x - a) ).2. Finding Point ( A ): - The tangent line intersects the x-axis at ( y = 0 ): [ 0 - a^3 = 3a^2(x - a) implies -a^3 = 3a^2x - 3a^3 implies 2a^3 = 3a^2x implies x = frac{2}{3}a ] - Thus, point ( A ) is ( left( frac{2}{3}a, 0 right) ).3. Isosceles Triangle Condition: - For ( triangle OAB ) to be isosceles with ( A ) as the apex, the distances ( OA ) and ( BA ) must be equal. - Calculate ( OA ) and ( BA ): - ( OA = left| frac{2}{3}a right| = frac{2}{3}|a| ) - ( BA = sqrt{left( a - frac{2}{3}a right)^2 + (a^3 - 0)^2} = sqrt{left( frac{1}{3}a right)^2 + a^6} )4. Equating Distances: - Set ( OA = BA ): [ frac{2}{3}|a| = sqrt{left( frac{a^2}{9} + a^6 right)} ] - Squaring both sides: [ left( frac{2}{3}a right)^2 = frac{a^2}{9} + a^6 implies frac{4}{9}a^2 = frac{1}{9}a^2 + a^6 implies frac{3}{9}a^2 = a^6 implies frac{1}{3}a^2 = a^6 implies a^4 = frac{1}{3} ] - Solving for ( a ): [ a = pm frac{1}{sqrt[4]{3}} ]5. Slope Calculation: - The slope of the tangent line at ( B ) is ( 3a^2 ): [ a^2 = frac{1}{sqrt{3}} implies 3a^2 = frac{3}{sqrt{3}} = sqrt{3} ] - The angle corresponding to slope ( sqrt{3} ) is ( arctan(sqrt{3}) = 60^circ ).Thus, the slope angle of the tangent line ( l ) is ( boxed{C} ).,<think>Okay, let me try to solve this problem step by step. So, we have a curve y = x³. There's a point B on this curve, and the tangent line at B intersects the x-axis at point A. The triangle OAB, with O being the origin, is an isosceles triangle with A as the apex. We need to find the slope angle of the tangent line l, and the options are 30°, 45°, 60°, or 120°.First, let me visualize this. The curve y = x³ is a cubic curve that passes through the origin and has an inflection point there. It's symmetric about the origin. If I take a point B on this curve, the tangent at B will have a certain slope, and where it intersects the x-axis is point A. Then, triangle OAB is formed with OA, AB, and OB as sides. Since it's isosceles with A as the apex, that means the two equal sides are OA and AB. Wait, apex at A means that the two equal sides are the ones meeting at A, right? So, in triangle OAB, vertex A is the apex, so the two equal sides must be AB and AO. Wait, but AO is from A to O, and AB is from A to B. So OA and AB must be equal in length.Wait, maybe I should confirm that. If A is the apex of the isosceles triangle, then the two legs meeting at A should be equal. So that would be sides OA and AB. So OA = AB. Alternatively, maybe it's AO and AB? Wait, OA and AB are sides, but AO is the same as OA, just in the opposite direction. So, the triangle has vertices O, A, B. If A is the apex, then the two equal sides are OA and AB. So OA = AB. So the lengths OA and AB must be equal.Okay, so that's the condition we need. So, we need to find the point B on the curve y = x³ such that the tangent at B intersects the x-axis at A, and OA = AB.Let me start by parametrizing point B. Let's let B be a point (a, a³) on the curve y = x³. Then, the derivative of y = x³ is y' = 3x², so the slope of the tangent line at B is 3a².Therefore, the equation of the tangent line at B is y - a³ = 3a²(x - a). To find where this tangent line intersects the x-axis, we set y = 0 and solve for x.So, 0 - a³ = 3a²(x - a)=> -a³ = 3a²x - 3a³Bring all terms to left side:- a³ - 3a²x + 3a³ = 0=> 2a³ - 3a²x = 0Wait, maybe better to solve step by step.Starting from:- a³ = 3a²(x - a)Divide both sides by a² (assuming a ≠ 0, which it can't be because if a=0, the tangent line is horizontal, and the triangle would collapse)So, -a = 3(x - a)=> -a = 3x - 3aBring 3a to the left:- a + 3a = 3x=> 2a = 3x=> x = (2a)/3Therefore, point A is at ((2a)/3, 0).So, coordinates:- O is (0,0)- A is (2a/3, 0)- B is (a, a³)Now, we need to compute OA and AB and set them equal.First, OA is the distance from O to A, which is simply the x-coordinate of A, since it's on the x-axis. So OA = |2a/3 - 0| = |2a/3|.AB is the distance from A to B. Let's compute that. Coordinates of A: (2a/3, 0), coordinates of B: (a, a³). The distance AB is sqrt[(a - 2a/3)^2 + (a³ - 0)^2] = sqrt[(a/3)^2 + (a³)^2] = sqrt[a²/9 + a^6].So, OA = |2a/3|, AB = sqrt(a²/9 + a^6). Since OA and AB must be equal, set them equal:|2a/3| = sqrt(a²/9 + a^6)Assuming a ≠ 0, we can square both sides to eliminate the square root:(2a/3)^2 = (a²/9 + a^6)=> 4a²/9 = a²/9 + a^6Subtract a²/9 from both sides:4a²/9 - a²/9 = a^6=> 3a²/9 = a^6Simplify:a²/3 = a^6Divide both sides by a² (again, a ≠ 0):1/3 = a^4So, a^4 = 1/3Therefore, a = ±(1/3)^{1/4} = ±(3^{-1/4}) = ±3^{-1/4}But let's compute 3^{-1/4} is 1 over 3^{1/4}, which is the fourth root of 3. Alternatively, a^4 = 1/3, so a = ±(1/3)^{1/4}But maybe we can write that as 3^{-1/4} or 3^{ -1/4 }, which is equivalent.But perhaps we can leave it as is for now.But now, we need to find the slope of the tangent line at B, which is 3a². Since the slope is 3a², and we need to find the angle that this slope makes with the x-axis. The angle θ satisfies tanθ = 3a².So, let's compute tanθ = 3a². Since we found that a^4 = 1/3, then a² = sqrt(1/3) = 1/√3. Wait, but a^4 = 1/3, so (a²)^2 = 1/3, so a² = ±sqrt(1/3). But since a² is a square, it must be non-negative, so a² = sqrt(1/3) = 1/√3.Therefore, tanθ = 3*(1/√3) = 3/√3 = √3.So tanθ = √3, which implies θ = 60°, since tan60° = √3.Therefore, the slope angle is 60°, which is option C.Wait, let me check this again. Because a^4 = 1/3, so a² = sqrt(1/3). Let's verify:If a^4 = 1/3, then take square roots: a² = ±sqrt(1/3). But since a² is positive, we have a² = sqrt(1/3) = 1/3^{1/2} = 3^{-1/2}. Therefore, a² = 3^{-1/2}.Therefore, 3a² = 3*(3^{-1/2}) = 3^{1 - 1/2} = 3^{1/2} = sqrt(3). So tanθ = sqrt(3). Which indeed corresponds to 60°, since tan(60°) = sqrt(3). Therefore, θ = 60°, so the answer is C.But let me double-check all steps because sometimes there could be a mistake.Starting again:1. Point B is (a, a³).2. The derivative at B is 3a², so slope is 3a².3. Equation of tangent: y - a³ = 3a²(x - a).4. Find intersection with x-axis (y=0):0 - a³ = 3a²(x - a)- a³ = 3a²x - 3a³Bring -3a³ to left:- a³ + 3a³ = 3a²x2a³ = 3a²xDivide both sides by 3a²:x = (2a³)/(3a²) = (2a)/3So point A is (2a/3, 0). Correct.Then OA is the distance from O to A: |2a/3|.AB is distance from A to B: sqrt[(a - 2a/3)^2 + (a³ - 0)^2] = sqrt[(a/3)^2 + a^6]. Correct.Set OA = AB:|2a/3| = sqrt( (a/3)^2 + a^6 )Square both sides:(4a²)/9 = (a²)/9 + a^6Subtract (a²)/9:3a²/9 = a^6 => a²/3 = a^6 => 1/3 = a^4Thus, a^4 = 1/3 => a² = 1/sqrt(3) = 3^{-1/2}Therefore, slope = 3a² = 3*(3^{-1/2}) = 3^{1 - 1/2} = 3^{1/2} = sqrt(3). So tanθ = sqrt(3), θ = 60°, correct.Therefore, the answer is C: 60°.Alternatively, just to check if there's another possible solution. For example, maybe the triangle could have OA and OB equal? But the problem states that A is the apex, so the two equal sides are at A, which would be OA and AB. If it were OA and OB, then the apex would be O, but the problem says A is the apex. So no, that's not the case.Alternatively, maybe BA and BO? But again, apex at A would mean sides at A are equal, so OA and AB.Therefore, conclusion is 60°, option C. Seems correct.Final Answerboxed{C}</think>Given the curve ( y = x^3 ), we need to find the slope angle of the tangent line at point ( B ) such that the tangent line intersects the x-axis at point ( A ), forming an isosceles triangle ( triangle OAB ) with ( O ) as the origin and ( A ) as the apex.1. Coordinates of Points: - Let ( B ) be a point on the curve ( y = x^3 ), so ( B ) has coordinates ( (a, a^3) ). - The derivative of ( y = x^3 ) is ( y' = 3x^2 ), so the slope of the tangent line at ( B ) is ( 3a^2 ).2. Equation of the Tangent Line: - The equation of the tangent line at ( B ) is ( y - a^3 = 3a^2(x - a) ).3. Intersection with the x-axis: - Setting ( y = 0 ) to find the x-intercept ( A ): [ 0 - a^3 = 3a^2(x - a) implies -a^3 = 3a^2x - 3a^3 implies 2a^3 = 3a^2x implies x = frac{2a}{3} ] - Thus, point ( A ) is ( left( frac{2a}{3}, 0 right) ).4. Distances: - Distance ( OA ) is ( left| frac{2a}{3} right| ). - Distance ( AB ) is ( sqrt{left( a - frac{2a}{3} right)^2 + (a^3 - 0)^2} = sqrt{left( frac{a}{3} right)^2 + a^6} ).5. Isosceles Triangle Condition: - Setting ( OA = AB ): [ left| frac{2a}{3} right| = sqrt{left( frac{a}{3} right)^2 + a^6} ] - Squaring both sides: [ left( frac{4a^2}{9} right) = left( frac{a^2}{9} + a^6 right) implies frac{4a^2}{9} = frac{a^2}{9} + a^6 implies frac{3a^2}{9} = a^6 implies frac{a^2}{3} = a^6 implies frac{1}{3} = a^4 ] - Solving for ( a^2 ): [ a^4 = frac{1}{3} implies a^2 = frac{1}{sqrt{3}} ]6. Slope Angle Calculation: - The slope of the tangent line is ( 3a^2 = 3 cdot frac{1}{sqrt{3}} = sqrt{3} ). - The angle ( theta ) with ( tan theta = sqrt{3} ) is ( 60^circ ).Thus, the slope angle of the tangent line is (boxed{C}).]

🔑:AnalysisThis problem mainly examines the application of derivatives, and being familiar with the geometric meaning of derivatives is key to solving this problem. It is a common type of question in college entrance examinations and is considered a medium-level question.SolutionFirst, differentiate the curve y=x^{3} to get y'=3x^{2}.Let the tangent point be B(x_{0},x_{0}^{3}). Then, the slope of the tangent line at point B is 3x_{0}^{2}.Therefore, the equation of the tangent line l is y-x_{0}^{3}=3x_{0}^{2}(x-x_{0}). Setting y=0, we find Aleft( frac{2}{3}x_{0},0right).Since |OA|=|AB|, we have |frac{2}{3}x_{0}|= sqrt{left( frac{x_{0}}{3}right)^{2}+(x_{0}^{3})^{2}}. Solving this equation yields x_{0}^{4}= frac{1}{3}.Therefore, the slope of the tangent line l is 3x_{0}^{2}= sqrt{3}. Hence, the slope angle of the tangent line l is 60^{circ}.Thus, the correct choice is boxed{C}.

🧠:[<think>Okay, let me try to figure out how to solve this vector problem. So, we have two vectors here: vector a is (2, 3) and vector b is (-1, 2). The question is asking for the magnitude of vector a minus twice vector b. Hmm, alright. Let me break this down step by step.First, I need to recall how vector subtraction and scalar multiplication work. If I remember correctly, when you subtract two vectors, you subtract their corresponding components. Similarly, when you multiply a vector by a scalar, you multiply each component by that scalar. So, the expression a - 2b means I should first compute 2 times vector b and then subtract that result from vector a.Let me write that down. Vector a is (2, 3), and vector b is (-1, 2). So, 2 times vector b would be 2 multiplied by each component of b. That would be 2*(-1) for the x-component and 2*2 for the y-component. Let me calculate that:2*(-1) is -2, and 2*2 is 4. So, 2b is (-2, 4). Got that part.Now, I need to subtract this result from vector a. Vector a is (2, 3). So, subtracting 2b which is (-2, 4) from a would be:(2 - (-2), 3 - 4). Wait, is that right? Let me double-check. Subtraction of vectors is component-wise, so the x-component of the result is the x-component of a minus the x-component of 2b, and similarly for the y-component.So, x-component: 2 (from a) minus (-2) (from 2b). Subtracting a negative is like adding the positive, so 2 + 2 = 4. Y-component: 3 (from a) minus 4 (from 2b). That would be 3 - 4 = -1. So, the resulting vector after a - 2b is (4, -1). Alright, that seems straightforward.Now, the next step is to find the magnitude of this resulting vector. The magnitude of a vector (x, y) is calculated using the formula sqrt(x² + y²). So, let me apply that to our vector (4, -1).First, square the x-component: 4 squared is 16. Then, square the y-component: (-1) squared is 1. Adding those together gives 16 + 1 = 17. Taking the square root of 17 gives the magnitude. Wait, but sqrt(17) is an irrational number. Let me confirm that I did all steps correctly. Starting from the beginning:Original vectors:a = (2, 3)b = (-1, 2)Compute 2b:2 * (-1) = -22 * 2 = 4So, 2b = (-2, 4)Subtract 2b from a:(2 - (-2), 3 - 4) = (2 + 2, -1) = (4, -1)Magnitude of (4, -1):sqrt(4² + (-1)²) = sqrt(16 + 1) = sqrt(17)Yes, that seems correct. But maybe I should check if there was a miscalculation in the components. Let me verify each step again.First component of 2b: -1 * 2 = -2. Second component: 2 * 2 = 4. Correct.Subtraction: For the x-component, 2 - (-2). Subtracting a negative is adding positive. 2 + 2 = 4. Y-component: 3 - 4 = -1. Correct.Then magnitude squared: 4^2 is 16, (-1)^2 is 1. 16 + 1 = 17. Square root of 17. Since 17 is a prime number, it doesn't have a square factor, so sqrt(17) is as simplified as it gets.Therefore, the answer should be sqrt(17). Let me check if there's another way this might be represented, but I don't think so. Alternatively, maybe they want a numerical approximation? But since the problem is presented with exact vectors and asks for the magnitude, I think sqrt(17) is the exact answer they're looking for.Alternatively, could I have made a mistake in the arithmetic? Let's see:2 - (-2) = 2 + 2 = 4. Correct.3 - 4 = -1. Correct.Then 4 squared is 16, -1 squared is 1. 16 + 1 is 17. Correct.So, sqrt(17) is definitely the magnitude. I think that's the answer. I don't see any mistakes here. Wait, let me recall the formula for magnitude once again. Yes, sqrt(x² + y²). So that's right.Alternatively, maybe I miscalculated 2b? Let's see: -1 times 2 is -2, 2 times 2 is 4. So, 2b is (-2, 4). Then a - 2b is (2 - (-2), 3 - 4) = (4, -1). Yep, that's correct.Another check: suppose I had added instead of subtracted. But no, the problem says a - 2b, so subtraction is the right operation.Alternatively, perhaps I confused the order of subtraction? If it's a - 2b, then it's vector a minus vector 2b, so components of a minus components of 2b. Which is exactly what I did.So, all steps seem correct. Therefore, the magnitude is sqrt(17). Since the problem is asking for the answer in boxed notation, I should present that as boxed{sqrt{17}}.Final Answerboxed{sqrt{17}}</think>Given vectors (overrightarrow{a} = (2, 3)) and (overrightarrow{b} = (-1, 2)), we need to find the magnitude of (overrightarrow{a} - 2overrightarrow{b}).First, compute (2overrightarrow{b}):[2overrightarrow{b} = 2 times (-1, 2) = (-2, 4)]Next, subtract (2overrightarrow{b}) from (overrightarrow{a}):[overrightarrow{a} - 2overrightarrow{b} = (2, 3) - (-2, 4) = (2 + 2, 3 - 4) = (4, -1)]Now, calculate the magnitude of the resulting vector ((4, -1)):[|overrightarrow{a} - 2overrightarrow{b}| = sqrt{4^2 + (-1)^2} = sqrt{16 + 1} = sqrt{17}]Thus, the final answer is (boxed{sqrt{17}}).,<think>Alright, let me try to figure out this vector problem. Okay, so we have vector a which is (2, 3) and vector b which is (-1, 2). The question is asking for the magnitude of vector a minus two times vector b. Hmm, right. So first, I need to remember how to subtract vectors and how to multiply a vector by a scalar. Then, after finding the resulting vector, I have to compute its magnitude. Let me break this down step by step.First, let's recall that when you multiply a vector by a scalar, you just multiply each component of the vector by that scalar. So, if I have 2 times vector b, which is (-1, 2), then 2b should be (2*(-1), 2*2) = (-2, 4). Let me check that again. Yes, multiplying each component by 2 gives (-2, 4). Okay, that seems right.Next, we need to subtract this result from vector a. Vector a is (2, 3) and we're subtracting 2b which is (-2, 4). So, subtracting vectors component-wise, we have (2 - (-2), 3 - 4). Wait, hold on. Let me make sure I'm doing this correctly. The operation is vector a minus 2 vector b, so each component of a minus twice the corresponding component of b. So, maybe I should re-express this.Alternatively, vector a is (2, 3) and 2 vector b is (-2, 4). So, subtracting 2b from a would be (2 - (-2), 3 - 4). Wait, 2 - (-2) is 2 + 2, which is 4, and 3 - 4 is -1. So the resulting vector is (4, -1). Is that correct? Let me verify.Original vectors: a = (2, 3); b = (-1, 2). 2b = 2*(-1, 2) = (-2, 4). Then, a - 2b = (2 - (-2), 3 - 4) = (2 + 2, 3 - 4) = (4, -1). Yeah, that seems right. So the resulting vector after the subtraction is (4, -1). Now, we need to find the magnitude of this vector.The magnitude of a vector (x, y) is given by the square root of (x squared plus y squared). So, for vector (4, -1), the magnitude should be sqrt(4² + (-1)²). Calculating that, 4 squared is 16, and (-1) squared is 1. So, 16 + 1 is 17. Therefore, the magnitude is sqrt(17). Let me just confirm each step again.First, scalar multiplication: 2 times vector b. Original components: -1 and 2. Multiply each by 2: -2 and 4. Correct. Then, subtract that from vector a. Vector a's components are 2 and 3. So, 2 - (-2) is 2 + 2 = 4. Then, 3 - 4 is -1. So, the resulting vector is (4, -1). Then, magnitude: sqrt(4² + (-1)²) = sqrt(16 + 1) = sqrt(17). That seems right.Hmm, sqrt(17) is an irrational number, approximately 4.1231. But since the question is asking for the exact value, we can leave it as sqrt(17). Let me check if there was a miscalculation in the components. Let me go through each step once more:1. Calculate 2b: - The original components of b are (-1, 2). - Multiply each component by 2: (-1*2, 2*2) = (-2, 4). Correct.2. Subtract 2b from a: - a is (2, 3), 2b is (-2, 4). - Subtract component-wise: (2 - (-2), 3 - 4) = (2 + 2, -1) = (4, -1). Correct.3. Find the magnitude of (4, -1): - Use the formula sqrt(x² + y²) = sqrt(4² + (-1)²) = sqrt(16 + 1) = sqrt(17). Correct.All steps check out. So, the answer should be sqrt(17). I don't see any mistakes here. Let me just make sure that the operations on vectors are indeed component-wise. Yes, when you add or subtract vectors, it's done component-wise, and scalar multiplication is also component-wise. The magnitude formula is standard as well. So, all the steps are following the correct vector operations. Therefore, the final answer should be the square root of 17.Final Answerboxed{sqrt{17}}</think>Given vectors (overrightarrow{a} = (2, 3)) and (overrightarrow{b} = (-1, 2)), we need to find the magnitude of (overrightarrow{a} - 2overrightarrow{b}).1. Scalar Multiplication: [ 2overrightarrow{b} = 2 times (-1, 2) = (-2, 4) ]2. Vector Subtraction: [ overrightarrow{a} - 2overrightarrow{b} = (2, 3) - (-2, 4) = (2 - (-2), 3 - 4) = (2 + 2, 3 - 4) = (4, -1) ]3. Magnitude Calculation: [ |overrightarrow{a} - 2overrightarrow{b}| = |(4, -1)| = sqrt{4^2 + (-1)^2} = sqrt{16 + 1} = sqrt{17} ]Thus, the final answer is (boxed{sqrt{17}}).]

🔑:To tackle the problem step-by-step, we start by understanding that the operation to be performed involves vector subtraction and multiplication by a scalar. Let's break down the solution accordingly:1. Subtracting 2overrightarrow{b} from overrightarrow{a}:Given vectors are overrightarrow{a}=(2,3) and overrightarrow{b}=(-1,2). To find overrightarrow{a}-2overrightarrow{b}, we first need to find 2overrightarrow{b} and then subtract it from overrightarrow{a}.- First, calculate 2overrightarrow{b}:2overrightarrow{b} = 2 cdot (-1,2) = (-2,4)- Next, subtract 2overrightarrow{b} from overrightarrow{a}:overrightarrow{a}-2overrightarrow{b} = (2,3) - (-2,4) = (2 + 2, 3 - 4) = (4, -1)2. Calculating the magnitude of overrightarrow{a}-2overrightarrow{b}:Now, with overrightarrow{a}-2overrightarrow{b} = (4, -1), the magnitude of this vector is found using the formula for the magnitude of a vector in two dimensions, sqrt{x^2 + y^2}, where (x, y) are the components of the vector.- Calculating the magnitude:|overrightarrow{a}-2overrightarrow{b}| = sqrt{4^2 + (-1)^2} = sqrt{16 + 1} = sqrt{17}Therefore, the magnitude of overrightarrow{a}-2overrightarrow{b} is sqrt{17}. Encapsulating the final answer as specified:boxed{sqrt{17}}

🧠:[<think>Okay, so I need to solve this system of equations:1. Cube root of (2 - x) plus cube root of (7 - y) equals -1.2. Square root of (x + y) equals 4.Hmm, let's start with the second equation because it looks simpler. The square root of (x + y) is 4. If I square both sides, that should get rid of the square root. Let me do that:√(x + y) = 4 Squaring both sides: (√(x + y))² = 4² x + y = 16So, that simplifies things. Now I know that x + y = 16. Maybe I can express one variable in terms of the other. Let me solve for y:y = 16 - xOkay, so now I can substitute y = 16 - x into the first equation. Let me rewrite the first equation with this substitution:Cube root of (2 - x) + Cube root of (7 - (16 - x)) = -1Simplify the expression inside the second cube root:7 - (16 - x) = 7 - 16 + x = -9 + xSo now the equation becomes:Cube root of (2 - x) + Cube root of (x - 9) = -1Hmm, that seems a bit complicated. Let me denote a = Cube root of (2 - x) and b = Cube root of (x - 9). Then the equation is a + b = -1.If I can find another equation involving a and b, maybe I can solve for them. Since a and b are cube roots, perhaps I can cube both sides of the equation a + b = -1. Let's try that.(a + b)³ = (-1)³ a³ + b³ + 3ab(a + b) = -1But I know what a³ and b³ are from their definitions:a³ = 2 - x b³ = x - 9So substituting those into the equation:(2 - x) + (x - 9) + 3ab(a + b) = -1Simplify the left side:2 - x + x - 9 + 3ab(-1) = -1 (2 - 9) + (-x + x) + (-3ab) = -1 -7 + 0 - 3ab = -1 -7 - 3ab = -1Now, solve for ab:-3ab = -1 + 7 -3ab = 6 ab = 6 / (-3) ab = -2So we have ab = -2. But remember, a = Cube root of (2 - x) and b = Cube root of (x - 9). So:[Cube root of (2 - x)] * [Cube root of (x - 9)] = -2I can write the product of cube roots as the cube root of the product:Cube root of [(2 - x)(x - 9)] = -2Then cube both sides to eliminate the cube root:[(2 - x)(x - 9)] = (-2)³ (2 - x)(x - 9) = -8Let me expand the left side:(2)(x - 9) - x(x - 9) = 2x - 18 - x² + 9x = (2x + 9x) - x² - 18 = 11x - x² - 18So:11x - x² - 18 = -8Rearrange terms:- x² + 11x - 18 + 8 = 0 - x² + 11x - 10 = 0Multiply both sides by -1 to make it a standard quadratic:x² - 11x + 10 = 0Now, factor the quadratic:Looking for two numbers that multiply to 10 and add to -11. Hmm, factors of 10 are 1 and 10, 2 and 5. Since the middle term is -11x, the numbers should be -10 and -1.So:(x - 10)(x - 1) = 0Therefore, x = 10 or x = 1Now, we need to find the corresponding y values using y = 16 - x.If x = 10:y = 16 - 10 = 6If x = 1:y = 16 - 1 = 15So the possible solutions are (10, 6) and (1, 15).But wait, I should check these solutions in the original equations to make sure they work, especially because sometimes when dealing with cube roots and squaring, extraneous solutions can pop up.First, check (10, 6):First equation: Cube root of (2 - 10) + Cube root of (7 - 6) = Cube root of (-8) + Cube root of (1) Cube root of -8 is -2, and cube root of 1 is 1. So -2 + 1 = -1, which matches the first equation.Second equation: √(10 + 6) = √16 = 4, which matches the second equation. So (10, 6) is valid.Now check (1, 15):First equation: Cube root of (2 - 1) + Cube root of (7 - 15) = Cube root of (1) + Cube root of (-8) Cube root of 1 is 1, cube root of -8 is -2. So 1 + (-2) = -1, which matches the first equation.Second equation: √(1 + 15) = √16 = 4, which is correct. So (1, 15) is also valid.Therefore, both solutions are valid. So the ordered pairs are (10, 6) and (1, 15).Final AnswerThe solutions are boxed{(1, 15)} and boxed{(10, 6)}.</think>We start with the system of equations:[left{begin{array}{l}sqrt[3]{2 - x} + sqrt[3]{7 - y} = -1 sqrt{x + y} = 4end{array}right.]First, we solve the second equation for (x + y):[sqrt{x + y} = 4 implies x + y = 16]Express (y) in terms of (x):[y = 16 - x]Substitute (y = 16 - x) into the first equation:[sqrt[3]{2 - x} + sqrt[3]{7 - (16 - x)} = -1]Simplify the second cube root:[sqrt[3]{2 - x} + sqrt[3]{x - 9} = -1]Let (a = sqrt[3]{2 - x}) and (b = sqrt[3]{x - 9}). Then (a + b = -1). Cubing both sides:[(a + b)^3 = (-1)^3 implies a^3 + b^3 + 3ab(a + b) = -1]Substitute (a^3 = 2 - x) and (b^3 = x - 9):[(2 - x) + (x - 9) + 3ab(-1) = -1]Simplify:[-7 - 3ab = -1 implies -3ab = 6 implies ab = -2]Thus,[sqrt[3]{(2 - x)(x - 9)} = -2]Cube both sides:[(2 - x)(x - 9) = -8]Expand and simplify:[11x - x^2 - 18 = -8 implies x^2 - 11x + 10 = 0]Factor the quadratic equation:[(x - 10)(x - 1) = 0 implies x = 10 text{ or } x = 1]Find corresponding (y) values:- If (x = 10), then (y = 16 - 10 = 6)- If (x = 1), then (y = 16 - 1 = 15)Check solutions in the original equations:- For ((10, 6)): - First equation: (sqrt[3]{-8} + sqrt[3]{1} = -2 + 1 = -1) - Second equation: (sqrt{16} = 4)- For ((1, 15)): - First equation: (sqrt[3]{1} + sqrt[3]{-8} = 1 - 2 = -1) - Second equation: (sqrt{16} = 4)Both solutions are valid. The ordered pairs are:[boxed{(1, 15)} text{ and } boxed{(10, 6)}],<think>Okay, let me try to figure out how to solve this system of equations. The equations are:1. Cube root of (2 - x) plus cube root of (7 - y) equals -1.2. Square root of (x + y) equals 4.Hmm. So the first equation is with cube roots and the second one is a square root. Let me start by dealing with the second equation because square roots might be simpler. If the square root of (x + y) is 4, then if I square both sides, I can get rid of the square root. Let's do that first.So, sqrt(x + y) = 4. Squaring both sides gives x + y = 16. That simplifies things a bit. So, we know that x + y = 16. That means we can express one variable in terms of the other. Maybe I can solve for y in terms of x, then substitute into the first equation. Let's see.From the second equation: y = 16 - x. Then we can substitute this into the first equation. The first equation is cube root of (2 - x) plus cube root of (7 - y) equals -1. Let's substitute y with 16 - x.So, substituting y: cube root(2 - x) + cube root(7 - (16 - x)) = -1. Let me simplify the second cube root's argument.7 - (16 - x) = 7 - 16 + x = x - 9. So the equation becomes:cube_root(2 - x) + cube_root(x - 9) = -1.Okay, so now we have an equation with cube roots of expressions involving x. Let's denote:Let’s let a = cube_root(2 - x) and b = cube_root(x - 9). Then the equation becomes a + b = -1.If we can find another relation between a and b, maybe we can solve for them. Remember that for cube roots, if we cube both sides, we can relate back to the original variables. Let's recall that a^3 = 2 - x and b^3 = x - 9.So, we have:a + b = -1,anda^3 + b^3 = (2 - x) + (x - 9) = 2 - x + x - 9 = -7.But also, there's a formula for a^3 + b^3: it's (a + b)(a^2 - ab + b^2). Let's use that.So, a^3 + b^3 = (a + b)(a^2 - ab + b^2) = (-1)(a^2 - ab + b^2) = -7.Therefore, (-1)(a^2 - ab + b^2) = -7, which simplifies to a^2 - ab + b^2 = 7.But we also know that a + b = -1. Let me see if we can express a^2 - ab + b^2 in terms of (a + b)^2. Because (a + b)^2 = a^2 + 2ab + b^2. So, a^2 + b^2 = (a + b)^2 - 2ab. Therefore, substituting into a^2 - ab + b^2, we get:(a^2 + b^2) - ab = [(a + b)^2 - 2ab] - ab = (a + b)^2 - 3ab.So, a^2 - ab + b^2 = (a + b)^2 - 3ab.We know that a + b = -1, so (a + b)^2 = 1. Therefore:1 - 3ab = 7 => -3ab = 6 => ab = -2.So now we have two equations:1. a + b = -12. ab = -2These are the sum and product of a and b, which reminds me of solving quadratic equations. If a and b are roots of the equation t^2 - (a + b)t + ab = 0, which would be t^2 + t - 2 = 0.Let me solve this quadratic equation:t^2 + t - 2 = 0Using quadratic formula: t = [-1 ± sqrt(1 + 8)] / 2 = [-1 ± 3]/2.So, the roots are t = (-1 + 3)/2 = 2/2 = 1, and t = (-1 - 3)/2 = -4/2 = -2.Therefore, a and b are either 1 and -2. But we need to check which one is which.But wait, a and b are defined as cube_root(2 - x) and cube_root(x - 9). Let's see. If a = 1, then cube_root(2 - x) = 1 => 2 - x = 1 => x = 1. Then, since x = 1, from the second equation, y = 16 - x = 15. Then check the first equation: cube_root(2 - 1) + cube_root(7 - 15) = cube_root(1) + cube_root(-8) = 1 + (-2) = -1, which matches. So that works.Alternatively, if a = -2, then cube_root(2 - x) = -2 => 2 - x = (-2)^3 = -8 => x = 2 - (-8) = 10. Then y = 16 - 10 = 6. Check the first equation: cube_root(2 - 10) + cube_root(7 - 6) = cube_root(-8) + cube_root(1) = -2 + 1 = -1, which also works.So both possibilities give valid solutions. Therefore, the solutions are (x, y) = (1, 15) and (10, 6).But wait, we should check if these solutions also satisfy the original equations, just to be sure.First solution: x = 1, y = 15.Check second equation: sqrt(1 + 15) = sqrt(16) = 4, which is correct.First equation: cube_root(2 - 1) + cube_root(7 - 15) = cube_root(1) + cube_root(-8) = 1 + (-2) = -1. Correct.Second solution: x = 10, y = 6.Check second equation: sqrt(10 + 6) = sqrt(16) = 4. Correct.First equation: cube_root(2 - 10) + cube_root(7 - 6) = cube_root(-8) + cube_root(1) = -2 + 1 = -1. Correct.Therefore, both solutions are valid.So the ordered pairs are (1, 15) and (10, 6). Are these the only solutions?Let me think. When we set up the equations, we squared the second equation, but that's okay because square root is non-negative, so we don't get extraneous solutions from there. Then, when we solved for a and b, we found all possible solutions for a and b, which gave us two possible x values. So I think these are the only solutions.Therefore, the solutions are (1, 15) and (10, 6).Final AnswerThe solutions are boxed{(1, 15)} and boxed{(10, 6)}.</think>Given the system of equations:[left{begin{array}{l}sqrt[3]{2 - x} + sqrt[3]{7 - y} = -1 sqrt{x + y} = 4end{array}right.]First, we simplify the second equation by squaring both sides:[sqrt{x + y} = 4 implies x + y = 16]We express ( y ) in terms of ( x ):[y = 16 - x]Substitute ( y = 16 - x ) into the first equation:[sqrt[3]{2 - x} + sqrt[3]{7 - (16 - x)} = -1]Simplify the argument of the second cube root:[7 - (16 - x) = x - 9]Thus, the equation becomes:[sqrt[3]{2 - x} + sqrt[3]{x - 9} = -1]Let ( a = sqrt[3]{2 - x} ) and ( b = sqrt[3]{x - 9} ). Then we have:[a + b = -1]We also know:[a^3 = 2 - x quad text{and} quad b^3 = x - 9]Adding these equations:[a^3 + b^3 = (2 - x) + (x - 9) = -7]Using the identity for the sum of cubes:[a^3 + b^3 = (a + b)(a^2 - ab + b^2)]Substituting ( a + b = -1 ):[-7 = (-1)(a^2 - ab + b^2) implies a^2 - ab + b^2 = 7]Expressing ( a^2 - ab + b^2 ) in terms of ( (a + b)^2 ):[a^2 - ab + b^2 = (a + b)^2 - 3ab]Substituting ( a + b = -1 ):[1 - 3ab = 7 implies -3ab = 6 implies ab = -2]We solve the quadratic equation with roots ( a ) and ( b ):[t^2 + t - 2 = 0]The roots are:[t = 1 quad text{and} quad t = -2]Thus, ( a ) and ( b ) can be 1 and -2 or -2 and 1.1. If ( a = 1 ): [ sqrt[3]{2 - x} = 1 implies 2 - x = 1 implies x = 1 ] [ y = 16 - 1 = 15 ]2. If ( a = -2 ): [ sqrt[3]{2 - x} = -2 implies 2 - x = -8 implies x = 10 ] [ y = 16 - 10 = 6 ]Both solutions satisfy the original equations. Therefore, the solutions are:[boxed{(1, 15)} quad text{and} quad boxed{(10, 6)}]]

🔑:## First solution.First, let's express the unknown y from the equation sqrt{x+y}=4.sqrt{x+y}=4 /^{2}x+y=16y=16-xBy substituting y=16-x into the above equation, we get the equationsqrt[3]{2-x}+sqrt[3]{x-9}=-1.1 pointsqrt[3]{2-x}+sqrt[3]{x-9}=-1 /{ }^{3}2-x+3 cdot sqrt[3]{(2-x)^{2}(x-9)}+3 cdot sqrt[3]{(2-x)(x-9)^{2}}+x-9=-13 cdot sqrt[3]{(2-x)^{2}(x-9)}+3 cdot sqrt[3]{(2-x)(x-9)^{2}}=6 /: 3sqrt[3]{(2-x)^{2}(x-9)}+sqrt[3]{(2-x)(x-9)^{2}}=2sqrt[3]{(2-x)(x-9)} cdot(sqrt[3]{2-x}+sqrt[3]{x-9})=2Notice that we can substitute the value of the expression sqrt[3]{2-x}+sqrt[3]{x-9}=-1 into the equation.sqrt[3]{(2-x)(x-9)} cdot(-1)=2sqrt[3]{(2-x)(x-9)}=-2 / 3-x^{2}+11 x-18=-8x^{2}-11 x+10=01 point(x-1)(x-10)=0The solutions to the equation are the numbers x_{1}=1 and x_{2}=10.1 pointLet's determine the corresponding values of the unknown y.y_{1}=16-1=15y_{2}=16-10=6The solutions to the system of equations are the ordered pairs (1,15) and (10,6).

🧠:[<think>Okay, let's see. The problem is about price increase and adjusting consumption to stay within a certain spending limit. Hmm. So the price goes up by 40%, and we can only spend 12% more money. We need to find out by what percentage we need to reduce our consumption. Alright, let me break this down.First, let me recall some basic concepts. If the price of a product increases, and your budget (the total amount you can spend) increases by a smaller percentage, you'll have to buy less of the product. The key here is understanding how price and quantity relate to total expenditure. Total expenditure is price multiplied by quantity, right? So if the price goes up, to keep the total expenditure the same or within a certain increase, the quantity purchased must adjust accordingly.Let me formalize this. Let's denote the original price as P and the original quantity consumed as Q. So initially, total expenditure is P * Q. Now, the price increases by 40%, so the new price is P + 40% of P, which is P * 1.4. Let's call the new price P_new = 1.4P.Our spending can only increase by 12%, so the new total expenditure is the original expenditure plus 12% of it. That would be P * Q * 1.12. Let's denote the new quantity consumed as Q_new. So the new total expenditure is also equal to P_new * Q_new.So, setting those equal:1.4P * Q_new = 1.12P * QWe can cancel out the original price P from both sides, which gives:1.4 * Q_new = 1.12 * QNow, solving for Q_new:Q_new = (1.12 / 1.4) * QLet me compute that division. 1.12 divided by 1.4. Hmm, 1.12 ÷ 1.4. Let's see. 1.4 goes into 1.12 how many times? 1.4 times 0.8 is 1.12, right? Because 1.4 * 0.8 = 1.12. So, Q_new = 0.8 * Q.So, the new quantity is 0.8 times the original quantity. That means the consumption has been reduced by 20%. Because 1 - 0.8 = 0.2, which is 20%.Wait, let me verify that again. If the original quantity is Q, and the new quantity is 0.8Q, then the reduction is Q - 0.8Q = 0.2Q, which is a 20% decrease. Yes, that seems right.So, in conclusion, if the price increases by 40% and we can only spend 12% more, we need to reduce our consumption by 20%.But wait, let me check this with an example to be sure. Let's assign some numbers. Suppose the original price is 100, and the original quantity is 10 units. Then original expenditure is 100 * 10 = 1000.After a 40% price increase, the new price is 140. The allowed expenditure is 12% more, so 1000 + 12% of 1000 = 1120.So, with the new price of 140, how many units can we buy with 1120? That's 1120 / 140 = 8 units. Original quantity was 10, now it's 8. So the reduction is 2 units, which is (2 / 10) * 100 = 20%. Yep, that checks out.Alternatively, if I take another example. Let's say original price is 50, original quantity is 20 units. Original expenditure is 50*20 = 1000. New price is 50*1.4 = 70. Allowed expenditure is 1000*1.12 = 1120. New quantity is 1120 / 70 = 16 units. Original quantity was 20, new is 16. Reduction is 4 units, which is 20% of 20. So again, 20%.Therefore, the answer should be 20% reduction.But just to make sure there's no step I missed. Let's see:Original expenditure: P*Q.After price increase: 1.4P.Desired expenditure: 1.12P*Q.Therefore, 1.4P * Q_new = 1.12P*Q.Cancel P: 1.4 Q_new = 1.12 Q.Q_new = 1.12 / 1.4 Q = 0.8 Q. So 80% of original Q, so 20% reduction.Yep, all steps make sense. So the percentage needed is 20%.Final AnswerThe consumption needs to be reduced by boxed{20%}.</think>The price of a product has increased by 40%, and we can only spend 12% more money on its purchase. To find the required percentage reduction in consumption, we start by noting the relationship between price, quantity, and total expenditure.Let ( P ) be the original price and ( Q ) be the original quantity consumed. The original total expenditure is ( P times Q ). After a 40% price increase, the new price is ( 1.4P ). The new total expenditure, which is 12% more than the original, is ( 1.12P times Q ).Setting the new total expenditure equal to the product of the new price and the new quantity consumed ( Q_{text{new}} ):[1.4P times Q_{text{new}} = 1.12P times Q]Canceling ( P ) from both sides:[1.4 times Q_{text{new}} = 1.12 times Q]Solving for ( Q_{text{new}} ):[Q_{text{new}} = frac{1.12}{1.4} times Q = 0.8 times Q]This shows that the new quantity consumed is 80% of the original quantity, indicating a 20% reduction in consumption. Verifying with examples confirms this result.Thus, the percentage reduction needed in consumption is (boxed{20%}).,<think>Okay, let's see. The problem is about the price of a product increasing by 40%, and we need to figure out by what percentage we should reduce our consumption so that we only spend 12% more money on it. Hmm, okay, so the price goes up, and we want to spend only a little more, so we have to use less of the product. I need to calculate the percentage reduction in consumption. First, let me recall the basic formula. The total expenditure is equal to the price multiplied by the quantity consumed. So, if we denote the original price as P and the original quantity consumed as Q, then the original expenditure is P*Q. Now, the price has increased by 40%, so the new price is P + 40% of P, which is 1.4P. Let's call the new quantity we can consume as Q'. The new expenditure would then be 1.4P * Q'. According to the problem, we can only spend 12% more money. That means the new expenditure is 12% more than the original expenditure. So, the original expenditure was P*Q, and 12% more would be P*Q + 12% of P*Q, which is 1.12P*Q. So, setting up the equation: 1.4P * Q' = 1.12P*Q. Hmm, okay, let me write that down step by step. Original expenditure: P * QNew price: 1.4PNew quantity: Q'New expenditure: 1.4P * Q'Desired new expenditure: 1.12P * QSo, 1.4P * Q' = 1.12P * QI can cancel out the P from both sides of the equation since P is not zero. So, dividing both sides by P:1.4 * Q' = 1.12 * QNow, solve for Q':Q' = (1.12 / 1.4) * QLet me calculate that. 1.12 divided by 1.4. Hmm, 1.12 divided by 1.4. Let's see. Since 1.4 is the same as 14/10, so 1.12 divided by 14/10 is 1.12 * 10/14. 1.12 * 10 = 11.211.2 / 14 = 0.8So, Q' = 0.8 * QThat means the new quantity is 0.8 times the original quantity. So, the consumption has been reduced to 80% of the original. Therefore, the percentage reduction is 100% - 80% = 20%.Wait, so that would mean we need to reduce consumption by 20%? Let me check that again. If the original quantity is Q, and the new quantity is 0.8Q, then the reduction is 0.2Q, which is 20% of Q. So yes, a 20% decrease in consumption. Let me verify with actual numbers to make sure. Suppose the original price is 100, and the original quantity is 10 units. Then original expenditure is 100 * 10 = 1000. After a 40% price increase, the new price is 140. If we can only spend 12% more, the new expenditure is 1000 * 1.12 = 1120. So, with the new price of 140, the quantity we can purchase is 1120 / 140 = 8 units. Original quantity was 10 units, new quantity is 8 units. So, the reduction is 2 units. 2/10 = 0.2, which is 20%. Okay, that checks out. So the answer is 20%. Wait, but just to make sure I didn't make any mistakes in the algebra. Let me go back. We had 1.4P * Q' = 1.12P * Q. Then dividing both sides by P gives 1.4 Q' = 1.12 Q. Then Q' = (1.12 / 1.4) Q. Calculating 1.12 divided by 1.4. Well, 1.4 times 0.8 is 1.12. So yes, 1.12 divided by 1.4 is 0.8, so Q' is 0.8 Q. Therefore, 20% reduction. Seems correct. Maybe another example. Original price 50, original quantity 20 units. Original expenditure is 50*20 = 1000. Price increases by 40%: new price is 50*1.4 = 70. Allowed expenditure is 12% more: 1000*1.12 = 1120. New quantity: 1120 / 70 = 16 units. Reduction from 20 to 16 is 4 units. 4/20 = 0.2, which is 20%. Same result. So definitely 20%. Alternatively, thinking in terms of percentages. If the price goes up by 40%, and we want to limit the expenditure increase to 12%, how much less do we need to consume? The expenditure is price times quantity. If price goes up by 40%, to keep expenditure the same, we would need to reduce quantity by a certain percentage. But since we allow expenditure to go up by 12%, the adjustment is a bit less. But the formula approach seems solid. The key idea is that the product of the new price and new quantity equals the desired expenditure. So, percentage change in quantity needed is (desired expenditure / new price) compared to original quantity. Alternatively, we can think in terms of inverse proportionality. If the price increases by 40%, then the quantity we can purchase with the same expenditure would decrease by 1/1.4 ≈ 0.714, which is a 28.57% decrease. But since we have 12% more money, we multiply that 0.714 by 1.12, which gives 0.714 * 1.12 ≈ 0.8, which is 80% of original quantity, so 20% decrease. Yes, that makes sense. So even if you approach it from different angles, you still get the same answer. Therefore, I feel confident that the answer is a 20% reduction in consumption.Final AnswerThe consumption needs to be reduced by boxed{20%}.</think>The price of a product has increased by 40%, and we need to determine the percentage reduction in consumption such that the total expenditure increases by only 12%.1. Original Expenditure: Let the original price be ( P ) and the original quantity consumed be ( Q ). The original expenditure is ( P times Q ).2. New Price: The price increases by 40%, so the new price is ( 1.4P ).3. Desired Expenditure: The new expenditure should be 12% more than the original expenditure, which is ( 1.12 times P times Q ).4. Setting Up the Equation: The new expenditure with the increased price and reduced consumption is ( 1.4P times Q' ), where ( Q' ) is the new quantity consumed. We equate this to the desired expenditure: [ 1.4P times Q' = 1.12P times Q ]5. Solving for ( Q' ): [ Q' = frac{1.12P times Q}{1.4P} = frac{1.12}{1.4} times Q = 0.8 times Q ]6. Percentage Reduction: The new quantity ( Q' ) is 80% of the original quantity ( Q ), indicating a 20% reduction in consumption.Verification with example values confirms the result. Therefore, the consumption needs to be reduced by (boxed{20%}).]

🔑:Originally, the unit price of the product was X Ft, and we spent Y Ft on it, which allowed us to buy frac{Y}{X} units of the product.If the unit price increases by 40 %, making it 1.4 X Ft, and our capital increases by 12 %, making it 1.12 Y Ft, then we can buy frac{1.12 Y}{1.4 X} units of the product. This is the percentage of the originally purchasable quantity:frac{1.12 Y}{1.4 X}:left(frac{Y}{X} cdot frac{1}{100}right)=frac{112}{1.4}=80percent, which means our consumption decreases by 20 %.