answer：Okay, so I have this problem with four statements about planes and lines in space, and I need to figure out how many of them are correct. The options are A:1, B:2, C:3, D:4. Let me go through each statement one by one.Starting with statement (1): "If two planes have three common points, then these two planes coincide." Hmm, I remember that two planes can either be parallel or intersect along a line. If they have three common points, does that mean they have to coincide? Well, if the three points are non-collinear, meaning they don't all lie on the same straight line, then yes, they would determine a unique plane. So if two planes share three non-collinear points, they must be the same plane. But wait, the statement doesn't specify whether the three points are non-collinear. If the three points are collinear, then both planes could still intersect along that line without coinciding. So, since the statement doesn't specify, I think it's not necessarily true. So I'm leaning towards statement (1) being incorrect.Moving on to statement (2): "Two lines can determine a plane." Okay, I recall that if two lines are either intersecting or parallel, they lie on the same plane. But if they are skew lines, which means they are not parallel and do not intersect, then they don't lie on the same plane. So the statement says "can determine a plane," which I think means that sometimes they can, but not always. So does the statement mean that two lines always determine a plane? If that's the case, then it's incorrect because skew lines don't lie on the same plane. But maybe the statement is saying that two lines can determine a plane, meaning it's possible, not necessarily always. Hmm, the wording is a bit ambiguous. Let me check the original statement again: "Two lines can determine a plane." I think it's implying that given two lines, you can determine a plane, which is only true if the lines are coplanar. Since not all two lines are coplanar, this statement isn't always true. So I think statement (2) is incorrect as well.Now, statement (3): "If M ∈ α, M ∈ β, and α ∩ β = l, then M ∈ l." So, point M is in both planes α and β, and the intersection of α and β is the line l. By the definition of the intersection of two planes, any point common to both planes must lie on their line of intersection. So M must be on l. That makes sense. So statement (3) is correct.Lastly, statement (4): "In space, three lines intersecting at the same point are in the same plane." Hmm, so if three lines all meet at a single point, does that mean they must lie on the same plane? I don't think so. For example, imagine three lines meeting at the origin in 3D space, each along the x, y, and z-axes. These three lines all intersect at the origin but are not all in the same plane. So statement (4) is incorrect.So, summarizing my thoughts:- Statement (1): Incorrect, unless the three points are non-collinear.- Statement (2): Incorrect, because two lines might be skew.- Statement (3): Correct.- Statement (4): Incorrect, as three lines can intersect at a point without being coplanar.Therefore, only statement (3) is correct. So the number of correct statements is 1, which corresponds to option A.**Final Answer**boxed{A}

answer：First, I need to determine the work rates of P and Q. Since P can complete the work in 20 days, P's work rate is ( frac{1}{20} ) of the work per day. Similarly, Q's work rate is ( frac{1}{12} ) of the work per day.Next, I'll calculate the amount of work P completes in the initial 4 days. P works alone during this period, so the work done is ( 4 times frac{1}{20} = frac{4}{20} = frac{1}{5} ) of the total work.This means that the remaining work after 4 days is ( 1 - frac{1}{5} = frac{4}{5} ) of the work.When Q joins P, their combined work rate is the sum of their individual rates: ( frac{1}{20} + frac{1}{12} = frac{3}{60} + frac{5}{60} = frac{8}{60} = frac{2}{15} ) of the work per day.To find out how many days they take to complete the remaining ( frac{4}{5} ) of the work together, I'll divide the remaining work by their combined rate: ( frac{4}{5} div frac{2}{15} = frac{4}{5} times frac{15}{2} = 6 ) days.Finally, adding the initial 4 days P worked alone to the 6 days they worked together, the total time taken to complete the work is ( 4 + 6 = 10 ) days.

answer：I need to factorize the expression (27x^9 - 512y^9) completely and then find the sum of all the integer coefficients in its factorized form.First, I notice that both terms are perfect cubes:- (27x^9 = (3x^3)^3)- (512y^9 = (8y^3)^3)So, the expression can be rewritten as:[(3x^3)^3 - (8y^3)^3]This is a difference of cubes, which can be factored using the formula:[a^3 - b^3 = (a - b)(a^2 + ab + b^2)]Applying this formula:[(3x^3 - 8y^3)(3x^3 + 8y^3)]Now, each of these cubic terms can be further factored. Let's start with (3x^3 - 8y^3):[3x^3 - 8y^3 = (x - 2y)(3x^2 + 6xy + 12y^2)]Similarly, factorizing (3x^3 + 8y^3):[3x^3 + 8y^3 = (x + 2y)(3x^2 - 6xy + 12y^2)]Combining all the factors, the complete factorization is:[27x^9 - 512y^9 = (x - 2y)(3x^2 + 6xy + 12y^2)(x + 2y)(3x^2 - 6xy + 12y^2)]Next, I'll list out all the integer coefficients from each factor:- From ((x - 2y)): 1, -2- From ((3x^2 + 6xy + 12y^2)): 3, 6, 12- From ((x + 2y)): 1, 2- From ((3x^2 - 6xy + 12y^2)): 3, -6, 12Adding these coefficients together:[1 + (-2) + 3 + 6 + 12 + 1 + 2 + 3 + (-6) + 12 = 32]So, the sum of all the integer coefficients in the complete factorization is 32.

answer：Okay, so I have this problem here where I need to find 100 times the difference between b and c, given some conditions about arithmetic and geometric progressions. Let me try to unpack this step by step.First, the problem says that 1, b, c is an arithmetic progression. I remember that in an arithmetic progression, the difference between consecutive terms is constant. So, that means the difference between b and 1 should be the same as the difference between c and b. Let me write that down:b - 1 = c - bHmm, if I rearrange this equation, I can solve for c in terms of b. Let me do that:b - 1 = c - b Adding b to both sides: 2b - 1 = cSo, c is equal to 2b minus 1. Got that.Now, the problem also says that 1, c, b is a geometric progression. In a geometric progression, the ratio between consecutive terms is constant. So, the ratio of c to 1 should be the same as the ratio of b to c. Let me write that as an equation:c / 1 = b / cSimplifying that, it becomes:c = b / cWait, that seems a bit confusing. Let me rewrite it:c / 1 = b / c So, cross-multiplying, I get:c^2 = bAh, okay, so b is equal to c squared. That makes more sense.Now, I have two equations:1. c = 2b - 12. b = c^2So, I can substitute the second equation into the first one. Let me do that:c = 2(c^2) - 1So, that's:c = 2c^2 - 1Let me rearrange this equation to bring all terms to one side:2c^2 - c - 1 = 0Now, I have a quadratic equation in terms of c. Let me try to solve this quadratic equation. I can use the quadratic formula, which is:c = [ -b ± sqrt(b^2 - 4ac) ] / (2a)In this equation, a = 2, b = -1, and c = -1. Plugging these into the formula:c = [ -(-1) ± sqrt( (-1)^2 - 4*2*(-1) ) ] / (2*2) Simplify that:c = [ 1 ± sqrt(1 + 8) ] / 4 c = [ 1 ± sqrt(9) ] / 4 sqrt(9) is 3, so:c = [ 1 ± 3 ] / 4So, that gives me two possible solutions for c:1. c = (1 + 3)/4 = 4/4 = 12. c = (1 - 3)/4 = (-2)/4 = -1/2Now, let's consider these two cases.Case 1: c = 1If c = 1, then from equation 2, b = c^2 = 1^2 = 1.But the problem states that b and c are not both equal to 1. So, this solution is invalid. We have to discard this case.Case 2: c = -1/2If c = -1/2, then from equation 2, b = c^2 = (-1/2)^2 = 1/4.So, b = 1/4 and c = -1/2.Now, the problem asks for 100 times (b - c). Let's compute that:b - c = (1/4) - (-1/2) = 1/4 + 1/2To add these, I need a common denominator. 1/2 is equal to 2/4, so:1/4 + 2/4 = 3/4Therefore, 100*(b - c) = 100*(3/4) = 75.So, the final answer is 75.Wait, let me double-check my steps to make sure I didn't make any mistakes.1. Arithmetic progression: 1, b, c. So, b - 1 = c - b, leading to c = 2b - 1. That seems correct.2. Geometric progression: 1, c, b. So, c/1 = b/c, leading to c^2 = b. That also seems correct.3. Substituted c = 2b - 1 into b = c^2, getting 2c^2 - c - 1 = 0. Wait, hold on, let me check that substitution again.Wait, if c = 2b - 1 and b = c^2, then substituting b into the first equation:c = 2*(c^2) - 1, which is 2c^2 - c - 1 = 0. Yes, that's correct.Solving the quadratic: c = [1 ± sqrt(1 + 8)] / 4 = [1 ± 3]/4. So, c = 1 or c = -1/2. Correct.Discarding c = 1 because b and c can't both be 1. So, c = -1/2, then b = ( -1/2 )^2 = 1/4. Correct.Calculating b - c: 1/4 - (-1/2) = 1/4 + 1/2 = 3/4. Multiply by 100: 75. Correct.So, all steps seem solid. I think 75 is the right answer.**Final Answer**boxed{75}

answer：Okay, so I have this problem here: the function f(x) is given as sin(ωx) + sqrt(3)cos(ωx), and it's mentioned that ω is greater than 0. The key piece of information is that the distance between two adjacent zeros of this function is 2. I need to find the sum of f(1) + f(2) + ... + f(9), and the options are A:1, B:-1, C:sqrt(3), D:-sqrt(3).Alright, let me start by understanding what the function looks like. It's a combination of sine and cosine functions with the same argument ωx. I remember that such functions can be rewritten in a single sine or cosine function using the amplitude-phase form. Maybe that would help me analyze the zeros more easily.So, f(x) = sin(ωx) + sqrt(3)cos(ωx). I think I can express this as a single sine function with some phase shift. The general formula for combining sine and cosine is A sin(ωx + φ), where A is the amplitude and φ is the phase shift. Let me recall how to find A and φ.The amplitude A is given by sqrt(a² + b²), where a and b are the coefficients of sin and cos respectively. In this case, a is 1 and b is sqrt(3). So, A = sqrt(1² + (sqrt(3))²) = sqrt(1 + 3) = sqrt(4) = 2. Okay, so the amplitude is 2.Now, the phase shift φ can be found using tanφ = b/a. Here, tanφ = sqrt(3)/1 = sqrt(3). So, φ is arctangent of sqrt(3). I remember that tan(π/3) = sqrt(3), so φ = π/3. Therefore, the function can be rewritten as f(x) = 2 sin(ωx + π/3).Alright, so f(x) = 2 sin(ωx + π/3). Now, the problem states that the distance between two adjacent zeros is 2. I need to find ω based on this information.First, let's recall that the zeros of a sine function occur where the argument is an integer multiple of π. So, for f(x) = 2 sin(ωx + π/3), the zeros occur when ωx + π/3 = nπ, where n is an integer.Let me solve for x: ωx + π/3 = nπ => ωx = nπ - π/3 => x = (nπ - π/3)/ω.So, the zeros are at x = (nπ - π/3)/ω for integer n. The distance between two adjacent zeros would be the difference between x for n and x for n+1.Calculating that: x_{n+1} - x_n = [( (n+1)π - π/3 ) / ω] - [ (nπ - π/3 ) / ω ] = ( (n+1)π - π/3 - nπ + π/3 ) / ω = (π)/ω.So, the distance between two adjacent zeros is π/ω. The problem states that this distance is 2. Therefore, π/ω = 2 => ω = π/2.Okay, so ω is π/2. Now, substituting back into f(x), we have f(x) = 2 sin( (π/2)x + π/3 ).Now, I need to compute the sum f(1) + f(2) + ... + f(9). Let's compute each term one by one.First, let's note that f(x) = 2 sin( (π/2)x + π/3 ). Let's compute f(1):f(1) = 2 sin( (π/2)(1) + π/3 ) = 2 sin( π/2 + π/3 ). Let's compute the angle: π/2 is 90 degrees, π/3 is 60 degrees, so together it's 150 degrees, which is 5π/6 radians.sin(5π/6) is 1/2. So, f(1) = 2*(1/2) = 1.Next, f(2):f(2) = 2 sin( (π/2)(2) + π/3 ) = 2 sin( π + π/3 ) = 2 sin(4π/3). Sin(4π/3) is -sqrt(3)/2. So, f(2) = 2*(-sqrt(3)/2) = -sqrt(3).f(3):f(3) = 2 sin( (π/2)(3) + π/3 ) = 2 sin( 3π/2 + π/3 ). Let's compute the angle: 3π/2 is 270 degrees, π/3 is 60 degrees, so together it's 330 degrees, which is 11π/6 radians.sin(11π/6) is -1/2. So, f(3) = 2*(-1/2) = -1.f(4):f(4) = 2 sin( (π/2)(4) + π/3 ) = 2 sin( 2π + π/3 ) = 2 sin(7π/3). Sin(7π/3) is sin(π/3) because 7π/3 is equivalent to π/3 (since 7π/3 - 2π = π/3). Sin(π/3) is sqrt(3)/2. So, f(4) = 2*(sqrt(3)/2) = sqrt(3).f(5):f(5) = 2 sin( (π/2)(5) + π/3 ) = 2 sin( 5π/2 + π/3 ). Let's compute the angle: 5π/2 is 450 degrees, which is equivalent to π/2 (since 5π/2 - 2π = π/2). So, 5π/2 + π/3 = π/2 + π/3 = 5π/6. Sin(5π/6) is 1/2. So, f(5) = 2*(1/2) = 1.f(6):f(6) = 2 sin( (π/2)(6) + π/3 ) = 2 sin( 3π + π/3 ) = 2 sin(10π/3). Sin(10π/3) is sin(4π/3) because 10π/3 - 2π = 4π/3. Sin(4π/3) is -sqrt(3)/2. So, f(6) = 2*(-sqrt(3)/2) = -sqrt(3).f(7):f(7) = 2 sin( (π/2)(7) + π/3 ) = 2 sin( 7π/2 + π/3 ). 7π/2 is 630 degrees, which is equivalent to 3π/2 (since 7π/2 - 2π = 3π/2). So, 7π/2 + π/3 = 3π/2 + π/3 = 11π/6. Sin(11π/6) is -1/2. So, f(7) = 2*(-1/2) = -1.f(8):f(8) = 2 sin( (π/2)(8) + π/3 ) = 2 sin( 4π + π/3 ) = 2 sin(13π/3). Sin(13π/3) is sin(π/3) because 13π/3 - 4π = π/3. Sin(π/3) is sqrt(3)/2. So, f(8) = 2*(sqrt(3)/2) = sqrt(3).f(9):f(9) = 2 sin( (π/2)(9) + π/3 ) = 2 sin( 9π/2 + π/3 ). 9π/2 is 810 degrees, which is equivalent to π/2 (since 9π/2 - 4π = π/2). So, 9π/2 + π/3 = π/2 + π/3 = 5π/6. Sin(5π/6) is 1/2. So, f(9) = 2*(1/2) = 1.Alright, so now I have all the f(x) values:f(1) = 1f(2) = -sqrt(3)f(3) = -1f(4) = sqrt(3)f(5) = 1f(6) = -sqrt(3)f(7) = -1f(8) = sqrt(3)f(9) = 1Now, let's add them up:f(1) + f(2) + f(3) + f(4) + f(5) + f(6) + f(7) + f(8) + f(9)= 1 + (-sqrt(3)) + (-1) + sqrt(3) + 1 + (-sqrt(3)) + (-1) + sqrt(3) + 1Let me compute term by term:Start with 1.1 + (-sqrt(3)) = 1 - sqrt(3)1 - sqrt(3) + (-1) = (1 - 1) - sqrt(3) = 0 - sqrt(3) = -sqrt(3)-sqrt(3) + sqrt(3) = 00 + 1 = 11 + (-sqrt(3)) = 1 - sqrt(3)1 - sqrt(3) + (-1) = (1 - 1) - sqrt(3) = 0 - sqrt(3) = -sqrt(3)-sqrt(3) + sqrt(3) = 00 + 1 = 1So, the total sum is 1.Wait, that seems a bit confusing. Let me try adding them in pairs to see if there's a pattern or cancellation.Looking at the terms:1, -sqrt(3), -1, sqrt(3), 1, -sqrt(3), -1, sqrt(3), 1Let me group them as follows:(1 - 1) + (-sqrt(3) + sqrt(3)) + (1 - 1) + (-sqrt(3) + sqrt(3)) + 1Wait, that might not be the best way. Alternatively, let's see:1 - sqrt(3) -1 + sqrt(3) +1 - sqrt(3) -1 + sqrt(3) +1Let me rearrange the terms:(1 -1 +1 -1 +1) + (-sqrt(3) + sqrt(3) - sqrt(3) + sqrt(3))Compute the constants: 1 -1 +1 -1 +1 = (1 -1) + (1 -1) +1 = 0 + 0 +1 =1Compute the sqrt(3) terms: -sqrt(3) + sqrt(3) - sqrt(3) + sqrt(3) = (-sqrt(3) + sqrt(3)) + (-sqrt(3) + sqrt(3)) = 0 + 0 =0So, total sum is 1 + 0 =1.Therefore, the sum is 1, which is option A.Wait, but let me double-check my calculations because sometimes when adding multiple terms, it's easy to make a mistake.Let me list all the terms:1, -sqrt(3), -1, sqrt(3), 1, -sqrt(3), -1, sqrt(3), 1Adding them step by step:Start with 0.Add 1: total =1Add -sqrt(3): total =1 - sqrt(3)Add -1: total =1 - sqrt(3) -1 = -sqrt(3)Add sqrt(3): total = -sqrt(3) + sqrt(3) =0Add 1: total =1Add -sqrt(3): total =1 - sqrt(3)Add -1: total =1 - sqrt(3) -1 = -sqrt(3)Add sqrt(3): total = -sqrt(3) + sqrt(3) =0Add 1: total =1Yes, same result. So, the sum is indeed 1.Alternatively, I can think about the periodicity of the function. Since ω = π/2, the period T = 2π / ω = 2π / (π/2) =4. So, the function repeats every 4 units.Looking at the sum from x=1 to x=9, that's 9 terms. Since the period is 4, the sequence of f(x) values will repeat every 4 terms. Let's see:f(1)=1, f(2)=-sqrt(3), f(3)=-1, f(4)=sqrt(3)Then f(5)=1, f(6)=-sqrt(3), f(7)=-1, f(8)=sqrt(3)Then f(9)=1So, the first 8 terms are two full periods: [1, -sqrt(3), -1, sqrt(3)] repeated twice.Sum of one period: 1 - sqrt(3) -1 + sqrt(3) =0So, two periods sum to 0, and then the ninth term is 1. So, total sum is 0 +1=1.That's another way to see it, which confirms the result.Therefore, the answer is 1, which is option A.

answer：Alright, so I'm trying to figure out this problem about the positional relationship between a line and a circle based on Ba Jin's description of the sunrise. The passage says, "Indeed, after a while, there appeared a small half of the sun's face, which was very red but not bright." Okay, so I need to translate this description into a geometric scenario involving a line and a circle.First, let's break down the passage. The sun is described as appearing with "a small half of its face." This makes me think of the sun as a circle, and the horizon as a line. So, the sun is just starting to rise above the horizon. The phrase "very red but not bright" might indicate that the sun is still partially below the horizon, hence only a small part is visible.Now, in geometric terms, when a circle (the sun) intersects a line (the horizon), there are a few possible positional relationships: the line can be tangent to the circle, separate from the circle, externally tangent, or intersecting the circle. - **Tangent:** This would mean the line just touches the circle at exactly one point. In the context of the sunrise, this would be the moment when the sun is just barely touching the horizon, appearing as a single point. But the passage mentions a "small half," which suggests more than just a single point is visible.- **Separate:** If the line and the circle are separate, they don't intersect at all. In the sunrise scenario, this would mean the sun is entirely below the horizon, which doesn't fit the description since part of the sun is visible.- **Externally tangent:** This term usually refers to two circles touching at one point externally, but in the context of a line and a circle, it might imply the line is tangent to the circle from the outside. However, this still seems similar to the tangent case, and the passage doesn't suggest the sun is just touching the horizon externally.- **Intersect:** This means the line cuts through the circle at two points. In the sunrise context, this would mean the sun is partially above and partially below the horizon, which aligns with the description of seeing a "small half" of the sun's face.So, based on this reasoning, the positional relationship described in the passage is that the line (horizon) intersects the circle (sun) at two points. Therefore, the correct answer should be **D: Intersect**.I think I've covered all the possibilities and matched them to the description. It makes sense because seeing a small half of the sun implies partial visibility, which geometrically means the line intersects the circle.