answer：Alright, so I have this problem where I need to find the probability that a randomly selected number ( k ) from the interval ([-2, 3]) will make the line ( y = kx + 3 ) intersect with the circle ( x^2 + (y + 2)^2 = 9 ). Hmm, okay, let me break this down step by step.First, I should recall what the equation of a circle represents. The general form is ( (x - h)^2 + (y - k)^2 = r^2 ), where ((h, k)) is the center and ( r ) is the radius. In this case, the circle is ( x^2 + (y + 2)^2 = 9 ), so the center is at ((0, -2)) and the radius is ( 3 ) because ( 9 = 3^2 ).Next, I need to figure out when the line ( y = kx + 3 ) intersects this circle. For a line to intersect a circle, the distance from the center of the circle to the line must be less than or equal to the radius of the circle. That makes sense because if the distance is greater than the radius, the line doesn't touch the circle at all.So, the formula for the distance from a point ((x_0, y_0)) to the line ( Ax + By + C = 0 ) is ( frac{|Ax_0 + By_0 + C|}{sqrt{A^2 + B^2}} ). Let me rewrite the line equation in standard form to make it easier. The line is ( y = kx + 3 ), which can be rewritten as ( kx - y + 3 = 0 ). So, ( A = k ), ( B = -1 ), and ( C = 3 ).The center of the circle is ((0, -2)). Plugging these into the distance formula, the distance ( d ) from the center to the line is:[d = frac{|k cdot 0 + (-1) cdot (-2) + 3|}{sqrt{k^2 + (-1)^2}} = frac{|0 + 2 + 3|}{sqrt{k^2 + 1}} = frac{5}{sqrt{k^2 + 1}}]Okay, so the distance is ( frac{5}{sqrt{k^2 + 1}} ). For the line to intersect the circle, this distance must be less than or equal to the radius, which is 3. So, I set up the inequality:[frac{5}{sqrt{k^2 + 1}} leq 3]Now, I need to solve this inequality for ( k ). Let me start by multiplying both sides by ( sqrt{k^2 + 1} ) to eliminate the denominator:[5 leq 3sqrt{k^2 + 1}]Next, I'll divide both sides by 3:[frac{5}{3} leq sqrt{k^2 + 1}]To get rid of the square root, I'll square both sides:[left(frac{5}{3}right)^2 leq k^2 + 1][frac{25}{9} leq k^2 + 1]Subtract 1 from both sides:[frac{25}{9} - 1 leq k^2][frac{25}{9} - frac{9}{9} = frac{16}{9} leq k^2]So, ( k^2 geq frac{16}{9} ). Taking the square root of both sides gives:[|k| geq frac{4}{3}]This means ( k leq -frac{4}{3} ) or ( k geq frac{4}{3} ).Now, I need to find the probability that a randomly selected ( k ) from the interval ([-2, 3]) satisfies either ( k leq -frac{4}{3} ) or ( k geq frac{4}{3} ).First, let's find the length of the interval ([-2, 3]). The total length is ( 3 - (-2) = 5 ).Next, let's find the lengths of the intervals where ( k leq -frac{4}{3} ) and ( k geq frac{4}{3} ).1. For ( k leq -frac{4}{3} ): - The interval is from (-2) to (-frac{4}{3}). - Length: ( -frac{4}{3} - (-2) = -frac{4}{3} + 2 = frac{2}{3} ).2. For ( k geq frac{4}{3} ): - The interval is from ( frac{4}{3} ) to ( 3 ). - Length: ( 3 - frac{4}{3} = frac{9}{3} - frac{4}{3} = frac{5}{3} ).Adding these two lengths together gives the total favorable length:[frac{2}{3} + frac{5}{3} = frac{7}{3}]Now, the probability is the favorable length divided by the total length:[text{Probability} = frac{frac{7}{3}}{5} = frac{7}{15}]So, the probability that the line intersects the circle is ( frac{7}{15} ).**Final Answer**boxed{A}

answer：Given the equation (sin theta + cos theta = frac{1}{3}), we aim to find the value of (sin 2theta).First, we square both sides of the equation:[(sin theta + cos theta)^2 = left(frac{1}{3}right)^2]Expanding the left side using the formula ((a + b)^2 = a^2 + 2ab + b^2), we get:[sin^2 theta + 2sin theta cos theta + cos^2 theta = frac{1}{9}]We know from the Pythagorean identity that (sin^2 theta + cos^2 theta = 1), so substituting this into the equation gives:[1 + 2sin theta cos theta = frac{1}{9}]Rearranging to isolate the term involving (sin 2theta), we subtract 1 from both sides:[2sin theta cos theta = frac{1}{9} - 1][2sin theta cos theta = -frac{8}{9}]Since (sin 2theta = 2sin theta cos theta), we find:[sin 2theta = -frac{8}{9}]Thus, the value of (sin 2theta) is (boxed{-frac{8}{9}}), which corresponds to option B.

answer：Okay, so I have this probability problem here where ξ follows a binomial distribution with parameters n=10 and p=1/2. The question is asking for P(ξ ≥ 2). Hmm, binomial distribution, right? That's when you have a fixed number of independent trials, each with two possible outcomes, and the probability of success is the same each time.Alright, so ξ ~ B(10, 1/2) means that we're dealing with 10 trials, each with a 50% chance of success. I need to find the probability that ξ is greater than or equal to 2. That is, the probability of getting 2 or more successes in these 10 trials.I remember that for binomial distributions, the probability of getting exactly k successes is given by the formula:P(ξ = k) = C(n, k) * p^k * (1-p)^(n-k)Where C(n, k) is the combination of n things taken k at a time. So, in this case, n=10 and p=1/2.But wait, the question isn't asking for exactly 2 successes; it's asking for 2 or more. That means I need to sum the probabilities from k=2 to k=10. That sounds like a lot of terms to add up. Maybe there's a smarter way to do this instead of calculating each term individually.Oh, right! I remember that sometimes it's easier to calculate the complement probability and subtract it from 1. The complement of P(ξ ≥ 2) is P(ξ < 2), which is the same as P(ξ = 0) + P(ξ = 1). So, if I can find those two probabilities and subtract their sum from 1, I'll get the desired probability.Let me write that down:P(ξ ≥ 2) = 1 - P(ξ < 2) = 1 - [P(ξ = 0) + P(ξ = 1)]Alright, so let's compute P(ξ = 0) and P(ξ = 1).Starting with P(ξ = 0):P(ξ = 0) = C(10, 0) * (1/2)^0 * (1/2)^10I know that C(10, 0) is 1 because there's only one way to choose nothing from 10 items. (1/2)^0 is also 1, since any number to the power of 0 is 1. So, this simplifies to:P(ξ = 0) = 1 * 1 * (1/2)^10 = (1/2)^10Calculating (1/2)^10, that's 1 divided by 2^10. 2^10 is 1024, so P(ξ = 0) is 1/1024.Now, moving on to P(ξ = 1):P(ξ = 1) = C(10, 1) * (1/2)^1 * (1/2)^9C(10, 1) is 10, since there are 10 ways to choose 1 item from 10. (1/2)^1 is 1/2, and (1/2)^9 is 1/512. So, multiplying these together:P(ξ = 1) = 10 * (1/2) * (1/512) = 10 * (1/1024) = 10/1024So, P(ξ = 1) is 10/1024.Now, adding P(ξ = 0) and P(ξ = 1) together:P(ξ = 0) + P(ξ = 1) = 1/1024 + 10/1024 = 11/1024Therefore, the probability of ξ being less than 2 is 11/1024.Subtracting this from 1 gives us the probability that ξ is greater than or equal to 2:P(ξ ≥ 2) = 1 - 11/1024 = (1024/1024) - (11/1024) = 1013/1024So, the probability is 1013/1024.Looking back at the options given:A: 1013/1024B: 11/1024C: 501/512D: 507/512So, option A is 1013/1024, which matches our calculation. Therefore, the correct answer should be A.Wait, let me double-check my calculations to make sure I didn't make any mistakes.First, P(ξ = 0):C(10,0) is 1, (1/2)^0 is 1, and (1/2)^10 is 1/1024. That seems right.P(ξ = 1):C(10,1) is 10, (1/2)^1 is 1/2, and (1/2)^9 is 1/512. Multiplying 10 * 1/2 * 1/512:10 * (1/2) is 5, and 5 * (1/512) is 5/512. Wait, hold on, that's different from what I had before. Did I make a mistake here?Wait, no, 10 * (1/2) * (1/512) is 10 * (1/1024) because (1/2)*(1/512) is 1/1024. So, 10 * (1/1024) is 10/1024. So, that part is correct.Adding 1/1024 and 10/1024 gives 11/1024, which is correct.Subtracting 11/1024 from 1 gives 1013/1024, which is option A. So, my initial calculation was correct.I think I confused myself for a second when I thought of 5/512, but that's not the case here. So, yeah, 10/1024 is correct.Therefore, I'm confident that the answer is A: 1013/1024.**Final Answer**boxed{A}

answer：Alright, so I have this question about correlation coefficients. It says, "If the degree of linear correlation between two variables is very high, then the absolute value of their correlation coefficient r should be close to (    )" and then gives options A: 0, B: 0.5, C: 2, D: 1.Okay, let's start by recalling what a correlation coefficient is. From what I remember, the correlation coefficient, usually denoted by r, measures the strength and direction of a linear relationship between two variables. The value of r ranges from -1 to 1. If r is close to 1, it means there's a strong positive linear relationship between the variables. That is, as one variable increases, the other variable tends to increase as well. On the other hand, if r is close to -1, there's a strong negative linear relationship, meaning as one variable increases, the other tends to decrease.Now, the question is asking about a very high degree of linear correlation. So, whether it's positive or negative, the absolute value of r should be close to 1 because that indicates the strongest possible linear relationship. Looking at the options, A is 0, which would mean no linear correlation. That's the opposite of what we want. B is 0.5, which indicates a moderate correlation, not very high. C is 2, but I remember that the correlation coefficient can't be more than 1 or less than -1, so 2 isn't a valid value for r. That leaves D: 1, which makes sense because a very high correlation would mean r is near 1 in absolute value.Wait, just to make sure I'm not missing anything. The question specifies the absolute value of r, so whether the correlation is positive or negative, the absolute value being close to 1 still indicates a very high correlation. Yeah, that seems right.I think I'm confident now. The correct answer should be D: 1.

answer：Okay, so I have this problem about soccer match points. Let me try to understand it step by step. The question is asking how many ways a team can score a total of 19 points from 14 matches. The scoring rules are 3 points for a win, 1 point for a draw, and 0 points for a loss. The options given are A: 3 ways, B: 4 ways, C: 5 ways, D: 6 ways.Hmm, so I need to figure out all the possible combinations of wins, draws, and losses that add up to 19 points in 14 matches. Let me break this down.First, let's denote the number of wins as W, draws as D, and losses as L. Since each match results in either a win, draw, or loss, the total number of matches should be W + D + L = 14.Also, the total points are calculated as 3W + D = 19 because each win gives 3 points and each draw gives 1 point. Losses give 0 points, so they don't contribute to the total points.So, I have two equations:1. W + D + L = 142. 3W + D = 19I need to find all non-negative integer solutions (W, D, L) that satisfy both equations.Let me try to express D from the second equation. From equation 2:D = 19 - 3WNow, substitute D into equation 1:W + (19 - 3W) + L = 14Simplify this:W + 19 - 3W + L = 14Combine like terms:-2W + L + 19 = 14Then,-2W + L = -5Which can be rewritten as:L = 2W - 5So, L must be equal to 2W - 5.But since the number of losses, L, can't be negative, 2W - 5 must be greater than or equal to 0.So,2W - 5 ≥ 0Which implies,2W ≥ 5Therefore,W ≥ 5/2Since W must be an integer (you can't have half a win), W must be at least 3.Also, since the total number of matches is 14, W can't be more than 14. But let's see the practical upper limit.From equation 2: D = 19 - 3WSince D can't be negative, 19 - 3W ≥ 0So,3W ≤ 19Therefore,W ≤ 19/3 ≈ 6.333Since W must be an integer, W can be at most 6.So, W can be 3, 4, 5, or 6.Let me list out each possible W and find the corresponding D and L.1. When W = 3:D = 19 - 3*3 = 19 - 9 = 10L = 2*3 - 5 = 6 - 5 = 1So, W=3, D=10, L=1. Total matches: 3+10+1=14. Points: 3*3 +10=9+10=19. This works.2. When W = 4:D = 19 - 3*4 = 19 - 12 = 7L = 2*4 - 5 = 8 - 5 = 3So, W=4, D=7, L=3. Total matches: 4+7+3=14. Points: 4*3 +7=12+7=19. This works.3. When W = 5:D = 19 - 3*5 = 19 - 15 = 4L = 2*5 - 5 = 10 - 5 = 5So, W=5, D=4, L=5. Total matches: 5+4+5=14. Points: 5*3 +4=15+4=19. This works.4. When W = 6:D = 19 - 3*6 = 19 - 18 = 1L = 2*6 - 5 = 12 - 5 = 7So, W=6, D=1, L=7. Total matches: 6+1+7=14. Points: 6*3 +1=18+1=19. This works.Wait, that's four different combinations. Let me check if W=2 is possible.If W=2:D = 19 - 3*2 = 19 - 6 = 13L = 2*2 -5 = 4 -5 = -1Negative losses don't make sense, so W=2 is invalid.Similarly, W=7:D =19 -21= -2, which is invalid.So, only W=3,4,5,6 are valid, giving four different ways.Therefore, the answer should be B:4 ways.Wait, but let me double-check if I missed any other possibilities.Is there a way where W=0? Let's see:If W=0:D=19-0=19But total matches would be 0+19+L=14, so L= -5, which is impossible.Similarly, W=1:D=19-3=16Total matches:1+16+L=14 => L= -3. Impossible.So, indeed, only W=3,4,5,6 are possible.Therefore, the number of ways is 4. So, option B.**Final Answer**boxed{B}

answer：Okay, so I have this triangle ABC, and angle A is 15 degrees smaller than angle B. That means if angle B is, say, x degrees, then angle A would be x minus 15 degrees. Got that part. Now, there's an altitude drawn from angle C, and it bisects angle C into two angles, C1 and C2. Hmm, so the altitude splits angle C into two equal parts? Wait, no, it just says it bisects C into C1 and C2, but doesn't necessarily say they're equal. Wait, actually, the term "bisects" usually means dividing into two equal parts, so maybe C1 and C2 are each half of angle C. But the problem also mentions that C2 is adjacent to the side opposite B. Hmm, so maybe C2 is next to side AC, since side opposite B is AC. Wait, let me visualize this. Triangle ABC, with angles A, B, and C. If I draw an altitude from C, it will hit the opposite side AB at some point, say D, forming two right triangles: triangle ADC and triangle BDC. So, angle C is split into two angles at point D: angle ADC is C1, and angle BDC is C2. Since the altitude is drawn, both ADC and BDC are right angles, so 90 degrees each.Wait, so in triangle ADC, angle at D is 90 degrees, angle at A is angle A, and angle at C is C1. Similarly, in triangle BDC, angle at D is 90 degrees, angle at B is angle B, and angle at C is C2. So, in triangle ADC: angle A + angle C1 + 90 = 180, so angle C1 = 90 - angle A. Similarly, in triangle BDC: angle B + angle C2 + 90 = 180, so angle C2 = 90 - angle B.So, angle C1 is 90 minus angle A, and angle C2 is 90 minus angle B. Therefore, the difference between C1 and C2 would be (90 - A) - (90 - B) = B - A. But wait, the problem states that angle A is 15 degrees smaller than angle B, so A = B - 15. Therefore, B - A = 15 degrees. So, C1 - C2 = 15 degrees. Wait, but the question asks for the relationship between C1 - C2 and A - B. Since A - B is negative 15 degrees, and C1 - C2 is positive 15 degrees, so C1 - C2 is equal to -(A - B). But the options are given as positive degrees, so maybe the absolute value? Or perhaps I have the difference reversed.Wait, let me double-check. If A = B - 15, then B = A + 15. So, C1 = 90 - A, and C2 = 90 - B = 90 - (A + 15) = 75 - A. Therefore, C1 - C2 = (90 - A) - (75 - A) = 15 degrees. So, regardless of the value of A, the difference is 15 degrees. So, the relationship between C1 - C2 and A - B is that C1 - C2 is equal to 15 degrees, which is the negative of A - B, since A - B is -15 degrees. But since the options are positive, the answer is 15 degrees.