🧠:[<think>Okay, so I need to find the value of sin α plus cos α, where the terminal side of angle α passes through the point (2, -1). Hmm, let me think. First, I remember that when a point (x, y) is on the terminal side of an angle in standard position, the trigonometric functions can be determined using that point. Specifically, sin α is y over r, and cos α is x over r, where r is the distance from the origin to the point. Right, so r is the hypotenuse of the right triangle formed by the coordinates. So, the formula for r is the square root of (x squared plus y squared). In this case, x is 2 and y is -1. Let me calculate r first. r = √(2² + (-1)²) = √(4 + 1) = √5. Okay, so r is the square root of 5. That seems right. Now, sin α is y/r. Since y here is -1, that would be sin α = -1/√5. Similarly, cos α is x/r, which is 2/√5. But wait, adding sin α and cos α together would be (-1/√5) + (2/√5). Since they have the same denominator, I can combine them. That would be (-1 + 2)/√5, which simplifies to 1/√5. Hmm, but usually, we rationalize the denominator, don't we? So, 1 divided by √5 is the same as √5 divided by 5. Because if you multiply numerator and denominator by √5, you get (1*√5)/(√5*√5) = √5/5. So, sin α + cos α is √5/5. Let me double-check my steps to make sure I didn't make a mistake. First, finding r: √(2² + (-1)²) is √(4 + 1) = √5. That's correct. Then, sin α = y/r = -1/√5 and cos α = 2/√5. Adding them: (-1 + 2)/√5 = 1/√5. Rationalizing gives √5/5. Yep, that seems right. Wait a second, is there any chance that the coordinates (2, -1) are in a different quadrant where the signs of sin and cos would vary? Let's see. The x-coordinate is positive and the y-coordinate is negative, so that's the fourth quadrant. In the fourth quadrant, cosine is positive and sine is negative, which matches what I found earlier. So that's consistent. Alternatively, maybe I could use the unit circle concept. If the terminal side passes through (2, -1), but that point isn't on the unit circle. So to relate it to the unit circle, we need to divide by r. So, the coordinates on the unit circle would be (x/r, y/r) which is (2/√5, -1/√5). Then sin α is the y-coordinate, -1/√5, and cos α is the x-coordinate, 2/√5. So, adding them gives the same result. I think that's solid. Another way to check is to consider the angle α. Maybe compute the angle using arctangent? The tangent of α is y/x, which is -1/2. So α is arctangent(-1/2). But since it's in the fourth quadrant, the angle would be 360 degrees minus arctangent(1/2) or in radians, 2π - arctan(1/2). But calculating sin and cos of that angle might be more complicated. Alternatively, maybe plug into the calculator. Let's see, if I compute sin(arctan(-1/2)) + cos(arctan(-1/2)). Wait, but angle is in the fourth quadrant. Let me think. Let's set θ = arctan(-1/2). But arctan(-1/2) would give me an angle in the fourth quadrant. Then, sin θ would be -1/√(1 + (2)^2) = -1/√5, and cos θ would be 2/√5. Wait, that's exactly the same result as before. So sin θ + cos θ is (-1 + 2)/√5 = 1/√5. So that checks out. Therefore, I think the answer is √5/5. But just to make sure, let's go through all steps once again. Given point (2, -1). To find r: sqrt(x² + y²). So sqrt(4 + 1) = sqrt(5). Then sin α = y/r = -1/sqrt(5). Cos α = x/r = 2/sqrt(5). Adding them: (-1 + 2)/sqrt(5) = 1/sqrt(5). Rationalizing denominator: sqrt(5)/5. Yep, that's consistent. Another possible approach: draw the point (2, -1) on the coordinate plane. The terminal side of angle α is the line from the origin to this point. The triangle formed has adjacent side 2, opposite side -1, hypotenuse sqrt(5). So, sine is opposite over hypotenuse, which is -1/sqrt(5); cosine is adjacent over hypotenuse, which is 2/sqrt(5). Sum is 1/sqrt(5), which is sqrt(5)/5. I think that's all. No mistakes noticed here. Seems straightforward. So the answer should be sqrt(5)/5. Final AnswerThe value of ( sin alpha + cos alpha ) is boxed{dfrac{sqrt{5}}{5}}.</think>To find the value of ( sin alpha + cos alpha ) where the terminal side of angle ( alpha ) passes through the point ( (2, -1) ):1. Calculate the distance ( r ) from the origin to the point ( (2, -1) ): [ r = sqrt{2^2 + (-1)^2} = sqrt{4 + 1} = sqrt{5} ]2. Determine ( sin alpha ) and ( cos alpha ): [ sin alpha = frac{y}{r} = frac{-1}{sqrt{5}}, quad cos alpha = frac{x}{r} = frac{2}{sqrt{5}} ]3. Add ( sin alpha ) and ( cos alpha ): [ sin alpha + cos alpha = frac{-1}{sqrt{5}} + frac{2}{sqrt{5}} = frac{-1 + 2}{sqrt{5}} = frac{1}{sqrt{5}} ]4. Rationalize the denominator: [ frac{1}{sqrt{5}} = frac{sqrt{5}}{5} ]Thus, the value of ( sin alpha + cos alpha ) is (boxed{dfrac{sqrt{5}}{5}}).,<think>Okay, let's see. I need to find the value of sin α plus cos α, where the terminal side of angle α passes through the point (2, -1). Hmm, right. So first, I remember that when dealing with trigonometric functions and points on the terminal side, the coordinates can help determine the sine and cosine values. Let me recall the exact relationships here.So, if a point (x, y) is on the terminal side of an angle in standard position, then we can find the trigonometric functions of that angle. Specifically, I think that sin α is equal to y divided by r, and cos α is equal to x divided by r, where r is the distance from the origin to the point (x, y). That distance r is calculated using the Pythagorean theorem: r = sqrt(x² + y²). Right? So in this case, the point given is (2, -1). Therefore, x is 2 and y is -1.Let me write that down step by step. First, find r. So x = 2, y = -1. Then r = sqrt(2² + (-1)²) = sqrt(4 + 1) = sqrt(5). Okay, so r is the square root of 5. That makes sense. So then, sin α would be y over r, which is (-1)/sqrt(5), and cos α is x over r, which is 2/sqrt(5). Then, the problem asks for sin α + cos α. So I need to add those two fractions together.Wait, but before I do that, I should remember that usually, we rationalize the denominator when presenting such answers. Let me check if that's necessary here. The question just asks for the value, so maybe it's okay to leave it as is, but maybe they want it rationalized. Hmm. Let me do both and see.First, adding them without rationalizing: (-1)/sqrt(5) + 2/sqrt(5). Since the denominators are the same, we can just add the numerators: (-1 + 2)/sqrt(5) = 1/sqrt(5). Then, if we rationalize the denominator, that becomes sqrt(5)/5. So, depending on how the answer is expected, either form could be acceptable. But in most math problems, rationalized form is preferred unless specified otherwise. So I should probably go with sqrt(5)/5.But let me double-check my steps. First, verifying r: x squared is 4, y squared is 1, sum is 5, so sqrt(5). Correct. Then, sin α is -1/sqrt(5). Wait, is that right? Since the point is (2, -1), that would be in the fourth quadrant, right? Quadrant IV. So in quadrant IV, sine is negative and cosine is positive. So sin α should be negative, which matches my calculation. Cos α is positive, which is 2/sqrt(5). Correct. Then adding them: -1/sqrt(5) + 2/sqrt(5) is indeed (1)/sqrt(5). Rationalizing gives sqrt(5)/5.Wait, let's make sure rationalization is done properly. 1/sqrt(5) multiplied by sqrt(5)/sqrt(5) is sqrt(5)/5. Yes, that's correct. So that's the simplified form.But maybe there's another way to approach this problem? Let me see. Alternatively, since we know the coordinates, maybe using the unit circle concept. But the point (2, -1) isn't on the unit circle because the distance from the origin isn't 1. So scaling down the coordinates by dividing by r, which is sqrt(5), gives the coordinates on the unit circle. Which would be (2/sqrt(5), -1/sqrt(5)). That corresponds to cos α and sin α, respectively. So adding those together gives the same result.Yeah, so either way, the answer should be sqrt(5)/5. Let me check once more to make sure I didn't make any sign errors. The y-coordinate is negative, so sine is negative. The x-coordinate is positive, so cosine is positive. Then adding a negative and a positive: 2/sqrt(5) + (-1)/sqrt(5) = 1/sqrt(5) = sqrt(5)/5. That seems right.Alternatively, if I plug this into a calculator, just to check the approximate decimal value. Let's see. sqrt(5) is approximately 2.236, so 2.236 divided by 5 is approximately 0.447. Let's check the original terms. 2/sqrt(5) is approximately 2/2.236 ≈ 0.894, and -1/sqrt(5) is approximately -0.447. Adding those gives 0.894 - 0.447 ≈ 0.447, which matches sqrt(5)/5. So that seems to check out.Therefore, the answer should be sqrt(5)/5. Is there any other consideration? The problem didn't specify whether to leave it in terms of sin and cos or to rationalize, but since they asked for the value, rationalizing is standard. So I think that's the way to go.Final AnswerThe value of (sin alpha + cos alpha) is boxed{dfrac{sqrt{5}}{5}}.</think>To find the value of (sin alpha + cos alpha) where the terminal side of angle (alpha) passes through the point ((2, -1)):1. Calculate the distance (r) from the origin to the point ((2, -1)) using the Pythagorean theorem: [ r = sqrt{x^2 + y^2} = sqrt{2^2 + (-1)^2} = sqrt{4 + 1} = sqrt{5} ]2. Determine (sin alpha) and (cos alpha) using the coordinates and (r): [ sin alpha = frac{y}{r} = frac{-1}{sqrt{5}}, quad cos alpha = frac{x}{r} = frac{2}{sqrt{5}} ]3. Add (sin alpha) and (cos alpha): [ sin alpha + cos alpha = frac{-1}{sqrt{5}} + frac{2}{sqrt{5}} = frac{-1 + 2}{sqrt{5}} = frac{1}{sqrt{5}} ]4. Rationalize the denominator: [ frac{1}{sqrt{5}} = frac{sqrt{5}}{5} ]Thus, the value of (sin alpha + cos alpha) is (boxed{dfrac{sqrt{5}}{5}}).]

🔑:To solve this problem, we need to apply the definition of sine and cosine functions in terms of the coordinates of the point lying on the terminal side of the angle in the standard position of the Cartesian plane.Given the point (2,-1) through which the terminal side of angle alpha passes, we can determine the lengths of the adjacent side (x-coordinate) and the opposite side (y-coordinate) of the right triangle formed with the radius (hypotenuse) of the circle in which the angle alpha lies.First, let's calculate the radius (r) of the hypotenuse, which is the distance of the point from the origin: r = sqrt{x^2 + y^2} = sqrt{2^2 + (-1)^2} = sqrt{4 + 1} = sqrt{5}. Now we can calculate the sine and cosine of angle alpha: sin alpha = frac{y}{r} = frac{-1}{sqrt{5}}, cos alpha = frac{x}{r} = frac{2}{sqrt{5}}. To simplify the expressions, we rationalize the denominators: sin alpha = -frac{1}{sqrt{5}} cdot frac{sqrt{5}}{sqrt{5}} = -frac{sqrt{5}}{5}, cos alpha = frac{2}{sqrt{5}} cdot frac{sqrt{5}}{sqrt{5}} = frac{2sqrt{5}}{5}. Now we add the sine and cosine of alpha together: sin alpha + cos alpha = -frac{sqrt{5}}{5} + frac{2sqrt{5}}{5} = frac{1sqrt{5}}{5}. The final expression is: sin alpha + cos alpha = frac{sqrt{5}}{5}. Thus, the correct option is D: boxed{frac{sqrt{5}}{5}}.

🧠:[<think>Okay, so I have this problem here about determining which method is correct for checking if a quadrilateral door frame is a rectangle. There are four options: A, B, C, D. Let me think through each one step by step.First, I remember that a rectangle is a quadrilateral with four right angles. But there are different properties and theorems related to rectangles that can help identify them. Let me recall some key points.Option A says to measure whether the diagonals are equal. Hmm, I remember that in a rectangle, the diagonals are congruent. That's one of the properties. So if the diagonals are equal in length, does that guarantee it's a rectangle? Wait, not necessarily. For example, an isosceles trapezoid also has equal diagonals. But an isosceles trapezoid isn't a rectangle. So maybe A isn't sufficient? Or is there another consideration here?Wait, but maybe in the context of a door frame, which is a quadrilateral, if it's a parallelogram with equal diagonals, then it is a rectangle. Because in a parallelogram, if the diagonals are equal, then it must be a rectangle. So maybe if we already know it's a parallelogram, then measuring the diagonals would confirm it's a rectangle. But does the problem state that the door frame is a parallelogram? The problem just says a quadrilateral door frame. So if it's any quadrilateral, just having equal diagonals isn't enough. So perhaps A is not the correct answer unless there's more context. Hmm.Option B: Measure whether the two sets of opposite sides are equal respectively. So that would check if it's a parallelogram. Because in a parallelogram, opposite sides are equal. But a parallelogram isn't necessarily a rectangle unless it has right angles. So measuring opposite sides equal would only confirm it's a parallelogram. To be a rectangle, it needs to have four right angles or congruent diagonals. So B alone isn't enough. So B is not the correct answer.Option C: Measure whether one set of opposite angles are both right angles. If one pair of opposite angles are right angles, does that make it a rectangle? Let me think. If in a quadrilateral, one pair of opposite angles are right angles, and if it's a parallelogram, then the consecutive angles would be supplementary, so the other angles would also be right angles. But if it's not a parallelogram, then having one pair of right angles doesn't ensure the others are. For example, a trapezoid with one pair of right angles. That's not a rectangle. So if we just check one set of opposite angles, that's not sufficient. So C is not correct.Option D: Measure whether three of the angles are right angles. Let's see. In a quadrilateral, the sum of the interior angles is 360 degrees. If three angles are right angles (90 degrees each), then the total for those three would be 270 degrees, so the fourth angle must be 90 degrees as well. Therefore, if three angles are right angles, the fourth has to be a right angle too. So measuring three right angles would guarantee that all four are right angles, which defines a rectangle. So D seems correct.Wait, but let me double-check. If three angles are 90 degrees, then yes, the fourth is 90. So regardless of the quadrilateral's type, as long as three angles are right angles, it's a rectangle. Because all four angles are right angles, and a quadrilateral with four right angles is a rectangle. So D is correct.Comparing with A, if we only check the diagonals, but if it's not a parallelogram, equal diagonals don't necessarily make it a rectangle. But if three angles are right angles, then it's definitely a rectangle. So D is the correct answer. Let me make sure.Yes, quadrilaterals have angle sums of 360. If three angles are 90, the fourth is 360 - 3*90 = 360 - 270 = 90. So four right angles. So regardless of sides, if it's a quadrilateral with four right angles, it's a rectangle. So D is correct. Therefore, the answer is D.Final Answerboxed{D}</think>To determine if a quadrilateral door frame is a rectangle, we need to consider the properties of a rectangle, which has four right angles. Let's evaluate each option:- Option A: Measuring if the diagonals are equal. While rectangles have equal diagonals, an isosceles trapezoid also has equal diagonals. Without knowing the door frame is a parallelogram, equal diagonals alone aren't sufficient. Thus, A is incorrect.- Option B: Measuring if the two sets of opposite sides are equal. This checks if the shape is a parallelogram, but a parallelogram isn't necessarily a rectangle unless it has right angles. Thus, B is incorrect.- Option C: Measuring if one set of opposite angles are right angles. A quadrilateral with one pair of right angles isn't necessarily a rectangle (e.g., a right trapezoid). Thus, C is incorrect.- Option D: Measuring if three angles are right angles. The sum of a quadrilateral's angles is 360°. If three angles are 90°, the fourth must also be 90°, ensuring all four are right angles. This definitively makes it a rectangle. Thus, D is correct.The correct answer is (boxed{D}).,<think>Okay, so I need to figure out which of the four options (A, B, C, D) is the correct method to determine if a quadrilateral door frame is a rectangle. Let me start by recalling the properties of rectangles and how to prove a quadrilateral is a rectangle.First off, a rectangle is defined as a quadrilateral with four right angles. So, all four angles must be 90 degrees. But sometimes, in practical situations, measuring all four angles might not be feasible, so there are other properties we can check based on the definitions and theorems in geometry.Let me go through each option one by one.Option A: Measure whether the diagonals are equalHmm, I remember that in a rectangle, the diagonals are congruent. This is a key property. However, is the converse true? If a quadrilateral has congruent diagonals, is it necessarily a rectangle? I think that's only true for parallelograms. Wait, so if a parallelogram has congruent diagonals, then it's a rectangle. But if the quadrilateral isn't a parallelogram, having equal diagonals doesn't guarantee it's a rectangle. For example, an isosceles trapezoid also has equal diagonals, but it's not a rectangle. So, if the door frame is a parallelogram (which would mean both pairs of opposite sides are equal), then checking the diagonals would confirm if it's a rectangle. But if we don't know it's a parallelogram first, just checking the diagonals might not be sufficient. But maybe in the case of a door frame, it's likely to be a parallelogram because opposite sides should be parallel for the door to function properly. Hmm, not sure. Maybe they assume that the door frame is already a parallelogram? Or is there a different reasoning here?Option B: Measure whether the two sets of opposite sides are equal respectivelyOkay, this checks if the quadrilateral is a parallelogram, because in a parallelogram, opposite sides are equal. But a parallelogram isn't necessarily a rectangle. So, if we just check that the opposite sides are equal, that would tell us it's a parallelogram, but not necessarily a rectangle. So, this alone isn't enough. So Option B is insufficient.Option C: Measure whether one set of opposite angles are both right anglesHmm. In a rectangle, all four angles are right angles. If we check that one set of opposite angles are right angles, does that ensure it's a rectangle? Let me think. Suppose we have a quadrilateral where one pair of opposite angles are 90 degrees. Is that enough?Wait, in a quadrilateral, if three angles are right angles, then the fourth must also be a right angle because the sum of angles in a quadrilateral is 360 degrees. So if three are 90, the last one is 360 - 3*90 = 360 - 270 = 90. So, if three angles are right angles, then it's a rectangle. But Option C is only checking one set of opposite angles, which is two angles. If two angles are right angles, does that make it a rectangle? No, because you can have a trapezoid with two right angles, for example. So, Option C is insufficient. So, if only two angles are right angles, it might not be a rectangle. Therefore, Option C is not sufficient.Option D: Measure whether three of the angles are right anglesAs I thought earlier, if three angles are right angles, the fourth has to be a right angle too. Therefore, this would confirm that all four angles are 90 degrees, making it a rectangle. So, Option D seems valid.Wait, but let me cross-check with the other options. Option A: If we check the diagonals. If the door frame is a parallelogram (which might not be the case), then equal diagonals would make it a rectangle. But how do we know it's a parallelogram? If it's a door frame, maybe the opposite sides are already parallel? But door frames are typically rectangles, but if they're out of shape, maybe they become a different quadrilateral. So, if someone wants to verify if it's a rectangle, they might need to check both whether it's a parallelogram and whether the diagonals are equal.But Option A is only checking the diagonals. If the diagonals are equal, but it's not a parallelogram, then it's not a rectangle. So, unless they already know it's a parallelogram, which they might not, Option A is not sufficient. So, if the door frame is not a parallelogram, equal diagonals could mean it's an isosceles trapezoid. Therefore, Option A is not necessarily correct unless you have prior knowledge that it's a parallelogram.But in the context of a door frame, maybe it's supposed to be a parallelogram? Because doors are usually rectangular, but if it's warped, maybe the opposite sides are still equal? Wait, but if it's a non-parallelogram with equal diagonals, like an isosceles trapezoid, the door wouldn't function properly. So, perhaps in the context of a door frame, they can assume it's a parallelogram, so checking diagonals would work. Hmm, this is a bit confusing. Wait, but in the problem statement, it's a math activity class on "judging whether a quadrilateral door frame is a rectangle." So, the students are supposed to figure out a method. The question is, which of the four methods is correct. So, according to geometric principles, which method is valid.From geometry, to prove a quadrilateral is a rectangle, one way is:1. Prove it's a parallelogram with one right angle. (Since in a parallelogram, adjacent angles are supplementary, so one right angle makes all angles right angles.)2. Prove it's a parallelogram with congruent diagonals.3. Prove that all four angles are right angles.Alternatively, if it's not a parallelogram, then having three right angles would force the fourth to be a right angle, making it a rectangle.So, let's see the options again:Option A: Diagonals equal. If it's a parallelogram, this would work. But if it's not a parallelogram, diagonals equal doesn't imply rectangle. So, unless they first confirm it's a parallelogram, this isn't sufficient. But the question doesn't mention checking if it's a parallelogram first.Option B: Two sets of opposite sides equal. That would confirm it's a parallelogram, but then you need an additional check for a right angle or equal diagonals. So, Option B alone is insufficient.Option C: One set of opposite angles are right angles. If it's a parallelogram, then one right angle is enough. But if it's not a parallelogram, two right angles don't guarantee it's a rectangle. So, again, unless it's a parallelogram, this isn't sufficient.Option D: Three right angles. As mentioned, three right angles force the fourth to be a right angle, regardless of being a parallelogram or not. So, this would confirm it's a rectangle.Therefore, Option D is correct because measuring three right angles ensures all four are right angles, hence a rectangle. Option A is only correct if the quadrilateral is a parallelogram. Since the problem doesn't state that the door frame is already known to be a parallelogram, we can't assume that. Therefore, the safest method is Option D.Wait, but maybe in practice, door frames are constructed as parallelograms? If the door is supposed to be rectangular, then even if it's warped, the opposite sides might still be equal? Not necessarily. If a door frame is warped, it could become a non-parallelogram quadrilateral. So, in reality, if you want to check if it's a rectangle, you can't assume it's a parallelogram. Hence, checking three angles is a direct way.Therefore, I think the correct answer is D.Final Answerboxed{D}</think>To determine if a quadrilateral door frame is a rectangle, we need to verify the properties of a rectangle. A rectangle has four right angles and congruent diagonals. Let's analyze each option:Option A: Measure whether the diagonals are equal While rectangles have equal diagonals, the converse is not necessarily true. A quadrilateral with equal diagonals could be an isosceles trapezoid, which is not a rectangle. Therefore, this method is insufficient unless the quadrilateral is already a parallelogram.Option B: Measure whether the two sets of opposite sides are equal respectively This confirms the quadrilateral is a parallelogram but not necessarily a rectangle. A parallelogram requires an additional check (like a right angle or equal diagonals) to be a rectangle. Thus, this method alone is insufficient.Option C: Measure whether one set of opposite angles are both right angles A rectangle requires all four angles to be right angles. Checking only one set of opposite angles (two right angles) is insufficient because other quadrilaterals, like a trapezoid, can have two right angles.Option D: Measure whether three of the angles are right angles If three angles are right angles, the fourth angle must also be a right angle (since the sum of angles in a quadrilateral is 360 degrees). This directly confirms the quadrilateral is a rectangle without needing prior knowledge of it being a parallelogram.Thus, the correct answer is (boxed{D}).]

🔑:A: If the diagonals are equal, the quadrilateral is not necessarily a rectangle, for example, an isosceles trapezoid; B: If the two sets of opposite sides are equal, the quadrilateral is not necessarily a rectangle, for example, a parallelogram; C: If one set of opposite angles are both right angles, the quadrilateral is not necessarily a rectangle, because the degrees of the other two angles are uncertain; D: According to the criteria for determining a rectangle, if three angles are right angles, the quadrilateral is a rectangle. Therefore, the correct choice is boxed{D}.

🧠:[<think>Okay, let's tackle this problem step by step. First, I need to understand what's being asked here. There are two parts: part (1) asks for the probability p that a single piece drawn is of second quality, given that when we draw twice with replacement, the probability that at most one is second quality is 0.96. Part (2) is about a batch of 100 pieces where two are drawn without replacement, and we need the probability that at least one is second quality.Starting with part (1). Let me parse the problem again. So we're drawing two products with replacement each time. The event A is "At most one of the two drawn products is of second quality," and P(A) = 0.96. They want to find the probability p that any one piece is second quality.First, since we're drawing with replacement, the two draws are independent. So the probability of getting a second quality product on each draw is the same, which is p. The event A is "at most one," which means either 0 or 1 second quality products in the two draws. So the complement of A would be "both products are second quality." Therefore, P(A) = 1 - P(both are second quality). Let me verify that.Yes, because "at most one" means 0 or 1, so the only case not included is 2. Therefore, P(A) = 1 - P(both second quality). Therefore, P(both second quality) = 1 - 0.96 = 0.04.Since the draws are independent, the probability that both are second quality is p * p = p². Therefore, p² = 0.04. To find p, we take the square root of 0.04. The square root of 0.04 is 0.2. So p = 0.2. Therefore, the probability that any one piece is second quality is 0.2. That seems straightforward. Let me check if there's another way to approach it in case I made a mistake.Alternatively, we can model the probability of event A directly. The probability of 0 second quality products is (1 - p)², since both draws must be non-second quality. The probability of exactly 1 second quality product is 2 * p * (1 - p), since there are two possible ways: first is second and second is not, or vice versa. Then, adding these two probabilities together should equal 0.96.So, (1 - p)² + 2p(1 - p) = 0.96.Let me compute that:First, expand (1 - p)²: 1 - 2p + p²Then, add 2p(1 - p): 2p - 2p²Adding them together: 1 - 2p + p² + 2p - 2p² = 1 - p²Therefore, 1 - p² = 0.96 => p² = 0.04 => p = 0.2. Same result. So that checks out. So part (1) answer is 0.2.Okay, moving on to part (2). Now, the batch has 100 pieces, and we draw two pieces randomly. They want the probability of event B: "At least one of the two drawn products is of second quality." So we need to compute P(B).First, let's note that in part (2), they mention 100 pieces in the batch. However, they don't specify how many of them are second quality. But wait, in part (1), we found that the probability p of a single piece being second quality is 0.2. If the batch has 100 pieces, then the number of second quality pieces should be 100 * p = 20. Because if each piece has a 20% chance of being second quality, but in reality, in a finite batch of 100, the number should be fixed, right? Wait, hold on. There might be a discrepancy here.Wait, in part (1), p is the probability that a single piece drawn is second quality. If the batch is large and we're drawing with replacement, p would correspond to the proportion of second quality products in the batch. So if the batch has N products, and K are second quality, then p = K/N. However, in part (1), we found p = 0.2. Then in part (2), assuming the same batch, so N = 100, K = 20. Therefore, there are 20 second quality pieces. Therefore, in part (2), when we draw two pieces without replacement (since it's a batch of 100, and we draw two randomly, probably without replacement unless stated otherwise). So the problem states: "If there are 100 pieces in this batch, and 2 pieces are randomly drawn, find the probability of event B: 'At least one of the two drawn products is of second quality'."So event B is the complement of "none are second quality." So P(B) = 1 - P(neither is second quality). Let's compute that.First, total number of ways to draw two pieces: C(100, 2) = 100*99/2 = 4950.Number of ways to draw two non-second quality pieces: C(80, 2) = 80*79/2 = 3160.Therefore, P(neither is second quality) = 3160 / 4950.Therefore, P(B) = 1 - (3160 / 4950).Let me compute that. First, simplify 3160/4950.Divide numerator and denominator by 10: 316/495.Check if they can be reduced. Let's see: 316 and 495. The prime factors of 316: 4*79. 495: 5*99 = 5*9*11. So no common factors. Therefore, 316/495 ≈ 0.6383. Therefore, P(B) ≈ 1 - 0.6383 = 0.3617. So approximately 36.17%.Alternatively, using probabilities step by step: The probability that the first piece is not second quality is 80/100. Then, given the first was not second quality, the second is 79/99. Therefore, P(neither) = (80/100)*(79/99) = (80*79)/(100*99) = 6320/9900 = 316/495, same as before. So 316/495 ≈ 0.6383, so P(B) ≈ 0.3616, which is approximately 36.16%.But let me check if there's another way. Alternatively, since the probability of at least one is 1 - P(none). Which we did. Alternatively, P(B) can be calculated as P(exactly one) + P(exactly two). Let's see.Number of ways to choose exactly one second quality: C(20,1)*C(80,1) = 20*80 = 1600.Number of ways to choose exactly two second quality: C(20,2) = 190.Total favorable: 1600 + 190 = 1790.Therefore, P(B) = 1790 / 4950. Let's compute that: 1790 divided by 4950. Divide numerator and denominator by 10: 179/495. Let's check if that's equal to 1 - 316/495. 1 is 495/495, so 495 - 316 = 179. Yes, 179/495. So same result. 179/495 ≈ 0.3616. So approximately 36.16%.Therefore, the exact probability is 179/495. Simplify that fraction. Let's see if 179 and 495 have any common factors. 179 is a prime number? Let me check. 179: it's not divisible by 2, 3 (1+7+9=17, not divisible by 3), 5, 7 (179 divided by 7 is ~25.57), 11 (179/11=16.27), 13 (13.769), 17 (10.529). So 179 is prime. 495 factors into 5*99=5*9*11. So no common factors. So 179/495 is the simplified form. As a decimal, approximately 0.3616.Alternatively, converting to decimal: 179 divided by 495.Let me compute that. 495 goes into 179.000...495 goes into 1790 three times (3*495=1485), subtract 1485 from 1790, get 305. Bring down a zero: 3050.495 goes into 3050 six times (6*495=2970), subtract, get 80. Bring down a zero: 800.495 goes into 800 once (495), subtract, get 305. So we start repeating: 3050, same as before. So the decimal is 0.3616..., so approximately 0.3616.Therefore, the probability is approximately 36.16%.But the question says "find the probability," so they might want an exact fraction or a decimal. Let me check if 179/495 can be simplified more. Since 179 is prime, and 495 is 5*9*11, which doesn't include 179, so no. So the exact probability is 179/495, which can also be written as 358/990, but 179/495 is simplest.Alternatively, if we compute using the initial p = 0.2, which was derived from part (1). Wait, in part (1), p was 0.2, which would mean that in the batch, 20% are second quality, so 20 out of 100. So yes, exactly. Therefore, the number of second quality pieces is 20. Therefore, the calculation above is correct.Alternatively, if we use the hypergeometric distribution, since we are drawing without replacement from a finite population. The probability of at least one success (second quality) is 1 - P(0 successes). Which is exactly what we did.Alternatively, using combinations:P(B) = 1 - [C(80,2)/C(100,2)] = 1 - [ (80*79/2) / (100*99/2) ] = 1 - [6320/9900] = 1 - 632/990 = (990 - 632)/990 = 358/990 = 179/495. Same result.So all methods lead to 179/495 ≈ 0.3616. Therefore, part (2)'s answer is 179/495 or approximately 0.3616, which can be written as 0.3616 or as a fraction.But the problem says "put your final answer within boxed{}". So likely they want the exact fraction. Let me confirm:179/495 simplifies to, as I saw before, 179 is a prime, so 179/495 is the simplest form. Alternatively, maybe the problem expects a decimal approximation? But unless specified, exact fraction is better.But let me check if I can divide 179 by 495:Wait, 495 divided by 5 is 99. 179 divided by 5 is 35.8, so no. 179 is prime. So 179/495 is the exact value.Alternatively, maybe 179/495 can be simplified by dividing numerator and denominator by GCD(179,495). Since 179 is prime and doesn't divide 495, GCD is 1. Therefore, 179/495 is the simplest form.Alternatively, maybe write it as a decimal with four decimal places: approximately 0.3616.But since 179 ÷ 495 is 0.361616..., so it's 0.3616... repeating.But the question doesn't specify, so maybe leave it as a fraction. Let me check the original problem statement. For part (2), it says "find the probability of event B". It doesn't specify the form, so safest to give the exact fraction, 179/495. Alternatively, maybe they want it as a decimal? But since 179/495 is exact, and the problem didn't specify, so probably as a fraction.Wait, but let me check: 179/495. Let me see if that's reducible. Let's check 179: prime. 495: 5 × 99 = 5 × 9 × 11. No, so yes, 179 is prime, so no common factors. So 179/495 is correct.Alternatively, maybe there's a different approach. Suppose instead of combinations, use probability steps. The probability that the first is second quality is 20/100 = 0.2. The probability that the second is second quality given the first wasn't is 20/99. Wait, no, if you want at least one, it's 1 - P(neither). Which is the same as 1 - (80/100)*(79/99). Let me compute that:80/100 is 0.8, 79/99 ≈ 0.798. Multiply them: 0.8 * 0.798 ≈ 0.6384. Therefore, 1 - 0.6384 ≈ 0.3616, same as before. So yes, 0.3616.Alternatively, maybe the problem expects an answer using the probability p=0.2 from part (1), but since in part (2), it's a finite population of 100, we can't directly use the binomial probability. Wait, but if they had said "with replacement", then it would be 1 - (0.8)^2 = 1 - 0.64 = 0.36. But in this case, it's without replacement, so 0.3616. So the answer is slightly higher than 0.36. So in part (2), the answer is approximately 0.3616, or exactly 179/495.But let me verify once again. If the batch has 100 products, 20 second quality. So total non-second quality is 80. The number of ways to choose 2 non-second quality is C(80,2). The total ways is C(100,2). Therefore, P(B) = 1 - [C(80,2)/C(100,2)].Compute C(80,2) = (80×79)/2 = 3160. C(100,2) = (100×99)/2 = 4950. So 3160/4950 = 632/990 = 316/495. Therefore, 1 - 316/495 = 179/495. So yes, exactly 179/495. So this is the exact probability.Alternatively, if the problem expects a decimal, they might want it rounded to a certain decimal place, but since it's not specified, 179/495 is exact. Therefore, part (2)'s answer is 179/495.But let me cross-verify once again. If there are 20 second quality items in 100, the probability of drawing at least one in two draws is 1 minus the probability of drawing two non-second quality. The number of non-second quality is 80. So first draw: 80/100, second draw: 79/99. Multiply them: (80/100)*(79/99) = (80*79)/(100*99) = 6320/9900 = 632/990 = 316/495. Therefore, 1 - 316/495 = 179/495. Correct.Another way to think: the expected number of second quality in two draws is 2*(20/100) = 0.4, but expectation is different from probability. Alternatively, use hypergeometric probability:P(X ≥ 1) = 1 - P(X = 0) = 1 - [C(20,0)C(80,2)/C(100,2)] = 1 - [1 * 3160 / 4950] = 1 - 316/495 = 179/495. So same result. So that's consistent.Therefore, I think 179/495 is the correct answer for part (2).So summarizing:(1) The probability p is 0.2.(2) The probability of event B is 179/495.But let me check if 179/495 can be simplified further. Since 179 is a prime number, and 495 ÷ 179 is approximately 2.76, which is not an integer. Therefore, 179 and 495 share no common factors besides 1. Therefore, 179/495 is in simplest terms.Alternatively, as a decimal, approximately 0.3616. Depending on the required form, but as the problem doesn't specify, and mathematical problems often prefer fractions, I think 179/495 is acceptable. But sometimes, probabilities are written as decimals. If the problem expects a decimal, maybe round to four decimal places: 0.3616. But perhaps they want it as a reduced fraction. Alternatively, maybe the problem expects an answer in terms of p from part (1). Wait, but in part (2), the context is different: it's a finite batch of 100 with exactly 20 second quality products, so the probability is based on the actual counts. So the answer is indeed 179/495. Alternatively, if they had said "use the probability from part (1)", but since they specified 100 pieces, we have to use the exact count.Alternatively, if the problem had not mentioned part (1), but was standalone, but since part (1) gives p=0.2, which would correspond to 20 out of 100, so it's connected. Therefore, 179/495 is the exact answer.Therefore, I think that's the solution.Final Answer(1) boxed{0.2}(2) boxed{dfrac{179}{495}}</think>Given a batch of products, we need to solve two probability problems:1. Find the probability ( p ) that any one piece drawn from this batch is of second quality: - Event ( A ): "At most one of the two drawn products is of second quality" has a probability ( P(A) = 0.96 ). - The complement of event ( A ) is both drawn products being of second quality, which has probability ( 1 - 0.96 = 0.04 ). - Since the draws are independent, the probability of both being second quality is ( p^2 ). Solving ( p^2 = 0.04 ) gives ( p = sqrt{0.04} = 0.2 ).2. Find the probability of event ( B ): "At least one of the two drawn products is of second quality" when there are 100 pieces in the batch: - The number of second quality products in the batch is ( 20 ) (since ( p = 0.2 )). - The probability of event ( B ) is the complement of the probability that neither of the two drawn products is of second quality. - The number of ways to draw two non-second quality products from 80 is ( binom{80}{2} ), and the total number of ways to draw two products from 100 is ( binom{100}{2} ). - Calculating these, we get: [ P(B) = 1 - frac{binom{80}{2}}{binom{100}{2}} = 1 - frac{frac{80 times 79}{2}}{frac{100 times 99}{2}} = 1 - frac{3160}{4950} = 1 - frac{316}{495} = frac{179}{495} ] Final Answer(1) boxed{0.2}(2) boxed{dfrac{179}{495}},<think>Okay, let's try to tackle this problem step by step. First, I need to understand what's being asked here. There are two parts: part (1) asks for the probability p that any one piece drawn is of second quality, given that when we draw twice with replacement, the probability of having at most one second quality product is 0.96. Part (2) changes the scenario to a batch of 100 pieces, and we draw two without replacement (I think, because it's not specified, but usually if it's not mentioned, it's without replacement). Then we need to find the probability that at least one of the two is second quality.Starting with part (1). Let's parse the problem again. We have a product being drawn twice with replacement. Each time we draw 1 piece. Event A is "At most one of the two drawn products is of second quality," and P(A) = 0.96. We need to find p, the probability that any one piece is of second quality.So, first, let's model this. Since we're drawing with replacement, the two draws are independent. The probability of a single product being second quality is p, so the probability of it being first quality is 1 - p.Event A: At most one is second quality. That means either 0 or 1 of the two drawn products are second quality. So, P(A) is the sum of the probabilities of getting 0 second quality products plus the probability of getting 1 second quality product.In probability terms, this would be:P(A) = P(0 second quality) + P(1 second quality)Since each draw is independent, P(0 second quality) is (1 - p)^2, because both draws need to be first quality.P(1 second quality) is the probability that one is second quality and the other is first. Since there are two possible ways this can happen (first draw is second and second is first, or vice versa), this probability is 2 * p * (1 - p).So, putting it together:P(A) = (1 - p)^2 + 2p(1 - p) = 0.96We can simplify this equation to solve for p.First, let's compute the left side:(1 - p)^2 + 2p(1 - p) = 1 - 2p + p^2 + 2p - 2p^2 = 1 - p^2Wait, that's interesting. Let me check that again.Expand (1 - p)^2: 1 - 2p + p^2Then 2p(1 - p): 2p - 2p^2Add them together:1 - 2p + p^2 + 2p - 2p^2 = 1 - p^2So indeed, P(A) = 1 - p^2 = 0.96Therefore, p^2 = 1 - 0.96 = 0.04So p = sqrt(0.04) = 0.2Therefore, the probability p is 0.2. That seems straightforward, but let me verify.Wait, if p = 0.2, then the probability of getting two second quality products is p^2 = 0.04, so the probability of at most one is 1 - 0.04 = 0.96, which matches the given P(A). Yup, that checks out.So part (1) answer is p = 0.2.Now moving on to part (2). The batch has 100 pieces, and we draw two pieces at random. We need to find the probability of event B: "At least one of the two drawn products is of second quality."First, note that in part (1), the probability p of a single piece being second quality is 0.2. But does this relate directly to part (2)? Wait, maybe part (2) is a separate scenario where the batch has 100 pieces, but we need to know how many are second quality? Wait, the problem statement says: "If there are 100 pieces in this batch, and 2 pieces are randomly drawn, find the probability of event B: 'At least one of the two drawn products is of second quality'."Wait, but is part (2) dependent on part (1)? Or is it a separate question? Let me check the original problem.The problem is structured as two separate questions:(1) Find the probability p that any one piece drawn from this batch is of second quality;(2) If there are 100 pieces in this batch, and 2 pieces are randomly drawn, find the probability of event B: "At least one of the two drawn products is of second quality".Wait, so (1) is about finding p given that when you draw with replacement, P(A) = 0.96. Then (2) is a different scenario where the batch has 100 pieces, but does that batch have the same p as found in (1)? Or is it a separate batch where we have 100 pieces, but we need to use the p from part (1)?Wait, the problem statement starts with "From a batch of products...", and then part (2) says "If there are 100 pieces in this batch...", so "this batch" likely refers to the same batch as in part (1). Therefore, if in part (1), the probability p is 0.2, then in the batch of 100 pieces, the number of second quality products would be 100 * p = 20. Therefore, there are 20 second quality products in the batch.Therefore, part (2) is equivalent to: In a batch of 100 products with 20 second quality, what is the probability that when you draw two without replacement, at least one is second quality.Alternatively, the problem might assume that each product has a 0.2 probability of being second quality, even in part (2). But since part (2) specifies there are 100 pieces in the batch, and we are drawing two pieces, it's more likely that we are dealing with a hypergeometric distribution, where the number of second quality items is fixed. But we need to confirm.Wait, the problem might be a bit ambiguous. Let me read again.Original problem:From a batch of products, a product is drawn twice with replacement, each time randomly drawing 1 piece. Assume event A: "At most one of the two drawn products is of second quality" has a probability P(A) = 0.96.(1) Find the probability p that any one piece drawn from this batch is of second quality;(2) If there are 100 pieces in this batch, and 2 pieces are randomly drawn, find the probability of event B: "At least one of the two drawn products is of second quality".So part (2) is under the same batch, which is now specified to have 100 pieces. Since part (1) found that the probability p of a single piece being second quality is 0.2, then in a batch of 100 pieces, the expected number of second quality products is 20. However, the problem does not specify whether the batch has exactly 20 second quality products or if each product independently has a 20% chance of being second quality. But in part (1), the probability p is a parameter of the batch, so likely, the batch has a proportion p=0.2 of second quality products. Therefore, in a batch of 100 pieces, there are 20 second quality products.Therefore, part (2) is a hypergeometric problem where we have 20 second quality and 80 first quality, drawing 2 without replacement, find the probability that at least one is second quality.Alternatively, if the draws are with replacement, but the problem says "randomly drawn" without specifying replacement, so in probability problems, unless stated otherwise, it's usually without replacement. So we can assume it's without replacement.Therefore, in part (2), we can model this as a hypergeometric distribution where N=100, K=20, n=2, and we want P(at least one success). Alternatively, it's easier to compute 1 - P(no successes).So, P(B) = 1 - P(both are first quality)Compute P(both are first quality):Number of ways to choose 2 first quality: C(80, 2)Number of ways to choose any 2: C(100, 2)Therefore, P(B) = 1 - [C(80, 2)/C(100, 2)]Compute C(80, 2) = 80*79/2 = 3160C(100, 2) = 100*99/2 = 4950Thus, P(B) = 1 - (3160/4950) = (4950 - 3160)/4950 = 1790/4950Simplify that fraction:Divide numerator and denominator by 10: 179/495Check if they can be reduced. Let's see, 179 is a prime number? Let's check. 179 divided by 2, no. 3: 1+7+9=17, not divisible by 3. 5, no. 7: 179/7≈25.57, not integer. 11: 179/11≈16.27, nope. 13: 13*13=169, 13*14=182, too big. So 179 is prime. 495 divided by 5 is 99. 99 is 9*11. So 179 and 495 share no common factors. Therefore, the fraction is 179/495 ≈ 0.3616.Alternatively, compute decimal value:179 ÷ 495 ≈ 0.3616So P(B) ≈ 0.3616Alternatively, to get an exact fraction, but maybe the problem expects an exact value.Alternatively, use the probability approach. If there are 20 second quality and 80 first quality, then the probability that the first is first quality is 80/100, then the second is 79/99. So:P(no second quality) = (80/100)*(79/99) = (80*79)/(100*99) = (6320)/9900 = 632/990 = 316/495, same as above. Then P(B) = 1 - 316/495 = 179/495.So the exact probability is 179/495, which simplifies to approximately 0.3616 or 36.16%.Alternatively, if we had considered that each product has a 20% chance of being second quality independently, even in part (2), but that would be a binomial scenario with replacement, but since the batch has 100 pieces, and we're drawing without replacement, it's more accurate to model it as hypergeometric. But the problem says "If there are 100 pieces in this batch", implying that the previous probability p=0.2 is now corresponding to 20 second quality products. So I think the hypergeometric approach is correct here.Therefore, the answer for part (2) is 179/495, which is approximately 0.3616.But let me confirm once again. If the batch has 100 products, 20 of which are second quality. Drawing two without replacement.Probability of at least one second quality: 1 - probability both are first quality.Number of first quality: 80. So first draw: 80/100, second draw: 79/99. Multiply them: (80/100)*(79/99) = (4/5)*(79/99) = (316/495) ≈ 0.638, so 1 - 0.638 ≈ 0.362, which matches.Yes, so 179/495 ≈ 0.3616 is correct.Alternatively, maybe express it as a decimal rounded to four decimal places: 0.3616, or as a fraction 179/495. Alternatively, if possible, simplify further, but 179 is prime.Alternatively, another way to compute P(at least one) is P(exactly 1) + P(exactly 2). Let's check:P(exactly 1) = [C(20,1)*C(80,1)] / C(100,2) = (20*80)/4950 = 1600/4950P(exactly 2) = C(20,2)/C(100,2) = 190/4950Then total P(B) = (1600 + 190)/4950 = 1790/4950 = 179/495, same as before. So that's correct.Therefore, the answers are:(1) p = 0.2(2) P(B) = 179/495 ≈ 0.3616But check if part (2) requires a fractional answer or a decimal. Since the problem says "put your final answer within boxed{}", and in math problems, fractions are often preferred unless specified otherwise. So 179/495 can be simplified? Wait, 179 is prime. 495 divided by 5 is 99, which is 9*11. So no common factors. Thus, 179/495 is the simplest form. Alternatively, the problem might accept 0.36 or 0.3616, but since 179/495 is exact, probably better to leave it as a fraction.Alternatively, maybe we can write it as a decimal with two decimal places: approximately 0.36. But since the exact value is 179/495, which is roughly 0.3616, perhaps write it as boxed{dfrac{179}{495}}.Alternatively, check if the answer can be simplified more. Let me check 179 and 495. GCD(179,495). Since 495 ÷ 179 = 2 with remainder 495 - 2*179 = 495 - 358 = 137. Then GCD(179,137). 179 ÷ 137 = 1, remainder 42. GCD(137,42). 137 ÷ 42 = 3, remainder 11. GCD(42,11). 42 ÷ 11 = 3, remainder 9. GCD(11,9). 11 ÷ 9 = 1, remainder 2. GCD(9,2). 9 ÷ 2 = 4, remainder 1. GCD(2,1). Which is 1. So GCD is 1. Therefore, 179/495 is reduced completely.Therefore, the answers are:1. boxed{0.2}2. boxed{dfrac{179}{495}}Alternatively, for part (2), if the problem expects a decimal approximation, maybe box 0.3616, but since the question is in Chinese (the original user might be Chinese), and Chinese math problems often prefer exact fractions, so I'll stick with 179/495.Wait, but let me check my calculation once again for part (2). If there are 100 products with 20 second quality, drawing two.Total ways: C(100,2) = 4950Number of ways to have at least one second quality: total - number of ways with none.Number of ways with none: C(80,2) = 3160Thus, number of ways with at least one: 4950 - 3160 = 1790Thus, probability is 1790/4950 = 179/495. Correct.Yes, that's correct.Therefore, the final answers are:(1) p = 0.2(2) P(B) = 179/495Final Answer(1) boxed{0.2}(2) boxed{dfrac{179}{495}}</think>Given a batch of products, we draw a product twice with replacement, each time randomly drawing 1 piece. Event A: "At most one of the two drawn products is of second quality" has a probability ( P(A) = 0.96 ).(1) To find the probability ( p ) that any one piece drawn from this batch is of second quality:- Event A corresponds to having 0 or 1 second quality products in two draws. The probability of this event is given by: [ P(A) = (1 - p)^2 + 2p(1 - p) = 0.96 ]- Simplifying the left-hand side: [ (1 - p)^2 + 2p(1 - p) = 1 - p^2 ]- Therefore: [ 1 - p^2 = 0.96 implies p^2 = 0.04 implies p = sqrt{0.04} = 0.2 ]Thus, the probability ( p ) is (boxed{0.2}).(2) If there are 100 pieces in this batch, and 2 pieces are randomly drawn, find the probability of event B: "At least one of the two drawn products is of second quality":- Given ( p = 0.2 ), there are 20 second quality products in the batch of 100.- The probability of drawing two first quality products (without replacement) is: [ frac{80}{100} times frac{79}{99} = frac{3160}{4950} ]- Therefore, the probability of event B is: [ P(B) = 1 - frac{3160}{4950} = frac{1790}{4950} = frac{179}{495} ]Thus, the probability of event B is (boxed{dfrac{179}{495}}).]

🔑:Solution: (1) Let A_0 represent the event "none of the two drawn products is of second quality", and A_1 represent the event "exactly one of the two drawn products is of second quality". Then A_0 and A_1 are mutually exclusive, and A=A_0+A_1, so P(A)=P(A_0+A_1) =P(A_0)+P(A_1) =(1-p)^2+C_2^1p(1-p) =1-p^2 Thus, 0.96=1-p^2. Solving this, we get p_1=0.2, p_2=-0.2 (discard this solution). (2) Let B_0 represent the event "none of the two drawn products is of second quality", then B= overline {B_{0}}. If there are 100 pieces in this batch, from (1) we know there are 100 times 0.2 = 20 second quality pieces, so P(B_{0})= frac { C_{ 80 }^{ 2 }}{ C_{ 100 }^{ 2 }}= frac {316}{495}.P(B)=P( overline {B_{0}})=1-P(B_{0})=1- frac {316}{495}= frac {179}{495}Therefore, the answers are: (1) The probability p that any one piece drawn from this batch is of second quality is boxed{0.2}. (2) The probability of event B: "At least one of the two drawn products is of second quality" is boxed{frac {179}{495}}.

🧠:[<think>Okay, let's see. I need to solve this hyperbola problem. The hyperbola is given by the equation x²/m - y²/3 = 1, where m is greater than 0. The eccentricity is 2, and F is the right focus. There's a moving point P on the right branch of the hyperbola, and point A is at (0,1). The question is asking for the maximum value of |PF| - |PA|. The options are A through D. Alright, let's start breaking this down step by step.First, let's recall some basic properties of hyperbolas. The standard form is x²/a² - y²/b² = 1, where a is the distance from the center to the vertices, b is related to the conjugate axis, and c is the distance from the center to each focus, with c² = a² + b². The eccentricity e is given by e = c/a. Given the equation here is x²/m - y²/3 = 1, so comparing to the standard form, a² = m and b² = 3. The eccentricity is given as 2, so e = c/a = 2. Therefore, c = 2a. But since c² = a² + b², substituting c = 2a gives (2a)² = a² + b² => 4a² = a² + b² => 3a² = b². Wait, but in the problem, b² is 3. So 3a² = 3 => a² = 1. But a² is given as m in the equation. Therefore, m must be 1. Let me verify that.Given e = 2, so e = c/a = 2. Then c = 2a. Then c² = 4a². But also, c² = a² + b². Therefore, 4a² = a² + b² => 3a² = b². But the hyperbola equation has b² = 3, so 3a² = 3 => a² = 1. Since in the equation, the denominator under x² is m, which is a². Therefore, m = a² = 1. So m is 1. Therefore, the hyperbola equation is x²/1 - y²/3 = 1, which simplifies to x² - y²/3 = 1. So that's the hyperbola.Then, the foci are at (±c, 0). Since c² = a² + b² = 1 + 3 = 4, so c = 2. Therefore, the foci are at (2,0) and (-2,0). The right focus F is (2,0).Now, point P is on the right branch of the hyperbola, so its x-coordinate is greater than or equal to a, which is 1. Since the right branch is for x ≥ a, so x ≥ 1.We need to find the maximum value of |PF| - |PA|. Let me think. So |PF| is the distance from point P to the focus F(2,0), and |PA| is the distance from P to point A(0,1). We need to find the maximum of |PF| - |PA| as P moves along the right branch.Hmm. How do I approach this? Maybe using properties of hyperbolas? Let me recall that for hyperbolas, the difference of distances from any point on the hyperbola to the two foci is constant and equal to 2a. Wait, but here we have |PF| - |PA|, not the difference between the two foci. Hmm. So maybe that property isn't directly applicable here.Alternatively, maybe express the coordinates of point P parametrically and then express |PF| - |PA| in terms of that parameter, then find its maximum.Let me consider parametrizing the hyperbola. For hyperbolas, a common parametrization is (a secθ, b tanθ). Since this hyperbola is x² - y²/3 = 1, so a = 1, b = sqrt(3). So the parametric equations would be x = secθ, y = sqrt(3) tanθ, where θ is a parameter. Then, point P can be written as (secθ, sqrt(3) tanθ). Since P is on the right branch, secθ ≥ 1, which is always true except at θ = π/2, 3π/2, etc., but those would be asymptotes.Alternatively, maybe use coordinates. Let me let P = (x, y) on the hyperbola x² - y²/3 = 1, with x ≥ 1. Then, |PF| is the distance from (x, y) to (2,0), which is sqrt[(x - 2)² + y²]. Similarly, |PA| is the distance from (x, y) to (0,1), which is sqrt[x² + (y - 1)²]. So we need to maximize sqrt[(x - 2)^2 + y^2] - sqrt[x^2 + (y - 1)^2].Hmm, that seems complicated. Maybe there's a smarter way. Let me think. Since we are dealing with hyperbola, maybe reflecting properties? Wait, for hyperbolas, the reflection property is about the difference of angles to the foci, but I don't know if that helps here.Alternatively, maybe use calculus. Let's set up the function to maximize: f(x, y) = sqrt[(x - 2)^2 + y^2] - sqrt[x^2 + (y - 1)^2], subject to the constraint x² - y²/3 = 1. Then, use Lagrange multipliers. But that might be messy. Alternatively, parameterize the hyperbola and then take the derivative with respect to the parameter. Let's try that.Using the parametric equations x = secθ, y = sqrt(3) tanθ. Then, substitute into the expression |PF| - |PA|.Compute |PF|: sqrt[(secθ - 2)^2 + (sqrt(3) tanθ)^2]Compute |PA|: sqrt[sec²θ + (sqrt(3) tanθ - 1)^2]So f(θ) = sqrt[(secθ - 2)^2 + 3 tan²θ] - sqrt[sec²θ + (sqrt(3) tanθ - 1)^2]This looks complicated, but maybe we can simplify it.Let me compute each term step by step.First term: sqrt[(secθ - 2)^2 + 3 tan²θ]Let's expand (secθ - 2)^2: sec²θ - 4 secθ + 4So adding 3 tan²θ gives sec²θ - 4 secθ + 4 + 3 tan²θBut sec²θ = 1 + tan²θ, so substituting:(1 + tan²θ) - 4 secθ + 4 + 3 tan²θ = 1 + tan²θ - 4 secθ + 4 + 3 tan²θ = 5 + 4 tan²θ - 4 secθSo first term becomes sqrt(5 + 4 tan²θ - 4 secθ)Second term: sqrt[sec²θ + (sqrt(3) tanθ - 1)^2]Compute (sqrt(3) tanθ - 1)^2: 3 tan²θ - 2 sqrt(3) tanθ + 1Add sec²θ: sec²θ + 3 tan²θ - 2 sqrt(3) tanθ + 1Again, sec²θ = 1 + tan²θ, so substituting:1 + tan²θ + 3 tan²θ - 2 sqrt(3) tanθ + 1 = 2 + 4 tan²θ - 2 sqrt(3) tanθSo the second term is sqrt(2 + 4 tan²θ - 2 sqrt(3) tanθ)Therefore, f(θ) = sqrt(5 + 4 tan²θ - 4 secθ) - sqrt(2 + 4 tan²θ - 2 sqrt(3) tanθ)Hmm, this still looks complicated. Maybe there's another substitution or identity we can use here.Alternatively, perhaps use trigonometric identities. Let me recall that tanθ = sinθ / cosθ, secθ = 1 / cosθ. Let's denote t = θ. Let me try to write everything in terms of cosθ and sinθ.But maybe it's better to use hyperbolic functions. Wait, hyperbola parametrization can also be done with hyperbolic functions: x = a cosh t, y = b sinh t. For hyperbola x²/a² - y²/b² = 1, so in this case, x = cosh t, y = sqrt(3) sinh t. Maybe this parametrization is easier. Let's try that.So let P = (cosh t, sqrt(3) sinh t), where t is a real number. Then, compute |PF| and |PA|.Compute |PF|: distance from (cosh t, sqrt(3) sinh t) to (2, 0):sqrt[(cosh t - 2)^2 + (sqrt(3) sinh t - 0)^2] = sqrt[(cosh t - 2)^2 + 3 sinh² t]Similarly, |PA|: distance from (cosh t, sqrt(3) sinh t) to (0,1):sqrt[(cosh t - 0)^2 + (sqrt(3) sinh t - 1)^2] = sqrt[cosh² t + (sqrt(3) sinh t - 1)^2]So f(t) = sqrt[(cosh t - 2)^2 + 3 sinh² t] - sqrt[cosh² t + (sqrt(3) sinh t - 1)^2]Again, this seems complicated. Maybe we can simplify these expressions.Let me first compute the first square root:sqrt[(cosh t - 2)^2 + 3 sinh² t]Expand (cosh t - 2)^2: cosh² t - 4 cosh t + 4Add 3 sinh² t: cosh² t - 4 cosh t + 4 + 3 sinh² tBut we know that cosh² t - sinh² t = 1, so sinh² t = cosh² t - 1. Therefore, substitute sinh² t:cosh² t - 4 cosh t + 4 + 3(cosh² t - 1) = cosh² t - 4 cosh t + 4 + 3 cosh² t - 3 = 4 cosh² t - 4 cosh t + 1Therefore, the first term becomes sqrt(4 cosh² t - 4 cosh t + 1)Similarly, compute the second square root:sqrt[cosh² t + (sqrt(3) sinh t - 1)^2]Compute (sqrt(3) sinh t - 1)^2: 3 sinh² t - 2 sqrt(3) sinh t + 1Add cosh² t: cosh² t + 3 sinh² t - 2 sqrt(3) sinh t + 1Again, using sinh² t = cosh² t - 1:cosh² t + 3(cosh² t - 1) - 2 sqrt(3) sinh t + 1 = cosh² t + 3 cosh² t - 3 - 2 sqrt(3) sinh t + 1 = 4 cosh² t - 2 sqrt(3) sinh t - 2Therefore, the second term is sqrt(4 cosh² t - 2 sqrt(3) sinh t - 2)So now f(t) = sqrt(4 cosh² t - 4 cosh t + 1) - sqrt(4 cosh² t - 2 sqrt(3) sinh t - 2)Hmm. Let me see if I can simplify sqrt(4 cosh² t - 4 cosh t + 1). Let me denote u = cosh t. Then, 4u² - 4u + 1. Let's see if this factors.4u² -4u +1. The discriminant is 16 - 16 = 0, so it's a perfect square: (2u -1)^2. Therefore, sqrt(4 cosh² t -4 cosh t +1) = |2 cosh t -1|. Since cosh t is always ≥1, so 2 cosh t -1 ≥ 2*1 -1 =1 ≥1. Therefore, sqrt(4 cosh² t -4 cosh t +1) = 2 cosh t -1.Wow, that's a nice simplification. So the first term simplifies to 2 cosh t -1.Now the second term is sqrt(4 cosh² t - 2 sqrt(3) sinh t - 2). Let me see if that can be simplified.Express 4 cosh² t - 2 sqrt(3) sinh t - 2. Maybe try to express this as a square or similar. Let me see:We know that cosh² t = 1 + sinh² t. Therefore, 4 cosh² t = 4 + 4 sinh² t. So substituting:4 cosh² t - 2 sqrt(3) sinh t - 2 = 4 + 4 sinh² t - 2 sqrt(3) sinh t - 2 = 2 + 4 sinh² t - 2 sqrt(3) sinh tLet me factor out a 2: 2[1 + 2 sinh² t - sqrt(3) sinh t]. Hmm, not sure if helpful. Maybe complete the square in terms of sinh t.Let me write the expression as 4 sinh² t - 2 sqrt(3) sinh t + 2. Wait, 4 sinh² t - 2 sqrt(3) sinh t + 2. Let's check:Wait, original expression after substitution was 2 + 4 sinh² t - 2 sqrt(3) sinh t. So 4 sinh² t -2 sqrt(3) sinh t +2. Let me factor it as:Let’s consider quadratic in sinh t: 4 sinh² t -2 sqrt(3) sinh t +2. Let me see if this quadratic can be written as (a sinh t + b)^2 + c.But 4 sinh² t -2 sqrt(3) sinh t +2 = 4 sinh² t -2 sqrt(3) sinh t + ( (sqrt(3)/2)^2 )* (4/3) + ... Hmm, maybe not straightforward. Alternatively, maybe complete the square.Take 4 sinh² t -2 sqrt(3) sinh t = 4[sinh² t - (sqrt(3)/2) sinh t]. To complete the square inside the brackets:sinh² t - (sqrt(3)/2) sinh t = sinh² t - (sqrt(3)/2) sinh t + (3/16) - (3/16) = [sinh t - (sqrt(3)/4)]^2 - 3/16.Therefore, 4[sinh² t - (sqrt(3)/2) sinh t] + 2 = 4[ [sinh t - sqrt(3)/4]^2 - 3/16 ] + 2 = 4[sinh t - sqrt(3)/4]^2 - 4*(3/16) + 2 = 4[sinh t - sqrt(3)/4]^2 - 3/4 + 2 = 4[sinh t - sqrt(3)/4]^2 + 5/4.Therefore, the second term becomes sqrt(4[sinh t - sqrt(3)/4]^2 + 5/4). Hmm, that's sqrt(4*(something)^2 + 5/4). Not sure if that helps. Alternatively, since sinh t can be expressed in terms of exponentials, but that might not help either.So now, f(t) = (2 cosh t -1) - sqrt(4 sinh² t - 2 sqrt(3) sinh t + 2)Wait, but maybe there's another way. Let's consider the expression inside the second square root: 4 sinh² t - 2 sqrt(3) sinh t +2. Let me check discriminant if it's a perfect square.Quadratic in sinh t: 4 sinh² t -2 sqrt(3) sinh t +2. Discriminant D = ( -2 sqrt(3) )² - 4*4*2 = 12 - 32 = -20 <0. So it's always positive, which makes sense as it's under a square root.Hmm. Alternatively, maybe express sinh t in terms of cosh t. Since sinh² t = cosh² t -1. Wait, but we already did that earlier.Alternatively, perhaps use numerical methods? But since it's a problem with options, maybe there's a geometric interpretation.Wait, going back to the original problem: maximize |PF| - |PA|. Let's think about triangle inequality or something. Maybe interpret |PF| - |PA| ≤ |FA|? But that would be if we use triangle inequality |PF| ≤ |PA| + |FA|, so |PF| - |PA| ≤ |FA|. But |FA| is the distance between F(2,0) and A(0,1), which is sqrt( (2)^2 + (-1)^2 ) = sqrt(4 +1)=sqrt(5). But sqrt(5) is option A, which is one of the choices. However, the question asks for the maximum value. But can this difference actually reach sqrt(5)?Wait, triangle inequality tells us that |PF| - |PA| ≤ |FA|. So the maximum possible value is |FA|, but this maximum is achieved when P lies on the line segment extending from F to A, but since P is constrained to be on the hyperbola, maybe it's achieved at some specific point. But since the hyperbola is on the right branch, maybe when PA is minimized? Hmm, not sure.Alternatively, maybe consider the reflection property of hyperbola. Wait, for hyperbola, the difference |PF1| - |PF2| = 2a for points on the right branch, where F1 and F2 are the two foci. But here we have |PF| - |PA|, not the difference between two foci. So maybe not directly applicable.Wait, but if we can somehow relate |PA| to another focus. Let me think. Suppose we consider a reflection. For ellipses, there's a reflection property involving foci, but hyperbolas have different properties. For hyperbola, the reflection property is that the tangent line at any point P makes equal angles with the lines from P to each focus. But I don't know if that helps here.Alternatively, maybe define another focus or use auxiliary points. Let me try to see.Wait, perhaps if we consider point A(0,1) and try to relate it to the hyperbola's properties. Let me check what the value of |PF| - |PA| would be. If we can express |PF| - |PA| as something related to the hyperbola's definition.But hyperbola's definition is |PF1 - PF2| = 2a. Here, F1 is (2,0), F2 is (-2,0). But we are dealing with |PF1| - |PA|.Alternatively, maybe introduce another focus. Wait, maybe not. Let's think differently.Since the hyperbola is x² - y²/3 =1, right focus F is (2,0). Point A is (0,1). Let's plot these points to get an idea.The hyperbola opens left and right, centered at the origin. The right focus is at (2,0). Point A is at (0,1). So point A is above the center.We need to find the maximum of |PF| - |PA| for P on the right branch. Let's consider if we can write this as |PF| - |PA| = (|PF| - |PA|). To maximize this, maybe we can consider |PF| as large as possible and |PA| as small as possible. But since P is on the hyperbola, as P moves towards infinity, |PF| and |PA| both go to infinity, but their difference might approach a limit.Alternatively, compute the limit as t approaches infinity of f(t) = (2 cosh t -1) - sqrt(4 sinh² t -2 sqrt(3) sinh t +2). Let's approximate for large t. cosh t ≈ sinh t ≈ (e^t)/2. So for large t, cosh t ≈ sinh t ≈ e^t / 2. Then:2 cosh t -1 ≈ 2*(e^t / 2) -1 = e^t -1sqrt(4 sinh² t -2 sqrt(3) sinh t +2) ≈ sqrt(4*(e^{2t}/4) -2 sqrt(3)*(e^t /2) +2) = sqrt(e^{2t} - sqrt(3) e^t + 2). For large t, the dominant term is e^{2t}, so sqrt(e^{2t} - sqrt(3) e^t +2) ≈ e^t - (sqrt(3)/2) + ... (using the expansion sqrt(e^{2t} - a e^t + b) ≈ e^t - a/2 + ... )Therefore, f(t) ≈ (e^t -1) - (e^t - sqrt(3)/2) ≈ -1 + sqrt(3)/2 ≈ -1 + 0.866 ≈ -0.134. So as t approaches infinity, f(t) approaches approximately -0.134, which is negative. So the maximum is not at infinity.Alternatively, as t approaches 0. Let me compute f(0). At t=0, cosh t =1, sinh t=0. So:First term: 2*1 -1 =1Second term: sqrt(4*0 -2 sqrt(3)*0 +2)=sqrt(2). So f(0)=1 - sqrt(2) ≈1 -1.414≈-0.414. Negative.Wait, so at t=0, it's negative. At t approaching infinity, it's approaching around -0.134. Maybe the maximum occurs somewhere in between?Alternatively, check at vertex. The vertex of the hyperbola is at (1,0). Let's compute |PF| - |PA| at P=(1,0).|PF| is distance from (1,0) to (2,0): sqrt((1-2)^2 +0)=1|PA| is distance from (1,0) to (0,1): sqrt(1 +1)=sqrt(2). So |PF| - |PA| =1 - sqrt(2)≈-0.414. Same as t=0.Wait, but the right vertex is (1,0). Hmm. Maybe check another point. Let's take P=(2, y). Let's find the y-coordinate when x=2 on the hyperbola. Plug x=2 into x² - y²/3=1: 4 - y²/3=1 ⇒ y²=9 ⇒ y=±3. So points (2,3) and (2,-3). Let's compute |PF| - |PA| for P=(2,3):|PF| is distance from (2,3) to (2,0): 3 units.|PA| is distance from (2,3) to (0,1): sqrt(4 +4)=sqrt(8)=2 sqrt(2). So |PF| - |PA|=3 - 2 sqrt(2)≈3 -2.828≈0.172. Positive.Similarly, for P=(2,-3):|PF| is distance to (2,0): 3 units.|PA| is distance to (0,1): sqrt(4 +16)=sqrt(20)=2 sqrt(5). So |PF| - |PA|=3 -2 sqrt(5)≈3 -4.472≈-1.472. Negative. So maximum here is 3 - 2 sqrt(2)≈0.172. But none of the answer options match this. Wait, the options are sqrt5≈2.236, sqrt5 -2≈0.236, 2sqrt2≈2.828, 2sqrt2 -2≈0.828. So 0.172 is closest to B: sqrt5 -2≈0.236, but still not matching.Hmm. Maybe there's another point where the maximum is achieved. Let's try another point. Let's take P=(sqrt(2), y). Plugging into hyperbola: (sqrt(2))² - y²/3=1 ⇒2 - y²/3=1 ⇒y²=3 ⇒y=±sqrt(3). Compute |PF| - |PA|.For P=(sqrt(2), sqrt(3)):|PF|: distance to (2,0): sqrt( (sqrt(2)-2)^2 + (sqrt(3))^2 )Compute (sqrt(2)-2)^2 = (2 -4 sqrt(2) +4)=6 -4 sqrt(2). Then add 3: total 9 -4 sqrt(2). sqrt(9 -4 sqrt(2)).Similarly, |PA|: distance to (0,1): sqrt( (sqrt(2))^2 + (sqrt(3)-1)^2 )=sqrt(2 + (sqrt(3)-1)^2 )Compute (sqrt(3)-1)^2=3 - 2 sqrt(3)+1=4 - 2 sqrt(3). So |PA|=sqrt(2 +4 -2 sqrt(3))=sqrt(6 -2 sqrt(3)).So |PF| - |PA|=sqrt(9 -4 sqrt(2)) - sqrt(6 -2 sqrt(3)). Let's approximate these values:sqrt(9 -4 sqrt(2))≈sqrt(9 -5.656)=sqrt(3.344)≈1.829sqrt(6 -2 sqrt(3))≈sqrt(6 -3.464)=sqrt(2.536)≈1.592So difference≈1.829 -1.592≈0.237. Hmm, this is very close to option B: sqrt5 -2≈2.236-2≈0.236. So maybe the maximum is sqrt5 -2. But how does that come?Wait, in the previous example, with P=(2,3), we had |PF| - |PA|≈0.172, which is less than 0.236. At P=(sqrt(2), sqrt(3)), we have≈0.237, which is almost equal to sqrt5 -2≈0.236. So maybe that's the maximum? Then the answer is B.But how to confirm this? Let's see. Maybe the maximum occurs when PA is minimized and PF is maximized? But since PA and PF are related through the position of P on the hyperbola, it's not straightforward.Alternatively, consider using calculus to find the maximum. Let's consider the parametrization with hyperbolic functions. Let me define f(t) = 2 cosh t -1 - sqrt(4 sinh² t -2 sqrt(3) sinh t +2)To find the maximum of f(t), take the derivative f’(t), set it to zero.First, compute f’(t):f(t) = 2 cosh t -1 - sqrt(4 sinh² t -2 sqrt(3) sinh t +2)Let me denote the second term as sqrt(g(t)), where g(t) =4 sinh² t -2 sqrt(3) sinh t +2.Then f’(t) = 2 sinh t - [ (1/(2 sqrt(g(t)))) * (8 sinh t cosh t -2 sqrt(3) cosh t) ]Simplify:f’(t)=2 sinh t - [ (8 sinh t cosh t -2 sqrt(3) cosh t) / (2 sqrt(g(t))) ]=2 sinh t - [ (4 sinh t cosh t - sqrt(3) cosh t) / sqrt(g(t)) ]Set this derivative equal to zero:2 sinh t - (4 sinh t cosh t - sqrt(3) cosh t)/sqrt(g(t)) =0Multiply both sides by sqrt(g(t)):2 sinh t sqrt(g(t)) -4 sinh t cosh t + sqrt(3) cosh t =0This is a complicated equation. Let me see if I can factor anything out.Factor out cosh t:cosh t [ -4 sinh t + sqrt(3) ] + 2 sinh t sqrt(g(t)) =0Not sure. Let me try substituting t such that sinh t = k, but maybe not helpful.Alternatively, let's assume that at the maximum, the derivative is zero, so:2 sinh t sqrt(g(t)) =4 sinh t cosh t - sqrt(3) cosh tDivide both sides by cosh t (since cosh t ≠0):2 sinh t sqrt(g(t))/cosh t =4 sinh t - sqrt(3)Let me write this as:2 tanh t sqrt(g(t)) =4 sinh t - sqrt(3)But sqrt(g(t)) is sqrt(4 sinh² t -2 sqrt(3) sinh t +2). Hmm, still complicated.Alternatively, maybe assume that the maximum occurs at a specific point where P, F, and A are colinear? Let me check. If P is on the line connecting F(2,0) and A(0,1), then P can be parametrized as (2 -2s, 0 +s) for s ≥0 (since P is on the right branch, so x=2-2s ≥1 ⇒2-2s ≥1 ⇒s ≤0.5. So s ∈ [0, 0.5]. Then, check if this line intersects the hyperbola.So substitute x=2-2s, y=s into the hyperbola equation x² - y²/3=1:(2 -2s)^2 - (s²)/3 =1Expand: 4 -8s +4s² - s²/3 =1Multiply through by 3 to eliminate denominator:12 -24s +12s² -s² =3Simplify: 12 -24s +11s² =3 ⇒11s² -24s +9=0Solve quadratic equation: s=(24 ±sqrt(576 - 396))/22=(24 ±sqrt(180))/22=(24 ±6 sqrt(5))/22=(12 ±3 sqrt(5))/11Compute numerical values:sqrt(5)≈2.236, so 3 sqrt(5)≈6.708s=(12 ±6.708)/11First solution: (12 +6.708)/11≈18.708/11≈1.700>0.5, which is outside the interval [0,0.5]. Second solution: (12 -6.708)/11≈5.292/11≈0.481≈0.481<0.5. So s≈0.481.Therefore, the line FA intersects the hyperbola at s≈0.481, P≈(2 -2*0.481, 0.481)=(2 -0.962, 0.481)=(1.038, 0.481). So point P≈(1.038, 0.481). Let's compute |PF| - |PA| here.Compute |PF|: distance from (1.038,0.481) to (2,0): sqrt( (1.038-2)^2 + (0.481)^2 )=sqrt( (-0.962)^2 +0.231 )=sqrt(0.925 +0.231)=sqrt(1.156)=1.075Compute |PA|: distance from (1.038,0.481) to (0,1): sqrt( (1.038)^2 + (-0.519)^2 )=sqrt(1.077 +0.269)=sqrt(1.346)=1.161So |PF| - |PA|≈1.075 -1.161≈-0.086. Negative. So even at the intersection point, the value is negative. Therefore, perhaps the maximum occurs elsewhere.Alternatively, maybe the maximum occurs when PA is minimized. The minimal PA would be the shortest distance from A to the hyperbola. Let's compute that. The minimal distance from A(0,1) to the hyperbola x² - y²/3=1.To find the minimal distance, we can set up the distance squared function: d² =x² + (y -1)^2, subject to x² - y²/3=1. Use Lagrange multipliers.Define L =x² + (y -1)^2 + λ(x² - y²/3 -1)Partial derivatives:∂L/∂x=2x +2λx=0 ⇒x(2 +2λ)=0 ⇒x=0 or λ=-1∂L/∂y=2(y -1) - (2λ y)/3=0∂L/∂λ=x² - y²/3 -1=0If x=0: Then from hyperbola equation: 0 - y²/3=1 ⇒y²=-3, which is impossible. So discard x=0. Therefore, λ=-1.Then, substitute λ=-1 into the second equation:2(y -1) - (2*(-1)y)/3=0 ⇒2(y -1) + (2y)/3=0Multiply through by 3: 6(y -1) +2y=0 ⇒6y -6 +2y=0 ⇒8y=6 ⇒y=6/8=3/4=0.75Then, from the hyperbola equation: x² - ( (3/4)^2 )/3 =1 ⇒x² - (9/16)/3=1 ⇒x² - 3/16=1 ⇒x²=19/16 ⇒x=±sqrt(19)/4≈±1.089. Since we are on the right branch, x= sqrt(19)/4≈1.089. Therefore, the closest point is (sqrt(19)/4, 3/4). Compute |PA| here.PA distance: sqrt( (sqrt(19)/4)^2 + (3/4 -1)^2 )=sqrt(19/16 + (-1/4)^2 )=sqrt(19/16 +1/16)=sqrt(20/16)=sqrt(5)/2≈1.118. So minimal |PA|≈1.118.Then, at this point P=(sqrt(19)/4, 3/4), compute |PF| - |PA|.|PF|: distance from P to F(2,0): sqrt( (sqrt(19)/4 -2)^2 + (3/4)^2 )Compute sqrt(19)/4≈1.089/4≈0.272. Wait, no. sqrt(19)=4.3589, so sqrt(19)/4≈1.089.Therefore, sqrt(19)/4 -2≈1.089 -2≈-0.911So (sqrt(19)/4 -2)^2≈(-0.911)^2≈0.830Plus (3/4)^2=0.5625. So total≈0.830 +0.5625≈1.3925. sqrt≈1.180Therefore, |PF|≈1.180, |PA|≈1.118. So |PF| - |PA|≈1.180 -1.118≈0.062. Positive, but small.But earlier, at P=(sqrt(2), sqrt(3)), we had≈0.237. Which is larger. So this isn't the maximum either.Hmm. Maybe the maximum occurs when the derivative is zero, which would be at some critical point. Since this is getting too involved, maybe we can use the parametric form and set up the derivative. Let's revisit the derivative:We had f’(t)=2 sinh t - [ (4 sinh t cosh t - sqrt(3) cosh t) / sqrt(g(t)) ] =0Let me rearrange:2 sinh t = (4 sinh t cosh t - sqrt(3) cosh t) / sqrt(g(t))Multiply both sides by sqrt(g(t)):2 sinh t sqrt(g(t)) =4 sinh t cosh t - sqrt(3) cosh tDivide both sides by cosh t (since cosh t >0):2 sinh t sqrt(g(t))/cosh t =4 sinh t - sqrt(3)Let me denote tanh t = sinh t / cosh t. Then,2 tanh t sqrt(g(t))=4 sinh t - sqrt(3)But g(t)=4 sinh² t -2 sqrt(3) sinh t +2. Let's see.Alternatively, let me square both sides to eliminate the square root. Let me define:Left side: 2 tanh t sqrt(g(t))Right side:4 sinh t - sqrt(3)Square both sides:4 tanh² t g(t) =16 sinh² t -8 sqrt(3) sinh t +3Express tanh² t = (sinh² t)/(cosh² t). So:4 (sinh² t / cosh² t) g(t) =16 sinh² t -8 sqrt(3) sinh t +3But g(t)=4 sinh² t -2 sqrt(3) sinh t +2. Substitute:4 (sinh² t / cosh² t)(4 sinh² t -2 sqrt(3) sinh t +2) =16 sinh² t -8 sqrt(3) sinh t +3This equation looks very complicated, but maybe substituting using hyperbolic identities. Remember that cosh² t - sinh² t =1.Let me compute left side:Left side=4 sinh² t / cosh² t * (4 sinh² t -2 sqrt(3) sinh t +2)Multiply numerator and denominator:=4 sinh² t (4 sinh² t -2 sqrt(3) sinh t +2)/cosh² tBut perhaps express everything in terms of sinh t. Let me let u = sinh t. Then cosh² t =1 + u². So:Left side=4 u² (4 u² -2 sqrt(3) u +2)/(1 + u²)Right side=16 u² -8 sqrt(3) u +3Therefore, equation becomes:4 u² (4 u² -2 sqrt(3) u +2)/(1 + u²) =16 u² -8 sqrt(3) u +3Multiply both sides by (1 + u²):4 u² (4 u² -2 sqrt(3) u +2) = (16 u² -8 sqrt(3) u +3)(1 + u²)Expand both sides:Left side:4 u² *4 u² +4 u²*(-2 sqrt(3) u) +4 u²*2=16 u^4 -8 sqrt(3) u^3 +8 u²Right side:16 u²*(1) +16 u²*u² -8 sqrt(3) u*(1) -8 sqrt(3) u*u² +3*(1) +3*u²=16 u² +16 u^4 -8 sqrt(3) u -8 sqrt(3) u^3 +3 +3 u²Simplify right side:16 u^4 -8 sqrt(3) u^3 + (16 u² +3 u²) -8 sqrt(3) u +3=16 u^4 -8 sqrt(3) u^3 +19 u² -8 sqrt(3) u +3Set left side equal to right side:16 u^4 -8 sqrt(3) u^3 +8 u² =16 u^4 -8 sqrt(3) u^3 +19 u² -8 sqrt(3) u +3Subtract left side from both sides:0 = (16 u^4 -8 sqrt(3) u^3 +19 u² -8 sqrt(3) u +3) - (16 u^4 -8 sqrt(3) u^3 +8 u²)Simplify:0= (0) + (0) + (19u² -8u²) -8 sqrt(3) u +3=11u² -8 sqrt(3) u +3Thus, we have:11u² -8 sqrt(3) u +3=0Solve quadratic equation for u:u=(8 sqrt(3) ±sqrt( (8 sqrt(3))^2 -4*11*3 ))/(2*11)Calculate discriminant:(8 sqrt(3))² -4*11*3= 64*3 -132=192 -132=60Thus:u=(8 sqrt(3) ±sqrt(60))/22= (8 sqrt(3) ±2 sqrt(15))/22=(4 sqrt(3) ±sqrt(15))/11So two solutions:u1=(4 sqrt(3) +sqrt(15))/11≈(6.928 +3.872)/11≈10.8/11≈0.981u2=(4 sqrt(3) -sqrt(15))/11≈(6.928 -3.872)/11≈3.056/11≈0.278So sinh t=u1≈0.981 or sinh t=u2≈0.278Compute t for each sinh t:For sinh t≈0.981: t≈sinh^{-1}(0.981)≈0.875For sinh t≈0.278: t≈sinh^{-1}(0.278)≈0.275Now, let's compute f(t) at these t values.First, t≈0.875:Compute cosh t≈cosh(0.875)= (e^{0.875} +e^{-0.875})/2≈(2.399 +0.416)/2≈1.407sinh t≈0.981Compute first term:2 cosh t -1≈2*1.407 -1≈2.814 -1=1.814Second term:sqrt(4 sinh² t -2 sqrt(3) sinh t +2)=sqrt(4*(0.981)^2 -2*1.732*0.981 +2)≈sqrt(4*0.962 -3.397 +2)=sqrt(3.848 -3.397 +2)=sqrt(2.451)=≈1.565Thus, f(t)=1.814 -1.565≈0.249Second, t≈0.275:Compute cosh t≈cosh(0.275)= (e^{0.275}+e^{-0.275})/2≈(1.316 +0.760)/2≈1.038sinh t≈0.278First term:2*1.038 -1≈2.076 -1=1.076Second term:sqrt(4*(0.278)^2 -2*1.732*0.278 +2)≈sqrt(4*0.077 -0.965 +2)=sqrt(0.308 -0.965 +2)=sqrt(1.343)≈1.159Thus, f(t)=1.076 -1.159≈-0.083Therefore, the maximum occurs at t≈0.875, giving f(t)≈0.249≈0.25, which is approximately sqrt(5)-2≈2.236-2≈0.236. Close, considering the approximations. So the exact value would be sqrt(5)-2, which is option B.Therefore, the answer is B: sqrt{5}-2.Final Answerboxed{B}</think>Given the hyperbola ( E: frac{x^2}{m} - frac{y^2}{3} = 1 ) with ( m > 0 ) and an eccentricity of 2, we need to find the maximum value of ( |PF| - |PA| ) where ( F ) is the right focus and ( A(0,1) ).1. Determine the value of ( m ): - The eccentricity ( e = 2 ). - For a hyperbola, ( e = frac{c}{a} ), where ( c = sqrt{a^2 + b^2} ). - Given ( e = 2 ), we have ( c = 2a ). - Using ( c^2 = a^2 + b^2 ), substituting ( c = 2a ) gives ( 4a^2 = a^2 + 3 ), leading to ( 3a^2 = 3 ) and ( a^2 = 1 ). Thus, ( m = 1 ).2. Identify the hyperbola equation and foci: - The hyperbola equation is ( x^2 - frac{y^2}{3} = 1 ). - The foci are at ( (pm 2, 0) ). The right focus ( F ) is ( (2, 0) ).3. Parametrize the hyperbola: - Use hyperbolic functions: ( x = cosh t ), ( y = sqrt{3} sinh t ).4. Express distances: - ( |PF| = sqrt{(cosh t - 2)^2 + 3 sinh^2 t} ) - ( |PA| = sqrt{cosh^2 t + (sqrt{3} sinh t - 1)^2} )5. Simplify expressions: - Simplify ( |PF| ) to ( 2 cosh t - 1 ). - Simplify ( |PA| ) to ( sqrt{4 sinh^2 t - 2 sqrt{3} sinh t + 2} ).6. Maximize the expression: - Use calculus and solve the derivative to find critical points. - Solve the quadratic equation for critical points, leading to the solution ( sqrt{5} - 2 ).The maximum value of ( |PF| - |PA| ) is found to be ( sqrt{5} - 2 ).boxed{B},<think>Okay, so I need to solve this hyperbola problem. Let me try to break it down step by step. The hyperbola is given by the equation (frac{x^2}{m} - frac{y^2}{3} = 1) where (m > 0), and it has an eccentricity of 2. The right focus is (F), and there's a moving point (P) on the right branch. We also have point (A(0,1)). The question is asking for the maximum value of (|PF| - |PA|), and the options are given as A to D.First, I remember that for hyperbolas, the standard form is (frac{x^2}{a^2} - frac{y^2}{b^2} = 1), where (a) is the distance from the center to the vertices, and (b) is related to the conjugate axis. The eccentricity (e) is given by (e = frac{c}{a}), where (c) is the distance from the center to each focus. The relationship between (a), (b), and (c) is (c^2 = a^2 + b^2).In this problem, the equation of the hyperbola is (frac{x^2}{m} - frac{y^2}{3} = 1), so comparing to the standard form, (a^2 = m) and (b^2 = 3). The eccentricity (e) is given as 2. So, using (e = frac{c}{a}), we can write (c = 2a). Then, using the relationship (c^2 = a^2 + b^2), substitute (c = 2a):[(2a)^2 = a^2 + b^2 4a^2 = a^2 + 3 quad text{since } b^2 = 3 4a^2 - a^2 = 3 3a^2 = 3 a^2 = 1 ]Since (a^2 = m), that means (m = 1). So the equation of the hyperbola simplifies to (frac{x^2}{1} - frac{y^2}{3} = 1), or (x^2 - frac{y^2}{3} = 1).Next, I need to find the coordinates of the right focus (F). For a hyperbola centered at the origin (since there are no shifts in the equation), the foci are located at ((pm c, 0)). We already found that (c = 2a), and since (a^2 = 1), (a = 1), so (c = 2*1 = 2). Therefore, the right focus (F) is at ((2, 0)).Now, the problem involves a moving point (P) on the right branch of the hyperbola. We need to find the maximum value of (|PF| - |PA|), where (A) is at ((0,1)).Hmm, okay. Let's think about how to approach this. The expression (|PF| - |PA|) is reminiscent of the definition of a hyperbola itself, where the difference of distances to two foci is constant. However, in this case, one of the points is a focus (F), and the other is an arbitrary point (A). So, it's not exactly the standard hyperbola property, but maybe there's a way to use similar reasoning.First, let's recall that for any point (P) on the hyperbola, the difference of distances to the two foci is (2a). For this hyperbola, since (a = 1), that difference is (2). But here, we're dealing with (|PF| - |PA|), which is a different combination. Maybe we can relate this expression to the hyperbola's properties or use some optimization techniques.Since we need to maximize (|PF| - |PA|), perhaps we can parametrize point (P) on the hyperbola and then express (|PF| and |PA| in terms of coordinates, then compute their difference and find its maximum.Alternatively, maybe we can use some geometric insights or triangle inequalities to find the maximum difference.Let me try parametrizing point (P). For hyperbolas, a common parametrization is using hyperbolic functions: (P(cosh t, sqrt{3}sinh t)), since (x = a cosh t) and (y = b sinh t) for the right branch. Here, (a = 1) and (b = sqrt{3}), so that parametrization should work.But maybe using trigonometric parametrization isn't suitable here because hyperbolas use hyperbolic functions. Wait, but in calculus, sometimes we parametrize hyperbolas with secant and tangent, but that might complicate things.Alternatively, since it's a hyperbola, we can express (x) in terms of (y) or vice versa, but that might not be the easiest path.Alternatively, let's denote point (P) as ((x, y)) on the hyperbola (x^2 - y^2/3 = 1), so (x geq 1) (since it's the right branch).We need to express (|PF| - |PA| in terms of (x) and (y), then maximize it.First, compute (|PF|: since (F) is at (2,0), the distance from (P(x,y)) to (F(2,0)) is (sqrt{(x - 2)^2 + (y - 0)^2}).Then, compute (|PA|: distance from (P(x,y)) to (A(0,1)) is (sqrt{(x - 0)^2 + (y - 1)^2}).Therefore, the expression to maximize is:[sqrt{(x - 2)^2 + y^2} - sqrt{x^2 + (y - 1)^2}]But (P(x,y)) lies on the hyperbola (x^2 - y^2/3 = 1), so we can express (y^2) in terms of (x^2): (y^2 = 3(x^2 - 1)).Substituting (y^2 = 3x^2 - 3) into the expressions:First term inside the first square root:[(x - 2)^2 + y^2 = (x^2 - 4x + 4) + (3x^2 - 3) = x^2 -4x +4 +3x^2 -3 = 4x^2 -4x +1]So (|PF| = sqrt{4x^2 -4x +1}).Second term inside the second square root:[x^2 + (y - 1)^2 = x^2 + y^2 - 2y +1 = x^2 + (3x^2 - 3) -2y +1 = 4x^2 -2 -2y]Therefore, (|PA| = sqrt{4x^2 - 2 -2y}).So now our expression to maximize becomes:[sqrt{4x^2 -4x +1} - sqrt{4x^2 -2 -2y}]But we still have both (x) and (y) here. However, since (y^2 = 3x^2 -3), we can express (y) in terms of (x), but it's going to involve square roots. So maybe we can parametrize (x) as a hyperbola parameter.Alternatively, let's use the parametrization with hyperbolic functions. Let me try that.Let’s parametrize the hyperbola (x^2 - y^2/3 =1) using hyperbolic cosine and sine:For the right branch, (x = cosh t), (y = sqrt{3} sinh t), where (t) is a real number.Then, substituting into the expressions:First, compute (|PF|:[sqrt{(cosh t - 2)^2 + ( sqrt{3} sinh t )^2 }]Similarly, (|PA|:[sqrt{ (cosh t)^2 + ( sqrt{3} sinh t - 1 )^2 }]So, let's compute these.Starting with (|PF|:[sqrt{ (cosh t - 2)^2 + 3 sinh^2 t }]Expand the terms inside:[(cosh t - 2)^2 + 3 sinh^2 t = cosh^2 t -4 cosh t +4 + 3 sinh^2 t]Recall that (cosh^2 t - sinh^2 t =1), so we can substitute:[= (cosh^2 t + 3 sinh^2 t) -4 cosh t +4]But (cosh^2 t + 3 sinh^2 t = cosh^2 t + sinh^2 t + 2 sinh^2 t = (1 + 2 sinh^2 t) + 2 sinh^2 t)? Wait, that might not be the right approach.Wait, perhaps express everything in terms of (cosh t). Remember that (sinh^2 t = cosh^2 t -1). So:[cosh^2 t -4 cosh t +4 + 3 (cosh^2 t -1 ) = cosh^2 t -4 cosh t +4 +3 cosh^2 t -3 = 4 cosh^2 t -4 cosh t +1]Therefore, (|PF| becomes (sqrt{4 cosh^2 t -4 cosh t +1}). Let me see if this can be simplified.Looking at the expression (4 cosh^2 t -4 cosh t +1), perhaps it's a perfect square. Let me check:Let’s suppose ( (a cosh t + b)^2 = a^2 cosh^2 t + 2ab cosh t + b^2 ). If we set this equal to 4 cosh²t -4 cosh t +1, then:(a^2 =4), so (a=2) or (a=-2)Then, 2ab = -4. Let’s take a=2: 2*2*b = -4 → 4b = -4 → b= -1.Then, (b^2 = (-1)^2 =1). So yes! Therefore,(4 cosh^2 t -4 cosh t +1 = (2 cosh t -1)^2)Therefore, (|PF| simplifies to ( |2 cosh t -1| ). Since ( cosh t geq1 ) for all real t, (2 cosh t -1 geq 2*1 -1 =1 >0 ). So absolute value can be dropped:( |PF| = 2 cosh t -1 )That's a nice simplification!Now, let's compute (|PA|:[sqrt{ cosh^2 t + ( sqrt{3} sinh t -1 )^2 }]Expanding the square term:[sqrt{ cosh^2 t + 3 sinh^2 t - 2 sqrt{3} sinh t +1 }]Combine the terms:First, (cosh^2 t + 3 sinh^2 t = cosh^2 t + sinh^2 t + 2 sinh^2 t = (cosh^2 t + sinh^2 t) + 2 sinh^2 t)But (cosh^2 t + sinh^2 t = 2 cosh^2 t -1), so substituting:[= (2 cosh^2 t -1) + 2 sinh^2 t = 2 cosh^2 t -1 + 2 (cosh^2 t -1) = 2 cosh^2 t -1 + 2 cosh^2 t -2 = 4 cosh^2 t -3]Therefore, inside the square root we have:[4 cosh^2 t -3 - 2 sqrt{3} sinh t +1 = 4 cosh^2 t -2 -2 sqrt{3} sinh t]So (|PA| = sqrt{4 cosh^2 t -2 -2 sqrt{3} sinh t})Therefore, the expression we need to maximize is:[(2 cosh t -1) - sqrt{4 cosh^2 t -2 -2 sqrt{3} sinh t}]This looks complicated. Maybe we can find a substitution or relate hyperbolic functions here. Alternatively, perhaps we can express (sinh t) in terms of (cosh t), but that might not lead us anywhere.Alternatively, let me consider that (cosh t = x), so (sinh t = sqrt{x^2 -1}), but since (x geq1). Wait, but that might not help because we still have the square roots.Alternatively, let's think of this as a function of t, so let's define:(f(t) = (2 cosh t -1) - sqrt{4 cosh^2 t -2 -2 sqrt{3} sinh t})To find its maximum, we can take the derivative with respect to t and set it to zero. But this might be quite involved. Let's attempt it.First, compute the derivative of f(t):(f'(t) = 2 sinh t - frac{1}{2} cdot frac{8 cosh t sinh t - 2 sqrt{3} cosh t}{2 sqrt{4 cosh^2 t -2 -2 sqrt{3} sinh t}}})Wait, let me do it step by step.First, derivative of (2 cosh t -1) is (2 sinh t).Now, derivative of (-sqrt{4 cosh^2 t -2 -2 sqrt{3} sinh t}):Let (g(t) = 4 cosh^2 t -2 -2 sqrt{3} sinh t). Then, the derivative is (- frac{1}{2 sqrt{g(t)}} cdot g'(t)).Compute (g'(t)):(g'(t) = 8 cosh t sinh t - 2 sqrt{3} cosh t)Therefore, derivative of the second term is:(- frac{8 cosh t sinh t - 2 sqrt{3} cosh t}{2 sqrt{g(t)}}})Thus, the total derivative:(f'(t) = 2 sinh t - frac{8 cosh t sinh t - 2 sqrt{3} cosh t}{2 sqrt{4 cosh^2 t -2 -2 sqrt{3} sinh t}}})Set this equal to zero for critical points:[2 sinh t = frac{8 cosh t sinh t - 2 sqrt{3} cosh t}{2 sqrt{4 cosh^2 t -2 -2 sqrt{3} sinh t}}}]Multiply both sides by (2 sqrt{g(t)}):[4 sinh t sqrt{g(t)} = 8 cosh t sinh t - 2 sqrt{3} cosh t]Divide both sides by 2:[2 sinh t sqrt{g(t)} = 4 cosh t sinh t - sqrt{3} cosh t]Factor out (cosh t) on the right side:[2 sinh t sqrt{g(t)} = cosh t (4 sinh t - sqrt{3})]This equation seems quite complicated. Maybe there's a substitution or a clever way to solve for t. Alternatively, perhaps we can assume that at maximum, certain conditions hold. Alternatively, maybe consider specific values of t that simplify the equation.Alternatively, maybe use Lagrange multipliers for constrained optimization. The problem is to maximize ( |PF| - |PA| ) subject to (x^2 - y^2/3 =1). Let's try that approach.Let me denote point (P(x, y)) on the hyperbola (x^2 - y^2/3 =1), with (x geq1).We need to maximize (f(x, y) = sqrt{(x -2)^2 + y^2} - sqrt{x^2 + (y -1)^2}) subject to (g(x, y) = x^2 - y^2/3 -1 =0).Using Lagrange multipliers, set up the system:[nabla f = lambda nabla g]Compute the gradients.First, compute (nabla f):Let’s denote:(f = sqrt{(x -2)^2 + y^2} - sqrt{x^2 + (y -1)^2})Compute partial derivatives.First, (frac{partial f}{partial x}):Derivative of first term: (frac{1}{2} cdot 2(x -2) / (2 sqrt{(x -2)^2 + y^2}) )) Wait, better to write it properly.Let me denote (f = f_1 - f_2), where (f_1 = sqrt{(x -2)^2 + y^2}), (f_2 = sqrt{x^2 + (y -1)^2}).Then,(frac{partial f_1}{partial x} = frac{(x -2)}{sqrt{(x -2)^2 + y^2}})(frac{partial f_1}{partial y} = frac{y}{sqrt{(x -2)^2 + y^2}})Similarly,(frac{partial f_2}{partial x} = frac{x}{sqrt{x^2 + (y -1)^2}})(frac{partial f_2}{partial y} = frac{(y -1)}{sqrt{x^2 + (y -1)^2}})Therefore, the gradient of f is:[nabla f = left( frac{x -2}{sqrt{(x -2)^2 + y^2}} - frac{x}{sqrt{x^2 + (y -1)^2}}, frac{y}{sqrt{(x -2)^2 + y^2}} - frac{y -1}{sqrt{x^2 + (y -1)^2}} right )]The gradient of g is:(g(x, y) = x^2 - frac{y^2}{3} -1)So,(nabla g = (2x, -2y/3))Thus, the Lagrange multiplier equations are:1. (frac{x -2}{sqrt{(x -2)^2 + y^2}} - frac{x}{sqrt{x^2 + (y -1)^2}} = lambda cdot 2x)2. (frac{y}{sqrt{(x -2)^2 + y^2}} - frac{y -1}{sqrt{x^2 + (y -1)^2}} = lambda cdot (-2y/3))3. (x^2 - frac{y^2}{3} =1)This system of equations looks quite complicated. Solving this might be challenging. Let's see if we can find some relationship between variables.Alternatively, perhaps there is a geometric interpretation here. Let me think about the problem again. We need to maximize (|PF| - |PA|. Is there a way to interpret this difference?In some optimization problems involving differences of distances, reflecting points can be helpful. For instance, in problems where you need to maximize |PA| - |PB|, reflecting one point over the other might help. Let me try to think in that direction.Suppose we consider the expression (|PF| - |PA|. To maximize this, perhaps we can consider the locus of points where (|PF| - |PA| = k), and find the maximum k such that this locus intersects the hyperbola.Alternatively, since we need the maximum value, maybe the maximum occurs when the derivative is zero, which we tried earlier but the equations were too messy.Alternatively, think about the triangle inequality. For any point P, |PF| - |PA| <= |PA - PF| <= |FA|. Wait, but that's not quite right. The triangle inequality says | |PF| - |PA| | <= |FA|. But in our case, since we are looking for |PF| - |PA|, the maximum possible value would be when |PF| - |PA| is as large as possible, which would be when PF is as large as possible and PA is as small as possible, but these might not occur at the same point.Alternatively, maybe the maximum of |PF| - |PA| is equal to |FA|, but I need to check.Compute |FA|: distance from F(2,0) to A(0,1):[sqrt{(2 -0)^2 + (0 -1)^2} = sqrt{4 +1} = sqrt{5} approx 2.236]Looking at the answer choices, sqrt(5) is option A. But one of the options is sqrt(5) - 2 (option B). So maybe it's not |FA|.Alternatively, maybe the maximum value is achieved when point P is such that PA is minimized and PF is maximized. But on the hyperbola, PF can be large as x increases, but PA also increases as x increases. So there might be a trade-off.Alternatively, since P is on the hyperbola, and hyperbola has the property that |PF1 - PF2| = 2a, where F1 and F2 are the foci. But in this problem, the two points are F (one focus) and A (another point). Not sure.Wait, the hyperbola's right branch is such that for any point P on it, |PF1 - PF2| = 2a = 2. Here, F1 is (2,0), F2 is (-2,0). But we have a different point A(0,1). Not directly related.Alternatively, perhaps we can use the fact that for the hyperbola, the right branch is parametrized with x >=1, so maybe we can express x as a function and try to maximize the expression.Wait, let's consider expressing the expression |PF| - |PA| in terms of x. Since we have y^2 = 3x^2 -3, so y = sqrt(3x^2 -3), but since hyperbola is symmetric, we can take y positive or negative. However, the maximum might occur at a specific y.But let's attempt to write |PF| - |PA| in terms of x.Earlier, we had:|PF| = sqrt(4x^2 -4x +1) = sqrt{(2x -1)^2} = |2x -1|. But since x >=1, 2x -1 >=1, so |PF| = 2x -1.Wait, hold on. Earlier, when we parametrized using hyperbolic functions, we found that |PF| = 2 cosh t -1. But cosh t = x, so if x = cosh t, then |PF| = 2x -1. But in the hyperbola equation, x^2 - y^2/3 =1, so x is indeed >=1.Wait, but in reality, x is equal to cosh t, which is always greater or equal to 1, so substituting x, |PF| = 2x -1. That's a linear function in x. Interesting.Similarly, |PA| was sqrt{4x^2 -2 -2y} as per earlier. Wait, but that was derived by substituting y^2 =3x^2 -3 into |PA|. Let me check again.Wait, when we calculated |PA|, we had:Original expression: sqrt(x^2 + (y -1)^2). Then, substituting y^2 =3x^2 -3, we expanded:x^2 + (y -1)^2 = x^2 + y^2 -2y +1 = x^2 + (3x^2 -3) -2y +1 =4x^2 -2 -2y. Therefore, |PA| = sqrt{4x^2 -2 -2y}But y here is sqrt{3x^2 -3} or -sqrt{3x^2 -3}. However, since we are looking for the maximum of |PF| - |PA|, and |PF| =2x -1, which increases with x, while |PA| might have a complex behavior. However, if we take y as positive, then y = sqrt{3x^2 -3}, so substituting that into |PA|:|PA| = sqrt{4x^2 -2 -2*sqrt{3x^2 -3}}Alternatively, if y is negative, |PA| would be sqrt{4x^2 -2 +2*sqrt{3x^2 -3}}. But since we are subtracting |PA|, to maximize |PF| - |PA|, we need |PA| to be as small as possible. Therefore, taking y negative might give a smaller |PA|. Wait, but let's check.If y is negative, then (y -1)^2 would be (negative number -1)^2, which is larger than (positive number -1)^2 if the number is positive. Wait, actually:If y is positive, then (y -1)^2 is (positive -1)^2. If y is negative, (y -1)^2 is (negative -1)^2 = ( - (y +1) )^2 = (y +1)^2. So depending on the value of y, this could be larger or smaller.But since y^2 =3x^2 -3, as x increases, y^2 increases, so |y| increases. For large x, y ~ sqrt(3)x, so whether y is positive or negative, (y -1)^2 ~ y^2 for large x. But perhaps for smaller x, the sign of y might affect |PA|.However, since we are trying to maximize |PF| - |PA|, which is (2x -1) - |PA|. To maximize this, we need to maximize 2x -1 and minimize |PA|.Since 2x -1 increases with x, and |PA| for large x is approximately sqrt{4x^2 -2 -2y} where y ~ sqrt(3)x. If y is negative, then sqrt{4x^2 -2 -2*(-sqrt{3}x)} = sqrt{4x^2 -2 + 2 sqrt{3} x}, which for large x behaves like sqrt{4x^2 + 2 sqrt{3}x} ~ 2x + (sqrt{3}/2). Therefore, 2x -1 - (2x + sqrt{3}/2 ) ~ -1 - sqrt{3}/2, which is negative. If y is positive, then |PA| ~ sqrt{4x^2 -2 -2 sqrt{3}x} ~ sqrt{(2x - sqrt{3}/2)^2} ~ 2x - sqrt{3}/2, so 2x -1 - (2x - sqrt{3}/2) = -1 + sqrt{3}/2 ~ -1 +0.866 ~ -0.134, which is also negative. So as x increases, the expression |PF| - |PA| tends to a negative limit. Therefore, the maximum might not be at infinity, but at some finite point.Alternatively, maybe the maximum occurs when the derivative is zero, which we attempted earlier but the equations were too complicated.Alternatively, maybe we can consider using reflection properties. For example, in ellipse problems, the reflection property is used, but hyperbola has a different reflection property. For hyperbola, the difference of distances to the foci is constant. Wait, but here we have a focus and another point. Maybe reflecting point A over the center or something?Alternatively, consider reflecting point A across the x-axis or y-axis to see if that helps. Let me think.Suppose we reflect point A(0,1) over the y-axis to get A'(0,1), which is the same point. Not helpful. If we reflect over the x-axis, we get A''(0,-1). Not sure.Alternatively, maybe the maximum of |PF| - |PA| is achieved when P is colinear with F and A in some way. But that might not hold.Alternatively, let's consider specific points on the hyperbola and compute |PF| - |PA| to see which one gives the maximum.First, take the vertex of the hyperbola, which is at (1,0). Compute |PF| - |PA|:|PF|: distance from (1,0) to (2,0) is 1.|PA|: distance from (1,0) to (0,1) is sqrt(1 +1) = sqrt(2).Thus, |PF| - |PA| =1 - sqrt(2) ≈1 -1.414≈ -0.414. Not the maximum.Another point: the other vertex, but since it's the right branch, only (1,0).What about the point where x=2? Let's check if x=2 is on the hyperbola.At x=2, y^2 =3*(4) -3=12 -3=9, so y=±3. So the point (2,3) and (2,-3).Compute |PF| - |PA| for (2,3):|PF|: distance from (2,3) to (2,0) is 3.|PA|: distance from (2,3) to (0,1) is sqrt(4 +4)=sqrt(8)=2sqrt(2)≈2.828.Thus, |PF| - |PA|≈3 -2.828≈0.172.For (2,-3):|PF|: distance from (2,-3) to (2,0) is 3.|PA|: distance from (2,-3) to (0,1) is sqrt(4 +16)=sqrt(20)=2sqrt(5)≈4.472.Thus, |PF| - |PA|≈3 -4.472≈-1.472. So taking y negative here gives a worse result.So at x=2, positive y gives a positive difference of 0.172, which is better than the vertex.Another point: x= sqrt( something ). Let's pick x= sqrt(2). Wait, x must be >=1. Let's take x=1.5.x=1.5, then y^2=3*(2.25) -3=6.75 -3=3.75, so y≈±1.936.Compute |PF| - |PA| for (1.5, 1.936):|PF|: sqrt( (1.5 -2)^2 + (1.936)^2 )=sqrt(0.25 +3.748)≈sqrt(3.998)≈1.999≈2.|PA|: sqrt( (1.5)^2 + (1.936 -1)^2 )=sqrt(2.25 +0.876)=sqrt(3.126)≈1.768.Thus, |PF| - |PA|≈2 -1.768≈0.232. Better than x=2.Another point: x=1. Let's already did that. x=1.25.x=1.25, y^2=3*(1.5625) -3=4.6875 -3=1.6875, y≈±1.299.Compute |PF| - |PA| for (1.25,1.299):|PF|: sqrt( (1.25 -2)^2 + (1.299)^2 )=sqrt(0.5625 +1.687)=sqrt(2.25)=1.5.|PA|: sqrt(1.5625 + (1.299 -1)^2 )=sqrt(1.5625 +0.089)=sqrt(1.6515)≈1.286.So |PF| - |PA|≈1.5 -1.286≈0.214. Less than x=1.5.Wait, so at x=1.5, the difference is ≈0.232, which is higher than at x=1.25. At x=2, it's ≈0.172.Wait, but at x=1.5, the difference is 0.232. Let's try x=1. Let's take x=1.2.x=1.2, y^2=3*(1.44) -3=4.32 -3=1.32, y≈±1.149.Compute |PF| - |PA|:|PF|: sqrt( (1.2 -2)^2 + (1.149)^2 )=sqrt(0.64 +1.32)=sqrt(1.96)=1.4.|PA|: sqrt(1.44 + (1.149 -1)^2 )=sqrt(1.44 +0.022)=sqrt(1.462)≈1.209.Thus, |PF| - |PA|≈1.4 -1.209≈0.191. Less than at x=1.5.Hmm, so x=1.5 gives a higher value. Let's try x=1.6.x=1.6, y^2=3*(2.56) -3=7.68 -3=4.68, y≈±2.164.Compute |PF| - |PA|:|PF|: sqrt( (1.6 -2)^2 + (2.164)^2 )=sqrt(0.16 +4.684)=sqrt(4.844)≈2.201.|PA|: sqrt(2.56 + (2.164 -1)^2 )=sqrt(2.56 +1.355)=sqrt(3.915)≈1.978.Difference: ≈2.201 -1.978≈0.223. Less than at x=1.5.Wait, so x=1.5 gives a better result. Let's try x=1.4.x=1.4, y^2=3*(1.96) -3=5.88 -3=2.88, y≈±1.697.Compute |PF| - |PA|:|PF|: sqrt( (1.4 -2)^2 + (1.697)^2 )=sqrt(0.36 +2.88)=sqrt(3.24)=1.8.|PA|: sqrt(1.96 + (1.697 -1)^2 )=sqrt(1.96 +0.485)=sqrt(2.445)≈1.563.Difference: ≈1.8 -1.563≈0.237. Better than x=1.5.Wait, higher. Let me check x=1.3.x=1.3, y²=3*(1.69)-3=5.07-3=2.07, y≈±1.439.Compute |PF| - |PA|:|PF|: sqrt( (1.3 -2)^2 + (1.439)^2 )=sqrt(0.49 +2.07)=sqrt(2.56)=1.6.|PA|: sqrt(1.69 + (1.439 -1)^2 )=sqrt(1.69 +0.193)=sqrt(1.883)≈1.372.Difference: ≈1.6 -1.372≈0.228. Less than at x=1.4.Hmm, so x=1.4 gives a higher value. Let's try x=1.45.x=1.45, y²=3*(2.1025) -3=6.3075 -3=3.3075, y≈±1.818.Compute |PF| - |PA|:|PF|: sqrt( (1.45 -2)^2 + (1.818)^2 )=sqrt(0.3025 +3.306)=sqrt(3.6085)≈1.899.|PA|: sqrt(2.1025 + (1.818 -1)^2 )=sqrt(2.1025 +0.669)=sqrt(2.7715)≈1.665.Difference: ≈1.899 -1.665≈0.234. Close to x=1.4.Wait, x=1.4 gives 0.237. Let's try x=1.35.x=1.35, y²=3*(1.8225) -3=5.4675 -3=2.4675, y≈±1.571.Compute |PF| - |PA|:|PF|: sqrt( (1.35 -2)^2 + (1.571)^2 )=sqrt(0.4225 +2.468)=sqrt(2.8905)≈1.7.|PA|: sqrt(1.8225 + (1.571 -1)^2 )=sqrt(1.8225 +0.327)=sqrt(2.1495)≈1.466.Difference: ≈1.7 -1.466≈0.234. So x=1.35 gives similar to x=1.4.Hmm. Maybe there's a maximum around x=1.4. But the options given are sqrt(5)≈2.236, sqrt(5)-2≈0.236, 2sqrt(2)≈2.828, and 2sqrt(2)-2≈0.828.Wait, the difference we calculated at x=1.4 is approximately 0.237, which is very close to sqrt(5)-2≈2.236-2≈0.236. So option B.Is that possible? The approximate value we found by testing points is around 0.236, which is exactly sqrt(5)-2. So probably option B is the answer.But how to confirm this? Maybe there's a way to find the exact maximum.Recall that when we parametrized using hyperbolic functions, we had:|PF| - |PA| = (2x -1) - sqrt{4x² -2 -2y}But since y²=3x² -3, we can express y in terms of x. Let's try to write |PA| in terms of x.From the expression, |PA| = sqrt{4x² -2 -2y}But since y = sqrt{3x² -3} (assuming y positive), substituting:|PA| = sqrt{4x² -2 -2 sqrt{3x² -3}}But we need to maximize (2x -1) - sqrt{4x² -2 -2 sqrt{3x² -3}}.Let me set z = sqrt{3x² -3}. Then, z² =3x² -3 ⇒3x² = z² +3 ⇒x² = (z² +3)/3.Substituting into the expression:|PF| - |PA| = 2x -1 - sqrt{4x² -2 -2z}But x in terms of z: x = sqrt{(z² +3)/3}Wait, but this might complicate things.Alternatively, let me set t = x. Then, the expression becomes:f(t) = 2t -1 - sqrt{4t² -2 -2 sqrt{3t² -3}}Let’s set u = sqrt{3t² -3}, then u = sqrt{3t² -3} ⇒u² =3t² -3 ⇒3t² =u² +3 ⇒t² = (u² +3)/3.But then, 4t² =4(u² +3)/3 = (4u² +12)/3So,4t² -2 -2u = (4u² +12)/3 -2 -2u = (4u² +12 -6 -6u)/3 = (4u² -6u +6)/3Therefore, sqrt{4t² -2 -2u} = sqrt{(4u² -6u +6)/3} = sqrt{(4u² -6u +6)/3}But this seems even more complicated.Alternatively, maybe consider that the maximum occurs when the derivative is zero, so go back to the derivative equation.Recall that when we did the parametrization with t, we arrived at an equation:2 sinh t sqrt{g(t)} = cosh t (4 sinh t - sqrt{3})Where g(t) =4 cosh²t -2 -2 sqrt{3} sinh t.This equation is still messy, but perhaps if we assume that sinh t = k, then cosh t = sqrt{k² +1}, but this substitution might not help.Alternatively, let's make a substitution: Let’s suppose that the maximum occurs when PA is minimized. Wait, but the expression is |PF| - |PA|, so maximizing this would involve maximizing |PF| and minimizing |PA|. So maybe there's a point P where PA is minimized, and PF is as large as possible given that PA is minimized.But the minimal PA for points on the hyperbola: the minimal distance from A(0,1) to the hyperbola.Let me compute the minimal distance from A(0,1) to the hyperbola x² - y²/3 =1.To find the minimal distance, set up the distance squared function:d² = x² + (y -1)^2 subject to x² - y²/3 =1.Using Lagrange multipliers again.Define L = x² + (y -1)^2 - λ(x² - y²/3 -1)Partial derivatives:∂L/∂x = 2x - λ*2x =0 ⇒2x(1 - λ)=0.∂L/∂y = 2(y -1) - λ*(-2y/3)=0 ⇒2(y -1) + (2λ y)/3 =0.∂L/∂λ = -(x² - y²/3 -1)=0 ⇒x² - y²/3=1.From ∂L/∂x: 2x(1 - λ)=0. So either x=0 or λ=1.But x² - y²/3 =1, so x=0 would imply -y²/3=1 ⇒ y²=-3, which is impossible. Thus, x ≠0, so λ=1.Thus, set λ=1.From ∂L/∂y: 2(y -1) + (2*1*y)/3 =0 ⇒2(y -1) + (2y)/3 =0.Multiply both sides by 3:6(y -1) +2y=0 ⇒6y -6 +2y=0 ⇒8y=6 ⇒y=6/8=3/4.Then, from the hyperbola equation x² - y²/3=1:x² - ( (3/4)^2 ) /3 =1 ⇒x² - (9/16)/3 =1 ⇒x² - 3/16=1 ⇒x²=19/16 ⇒x=±sqrt(19)/4≈±1.089.But since we are considering the right branch, x>0, so x= sqrt(19)/4≈1.089.So the minimal distance from A(0,1) to the hyperbola is at point P≈(1.089, 3/4).Compute |PF| - |PA| at this point:|PF|: distance from (1.089, 0.75) to (2,0):sqrt( (1.089 -2)^2 + (0.75 -0)^2 ) = sqrt( (-0.911)^2 +0.5625 )≈sqrt(0.829 +0.5625)=sqrt(1.3915)≈1.179.|PA|: distance from (1.089,0.75) to (0,1):sqrt(1.089² + (0.75 -1)^2 )≈sqrt(1.186 +0.0625)=sqrt(1.2485)≈1.117.Thus, |PF| - |PA|≈1.179 -1.117≈0.062. Which is much lower than the previous values. So minimizing PA does not maximize the difference. Therefore, the maximum occurs elsewhere.Alternatively, maybe the maximum of |PF| - |PA| is achieved when the derivative is zero, which corresponds to the critical point we found earlier. But solving that equation is complicated.Alternatively, let's consider that we have |PF| - |PA| = (2x -1) - sqrt(4x² -2 -2y). But since y = sqrt{3x² -3}, we can substitute:sqrt(4x² -2 -2*sqrt{3x² -3})Let me denote z = sqrt{3x² -3}, then z = sqrt{3x² -3} ⇒z²=3x² -3 ⇒x²=(z² +3)/3.Then, substitute into the expression:sqrt(4*( (z² +3)/3 ) -2 -2z )= sqrt( (4z² +12)/3 -2 -2z )= sqrt( (4z² +12 -6 -6z)/3 )= sqrt( (4z² -6z +6)/3 )So, |PA|=sqrt( (4z² -6z +6)/3 )But |PF| - |PA| = 2x -1 - sqrt( (4z² -6z +6)/3 )But x= sqrt( (z² +3)/3 )So,2x -1 = 2*sqrt( (z² +3)/3 ) -1Therefore, the expression becomes:2*sqrt( (z² +3)/3 ) -1 - sqrt( (4z² -6z +6)/3 )This is a function of z. Let's denote this function as f(z).To find its maximum, take derivative with respect to z and set to zero.But this is getting more complicated. Maybe there's a smarter substitution.Alternatively, perhaps consider writing the expression inside the square roots as squares.Wait, let's see:Looking at 4x² -4x +1, which is |PF| squared. Wait, earlier we found that |PF|=2x -1.Wait, but |PF| =2x -1. That's a linear function.So, if we can express |PA| in terms of x, then we have:|PF| - |PA| =2x -1 - sqrt(4x² -2 -2 sqrt{3x² -3})Let’s consider setting sqrt{3x² -3}=t. Then t= sqrt{3x² -3} ⇒t²=3x² -3 ⇒x²=(t² +3)/3.So, substitute into |PA|:sqrt{4*( (t² +3)/3 ) -2 -2t } = sqrt{ (4t² +12)/3 -2 -2t } = sqrt{ (4t² +12 -6 -6t)/3 }= sqrt{ (4t² -6t +6)/3 }Therefore, the expression becomes:2x -1 - sqrt{ (4t² -6t +6)/3 }But x= sqrt{ (t² +3)/3 }So,2x= 2*sqrt{ (t² +3)/3 }Thus, expression is:2*sqrt{ (t² +3)/3 } -1 - sqrt{ (4t² -6t +6)/3 }Factor out sqrt(1/3):sqrt(1/3)*[ 2*sqrt(t² +3) - sqrt(4t² -6t +6) ] -1This seems still complicated. Maybe let’s set u = t, and express everything in terms of u.But perhaps instead of symbolic manipulation, we can look for when the expression inside the square roots becomes a perfect square.Looking at 4t² -6t +6. Let me see if this can be written as (something)^2.Suppose 4t² -6t +6 = (at +b)^2 = a²t² +2abt +b².Then,a²=4 ⇒a=2 or -22ab= -6. If a=2, then 2*2*b= -6 ⇒4b= -6 ⇒b= -6/4= -3/2.Then, b²=( -3/2 )²=9/4.Thus, (2t -3/2)^2=4t² -6t +9/4. But 4t² -6t +6 is different from this by 6 -9/4=15/4. So not equal.Therefore, it's not a perfect square.Alternatively, maybe completing the square:4t² -6t +6 =4(t² - (3/2)t ) +6=4[ t² - (3/2)t + (9/16) ] -4*(9/16) +6=4(t - 3/4)^2 - 9/4 +6=4(t - 3/4)^2 +15/4.Thus, sqrt(4t² -6t +6)=sqrt{4(t - 3/4)^2 +15/4}=sqrt{(2(t - 3/4))^2 +15/4}.Not helpful.Alternatively, maybe there's a value of t that minimizes the expression 2*sqrt(t² +3) - sqrt(4t² -6t +6). But this is getting too abstract.Alternatively, think about the maximum of the expression |PF| - |PA| =2x -1 - sqrt(4x² -2 -2 sqrt{3x² -3}).Let’s denote s= sqrt{3x² -3}, then s≥0, and x= sqrt{(s² +3)/3}Thus, 2x -1 =2*sqrt{(s² +3)/3} -1And the expression becomes:2*sqrt{(s² +3)/3} -1 - sqrt{4*( (s² +3)/3 ) -2 -2s }= 2*sqrt{(s² +3)/3} -1 - sqrt{ (4s² +12 -6 -6s)/3 }= 2*sqrt{(s² +3)/3} -1 - sqrt{ (4s² -6s +6)/3 }Factor out sqrt(1/3):= sqrt(1/3) [2 sqrt(s² +3) - sqrt(4s² -6s +6) ] -1Let’s set f(s) = sqrt(1/3)[2 sqrt(s² +3) - sqrt(4s² -6s +6) ] -1We need to maximize f(s) over s≥0.Let’s compute derivative of f(s) with respect to s:df/ds= sqrt(1/3)[ (2*(1/(2 sqrt(s² +3)))*2s ) - (1/(2 sqrt(4s² -6s +6)))*(8s -6) ) ] -0= sqrt(1/3)[ ( (2s)/sqrt(s² +3) ) - ( (8s -6)/(2 sqrt(4s² -6s +6)) ) ]Set derivative to zero:(2s)/sqrt(s² +3) - (8s -6)/(2 sqrt(4s² -6s +6)) =0Multiply both sides by 2 sqrt(s² +3) sqrt(4s² -6s +6)):4s sqrt(4s² -6s +6) - (8s -6) sqrt(s² +3) =0This is quite a complicated equation. Let’s try plugging s=1:Left side:4*1*sqrt(4 -6 +6) - (8 -6)sqrt(1 +3)=4*sqrt(4) -2*sqrt(4)=4*2 -2*2=8 -4=4≠0.s= sqrt(3):Left side:4*sqrt(3)*sqrt(4*3 -6*sqrt(3) +6) - (8*sqrt(3) -6)sqrt(3 +3)Hmm, this is messy.Alternatively, perhaps make a substitution. Let’s denote sqrt(s² +3)=u, sqrt(4s² -6s +6)=v.But I’m not sure.Alternatively, suppose that 4s sqrt(4s² -6s +6) = (8s -6) sqrt(s² +3)Square both sides:16s² (4s² -6s +6) = (8s -6)^2 (s² +3)Expand both sides:Left side:16s²(4s² -6s +6)=64s⁴ -96s³ +96s²Right side:(64s² -96s +36)(s² +3)=64s²(s² +3) -96s(s² +3) +36(s² +3)=64s⁴ +192s² -96s³ -288s +36s² +108=64s⁴ -96s³ +228s² -288s +108Set left = right:64s⁴ -96s³ +96s² =64s⁴ -96s³ +228s² -288s +108Subtract left from right:0=0 +0 +132s² -288s +108Thus,132s² -288s +108=0Divide by 12:11s² -24s +9=0Solve quadratic equation:s=(24±sqrt(576 -396))/22=(24±sqrt(180))/22=(24±6*sqrt(5))/22=(12±3*sqrt(5))/11Compute approximate values:sqrt(5)≈2.236, so:s=(12 +3*2.236)/11≈(12 +6.708)/11≈18.708/11≈1.700s=(12 -3*2.236)/11≈(12 -6.708)/11≈5.292/11≈0.481So possible critical points at s≈1.700 and s≈0.481.Now, we need to check these solutions in the original equation, because squaring can introduce extraneous solutions.First, check s≈1.700:Compute left side:4s sqrt(4s² -6s +6)s=1.7:4*1.7=6.84s²=4*(2.89)=11.564s² -6s +6=11.56 -10.2 +6=7.36sqrt(7.36)≈2.713Thus, left side=6.8*2.713≈18.45Right side: (8s -6)sqrt(s² +3)8s=13.6, 13.6 -6=7.6s² +3=2.89 +3=5.89, sqrt≈2.427Thus, right side=7.6*2.427≈18.45So equality holds. Valid solution.Now, check s≈0.481:s=0.481:Left side:4*0.481≈1.9244s²=4*(0.231)=0.9244s² -6s +6=0.924 -2.886 +6≈4.038sqrt(4.038)≈2.009Left side≈1.924*2.009≈3.866Right side:8s -6=3.848 -6=-2.152sqrt(s² +3)=sqrt(0.231 +3)=sqrt(3.231)≈1.797Thus, right side=-2.152*1.797≈-3.868Left side≈3.866 ≈-right side≈3.868. But since the original equation was 4s sqrt(...) - (8s -6) sqrt(...)=0, and substituting s≈0.481 gives approximately 3.866 - (-3.868)=7.734≠0. Therefore, this solution is extraneous.Thus, only s≈1.700 is valid.Therefore, the critical point occurs at s≈1.700.Recall that s= sqrt{3x² -3} ⇒x= sqrt{(s² +3)/3}So when s≈1.700:s²≈2.89x= sqrt{(2.89 +3)/3}=sqrt{5.89/3}=sqrt{1.963}≈1.401Which aligns with our earlier trial at x≈1.4.Thus, the maximum occurs at x≈1.401.Compute |PF| - |PA|:|PF|=2x -1≈2*1.401 -1≈2.802 -1≈1.802|PA|=sqrt{4x² -2 -2y} where y= sqrt{3x² -3}≈1.700.Thus,4x²≈4*(1.963)≈7.8527.852 -2 -2*1.700≈7.852 -2 -3.4≈2.452sqrt(2.452)≈1.566Thus, |PF| - |PA|≈1.802 -1.566≈0.236, which is approximately sqrt(5) -2≈2.236 -2≈0.236.Therefore, the maximum value is sqrt(5)-2, which is option B.Final Answerboxed{B}</think>Given the hyperbola (E: frac{x^2}{m} - frac{y^2}{3} = 1) with (m > 0) and an eccentricity of 2, we need to find the maximum value of (|PF| - |PA|) where (F) is the right focus and (A(0,1)).1. Determine the value of (m): - The eccentricity (e = 2) implies (e = frac{c}{a} = 2). - For hyperbolas, (c^2 = a^2 + b^2). Given (b^2 = 3), we solve: [ c = 2a implies (2a)^2 = a^2 + 3 implies 4a^2 = a^2 + 3 implies 3a^2 = 3 implies a^2 = 1 implies m = 1 ] - The hyperbola equation simplifies to (x^2 - frac{y^2}{3} = 1).2. Find the coordinates of the right focus (F): - The distance to the foci is (c = 2a = 2), so (F(2, 0)).3. Parametrize the hyperbola: - For a point (P) on the right branch, use hyperbolic functions: (P(cosh t, sqrt{3} sinh t)).4. Express distances (|PF|) and (|PA|): - (|PF| = sqrt{(x - 2)^2 + y^2}) simplifies to (2 cosh t - 1). - (|PA| = sqrt{x^2 + (y - 1)^2}) simplifies to (sqrt{4 cosh^2 t - 2 - 2sqrt{3} sinh t}).5. Maximize the expression (|PF| - |PA|): - Using Lagrange multipliers and critical point analysis, we find the maximum value occurs when (s = sqrt{3x^2 - 3} approx 1.700). - Substituting back, the maximum value is found to be (sqrt{5} - 2).Thus, the maximum value of (|PF| - |PA|) is (boxed{B}).]

🔑:To solve this problem, we follow the steps closely related to the given solution:1. Determine the value of m using the eccentricity of the hyperbola: The eccentricity e of the hyperbola frac{x^2}{m}-frac{y^2}{3}=1 is given by e = frac{sqrt{m+3}}{sqrt{m}} = 2. Solving for m, we get [ frac{sqrt{m+3}}{sqrt{m}} = 2 implies sqrt{m+3} = 2sqrt{m} implies m+3 = 4m implies 3m = 3 implies m = 1. ]2. Identify the parameters of the hyperbola: With m=1, we find the semi-major axis a = sqrt{m} = 1, the semi-minor axis b = sqrt{3}, and the focal distance c = sqrt{a^2+b^2} = sqrt{1+3} = 2.3. Locate the left focus E of the hyperbola: The left focus E is at (-c,0) = (-2,0).4. Calculate the maximum value of |PF|-|PA|: Given point A(0,1), we can express |PF|-|PA| as |PE|-2a-|PA|. Since |PE|-|PA| is the distance difference between points E and A to any point P on the hyperbola, the maximum difference occurs when E, A, and P are collinear. Thus, [ |PF|-|PA| = |PE|-2a-|PA| leq |AE|-2. ] Calculating |AE|, we find |AE| = sqrt{(-2-0)^2+(0-1)^2} = sqrt{4+1} = sqrt{5}. Therefore, [ |PF|-|PA| leq sqrt{5}-2. ] The equality holds when E, A, and P are collinear, indicating that sqrt{5}-2 is the maximum value.Hence, the maximum value of |PF|-|PA| is boxed{sqrt{5}-2}, which corresponds to option boxed{B}.

🧠:[<think>Okay, let's see here. The problem is about sets, right? So we have two sets, M and N. M is {0, 1, 2, 3, 4} and N is {1, 3, 5}. Then we need to find P, which is the intersection of M and N, and then figure out how many subsets P has. Hmm, okay. Let me break this down step by step.First, I remember that the intersection of two sets, M ∩ N, consists of all elements that are in both M and N. So I need to look at the elements in M and see which ones are also in N. Let's list out the elements of both sets again to make sure I have them right.Set M is {0, 1, 2, 3, 4}. So those are the numbers 0 through 4. Set N is {1, 3, 5}. So the numbers 1, 3, and 5. Now, the intersection P would be elements common to both. Let me go through each element of M and check if it's in N.Starting with 0: Is 0 in N? N has 1, 3, 5. Nope, so 0 isn't in the intersection. Next, 1: Is 1 in N? Yes, 1 is the first element of N. So 1 is in P. Then 2: Is 2 in N? Looking at N again, no, there's no 2. So 2 is out. Next, 3: Is 3 in N? Yes, 3 is in N. So 3 is in P. Then 4: Is 4 in N? N has 1, 3, 5. No, so 4 isn't in the intersection. So the elements common to both M and N are 1 and 3. Therefore, P = {1, 3}.Now, the next part is to find how many subsets P has. Hmm, subsets. I remember that the number of subsets of a set is given by 2 to the power of the number of elements in the set. The formula is 2^n, where n is the number of elements. So first, let's confirm how many elements are in P. P is {1, 3}, so that's two elements. Therefore, the number of subsets should be 2^2 = 4. Let me verify this by listing all subsets to make sure.The subsets of a set with two elements are: the empty set, the set containing the first element, the set containing the second element, and the set containing both elements. So for P = {1, 3}, the subsets are:1. {} (the empty set)2. {1}3. {3}4. {1, 3}Yep, that's four subsets. So 2^2 is indeed 4. Therefore, the number of subsets of P is 4.But wait, just to be thorough, let me make sure that I didn't make a mistake in finding the intersection. Let me check again. M has 0,1,2,3,4. N has 1,3,5. The common elements are 1 and 3. That's correct. So P is indeed {1,3}. Then the number of elements in P is 2, so 2^2 is 4. Yep, that seems solid.Another way to think about subsets: Each element can be either included or excluded from a subset. So for each element, there are two choices. So for n elements, it's 2 * 2 * ... * 2 (n times), which is 2^n. So with two elements, 2*2=4. That matches. So that formula holds here.Is there a chance I counted wrong? Let me try another example. If a set has 3 elements, say {a, b, c}, then the number of subsets should be 8. Let's see: {}, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}. Yes, that's 8. So 2^3=8. So the formula works. Therefore, applying it to P with 2 elements gives 4. Which matches when I listed them out.So I think that's correct. The key steps were finding the intersection correctly, then applying the subset formula. Let me just recap to make sure I didn't skip any steps.First, M ∩ N: Compare each element of M with N. 0 not in N, 1 yes, 2 no, 3 yes, 4 no. So intersection is {1,3}. Then, number of subsets: 2 elements, so 2^2=4. Therefore, the answer is 4. Yep, that all checks out.I guess the possible mistakes could be either in the intersection step or in the subset calculation. But both steps seem straightforward here. Maybe if the intersection was empty? But in this case, it's not. If P was empty, then it would have 1 subset (the empty set itself). But here, P has two elements, so no issues there.Alternatively, if someone thought that the number of subsets is equal to the number of elements, which is a common mistake, but in this case, 2 vs. 4. But since the question specifically asks for subsets, not proper subsets or something else, it's definitely 2^n. So the answer should be 4.Yeah, I think that's solid. I can't see any errors in my reasoning. So final answer is 4 subsets.Final AnswerThe number of subsets of ( P ) is boxed{4}.</think>Given sets ( M = {0, 1, 2, 3, 4} ) and ( N = {1, 3, 5} ), we need to find ( P = M cap N ) and determine how many subsets ( P ) has.First, we find the intersection ( P ) of sets ( M ) and ( N ). The intersection consists of elements common to both sets. Checking each element of ( M ) against ( N ):- ( 0 ) is not in ( N )- ( 1 ) is in ( N )- ( 2 ) is not in ( N )- ( 3 ) is in ( N )- ( 4 ) is not in ( N )Thus, the intersection ( P ) is ( {1, 3} ).Next, we calculate the number of subsets of ( P ). A set with ( n ) elements has ( 2^n ) subsets. Since ( P ) has 2 elements, the number of subsets is ( 2^2 = 4 ).Listing all subsets of ( P ):1. The empty set ( emptyset )2. ( {1} )3. ( {3} )4. ( {1, 3} )Therefore, the number of subsets of ( P ) is (boxed{4}).,<think>Okay, so I need to find out how many subsets the set P has, where P is the intersection of sets M and N. The problem gives me M as {0, 1, 2, 3, 4} and N as {1, 3, 5}. Hmm, let me start by recalling what the intersection of two sets means. The intersection, denoted by the symbol ∩, includes all elements that are common to both sets. So, P = M ∩ N should be a set containing every element that is in both M and N. Let me check the elements one by one.Looking at set M: 0, 1, 2, 3, 4. Set N: 1, 3, 5. Let me compare each element. Starting with the first element of M, which is 0. Is 0 in N? No, because N has 1, 3, 5. So, 0 is not in the intersection. Next element in M is 1. Is 1 in N? Yes, N has 1. So, 1 should be included in P. Then 2 in M, is 2 in N? Nope. Then 3 in M, which is also in N (since N has 3). So, 3 is part of the intersection. Next, 4 in M. Is 4 in N? No. So, the only common elements between M and N are 1 and 3. Therefore, P = {1, 3}.Now, the question is how many subsets does P have. I remember that the number of subsets of a set is given by 2 raised to the power of the number of elements in the set. So if a set has n elements, it has 2^n subsets. Let me confirm this formula. For example, a set with 1 element has 2 subsets: the empty set and the set itself. For a set with 2 elements, it has 4 subsets: empty set, each individual element, and the set itself. Yeah, that seems right.So, since P has two elements, the number of subsets should be 2^2 = 4. Let me list them out to make sure I haven't missed anything. The subsets of P = {1, 3} are:1. The empty set: {}2. {1}3. {3}4. {1, 3}Yep, that's four subsets. So the calculation checks out. Therefore, the answer should be 4, which is 2 squared, since there are two elements in P. Wait, just to make sure I didn't make a mistake in the intersection. Let me go back. M is {0,1,2,3,4}, N is {1,3,5}. So overlapping elements are 1 and 3. So intersection is {1,3}. Then subsets: 4. That seems right. Maybe I can also recall that the formula includes all possible combinations, so each element can be either included or excluded. So for each element, two choices, hence 2*2=4. Yep, that makes sense.Alternatively, if P had 0 elements, it would have 1 subset (the empty set). If it had 1 element, 2 subsets. 2 elements, 4 subsets, and so on. So with P confirmed to have 2 elements, the answer is 4. I think that's solid. No mistakes in the reasoning. The key steps were correctly finding the intersection and then applying the subset formula. Let me just recap to ensure:1. Find P = M ∩ N: elements common to both M and N. Check each element of M against N. Found 1 and 3. So P = {1,3}.2. Number of subsets of P: 2^n where n is the number of elements in P. n=2, so 2^2=4.All steps make sense. I don't think I overlooked anything. Maybe a common mistake could be to list the subsets and miscount, but listing them here shows clearly there are 4. Another possible error could be miscalculating the intersection, but double-checking confirms that 1 and 3 are indeed in both sets. 5 is only in N, and 0,2,4 are only in M. So no other elements are shared.Therefore, I'm confident that the number of subsets of P is 4.Final AnswerThe number of subsets of ( P ) is boxed{4}.</think>Given sets ( M = {0, 1, 2, 3, 4} ) and ( N = {1, 3, 5} ), we need to find the number of subsets of ( P = M cap N ).1. Finding the intersection ( P ): - The intersection ( M cap N ) includes all elements common to both ( M ) and ( N ). - Comparing elements of ( M ) and ( N ), the common elements are 1 and 3. - Therefore, ( P = {1, 3} ).2. Calculating the number of subsets: - The number of subsets of a set with ( n ) elements is given by ( 2^n ). - The set ( P ) has 2 elements, so the number of subsets is ( 2^2 = 4 ).Listing the subsets of ( P ):- The empty set: ( {} )- ( {1} )- ( {3} )- ( {1, 3} )Thus, the number of subsets of ( P ) is (boxed{4}).]

🔑:Since M = {0, 1, 2, 3, 4} and N = {1, 3, 5}, we find the intersection of M and N to determine set P: P = M cap N = {1, 3} Now we want to find all the subsets of P. Remember that a set with n elements has 2^n subsets (including the empty set and the set itself). So for set P which has 2 elements, we would have: 2^n = 2^2 = 4 text{ subsets} Listing the subsets of P:1. The empty set: emptyset2. The set containing just 1: {1}3. The set containing just 3: {3}4. The set containing both 1 and 3: {1, 3}In total, there are 4 subsets of P. Thus, the answer is: boxed{4}

🧠:[<think>Okay, let me try to figure this problem out. So, the question is about probabilities of getting heads and tails when flipping a coin. They mention forming an equation with combinations, specifically C(10,7) p^7 q^3 equals C(10,6) p^6 q^4. Then they solve for p, getting 7/11. Hmm, I need to understand how they arrived at that solution.First, let me recall what C(n,k) means. That's the combination formula, right? It's the number of ways to choose k successes out of n trials. So, C(10,7) would be the number of ways to get 7 heads out of 10 flips, and C(10,6) is the number of ways to get 6 heads out of 10 flips. The equation given is equating the probabilities of getting exactly 7 heads and exactly 6 heads. So, the equation is:C(10,7) * p^7 * q^3 = C(10,6) * p^6 * q^4Where p is the probability of heads and q is the probability of tails, which is 1 - p.The next step they show is simplifying this equation to 120p = 210q. Then, from that, they find p/q = 7/4, leading to p = 7/11. I need to verify each step here.First, let me compute C(10,7) and C(10,6). The combination formula is n! / (k! (n - k)! )So, C(10,7) = 10! / (7! 3!) = (10*9*8)/(3*2*1) = 120. Similarly, C(10,6) = 10! / (6! 4!) = (10*9*8*7)/(4*3*2*1) = 210. That matches the coefficients in the equation 120p = 210q. So, when we divide both sides by p^6 q^3, we get:C(10,7) * p = C(10,6) * qWhich is 120p = 210q. Then, dividing both sides by 210 and by p, we get:(120/210) = q/pWait, hold on. Let's see. If 120p = 210q, then dividing both sides by 210p, we get:(120/210) = q/pBut 120/210 simplifies to 4/7, right? Because both divide by 30: 120 ÷ 30 = 4, 210 ÷ 30 = 7. So 4/7 = q/p. Therefore, q = (4/7)p. But since q = 1 - p, we can substitute:1 - p = (4/7)pSo, let's solve for p. Bring all terms to one side:1 = (4/7)p + p = (4/7 + 7/7)p = (11/7)pMultiply both sides by 7/11:p = 7/11Oh, okay, that makes sense. Wait, but in the original solution they wrote p/q = 7/4. So maybe they rearranged 120p = 210q as p/q = 210/120 = 7/4. That's right. 210 divided by 120 is 7/4. So, p/q = 7/4, which means p = (7/4)q. Then, since q = 1 - p, substituting:p = (7/4)(1 - p)Multiply both sides by 4:4p = 7(1 - p)4p = 7 - 7p4p + 7p = 711p = 7p = 7/11Same result. So either way you approach it, you end up with p = 7/11. Let me check if all the steps are valid.Starting with the equation of the two probabilities: C(10,7)p^7 q^3 = C(10,6)p^6 q^4. Dividing both sides by p^6 q^3 gives C(10,7)p = C(10,6)q. Then substituting the combination numbers: 120p = 210q. Simplifying the ratio p/q = 210/120 = 7/4. Therefore, p = (7/4)q. Since q = 1 - p, substitution gives p = (7/4)(1 - p). Solving that equation gives p = 7/11. All steps check out.Is there another way to think about this problem? Maybe intuitively. If the probability of getting 7 heads is equal to the probability of getting 6 heads in 10 flips, then the ratio of these probabilities is 1. The ratio of the probabilities is [C(10,7)/C(10,6)] * (p/q) = 1. Which gives (120/210)*(p/q) = 1 => (4/7)*(p/q) = 1 => p/q = 7/4. So same result. That's another way to see it.Alternatively, in general, for a binomial distribution, if the probability of k successes equals the probability of k-1 successes, you can set up the equation C(n,k)p^k q^{n -k} = C(n, k -1)p^{k -1} q^{n -k +1} and solve for p. The ratio would give you [C(n,k)/C(n, k -1)] * p/q = 1. Then p/q = C(n, k -1)/C(n,k). Which simplifies to [n! / ((k -1)! (n - k + 1)!))] / [n! / (k! (n - k)!))] = [k / (n - k + 1)]. So, p/q = k / (n - k + 1). Therefore, p = [k / (n - k +1 + k)] = k / (n +1). Wait, that can't be. Wait, let me recast.Wait, if p/q = k / (n - k +1), then p = [k / (n - k +1 + k)] * (1). Wait, no. If p/q = k / (n - k +1), then since p + q =1, so q =1 - p. Then p/(1 - p) = k/(n -k +1). Solving for p:p = [k/(n -k +1)]*(1 - p)p = [k/(n -k +1)] - [k/(n -k +1)]pBring terms together:p + [k/(n -k +1)]p = k/(n -k +1)p[1 + k/(n -k +1)] = k/(n -k +1)Multiply numerator and denominator:p[(n -k +1 +k)/(n -k +1)] = k/(n -k +1)Simplify numerator inside the brackets:(n +1)/(n -k +1)Therefore:p * (n +1)/(n -k +1) = k/(n -k +1)Multiply both sides by (n -k +1)/(n +1):p = k/(n +1)Ah, so in general, if the probabilities of k and k-1 successes are equal, then p = k/(n +1). That's a general result. So in this problem, n =10, k =7. Then p =7/(10 +1)=7/11. Exactly! So that's a quicker way. So instead of computing combinations, you can use that formula. But it's good to verify with the combinations.So, in this case, setting k =7, n=10, so p=7/11. So that's the answer. Therefore, the probability of heads is 7/11. Let me check with another example to see if this general formula holds. Suppose n=2, k=1. Then according to the formula, p=1/(2+1)=1/3. Let's check. The equation would be C(2,1)p(1-p) = C(2,0)(1-p)^2. Which is 2p(1-p) =1*(1-p)^2. Dividing both sides by (1-p), assuming p ≠1, gives 2p =1 -p, so 3p=1, p=1/3. Yep, that works. So the general formula holds.Therefore, the answer here is indeed p=7/11. The initial steps are correct, and the general formula corroborates it. So I feel confident that the solution is correct.Final AnswerThe probability of heads is boxed{dfrac{7}{11}}.</think>Given the problem of finding the probability of heads ( p ) such that the probability of getting 7 heads in 10 flips is equal to the probability of getting 6 heads in 10 flips, we start by setting up the equation using the binomial probability formula:[C_{10}^{7} p^{7} q^{3} = C_{10}^{6} p^{6} q^{4}]where ( q = 1 - p ). Simplifying this equation, we divide both sides by ( p^6 q^3 ):[C_{10}^{7} p = C_{10}^{6} q]Calculating the combinations, we get ( C_{10}^{7} = 120 ) and ( C_{10}^{6} = 210 ). Substituting these values into the equation, we have:[120 p = 210 q]This simplifies to:[frac{p}{q} = frac{210}{120} = frac{7}{4}]Thus, ( p = frac{7}{4} q ). Since ( q = 1 - p ), substituting ( q ) gives:[p = frac{7}{4} (1 - p)]Solving for ( p ):[4p = 7 - 7p][11p = 7][p = frac{7}{11}]This result can also be verified using the general formula for the probability ( p ) when the probabilities of ( k ) and ( k-1 ) successes are equal, which gives ( p = frac{k}{n + 1} ). For ( n = 10 ) and ( k = 7 ):[p = frac{7}{10 + 1} = frac{7}{11}]Thus, the probability of heads is (boxed{dfrac{7}{11}}).,<think>Okay, so I need to solve this probability problem where they give an equation involving combinations and probabilities of heads and tails. Let me try to understand what's going on here.The problem states that we form the equation:C_{10}^{7} p^{7} q^{3} = C_{10}^{6} p^{6} q^{4}And from that, they conclude that 120p = 210q, then p/q = 7/4, leading to p = 7/11. Hmm, let me break this down step by step.First, I know that in probability, when dealing with coins, p is the probability of heads and q is tails, and since they are complementary, q = 1 - p. The notation C_{10}^{7} is the combination of 10 things taken 7 at a time, which is also written as (binom{10}{7}). Similarly, C_{10}^{6} is (binom{10}{6}).The equation given looks like the probability of getting 7 heads and 3 tails in 10 flips equals the probability of getting 6 heads and 4 tails. So, they set these two probabilities equal and solve for p. That makes sense if they want to find the value of p where these two probabilities are the same.Let me verify the combinations first. (binom{10}{7}) is equal to 120, right? Because (binom{n}{k} = frac{n!}{k!(n - k)!}). So, (binom{10}{7} = frac{10!}{7!3!} = frac{10 times 9 times 8}{3 times 2 times 1} = 120). Similarly, (binom{10}{6} = frac{10!}{6!4!} = frac{10 times 9 times 8 times 7}{4 times 3 times 2 times 1} = 210). So, that part checks out: 120 and 210 are the correct coefficients.So substituting these into the equation:120 * p^7 * q^3 = 210 * p^6 * q^4Now, let's simplify this equation. Let's divide both sides by p^6 * q^3 to reduce the exponents. That would leave:120 * p = 210 * qYes, that seems right. Because p^7 divided by p^6 is p, and q^3 divided by q^3 is 1, and on the right side, q^4 divided by q^3 is q. So the equation becomes 120p = 210q.Then, since q = 1 - p, we can substitute that in. But before that, maybe they just divided both sides by q and then by 120? Let me see.If I have 120p = 210q, then dividing both sides by 210 gives (120/210)p = q. Simplifying 120/210, both are divisible by 30: 120 ÷ 30 = 4, 210 ÷ 30 = 7. So 4/7 p = q. But since q = 1 - p, we can write 4/7 p = 1 - p.Alternatively, they might have taken the ratio p/q. Let's see. From 120p = 210q, dividing both sides by q and by 120 gives p/q = 210/120. Simplify 210/120: divide numerator and denominator by 30, which gives 7/4. So p/q = 7/4. Therefore, p = (7/4) q. But since q = 1 - p, substitute:p = (7/4)(1 - p)Multiply both sides by 4 to eliminate the denominator:4p = 7(1 - p)4p = 7 - 7pAdd 7p to both sides:11p = 7Therefore, p = 7/11. So that's how they get p = 7/11. That makes sense.Let me just check all the steps again to make sure I didn't make a mistake. Starting from the original equation:(binom{10}{7} p^7 q^3 = binom{10}{6} p^6 q^4)Calculating the combinations: 120 and 210, correct.Simplify by dividing both sides by p^6 q^3:120p = 210qThen, ratio p/q = 210/120 = 7/4. Hence, p = (7/4)q. Then substitute q = 1 - p:p = (7/4)(1 - p)Multiply both sides by 4:4p = 7 - 7p11p = 7 => p = 7/11. All steps seem valid.Alternatively, if I didn't take the ratio, and instead substituted q = 1 - p directly into 120p = 210q:120p = 210(1 - p)120p = 210 - 210p120p + 210p = 210330p = 210p = 210 / 330 = 7/11. Same result. So that's another way to do it. Either approach works.So, the key steps are recognizing the binomial probabilities, setting them equal, simplifying the equation by canceling common terms, then solving for p either by substitution or ratio. Either method leads to p = 7/11.Just to think about it intuitively, if getting 7 heads is equally likely as getting 6 heads in 10 flips, then the coin must be biased towards heads because you need more heads to balance out the combinations which are fewer? Wait, no, the combinations for 7 heads is 120 and for 6 heads is 210, which is higher. So actually, even though 210 is higher, if p is higher, the probability of 7 heads would be higher. So to make them equal, p has to compensate for the higher combination number for 6 heads. Since 210 is higher than 120, the probability of 6 heads would naturally be higher unless p is higher to make 7 heads more probable. So the balance is such that p needs to be higher to make 7 heads as likely as 6 heads despite the lower combination count. So, p = 7/11 is approximately 0.636, which is higher than 0.5, so that makes sense.Yes, that checks out. If the coin were fair, p = 0.5, then getting 6 heads would be more probable than 7 heads because there are more combinations for 6 heads. But if we have a biased coin where heads are more likely, the probability of 7 heads can catch up to 6 heads. So p = 7/11 ≈ 0.636 is the point where those two probabilities are equal. That seems reasonable.I don't see any mistakes in the reasoning. The steps are logically consistent, and the algebra checks out. So, the answer is p = 7/11.Final AnswerThe probability of heads is boxed{dfrac{7}{11}}.</think>Given the problem where we need to find the probability of heads ( p ) such that the probability of getting 7 heads and 3 tails in 10 flips equals the probability of getting 6 heads and 4 tails, we start with the equation:C_{10}^{7} p^{7} q^{3} = C_{10}^{6} p^{6} q^{4}First, we calculate the combinations:- ( binom{10}{7} = 120 )- ( binom{10}{6} = 210 )Substituting these values into the equation, we get:120 p^7 q^3 = 210 p^6 q^4We simplify by dividing both sides by ( p^6 q^3 ):120 p = 210 qNext, we express ( q ) in terms of ( p ) (since ( q = 1 - p )) and solve for ( p ):frac{p}{q} = frac{210}{120} = frac{7}{4}Thus, ( p = frac{7}{4} q ). Substituting ( q = 1 - p ) into this equation:p = frac{7}{4} (1 - p)Multiplying both sides by 4 to clear the fraction:4p = 7 - 7pCombining like terms:11p = 7Solving for ( p ):p = frac{7}{11}Thus, the probability of heads is boxed{dfrac{7}{11}}.]

🔑:Answer: frac{7}{11}.Grading Criteria| Correct and justified solution | 2 points || :--- | :---: || Correct equation is set up, but an error is made or the solution is not completed | 1 point || Solution is incorrect or missing (including only providing the answer) | 0 points |