answer：Okay, let's try to figure out this problem step by step. So, we have two propositions, p and q, and we need to find the range of the real number a such that p is true and q is false. First, let's understand proposition p: The function f(x) = 2a x² - x - 1 (where a ≠ 0) has exactly one zero in the interval (0, 1). Hmm, okay. So, this is a quadratic function, and we need it to cross the x-axis exactly once between 0 and 1. I remember that for a quadratic function ax² + bx + c, the discriminant Δ = b² - 4ac tells us about the nature of the roots. If Δ > 0, there are two distinct real roots; if Δ = 0, there's exactly one real root; and if Δ < 0, there are no real roots. But in this case, we need exactly one zero in the interval (0, 1). So, it's not just about the number of roots, but where they lie.I think I need to use the Intermediate Value Theorem here. If the function changes sign over the interval (0, 1), then by the theorem, there must be at least one root in that interval. But since it's a quadratic, it can have at most two roots. So, to have exactly one root in (0, 1), the function must cross the x-axis once in that interval and not cross it again elsewhere.Wait, but quadratics can only have two roots at most, so if one is inside (0, 1), the other could be outside. So, maybe we need to ensure that f(0) and f(1) have opposite signs, which would guarantee a root in between. But also, we need to make sure that the quadratic doesn't have both roots in (0, 1). So, maybe we need to check the discriminant as well.Let me write down the conditions:1. The function f(x) must have exactly one root in (0, 1). So, f(0) and f(1) must have opposite signs. That is, f(0) * f(1) < 0.2. Also, since it's a quadratic, the discriminant must be positive to have two real roots. But since we want exactly one in (0, 1), the other root must be outside of (0, 1). So, maybe we need to ensure that the quadratic doesn't have both roots in (0, 1). Wait, actually, if the quadratic has two roots, and we want exactly one in (0, 1), then the other root must be either less than or equal to 0 or greater than or equal to 1. So, perhaps we can use the conditions on the roots.Alternatively, maybe using calculus to check the behavior of the function. Let me think.First, let's compute f(0) and f(1):f(0) = 2a*(0)^2 - 0 - 1 = -1.f(1) = 2a*(1)^2 - 1 - 1 = 2a - 2.So, f(0) = -1 and f(1) = 2a - 2.For f(0) * f(1) < 0, we need (-1)*(2a - 2) < 0.That simplifies to (2a - 2) > 0, because multiplying by -1 reverses the inequality.So, 2a - 2 > 0 => 2a > 2 => a > 1.So, that's one condition: a > 1.But we also need to make sure that the quadratic doesn't have both roots in (0, 1). So, let's find the roots of f(x):f(x) = 2a x² - x - 1.Using the quadratic formula:x = [1 ± sqrt(1 + 8a)] / (4a).So, the roots are [1 + sqrt(1 + 8a)] / (4a) and [1 - sqrt(1 + 8a)] / (4a).We need to ensure that only one of these roots is in (0, 1). Let's analyze the roots.First, since a ≠ 0, and from the earlier condition, a > 1, so a is positive.So, sqrt(1 + 8a) is positive, so both numerator terms are positive.So, the larger root is [1 + sqrt(1 + 8a)] / (4a), and the smaller root is [1 - sqrt(1 + 8a)] / (4a).Wait, but sqrt(1 + 8a) is greater than 1, since 8a is positive, so 1 + 8a > 1, so sqrt(1 + 8a) > 1.Therefore, the smaller root is [1 - something greater than 1] / (4a), which is negative. So, the smaller root is negative, which is less than 0.Therefore, the only root in (0, 1) is the larger root, [1 + sqrt(1 + 8a)] / (4a). So, we need this root to be less than 1.So, let's set [1 + sqrt(1 + 8a)] / (4a) < 1.Multiply both sides by 4a (since a > 1, 4a is positive, so inequality sign remains):1 + sqrt(1 + 8a) < 4a.Subtract 1:sqrt(1 + 8a) < 4a - 1.Now, square both sides (since both sides are positive, because a > 1, so 4a -1 > 3 > 0):1 + 8a < (4a - 1)^2.Compute the right side:(4a - 1)^2 = 16a² - 8a + 1.So, 1 + 8a < 16a² - 8a + 1.Subtract 1 + 8a from both sides:0 < 16a² - 16a.Factor:0 < 16a(a - 1).Since a > 1, both 16a and (a - 1) are positive, so 16a(a - 1) > 0.Therefore, the inequality holds for a > 1.Wait, so that condition is automatically satisfied for a > 1. So, does that mean that as long as a > 1, the larger root is less than 1?Wait, let me test with a = 2.Compute [1 + sqrt(1 + 16)] / 8 = [1 + sqrt(17)] / 8 ≈ (1 + 4.123)/8 ≈ 5.123/8 ≈ 0.64, which is less than 1.Similarly, for a = 1.5:sqrt(1 + 12) = sqrt(13) ≈ 3.606, so [1 + 3.606]/6 ≈ 4.606/6 ≈ 0.768, which is less than 1.For a approaching 1 from above, say a = 1.1:sqrt(1 + 8.8) = sqrt(9.8) ≈ 3.13, so [1 + 3.13]/4.4 ≈ 4.13/4.4 ≈ 0.939, still less than 1.So, it seems that for a > 1, the larger root is always less than 1. Therefore, the only root in (0, 1) is the larger one, and it's less than 1.Therefore, the condition for proposition p is a > 1.Okay, so that's p.Now, let's look at proposition q: The function y = x^{2 - a} is decreasing on the interval (0, +∞).We need to find when q is false, so the negation of q is true. So, when is y = x^{2 - a} not decreasing on (0, +∞)?First, let's recall that for the function y = x^k, it is decreasing on (0, +∞) if and only if k < 0. Because the derivative y' = k x^{k - 1}, which is negative when k < 0.So, for y = x^{2 - a} to be decreasing, we need 2 - a < 0 => a > 2.Therefore, proposition q is true when a > 2, and false when a ≤ 2.But we need the negation of q to be true, which is when a ≤ 2.So, putting it all together: proposition p is true when a > 1, and the negation of q is true when a ≤ 2. Therefore, the range of a where both p is true and q is false is the intersection of a > 1 and a ≤ 2, which is 1 < a ≤ 2.So, the correct answer should be option C: 1 < a ≤ 2.Let me double-check.For p: a > 1.For ¬q: a ≤ 2.Intersection: 1 < a ≤ 2. Yep, that seems right.I think that's solid. I don't see any mistakes in the reasoning.

answer：Okay, so I have this problem about a right triangle ABC where AC is 16 units and BC is 12 units. There's a circle centered at B with radius BC, which is 12 units. Then, there's a tangent to this circle that's parallel to the hypotenuse AB, and this tangent is on the opposite side of AB from the triangle. I need to figure out how much the leg BC is extended to meet this tangent.First, I should probably sketch the triangle to visualize it better. Let me imagine triangle ABC with right angle at C. So, AC is one leg, BC is the other leg, and AB is the hypotenuse. AC is 16, BC is 12, so I can use the Pythagorean theorem to find AB.Calculating AB: AB² = AC² + BC² = 16² + 12² = 256 + 144 = 400. So AB is √400, which is 20 units. Got that.Now, there's a circle centered at B with radius BC, which is 12. So, the circle will pass through point C because BC is the radius. Now, a tangent is drawn to this circle parallel to AB, and this tangent is on the opposite side of AB from the triangle. So, if the triangle is on one side of AB, the tangent is on the other side.I need to find where this tangent intersects the extension of BC. Let me denote the point where the tangent touches the circle as M and the point where the tangent intersects the extension of BC as K. So, K is beyond C on the line BC extended.Since the tangent MK is parallel to AB, the triangles BMK and ABC should be similar because they have the same angles. Let me check that. Both triangles have a right angle at M and C respectively, and they share angle at B. So, yes, by AA similarity, triangles BMK and ABC are similar.Therefore, the ratio of corresponding sides should be equal. So, BM / BC = BK / AB.Wait, BM is the radius of the circle, which is BC, so BM = 12. BC is also 12. So, BM / BC = 12 / 12 = 1. But that would imply BK / AB = 1, so BK = AB = 20. But that doesn't make sense because BK is an extension beyond BC, which is already 12. So, BK should be longer than 12, but 20 is longer than 12, but I think I might have messed up the ratio.Wait, maybe I got the ratio wrong. Let's think again. Since triangle BMK is similar to triangle ABC, the ratio should be BM / AB = BK / AC.Wait, no, let's define the similarity properly. In triangle ABC, the sides are AB = 20, BC = 12, AC = 16. In triangle BMK, the sides are BM = 12, BK (which is BC extended to K, so BK = BC + CK = 12 + x, where x is the extension we need to find), and MK.Since the triangles are similar, the ratio of corresponding sides should be equal. So, BM / BC = BK / AB.Wait, BM is 12, BC is 12, so BM / BC = 1. Then BK / AB = 1, so BK = AB = 20. But BK is the length from B to K, which is BC extended by x, so BK = 12 + x. Therefore, 12 + x = 20, so x = 8. Hmm, but I thought the answer was 15. Maybe I'm missing something.Wait, maybe I got the similar triangles wrong. Let me think again. The tangent MK is parallel to AB, so the angle at B is common, and both have a right angle. So, triangle BMK is similar to triangle BAC, not ABC. Wait, no, triangle ABC is the original triangle, and triangle BMK is the smaller one.Wait, maybe I should use the fact that the tangent is parallel to AB, so the slope of MK is the same as AB. Let me assign coordinates to the triangle to make it easier.Let me place point C at (0,0), point B at (0,12), and point A at (16,0). Then, AB is the hypotenuse from (16,0) to (0,12). The circle centered at B (0,12) with radius 12 will have the equation x² + (y - 12)² = 12².Now, the tangent to this circle parallel to AB. The slope of AB is (12 - 0)/(0 - 16) = 12 / (-16) = -3/4. So, the tangent line will have the same slope, -3/4.The equation of the tangent line can be written as y = (-3/4)x + c. Since it's tangent to the circle, the distance from the center B (0,12) to this line should be equal to the radius, which is 12.The distance from a point (x0,y0) to the line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a² + b²). So, let's write the tangent line in standard form: (3/4)x + y - c = 0. So, a = 3/4, b = 1, c = -c.The distance from (0,12) to this line is |(3/4)(0) + (1)(12) - c| / sqrt((3/4)² + 1²) = |12 - c| / sqrt(9/16 + 1) = |12 - c| / sqrt(25/16) = |12 - c| / (5/4) = (4/5)|12 - c|.This distance should be equal to the radius, which is 12. So, (4/5)|12 - c| = 12. Therefore, |12 - c| = (12 * 5)/4 = 15. So, 12 - c = 15 or 12 - c = -15.If 12 - c = 15, then c = 12 - 15 = -3.If 12 - c = -15, then c = 12 + 15 = 27.Now, since the tangent is on the opposite side of AB from the triangle, which is in the lower part, the tangent should be above the triangle. So, the y-intercept c should be higher than 12. So, c = 27.Therefore, the equation of the tangent line is y = (-3/4)x + 27.Now, we need to find where this tangent intersects the extension of BC. Since BC is the line from (0,12) to (0,0), which is the y-axis. So, the extension of BC beyond C is the line x = 0, y < 0. Wait, but the tangent line is y = (-3/4)x + 27. To find the intersection with BC extended, which is x=0, plug x=0 into the tangent line: y = 0 + 27 = 27. So, the point K is at (0,27).But wait, BC is from (0,12) to (0,0). Extending it beyond C would go towards negative y, but the tangent intersects BC extended at (0,27), which is above B. That doesn't make sense because the tangent is supposed to be on the opposite side of AB from the triangle. Maybe I made a mistake in determining the side.Wait, the triangle is in the lower part, so the opposite side of AB would be above the triangle. So, the tangent is above the triangle, which is why c=27 is correct. But then, the intersection point K is at (0,27), which is above B at (0,12). So, the extension of BC beyond B is upwards, but BC is from B to C, so extending BC beyond C would be downwards. Wait, maybe I got the direction wrong.Wait, BC is from B (0,12) to C (0,0). So, extending BC beyond C would go towards negative y, but the tangent is above the triangle, so it intersects BC extended beyond B, not beyond C. So, the intersection point K is at (0,27), which is 15 units above B. So, the extension is from B to K, which is 15 units.Wait, but the problem says "the leg BC is extended to intersect this tangent." So, BC is from B to C, so extending it beyond C would go towards negative y, but the tangent is above the triangle, so it intersects the extension beyond B, not beyond C. So, maybe the extension is from B upwards to K, which is 15 units beyond B. So, the length by which BC is extended is 15 units.Wait, but in the problem statement, it says "the leg BC is extended to intersect this tangent." So, BC is the leg from B to C, so extending it beyond C would be in the direction away from B, but the tangent is on the opposite side of AB from the triangle, which is above AB, so the tangent is above the triangle, so the intersection point K is above B, not below C. Therefore, the extension is from B upwards to K, which is 15 units beyond B. So, the length by which BC is extended is 15 units.Wait, but in the coordinate system, BC is from (0,12) to (0,0). Extending it beyond C would go to (0,-something), but the tangent intersects BC extended beyond B, which is upwards to (0,27). So, the extension is from B to K, which is 15 units. So, the length by which BC is extended is 15 units.Alternatively, maybe the extension is from C beyond C, but in that case, the tangent is above the triangle, so it wouldn't intersect the extension beyond C. So, the correct extension is from B upwards to K, which is 15 units beyond B. So, the answer is 15.Wait, but earlier I thought the ratio gave me 8, but that was probably because I messed up the similar triangles. Using coordinates and equations, I got 15, which seems correct. So, I think the answer is 15.

answer：Alright, so I've got this problem here about a store that received some plates. They got fewer than 600 but more than 500 plates. Okay, so the number of plates is somewhere between 501 and 599. Got that down.Now, when they tried to arrange the plates in groups of ten, three plates were missing to make complete groups of ten. Hmm, let me think about that. So, if they're trying to make groups of ten and they're three plates short, that means the total number of plates isn't a multiple of ten. Instead, it's three less than a multiple of ten. So, mathematically, if I let the number of plates be ( n ), then ( n ) is congruent to -3 modulo 10. But since we're dealing with positive numbers, -3 modulo 10 is the same as 7 modulo 10. So, ( n equiv 7 mod 10 ). That means when you divide ( n ) by 10, the remainder is 7. So, the number of plates ends with a 7. Got it.Next, when they tried to arrange them in dozens, which is 12 plates, 7 plates remained. So, similar to before, this means that when ( n ) is divided by 12, the remainder is 7. So, ( n equiv 7 mod 12 ). Okay, so now I have two congruences:1. ( n equiv 7 mod 10 )2. ( n equiv 7 mod 12 )And I also know that ( 500 < n < 600 ).So, I need to find a number between 501 and 599 that leaves a remainder of 7 when divided by both 10 and 12. Hmm, interesting. Since both congruences have the same remainder, maybe there's a way to combine these.Let me think about the Chinese Remainder Theorem or something like that. But I'm not sure if I need to go that deep. Maybe I can just list out numbers that satisfy one condition and see which ones satisfy the other.Starting with the first condition: ( n equiv 7 mod 10 ). So, numbers between 501 and 599 that end with 7. Let's list them:507, 517, 527, 537, 547, 557, 567, 577, 587, 597.Okay, that's all the numbers between 501 and 599 that end with 7. Now, from these, I need to find which ones also satisfy ( n equiv 7 mod 12 ).So, let's check each one:1. 507 divided by 12: 12*42 = 504, so 507 - 504 = 3. Remainder is 3. Not 7.2. 517 divided by 12: 12*43 = 516, so 517 - 516 = 1. Remainder is 1. Not 7.3. 527 divided by 12: 12*43 = 516, 527 - 516 = 11. Remainder is 11. Not 7.4. 537 divided by 12: 12*44 = 528, 537 - 528 = 9. Remainder is 9. Not 7.5. 547 divided by 12: 12*45 = 540, 547 - 540 = 7. Remainder is 7. Bingo!6. Let's check the rest just to be thorough.7. 557 divided by 12: 12*46 = 552, 557 - 552 = 5. Remainder is 5. Not 7.8. 567 divided by 12: 12*47 = 564, 567 - 564 = 3. Remainder is 3. Not 7.9. 577 divided by 12: 12*48 = 576, 577 - 576 = 1. Remainder is 1. Not 7.10. 587 divided by 12: 12*48 = 576, 587 - 576 = 11. Remainder is 11. Not 7.11. 597 divided by 12: 12*49 = 588, 597 - 588 = 9. Remainder is 9. Not 7.So, the only number that satisfies both conditions is 547. Let me double-check:547 divided by 10 is 54 with a remainder of 7. That matches the first condition.547 divided by 12 is 45 with a remainder of 7. That matches the second condition.And 547 is indeed between 500 and 600. Perfect.I think that's the answer. It makes sense because when you have two congruences with the same remainder, the solution is the least common multiple of the moduli plus the remainder. But since we're dealing with a specific range, listing out the possibilities was manageable.Another way to approach it could be to set up the equations:Let ( n = 10k + 7 ) and ( n = 12m + 7 ). So, ( 10k + 7 = 12m + 7 ), which simplifies to ( 10k = 12m ) or ( 5k = 6m ). This means that ( k ) must be a multiple of 6, and ( m ) must be a multiple of 5. Let ( k = 6a ) and ( m = 5a ). Then, ( n = 10*6a + 7 = 60a + 7 ).Now, since ( 500 < n < 600 ), we have ( 500 < 60a + 7 < 600 ). Subtracting 7 gives ( 493 < 60a < 593 ). Dividing by 60, we get ( 8.2167 < a < 9.8833 ). Since ( a ) must be an integer, ( a = 9 ). Plugging back in, ( n = 60*9 + 7 = 547 ).Yep, same result. So, whether I list them out or solve it algebraically, I end up with 547. That must be the correct answer.

answer：Okay, so I've got this quadratic equation: x² + 2x - 5 = 0. The roots are m and n. I need to find the value of m² + mn + 2m. Hmm, let's see how to approach this.First, I remember that for a quadratic equation ax² + bx + c = 0, the sum of the roots is -b/a and the product is c/a. So in this case, a is 1, b is 2, and c is -5. That means m + n = -2/1 = -2, and mn = -5/1 = -5. Okay, so I have m + n = -2 and mn = -5.Now, I need to find m² + mn + 2m. Let me break this down. I can see that m² + 2m is part of the expression. Wait a second, the original equation is x² + 2x - 5 = 0. If I plug m into the equation, it should satisfy it, right? So, m² + 2m - 5 = 0. That means m² + 2m = 5. Cool, so m² + 2m is 5.So, going back to the expression m² + mn + 2m, I can substitute m² + 2m with 5. That gives me 5 + mn. But I already know mn is -5. So, 5 + (-5) = 0. So, the value of m² + mn + 2m is 0.Wait, let me double-check that. If m² + 2m = 5, and mn = -5, then adding them together gives 5 + (-5) = 0. Yeah, that seems right. I don't think I made any mistakes here. So, the answer should be 0, which is option C.Just to make sure, maybe I can try plugging in the actual roots. Let me solve the quadratic equation to find m and n. Using the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a. Plugging in the values, x = [-2 ± √(4 + 20)] / 2 = [-2 ± √24] / 2 = [-2 ± 2√6] / 2 = -1 ± √6. So, m and n are -1 + √6 and -1 - √6.Let's pick m = -1 + √6. Then, m² = (-1 + √6)² = 1 - 2√6 + 6 = 7 - 2√6. mn = (-1 + √6)(-1 - √6) = 1 + √6 - √6 - 6 = -5. And 2m = 2*(-1 + √6) = -2 + 2√6.So, m² + mn + 2m = (7 - 2√6) + (-5) + (-2 + 2√6). Let's compute this: 7 - 5 - 2 = 0, and -2√6 + 2√6 = 0. So, it all adds up to 0. Yep, that confirms it. The value is indeed 0.

answer：First, I need to simplify the given expression by distributing the negative sign and combining like terms.I'll start by rewriting the expression:[(10x^3 + 45x^2 - 5x) - (-5x^3 + 10x^2 - 5x)]Next, I'll distribute the negative sign to each term inside the second parenthesis:[10x^3 + 45x^2 - 5x + 5x^3 - 10x^2 + 5x]Now, I'll combine the like terms:- For the (x^3) terms: (10x^3 + 5x^3 = 15x^3)- For the (x^2) terms: (45x^2 - 10x^2 = 35x^2)- For the (x) terms: (-5x + 5x = 0)This simplifies the expression to:[15x^3 + 35x^2]Now, I'll factor out the greatest common factor (GCF) from the simplified expression. The GCF of (15x^3) and (35x^2) is (5x^2).Factoring out (5x^2), I get:[5x^2(3x + 7)]So, the completely factored form of the expression is:[boxed{5x^2(3x + 7)}]

answer：Okay, so I have this problem about a museum with 16 halls arranged in some way, and half of them have paintings while the other half have sculptures. The tour starts at hall A, which has paintings, and ends at hall B. The halls alternate between paintings and sculptures as you move from one to another. First, I need to figure out which halls have paintings. Since half of the 16 halls have paintings, that means 8 halls have paintings and 8 have sculptures. The tour starts at hall A, which is a painting hall, and alternates from there. So, if I can figure out the pattern of alternation, I can mark all the painting halls with crosses.But wait, the problem says that the solution can be seen in the diagram, but I don't have the diagram. Hmm, maybe I can figure it out without the diagram. Let's think about the arrangement. If it's a typical museum layout, maybe it's a grid or some symmetrical shape. Since there are 16 halls, it could be a 4x4 grid. In a 4x4 grid, each hall is connected to its adjacent halls (up, down, left, right). So, starting from hall A, which is a painting hall, the next hall must be a sculpture hall, then back to painting, and so on. If the grid is colored like a chessboard, with alternating colors, then painting halls would be on one color and sculpture halls on the other. Assuming hall A is on a corner, like the top-left corner, then the painting halls would be all the halls on the same color as A. So, in a 4x4 grid, that would be 8 halls. I can mark those with crosses. Now, part b asks about the maximum number of halls a tourist can visit without repeating any, starting at A and ending at B. Since the halls alternate between paintings and sculptures, the path must alternate as well. So, starting at a painting hall, the next is sculpture, then painting, and so on. To maximize the number of halls visited, the path should be as long as possible without repeating any halls. In graph theory terms, this is like finding the longest path in a bipartite graph, which is a graph where nodes are divided into two sets (painting and sculpture halls) and edges only go between sets, not within. In a bipartite graph, the longest path is limited by the number of nodes in the smaller set. Since both sets have 8 halls each, the maximum path length would be 15 halls, alternating between the two sets. However, since the tour starts and ends at specific halls (A and B), both of which are painting halls, the path must start and end in the same set. This means that the number of painting halls visited must be one more than the number of sculpture halls visited. So, if there are 8 painting halls and 8 sculpture halls, the maximum number of halls visited would be 8 (painting) + 7 (sculpture) = 15 halls. To prove that 15 is the maximum, consider that if you try to visit all 16 halls, you would have to alternate perfectly, but since both A and B are painting halls, you can't end on a sculpture hall. Therefore, visiting all 16 halls would require starting and ending on different types, which isn't the case here. Hence, 15 is the maximum number of halls that can be visited without repetition.