answer：Alright, so I have this geometry problem here. Let me try to figure it out step by step. First, let me understand the problem.We have a diagram with points A, B, C, D. The coordinates are given in the Asymptote code, but I think I can visualize it without looking at the exact coordinates. AB is 36 units long, and angle ADB is 90 degrees. So, triangle ADB is a right triangle with the right angle at D. Also, we're given that sin A is 3/5 and sin C is 1/4. The question is asking for the length of DC.Hmm, okay. Let me try to sketch this in my mind. Point A is at the origin, I think, based on the Asymptote code. Point D is somewhere along the x-axis, and point B is somewhere above D, making triangle ADB a right triangle. Point C is further along the x-axis from D, connected to B and A.So, AB is 36, and angle at D is 90 degrees. So, triangle ADB is right-angled at D. We also have triangle BCD, which is another right triangle because angle at D is 90 degrees as well? Wait, no. Wait, angle ADB is 90 degrees, but angle CDB isn't necessarily 90 degrees. Hmm, maybe I need to clarify that.Wait, looking back at the Asymptote code, it draws D connected to B, B connected to A, A connected to C, and C connected to B. So, the figure is a quadrilateral with points A, B, C, D, where A is connected to B and C, B is connected to D, and D is connected to C. Also, there's a right angle at D between B and A, meaning angle ADB is 90 degrees.So, triangle ABD is a right triangle with right angle at D, and triangle BCD is another triangle connected to it. But I don't know if triangle BCD is right-angled. Let me check the Asymptote code again. It says draw(D--B--A--C--B); so, it's connecting D to B, B to A, A to C, and C to B. So, the figure is a quadrilateral with two triangles sharing the side AB.Wait, no, actually, AB is 36, and AD is connected to D, which is connected to C. So, maybe triangle ABD is a right triangle, and then triangle BCD is another triangle attached to it.Given that, we have two triangles: triangle ABD and triangle BCD. We know AB is 36, angle ADB is 90 degrees, sin A is 3/5, and sin C is 1/4. We need to find DC.Alright, let's break this down. First, in triangle ABD, which is a right triangle at D, we can use the sine of angle A to find some sides.Given sin A = 3/5. In triangle ABD, angle A is at point A, so the sine of angle A is opposite over hypotenuse. The opposite side to angle A is BD, and the hypotenuse is AB, which is 36.So, sin A = BD / AB = BD / 36 = 3/5.Therefore, BD = (3/5) * 36 = (3 * 36)/5 = 108/5 = 21.6.So, BD is 21.6 units.Now, since triangle ABD is a right triangle at D, we can find AD using the Pythagorean theorem.AB^2 = AD^2 + BD^2.We know AB is 36, BD is 21.6, so:36^2 = AD^2 + 21.6^2.Calculating 36 squared: 36 * 36 = 1296.21.6 squared: Let's compute that. 21.6 is 21 and 3/5, which is 21.6. So, 21.6 * 21.6.Let me compute 21 * 21 = 441, 21 * 0.6 = 12.6, 0.6 * 21 = 12.6, and 0.6 * 0.6 = 0.36.So, (21 + 0.6)^2 = 21^2 + 2*21*0.6 + 0.6^2 = 441 + 25.2 + 0.36 = 441 + 25.2 is 466.2 + 0.36 is 466.56.So, 21.6^2 = 466.56.Therefore, AD^2 = AB^2 - BD^2 = 1296 - 466.56 = 829.44.So, AD = sqrt(829.44). Let me compute that.Well, sqrt(829.44). Let's see, 28 squared is 784, 29 squared is 841. So, it's between 28 and 29.Compute 28.8 squared: 28 * 28 = 784, 28 * 0.8 = 22.4, 0.8 * 28 = 22.4, 0.8 * 0.8 = 0.64.So, (28 + 0.8)^2 = 28^2 + 2*28*0.8 + 0.8^2 = 784 + 44.8 + 0.64 = 784 + 44.8 is 828.8 + 0.64 is 829.44.So, sqrt(829.44) is 28.8.Therefore, AD is 28.8 units.Alright, so now we have AD = 28.8, BD = 21.6, AB = 36.Now, we need to find DC. Looking at the figure, DC is a segment from D to C. Since D is connected to C, and A is connected to C, I think triangle ADC is also a triangle, but I'm not sure if it's right-angled.Wait, in the Asymptote code, point C is at (12*sqrt(13) + 108*sqrt(2), 0), so it's on the x-axis, same as D. So, D and C are both on the x-axis, with D at (12*sqrt(13), 0) and C at (12*sqrt(13) + 108*sqrt(2), 0). So, DC is the distance between D and C on the x-axis, which is 108*sqrt(2). But wait, the problem is asking for DC, so maybe that's the answer? But in the problem statement, they don't give coordinates, so I think I need to solve it without relying on the Asymptote code's coordinates.Wait, maybe I should think of triangle BCD. Since we have point B connected to C, and we know sin C is 1/4. So, in triangle BCD, angle at C is given, sin C = 1/4. So, perhaps we can find BC and then use the Pythagorean theorem to find DC.Wait, but I don't know if triangle BCD is right-angled. Hmm.Wait, in the Asymptote code, point D is at (12*sqrt(13), 0), and point C is at (12*sqrt(13) + 108*sqrt(2), 0). So, DC is 108*sqrt(2). But in the problem, they don't give coordinates, so I think I need to find DC using the given information: AB = 36, sin A = 3/5, sin C = 1/4.So, perhaps I need to consider triangles ABD and BCD.We already found BD = 21.6 in triangle ABD.Now, in triangle BCD, we have angle at C, sin C = 1/4. So, in triangle BCD, sin C = opposite side over hypotenuse. The opposite side to angle C is BD, which is 21.6, and the hypotenuse is BC.So, sin C = BD / BC = 1/4.Therefore, BD / BC = 1/4, so BC = 4 * BD = 4 * 21.6 = 86.4.So, BC is 86.4 units.Now, in triangle BCD, we have BC = 86.4, BD = 21.6, and we need to find DC.Since triangle BCD is a triangle with sides BC, BD, and DC, and we know two sides and the angle at C, perhaps we can use the Pythagorean theorem if it's a right triangle. But is triangle BCD a right triangle?Wait, in the Asymptote code, point D is at (12*sqrt(13), 0), point C is at (12*sqrt(13) + 108*sqrt(2), 0), and point B is at (12*sqrt(13), 27). So, point B is directly above point D at (12*sqrt(13), 27). So, BD is vertical, from (12*sqrt(13), 0) to (12*sqrt(13), 27). So, BD is 27 units, but wait, earlier we calculated BD as 21.6. Hmm, that's conflicting.Wait, hold on, maybe I made a mistake earlier. Let me check.Wait, in the Asymptote code, point B is at (12*sqrt(13), 27), so BD is from (12*sqrt(13), 0) to (12*sqrt(13), 27), so BD is 27 units. But earlier, I calculated BD as 21.6. That's a discrepancy. So, perhaps my initial assumption was wrong.Wait, the problem says AB = 36, and in the Asymptote code, point A is at (0,0), point B is at (12*sqrt(13), 27). So, let's compute AB's length.AB is from (0,0) to (12*sqrt(13), 27). So, the distance is sqrt[(12*sqrt(13))^2 + 27^2].Compute (12*sqrt(13))^2: 144 * 13 = 1872.27^2 = 729.So, AB^2 = 1872 + 729 = 2601.sqrt(2601) = 51.Wait, but the problem says AB = 36. So, that's conflicting. So, the Asymptote code must have some scaling or something. So, maybe the coordinates are not to scale, and I shouldn't rely on them. So, I need to solve it purely based on the given information: AB = 36, angle ADB = 90 degrees, sin A = 3/5, sin C = 1/4.Alright, let's forget the coordinates for now.So, we have triangle ABD, right-angled at D, with AB = 36, sin A = 3/5.So, sin A = opposite / hypotenuse = BD / AB = 3/5.Therefore, BD = (3/5)*36 = 21.6.Then, AD can be found using Pythagoras: AD = sqrt(AB^2 - BD^2) = sqrt(36^2 - 21.6^2).Compute 36^2 = 1296.21.6^2: Let's compute 21.6 * 21.6.21 * 21 = 441.21 * 0.6 = 12.6.0.6 * 21 = 12.6.0.6 * 0.6 = 0.36.So, (21 + 0.6)^2 = 21^2 + 2*21*0.6 + 0.6^2 = 441 + 25.2 + 0.36 = 466.56.Therefore, AD^2 = 1296 - 466.56 = 829.44.So, AD = sqrt(829.44). Let's compute that.Well, 28^2 = 784, 29^2 = 841. So, sqrt(829.44) is between 28 and 29.Compute 28.8^2: 28^2 = 784, 2*28*0.8 = 44.8, 0.8^2 = 0.64. So, 784 + 44.8 + 0.64 = 829.44. So, sqrt(829.44) = 28.8.Therefore, AD = 28.8.So, now, we have triangle ABD with AD = 28.8, BD = 21.6, AB = 36.Now, we need to find DC. Looking at the figure, DC is a segment from D to C. Since D is connected to C, and A is connected to C, I think triangle ADC is also a triangle, but I'm not sure if it's right-angled.Wait, but in the Asymptote code, D and C are both on the x-axis, so DC is a horizontal line. So, if I consider triangle BCD, which is connected to point B, which is above D, then triangle BCD is another triangle.Given that, in triangle BCD, we have angle at C, sin C = 1/4. So, in triangle BCD, sin C = opposite / hypotenuse = BD / BC = 1/4.Wait, BD is 21.6, so 21.6 / BC = 1/4, so BC = 21.6 * 4 = 86.4.So, BC = 86.4.Now, in triangle BCD, we have BC = 86.4, BD = 21.6, and we need to find DC.Assuming triangle BCD is a right triangle, but we don't know where the right angle is. Wait, in the Asymptote code, point D is at (12*sqrt(13), 0), point C is at (12*sqrt(13) + 108*sqrt(2), 0), and point B is at (12*sqrt(13), 27). So, BD is vertical, DC is horizontal, and BC is the hypotenuse.So, in reality, triangle BCD is a right triangle at D, because BD is vertical and DC is horizontal, so angle at D is 90 degrees. Therefore, triangle BCD is right-angled at D.Wait, but in the problem statement, it's only given that angle ADB is 90 degrees. So, maybe triangle BCD is not necessarily right-angled. Hmm, conflicting information.Wait, in the Asymptote code, it's drawn as D connected to B, B connected to A, A connected to C, and C connected to B. So, the figure is a quadrilateral with two triangles: ABD and BCD, sharing the side BD. Since angle ADB is 90 degrees, triangle ABD is right-angled at D. But triangle BCD is connected to D and C, but we don't know if it's right-angled.However, in the Asymptote code, point C is on the x-axis, same as D, so DC is horizontal, and BD is vertical, so BC would be the hypotenuse of a right triangle at D. So, in reality, triangle BCD is right-angled at D.But in the problem statement, it's only given that angle ADB is 90 degrees. So, maybe in the problem, triangle BCD is not necessarily right-angled. Hmm, this is confusing.Wait, maybe I can assume that triangle BCD is right-angled at D because in the Asymptote code, it's drawn that way. But the problem statement doesn't specify that. So, perhaps I shouldn't assume that.Alternatively, maybe I can use the Law of Sines or Cosines in triangle BCD.Wait, in triangle BCD, we know angle at C, which is sin C = 1/4, and we know side BD = 21.6, which is opposite to angle C. So, using the Law of Sines, we can find other sides or angles.Wait, Law of Sines says that in any triangle, a / sin A = b / sin B = c / sin C.In triangle BCD, let's denote:- angle at C: angle C, sin C = 1/4.- side opposite angle C: BD = 21.6.- side opposite angle B: DC, which we need to find.- side opposite angle D: BC, which we found earlier as 86.4.Wait, but if we use the Law of Sines, we have:BD / sin C = BC / sin D = DC / sin B.But we don't know angle D or angle B. Hmm.Alternatively, maybe we can use the Law of Cosines.Wait, but without knowing any angles except angle C, it might be difficult.Wait, but in triangle BCD, we have side BD = 21.6, side BC = 86.4, and we need to find DC.If we can find angle at D or angle at B, we can use the Law of Cosines.Alternatively, if triangle BCD is right-angled at D, then DC = sqrt(BC^2 - BD^2).But since in the Asymptote code, it's right-angled at D, but in the problem statement, it's not specified. Hmm.Wait, maybe I can check if triangle BCD is right-angled at D.If it is, then DC = sqrt(BC^2 - BD^2) = sqrt(86.4^2 - 21.6^2).Compute 86.4^2: 86.4 * 86.4.Let me compute 80^2 = 6400, 6.4^2 = 40.96, and 2*80*6.4 = 1024.So, (80 + 6.4)^2 = 80^2 + 2*80*6.4 + 6.4^2 = 6400 + 1024 + 40.96 = 6400 + 1024 is 7424 + 40.96 is 7464.96.Similarly, 21.6^2 is 466.56, as computed earlier.So, DC^2 = 7464.96 - 466.56 = 6998.4.Therefore, DC = sqrt(6998.4). Let's compute that.Well, sqrt(6998.4). Let's see, 83^2 = 6889, 84^2 = 7056. So, it's between 83 and 84.Compute 83.6^2: 83^2 = 6889, 2*83*0.6 = 99.6, 0.6^2 = 0.36. So, 6889 + 99.6 + 0.36 = 6989.96.Hmm, that's still less than 6998.4.Compute 83.7^2: 83^2 = 6889, 2*83*0.7 = 116.2, 0.7^2 = 0.49. So, 6889 + 116.2 + 0.49 = 7005.69.That's more than 6998.4.So, sqrt(6998.4) is between 83.6 and 83.7.Compute 83.65^2: Let's compute 83 + 0.65.(83 + 0.65)^2 = 83^2 + 2*83*0.65 + 0.65^2 = 6889 + 107.9 + 0.4225 = 6889 + 107.9 is 6996.9 + 0.4225 is 6997.3225.Still less than 6998.4.Compute 83.66^2: 83.65 + 0.01.(83.65 + 0.01)^2 = 83.65^2 + 2*83.65*0.01 + 0.01^2 = 6997.3225 + 1.673 + 0.0001 = 6997.3225 + 1.673 is 7000.0 approximately.Wait, that can't be. Wait, 83.65^2 is 6997.3225, adding 2*83.65*0.01 = 1.673, so 6997.3225 + 1.673 = 6999.0 approximately, and then +0.0001 is 6999.0001.Wait, but 83.65^2 is 6997.3225, and 83.66^2 is approximately 6999.0001. So, 6998.4 is between 83.65 and 83.66.Compute how much more: 6998.4 - 6997.3225 = 1.0775.Since each 0.01 increase in x increases x^2 by approximately 1.673 (from 83.65 to 83.66). So, to get an increase of 1.0775, we need approximately 1.0775 / 1.673 ≈ 0.644 of 0.01, which is 0.00644.So, sqrt(6998.4) ≈ 83.65 + 0.00644 ≈ 83.6564.So, approximately 83.66.But since the problem is likely expecting an exact value, not a decimal approximation, maybe I made a wrong assumption earlier.Wait, let's see. If triangle BCD is right-angled at D, then DC = sqrt(BC^2 - BD^2) = sqrt(86.4^2 - 21.6^2) = sqrt(7464.96 - 466.56) = sqrt(6998.4). Hmm, 6998.4 is 69984/10, which is 69984 divided by 10. Let me see if 69984 is a perfect square.Compute sqrt(69984). Let's see, 264^2 = 69696, 265^2 = 70225. So, 264^2 = 69696, which is less than 69984. 264.5^2 = ?Wait, 264^2 = 69696, 264.5^2 = (264 + 0.5)^2 = 264^2 + 2*264*0.5 + 0.5^2 = 69696 + 264 + 0.25 = 69960.25.Still less than 69984.264.7^2: Let's compute 264.5 + 0.2.(264.5 + 0.2)^2 = 264.5^2 + 2*264.5*0.2 + 0.2^2 = 69960.25 + 105.8 + 0.04 = 69960.25 + 105.8 is 70066.05 + 0.04 is 70066.09.Wait, that's over. Wait, 264.5^2 = 69960.25, 264.6^2 = ?264.6^2 = (264 + 0.6)^2 = 264^2 + 2*264*0.6 + 0.6^2 = 69696 + 316.8 + 0.36 = 69696 + 316.8 is 70012.8 + 0.36 is 70013.16.Still over.Wait, 264.5^2 = 69960.25, 264.5 + x)^2 = 69984.So, 69960.25 + 2*264.5*x + x^2 = 69984.Assuming x is small, x^2 is negligible.So, 2*264.5*x ≈ 69984 - 69960.25 = 23.75.So, x ≈ 23.75 / (2*264.5) ≈ 23.75 / 529 ≈ 0.0448.So, sqrt(69984) ≈ 264.5 + 0.0448 ≈ 264.5448.Therefore, sqrt(69984) ≈ 264.5448.But 69984 is 10 times 6998.4, so sqrt(69984) = sqrt(10 * 6998.4) = sqrt(10) * sqrt(6998.4).Wait, that's not helpful.Alternatively, 6998.4 = 69984 / 10, so sqrt(6998.4) = sqrt(69984 / 10) = sqrt(69984)/sqrt(10) ≈ 264.5448 / 3.1623 ≈ 83.65.So, DC ≈ 83.65.But since the problem is likely expecting an exact value, maybe I need to express it in terms of sqrt.Wait, 6998.4 is equal to 69984/10. Let's factor 69984.69984 divided by 16 is 4374.4374 divided by 9 is 486.486 divided by 9 is 54.54 divided by 9 is 6.So, 69984 = 16 * 9 * 9 * 9 * 6.Wait, 16 * 9 = 144, 144 * 9 = 1296, 1296 * 9 = 11664, 11664 * 6 = 69984.So, 69984 = 16 * 9^3 * 6.So, sqrt(69984) = sqrt(16 * 9^3 * 6) = 4 * 9^(3/2) * sqrt(6) = 4 * 9 * sqrt(9) * sqrt(6) = 4 * 9 * 3 * sqrt(6) = 108 * sqrt(6).Wait, let me check:sqrt(16) = 4,sqrt(9^3) = 9^(3/2) = 9 * sqrt(9) = 9 * 3 = 27,sqrt(6) remains.So, sqrt(16 * 9^3 * 6) = 4 * 27 * sqrt(6) = 108 * sqrt(6).Therefore, sqrt(69984) = 108 * sqrt(6).Therefore, sqrt(6998.4) = sqrt(69984 / 10) = (108 * sqrt(6)) / sqrt(10) = (108 / sqrt(10)) * sqrt(6) = (108 * sqrt(6)) / sqrt(10).Simplify sqrt(6)/sqrt(10) = sqrt(6/10) = sqrt(3/5).So, sqrt(6998.4) = 108 * sqrt(3/5).But sqrt(3/5) can be rationalized as sqrt(15)/5.So, sqrt(6998.4) = 108 * sqrt(15)/5 = (108/5) * sqrt(15) = 21.6 * sqrt(15).Wait, 108 divided by 5 is 21.6.So, sqrt(6998.4) = 21.6 * sqrt(15).But 21.6 is 108/5, so sqrt(6998.4) = (108/5) * sqrt(15).Alternatively, 108 * sqrt(15)/5.So, DC = sqrt(6998.4) = (108 * sqrt(15))/5.But let me check if that's correct.Wait, 108 * sqrt(15)/5 squared is (108^2 * 15)/25 = (11664 * 15)/25 = (174960)/25 = 6998.4. Yes, that's correct.So, DC = (108 * sqrt(15))/5.Simplify 108/5: 108 divided by 5 is 21.6, so DC = 21.6 * sqrt(15).But 21.6 is 108/5, so it's better to write it as (108/5) * sqrt(15).Alternatively, factor 108: 108 = 36 * 3, so (36 * 3 * sqrt(15))/5 = 36 * (3 * sqrt(15))/5.But 3 * sqrt(15) is sqrt(135), so 36 * sqrt(135)/5.But sqrt(135) = sqrt(9*15) = 3*sqrt(15), so that brings us back.Alternatively, leave it as (108 * sqrt(15))/5.But let me see if 108 and 5 can be simplified. 108 and 5 have no common factors, so that's the simplest form.Alternatively, factor 108 as 12*9, so 12*9*sqrt(15)/5 = 12*sqrt(15)*9/5.But that doesn't help much.Alternatively, write it as 108/5 * sqrt(15) = 21.6 * sqrt(15).But perhaps the problem expects the answer in terms of sqrt(15), so 108√15 / 5.Alternatively, maybe I made a mistake in assuming triangle BCD is right-angled at D. Because in the problem statement, only angle ADB is given as 90 degrees. So, perhaps triangle BCD is not right-angled, and I need to use the Law of Sines or Cosines.Wait, let's try that approach.In triangle BCD, we have:- angle at C: sin C = 1/4.- side opposite angle C: BD = 21.6.- side BC: which we found earlier as 86.4.Wait, how did we find BC? Because we assumed that in triangle BCD, sin C = BD / BC = 1/4, which led us to BC = 86.4.But if triangle BCD is not right-angled, then BD / BC = sin C is only true if angle at C is opposite to BD, which it is, but we need to ensure that the triangle is defined correctly.Wait, in triangle BCD, angle at C is opposite to BD, which is 21.6, and sin C = 1/4.So, using the Law of Sines:BD / sin C = BC / sin D = DC / sin B.We have BD = 21.6, sin C = 1/4, BC = 86.4.So, BD / sin C = 21.6 / (1/4) = 86.4.Similarly, BC / sin D = 86.4 / sin D.Therefore, 86.4 / sin D = 86.4, so sin D = 1.Wait, sin D = 1 implies that angle D is 90 degrees.Therefore, triangle BCD is right-angled at D.So, that confirms that triangle BCD is right-angled at D.Therefore, DC can be found using Pythagoras:DC = sqrt(BC^2 - BD^2) = sqrt(86.4^2 - 21.6^2) = sqrt(7464.96 - 466.56) = sqrt(6998.4) = (108 * sqrt(15))/5.So, DC = (108√15)/5.But let me check if 108/5 is reducible. 108 divided by 5 is 21.6, which is 21 and 3/5. So, 108/5 is 21.6.Alternatively, 108√15 /5 is the exact value.But let me see if the problem expects a decimal or an exact value. The problem didn't specify, but since it's a competition problem, it's likely expecting an exact value.So, DC = (108√15)/5.Alternatively, simplifying 108/5: 108 divided by 5 is 21.6, so DC = 21.6√15.But 21.6 is 108/5, so it's better to write it as (108√15)/5.Alternatively, factor 108: 108 = 36 * 3, so (36 * 3 * √15)/5 = 36 * (3√15)/5.But 3√15 is √135, so 36√135 /5.But that's not simpler.Alternatively, leave it as (108√15)/5.So, the exact value is (108√15)/5, which is approximately 83.65.But since the problem is likely expecting an exact value, I'll go with (108√15)/5.Wait, but let me check the calculation again.We have triangle BCD, right-angled at D, so DC = sqrt(BC^2 - BD^2).BC = 86.4, BD = 21.6.So, BC^2 = 86.4^2 = 7464.96.BD^2 = 21.6^2 = 466.56.So, DC^2 = 7464.96 - 466.56 = 6998.4.Therefore, DC = sqrt(6998.4).As we computed earlier, sqrt(6998.4) = (108√15)/5.Yes, that's correct.So, the exact value is (108√15)/5.Alternatively, if we rationalize or simplify further, but I think that's the simplest form.Therefore, DC = (108√15)/5.But let me check if 108 and 5 can be simplified. 108 is divisible by 3, 5 is prime, so no common factors. So, that's the simplest form.Alternatively, 108/5 is 21.6, so DC = 21.6√15.But 21.6 is 108/5, so both forms are acceptable.Therefore, the answer is (108√15)/5.But let me see if the problem expects it in a different form.Alternatively, perhaps I can write it as (108/5)√15.Yes, that's the same thing.So, DC = (108√15)/5.Alternatively, factor 108 as 12*9, so (12*9√15)/5 = 12*(9√15)/5.But that's not simpler.Alternatively, leave it as is.So, I think (108√15)/5 is the exact value, which is approximately 83.65.Therefore, the length of DC is (108√15)/5 units.**Final Answer**The length of ( DC ) is (boxed{dfrac{108sqrt{15}}{5}}).

answer：Alright, so I have this problem here. It says, let a be the integral from 0 to pi of (cos x minus sin x) dx. Then, I need to find the coefficient of the x cubed term in the expansion of (x squared plus a over x) to the sixth power. The options are A: -160, B: -20, C: 20, D: 160.Okay, first step is to figure out what a is. So, I need to compute the integral of (cos x - sin x) from 0 to pi. Let me recall how to integrate cos x and sin x. The integral of cos x is sin x, and the integral of sin x is -cos x. So, integrating (cos x - sin x) should be sin x + cos x, right? Because the integral of cos x is sin x, and the integral of -sin x is cos x.So, let me write that down:a = ∫₀^π (cos x - sin x) dx = [sin x + cos x] from 0 to π.Now, plugging in the limits. At pi, sin pi is 0, and cos pi is -1. So, at pi, it's 0 + (-1) = -1. At 0, sin 0 is 0, and cos 0 is 1. So, at 0, it's 0 + 1 = 1. Therefore, the integral is (-1) - (1) = -2. Wait, hold on, that's not what I thought earlier. I thought it was zero, but now I'm getting -2.Wait, maybe I made a mistake earlier. Let me double-check. The integral of cos x is sin x, and the integral of -sin x is cos x. So, the antiderivative is sin x + cos x. Evaluating from 0 to pi:At pi: sin(pi) + cos(pi) = 0 + (-1) = -1.At 0: sin(0) + cos(0) = 0 + 1 = 1.So, subtracting, it's (-1) - (1) = -2. So, a is -2.Wait, so earlier I thought a was zero, but that was a mistake. So, a is actually -2.Okay, so now, the expression we're looking at is (x² + a/x)^6, which is (x² + (-2)/x)^6. So, that's (x² - 2/x)^6.Now, I need to find the coefficient of the x³ term in the expansion of this expression.Alright, so to find the coefficient of a specific term in a binomial expansion, we can use the binomial theorem. The binomial theorem says that (A + B)^n = Σ (from k=0 to n) [C(n, k) * A^(n - k) * B^k], where C(n, k) is the combination of n things taken k at a time.In this case, A is x², B is (-2)/x, and n is 6.So, the general term in the expansion is C(6, k) * (x²)^(6 - k) * (-2/x)^k.Simplify that:C(6, k) * x^(12 - 2k) * (-2)^k * x^(-k) = C(6, k) * (-2)^k * x^(12 - 3k).We need the exponent of x to be 3. So, set 12 - 3k = 3.Solving for k:12 - 3k = 3Subtract 3 from both sides:9 - 3k = 0Add 3k to both sides:9 = 3kDivide both sides by 3:k = 3.So, the term where k=3 will have x^3.Therefore, the coefficient is C(6, 3) * (-2)^3.Compute C(6, 3): that's 6! / (3! * (6 - 3)!) = (720) / (6 * 6) = 720 / 36 = 20.Wait, hold on, 6! is 720, 3! is 6, so 720 / (6 * 6) is 720 / 36 = 20. Yes, that's correct.Then, (-2)^3 is -8.So, the coefficient is 20 * (-8) = -160.Therefore, the coefficient of x^3 is -160, which is option A.Wait, but let me double-check everything to make sure I didn't make a mistake.First, computing a: integral of cos x - sin x from 0 to pi.Antiderivative is sin x + cos x.At pi: sin(pi) = 0, cos(pi) = -1, so total is -1.At 0: sin(0) = 0, cos(0) = 1, so total is 1.Subtracting, -1 - 1 = -2. So, a is -2. That seems correct.Then, the expression becomes (x² - 2/x)^6.Using binomial theorem, general term is C(6, k)*(x²)^(6 - k)*(-2/x)^k.Simplify exponents: x^(12 - 2k) * x^(-k) = x^(12 - 3k).Set 12 - 3k = 3, so k = 3.Compute C(6, 3) = 20, (-2)^3 = -8, so 20*(-8) = -160.Yes, that seems correct.So, the answer should be A: -160.I think that's solid. I don't see any mistakes in the calculations.**Final Answer**boxed{A}

answer：Okay, so I have this problem about an arithmetic sequence. It says that the 10th term, a₁₀, is 30, and the 20th term, a₂₀, is 50. I need to find the general term of the sequence, which is part (1). Then, for part (2), I have to define another sequence bₙ based on aₙ and find the sum of the first n terms of bₙ, denoted as Sₙ.Alright, let's start with part (1). I remember that an arithmetic sequence has a common difference, d, between consecutive terms. The general formula for the nth term of an arithmetic sequence is aₙ = a₁ + (n - 1)d, where a₁ is the first term.Given that a₁₀ = 30, I can plug n = 10 into the formula: a₁₀ = a₁ + 9d = 30. Similarly, for a₂₀ = 50, plugging n = 20 gives a₂₀ = a₁ + 19d = 50.So now I have two equations:1) a₁ + 9d = 302) a₁ + 19d = 50I can solve this system of equations to find a₁ and d. Let me subtract equation 1 from equation 2 to eliminate a₁.(a₁ + 19d) - (a₁ + 9d) = 50 - 30Simplifying:a₁ + 19d - a₁ - 9d = 2010d = 20So, d = 2.Now that I have d, I can plug it back into equation 1 to find a₁.a₁ + 9*2 = 30a₁ + 18 = 30a₁ = 30 - 18 = 12.Okay, so the first term a₁ is 12, and the common difference d is 2. Therefore, the general term aₙ is:aₙ = a₁ + (n - 1)d = 12 + (n - 1)*2.Let me simplify that:aₙ = 12 + 2n - 2 = 2n + 10.Wait, let me check that. If n = 10, a₁₀ should be 30. Plugging in n = 10:a₁₀ = 2*10 + 10 = 20 + 10 = 30. That's correct. Similarly, a₂₀ = 2*20 + 10 = 40 + 10 = 50. Perfect, that matches the given terms.So, part (1) is done. The general term is aₙ = 2n + 10.Now, moving on to part (2). I need to define bₙ as 4 divided by (aₙ - 10)(aₙ - 8). So, bₙ = 4 / [(aₙ - 10)(aₙ - 8)].First, let's substitute aₙ with the expression we found in part (1). So, aₙ = 2n + 10.Therefore, aₙ - 10 = (2n + 10) - 10 = 2n.Similarly, aₙ - 8 = (2n + 10) - 8 = 2n + 2.So, substituting back into bₙ:bₙ = 4 / [2n * (2n + 2)].Let me simplify the denominator:2n * (2n + 2) = 2n * 2(n + 1) = 4n(n + 1).So, bₙ = 4 / [4n(n + 1)] = 1 / [n(n + 1)].Wait, that simplifies nicely. So, bₙ = 1 / [n(n + 1)].Hmm, I remember that 1 / [n(n + 1)] can be expressed as a telescoping series. It can be written as 1/n - 1/(n + 1). Let me verify that:1/n - 1/(n + 1) = (n + 1 - n) / [n(n + 1)] = 1 / [n(n + 1)]. Yes, that's correct.So, bₙ = 1/n - 1/(n + 1).Now, I need to find the sum of the first n terms of bₙ, which is Sₙ = b₁ + b₂ + ... + bₙ.Since each bₖ = 1/k - 1/(k + 1), the sum Sₙ will be:Sₙ = (1/1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + ... + (1/n - 1/(n + 1)).Let me write that out:Sₙ = (1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + ... + (1/n - 1/(n + 1)).I notice that in this series, most of the terms cancel out. The -1/2 cancels with the +1/2, the -1/3 cancels with the +1/3, and so on, up to the -1/n cancels with the +1/n.So, what's left is the first term of the first expression, which is 1, and the last term of the last expression, which is -1/(n + 1).Therefore, Sₙ = 1 - 1/(n + 1).Simplifying that, Sₙ = (n + 1)/(n + 1) - 1/(n + 1) = [n + 1 - 1]/(n + 1) = n/(n + 1).So, the sum of the first n terms of the sequence {bₙ} is Sₙ = n/(n + 1).Let me double-check my steps to make sure I didn't make a mistake.1. Found aₙ correctly by solving the system of equations, got aₙ = 2n + 10. Checked with n=10 and n=20, works.2. Substituted aₙ into bₙ, simplified correctly to 1/[n(n + 1)].3. Expressed 1/[n(n + 1)] as 1/n - 1/(n + 1), which is correct.4. Summed the series, noticed the telescoping nature, correctly canceled out intermediate terms, ended up with Sₙ = 1 - 1/(n + 1) = n/(n + 1).Everything seems to check out. So, I think that's the correct answer.**Final Answer**(1) The general term is boxed{2n + 10}.(2) The sum of the first ( n ) terms is boxed{dfrac{n}{n + 1}}.

answer：Okay, so I have this problem with two propositions, p and q, and I need to find the range of values for the real number a such that "p or q" is true and "p and q" is false. Hmm, let me try to break this down step by step.First, let me understand what each proposition means.Proposition p: The inequality (a - 2)x² + 2(a - 2)x - 4 < 0 holds true for any real number x.Proposition q: The function y = log_a(X) is strictly decreasing on its domain.I need to figure out the conditions on a for each proposition and then combine them according to the logical connectives given.Starting with proposition p. The inequality is quadratic in x: (a - 2)x² + 2(a - 2)x - 4 < 0. For this inequality to hold for all real x, the quadratic must be negative definite. That means a few things:1. The leading coefficient must be negative. So, (a - 2) < 0. That implies a < 2.2. The quadratic must not cross the x-axis, meaning it has no real roots. So, the discriminant must be negative.The discriminant D of a quadratic ax² + bx + c is D = b² - 4ac.Let me compute the discriminant for our quadratic:D = [2(a - 2)]² - 4*(a - 2)*(-4)Simplify that:D = 4(a - 2)² + 16(a - 2)Factor out 4(a - 2):D = 4(a - 2)[(a - 2) + 4] = 4(a - 2)(a + 2)So, D = 4(a - 2)(a + 2)For the quadratic to have no real roots, D must be less than 0:4(a - 2)(a + 2) < 0Divide both sides by 4 (which is positive, so inequality sign doesn't change):(a - 2)(a + 2) < 0This product is negative when one factor is positive and the other is negative. So, let's find the intervals where this is true.The critical points are a = -2 and a = 2.Test intervals:1. a < -2: Let's pick a = -3. (-3 - 2)(-3 + 2) = (-5)(-1) = 5 > 02. -2 < a < 2: Let's pick a = 0. (0 - 2)(0 + 2) = (-2)(2) = -4 < 03. a > 2: Let's pick a = 3. (3 - 2)(3 + 2) = (1)(5) = 5 > 0So, the discriminant is negative only when -2 < a < 2.But remember, for the quadratic to be negative definite, we also need the leading coefficient (a - 2) < 0, which is already satisfied when a < 2. So, combining both conditions, the inequality holds for all x when -2 < a < 2.Wait, but what about a = 2? If a = 2, the quadratic becomes (2 - 2)x² + 2(2 - 2)x - 4 = 0x² + 0x - 4 = -4. So, the inequality becomes -4 < 0, which is always true. So, a = 2 is also acceptable.Therefore, proposition p is true when -2 < a ≤ 2.Now, moving on to proposition q: The function y = log_a(X) is strictly decreasing on its domain.I remember that logarithmic functions are strictly decreasing if their base is between 0 and 1. If the base is greater than 1, they are strictly increasing. Also, the base must be positive and not equal to 1.So, for y = log_a(X) to be strictly decreasing, we need 0 < a < 1.Therefore, proposition q is true when 0 < a < 1.Now, the problem states that "p or q" is true and "p and q" is false. Let me translate this into logical terms."p or q" is true means that at least one of p or q is true."p and q" is false means that both p and q cannot be true at the same time.So, we need the set of a where either p is true or q is true, but not both.In other words, the solution set is (p ∨ q) (p ∧ q), which is the symmetric difference of p and q.Mathematically, this is equivalent to (p q) ∪ (q p).So, let's find the ranges where p is true and q is false, and where q is true and p is false.First, let's find p q: p is true, q is false.p is true when -2 < a ≤ 2.q is false when a ≤ 0 or a ≥ 1.So, p q is the intersection of (-2 < a ≤ 2) and (a ≤ 0 or a ≥ 1).So, that would be (-2 < a ≤ 0) ∪ (1 ≤ a ≤ 2).Next, q p: q is true, p is false.q is true when 0 < a < 1.p is false when a ≤ -2 or a > 2.So, q p is the intersection of (0 < a < 1) and (a ≤ -2 or a > 2).But 0 < a < 1 and a ≤ -2 is impossible, and 0 < a < 1 and a > 2 is also impossible. So, q p is empty.Therefore, the solution set is just p q, which is (-2 < a ≤ 0) ∪ (1 ≤ a ≤ 2).Wait, but let me double-check. When a is between 1 and 2, p is true and q is false because q requires 0 < a < 1. So, that's correct. Similarly, when a is between -2 and 0, p is true (since p is true up to 2) and q is false because q requires 0 < a < 1. So, that's correct too.But what about a = 1? At a = 1, p is true because 1 is within (-2, 2], and q is false because q requires a < 1. So, a = 1 is included in p q.Similarly, a = 2 is included in p q because p is true at a = 2, and q is false.And a = -2 is not included because p is true only when a > -2.Wait, but in the initial analysis, p is true when -2 < a ≤ 2. So, a = -2 is not included because at a = -2, the discriminant D becomes zero, right?Wait, let me check that. When a = -2, the discriminant D = 4(a - 2)(a + 2) = 4(-4)(0) = 0. So, the quadratic would have a repeated root, meaning it touches the x-axis. Therefore, the inequality (a - 2)x² + 2(a - 2)x - 4 < 0 would not hold for all x because at the root, it's equal to zero. So, a = -2 is excluded.Therefore, p is true when -2 < a ≤ 2.So, the solution set is indeed (-2 < a ≤ 0) ∪ (1 ≤ a ≤ 2).Wait, but let me think again about the case when a is between 0 and 1. In that interval, q is true because 0 < a < 1, but p is also true because 0 < a < 1 is within -2 < a ≤ 2. So, in that interval, both p and q are true, which would make "p and q" true, but we need "p and q" to be false. Therefore, that interval (0,1) is excluded from our solution set.Therefore, the correct solution is (-2 < a ≤ 0) ∪ (1 ≤ a ≤ 2).So, to write this as a range, it's two intervals: from -2 to 0, not including -2, and from 1 to 2, including both endpoints.I think that's the correct answer.

answer：First, I need to determine the miles per gallon (mpg) for Car Z when driven at 45 miles per hour. According to the problem, Car Z achieves 50 mpg at this speed.Next, I'll calculate the mpg for Car Z when driven at 60 miles per hour. The problem states that Car Z travels 400 miles using 10 gallons of gasoline at this speed. To find the mpg, I'll divide the total miles traveled by the total gallons used:mpg at 60 mph = 400 miles / 10 gallons = 40 mpgNow, I have the mpg values for both speeds:- At 45 mph: 50 mpg- At 60 mph: 40 mpgTo find the percentage decrease in mpg when the car is driven at 60 mph compared to 45 mph, I'll use the percentage decrease formula:Percentage decrease = [(Original mpg - New mpg) / Original mpg] * 100Plugging in the values:Percentage decrease = [(50 mpg - 40 mpg) / 50 mpg] * 100Percentage decrease = [10 mpg / 50 mpg] * 100Percentage decrease = 0.2 * 100Percentage decrease = 20%Therefore, the percentage decrease in miles per gallon when the car is driven at 60 miles per hour compared to 45 miles per hour is 20%.

answer：Alright, so I need to find the millionth digit after the decimal point in the decimal expansion of 1/41. Hmm, okay, let's break this down step by step. First, I remember that when you divide 1 by a number, you get a decimal that either terminates or repeats. Since 41 is a prime number and doesn't divide into 10 evenly, I think the decimal expansion of 1/41 will be repeating. So, it's a repeating decimal. Now, I need to figure out the pattern of this repeating decimal. To do that, I guess I can perform the long division of 1 divided by 41. Let me try that. So, 41 goes into 1 zero times. Then, I add a decimal point and a zero, making it 10. 41 goes into 10 zero times again. Add another zero, making it 100. Now, 41 goes into 100 two times because 41*2 is 82. Subtract 82 from 100, and I get 18. Bring down another zero, making it 180. 41 goes into 180 four times because 41*4 is 164. Subtract 164 from 180, and I get 16. Bring down another zero, making it 160. 41 goes into 160 three times because 41*3 is 123. Subtract 123 from 160, and I get 37. Bring down another zero, making it 370. 41 goes into 370 nine times because 41*9 is 369. Subtract 369 from 370, and I get 1. Now, I'm back to where I started with a remainder of 1. So, the decimal repeats from here. Putting it all together, the decimal expansion of 1/41 is 0.0243902439..., where "02439" repeats indefinitely. So, the repeating sequence is 5 digits long: 0, 2, 4, 3, 9.Now, I need to find the millionth digit after the decimal point. Since the sequence repeats every 5 digits, I can think of the decimal as blocks of "02439" repeating over and over. To find the millionth digit, I can determine where the millionth position falls within these blocks. Since each block is 5 digits, I can divide 1,000,000 by 5 to see how many complete blocks there are and what the remainder is. Let's do that: 1,000,000 divided by 5 is 200,000 with a remainder of 0. Hmm, a remainder of 0 means that the millionth digit is the last digit of the 200,000th block. Looking back at the repeating sequence "02439", the last digit is 9. Therefore, the millionth digit after the decimal point in the decimal expansion of 1/41 is 9.Wait, let me double-check that. If I have 200,000 blocks of 5 digits each, that's exactly 1,000,000 digits. So, the last digit of the last block is indeed the millionth digit. And since each block ends with a 9, that should be correct.Yeah, I think that makes sense. So, the millionth digit is 9.