answer：Okay, so I have this problem here where I'm given that α is between π/2 and π, which means it's in the second quadrant. I also know that sin α is 3/5. I need to find tan(α + π/4). Hmm, okay, let's break this down step by step.First, since α is in the second quadrant, I remember that sine is positive there, which matches with sin α being 3/5. But tangent is negative in the second quadrant because tangent is sine over cosine, and cosine is negative there. So, I need to find tan α, but I only have sin α. I guess I can use the Pythagorean identity to find cos α.The Pythagorean identity is sin²α + cos²α = 1. So, if sin α is 3/5, then sin²α is (3/5)² = 9/25. Plugging that into the identity, we get 9/25 + cos²α = 1. Solving for cos²α, that would be 1 - 9/25 = 16/25. So, cos α is either 4/5 or -4/5. But wait, since α is in the second quadrant, cosine should be negative. So, cos α is -4/5.Now, tan α is sin α over cos α, which is (3/5) / (-4/5) = -3/4. Okay, so tan α is -3/4. Got that.Next, I need to find tan(α + π/4). I remember there's a formula for the tangent of a sum of two angles: tan(A + B) = (tan A + tan B) / (1 - tan A tan B). In this case, A is α and B is π/4. I know that tan(π/4) is 1 because tan of 45 degrees is 1. So, plugging into the formula, tan(α + π/4) = (tan α + 1) / (1 - tan α * 1).Substituting tan α as -3/4, we get: (-3/4 + 1) / (1 - (-3/4)). Let's compute the numerator and denominator separately.Numerator: -3/4 + 1. Hmm, 1 is 4/4, so 4/4 - 3/4 = 1/4.Denominator: 1 - (-3/4) = 1 + 3/4. Again, 1 is 4/4, so 4/4 + 3/4 = 7/4.So, putting it together, tan(α + π/4) = (1/4) / (7/4). Dividing fractions is the same as multiplying by the reciprocal, so (1/4) * (4/7) = 1/7.Wait a second, that's positive 1/7. But let me think about the angle α + π/4. Since α is between π/2 and π, adding π/4 would make it between 3π/4 and 5π/4. So, the angle α + π/4 is in the third quadrant where tangent is positive because both sine and cosine are negative, making tangent positive. So, 1/7 is positive, which makes sense.But hold on, let me double-check my calculations to make sure I didn't make any mistakes. Starting from tan α = -3/4, that's correct because in the second quadrant, tangent is negative. Then, using the tangent addition formula, I substituted correctly: (tan α + tan π/4) / (1 - tan α tan π/4). Since tan π/4 is 1, that simplifies to (tan α + 1) / (1 - tan α).Plugging in tan α = -3/4: (-3/4 + 1) / (1 - (-3/4)) = (1/4) / (7/4) = 1/7. Yep, that seems right.So, the answer should be 1/7, which is option A. But wait, let me check the options again to make sure. The options are A: 1/7, B: 7, C: -1/7, D: -7. So, 1/7 is option A.Hmm, but I just want to make sure I didn't make a mistake in determining the sign. Since α is in the second quadrant, adding π/4 would take it to the third quadrant, where tangent is positive, so 1/7 is correct. If I had gotten a negative value, that would have been incorrect because tangent in the third quadrant is positive.Alternatively, I could think about it another way. Maybe using sine and cosine addition formulas to compute tan(α + π/4). Let's see.tan(α + π/4) = sin(α + π/4) / cos(α + π/4). Using the sine and cosine addition formulas:sin(α + π/4) = sin α cos π/4 + cos α sin π/4cos(α + π/4) = cos α cos π/4 - sin α sin π/4We know sin α = 3/5, cos α = -4/5, and cos π/4 = sin π/4 = √2/2.So, sin(α + π/4) = (3/5)(√2/2) + (-4/5)(√2/2) = (3√2/10) - (4√2/10) = (-√2)/10.cos(α + π/4) = (-4/5)(√2/2) - (3/5)(√2/2) = (-4√2/10) - (3√2/10) = (-7√2)/10.Therefore, tan(α + π/4) = sin(α + π/4) / cos(α + π/4) = (-√2/10) / (-7√2/10) = (1/7). The negatives cancel out, and the √2 and 10 terms cancel too, leaving 1/7.So, that confirms it again. The answer is 1/7, which is option A.I think I'm confident now that option A is correct. I don't see any mistakes in my calculations, and both methods—using the tangent addition formula directly and using the sine and cosine addition formulas—give me the same result. So, yeah, I think that's solid.

answer：First, I need to convert the speeds of both trains from kilometers per hour to meters per second to ensure the units are consistent with the given length and time.For the first train:[text{Speed} = 120 , text{km/h} times frac{1000 , text{m}}{1 , text{km}} times frac{1 , text{h}}{3600 , text{s}} = 33.33 , text{m/s}]For the second train:[text{Speed} = 150 , text{km/h} times frac{1000 , text{m}}{1 , text{km}} times frac{1 , text{h}}{3600 , text{s}} = 41.67 , text{m/s}]Since the trains are moving in opposite directions, their relative speed is the sum of their individual speeds:[text{Relative Speed} = 33.33 , text{m/s} + 41.67 , text{m/s} = 75 , text{m/s}]The total distance covered while crossing each other is the sum of the lengths of both trains. Let ( L ) be the length of the second train:[text{Distance} = 360 , text{m} + L]Using the formula (text{Distance} = text{Speed} times text{Time}), we can set up the equation:[75 , text{m/s} times 6 , text{s} = 360 , text{m} + L]Solving for ( L ):[450 , text{m} = 360 , text{m} + L L = 450 , text{m} - 360 , text{m} L = 90 , text{m}]Therefore, the length of the other train is 90 meters.

answer：First, I need to convert the decimal number 215 to base eight. To do this, I'll determine the highest power of 8 that fits into 215. Calculating the powers of 8:- (8^0 = 1)- (8^1 = 8)- (8^2 = 64)- (8^3 = 512)Since (8^3 = 512) is greater than 215, the highest power I'll use is (8^2 = 64). Next, I'll find out how many times 64 fits into 215. Dividing 215 by 64 gives 3 with a remainder of 23. Then, I'll determine how many times 8 fits into the remainder of 23. Dividing 23 by 8 gives 2 with a remainder of 7. Finally, the remaining 7 is the units place in base eight. Combining these values, the base eight equivalent of 215 in decimal is 327.

answer：First, I need to find the ending number ( N ) such that the average of all even numbers between 11 and ( N ) is 16.The first even number after 11 is 12. If ( N ) is even, the sequence of even numbers starts at 12 and ends at ( N ). If ( N ) is odd, the sequence ends at ( N - 1 ).The average of an evenly spaced sequence is the average of the first and last numbers. Therefore, if ( N ) is even, the average is ( frac{12 + N}{2} ). If ( N ) is odd, the average is ( frac{12 + (N - 1)}{2} ).Given that the average is 16, I set up the equation for both cases:1. If ( N ) is even:[frac{12 + N}{2} = 16]Multiplying both sides by 2:[12 + N = 32]Subtracting 12 from both sides:[N = 20]2. If ( N ) is odd:[frac{12 + (N - 1)}{2} = 16]Multiplying both sides by 2:[12 + N - 1 = 32]Simplifying:[N - 1 = 20]Adding 1 to both sides:[N = 21]Since we are considering even numbers, the ending number must be even. Therefore, the correct ending number is 20.

answer：Okay, so I have this differential equation to solve: (dy/dx) cos x + y sin x = 1. Hmm, let's see. I remember that differential equations can sometimes be tricky, but maybe I can simplify this one somehow. First, I notice that there's a cos x multiplied by dy/dx and a sin x multiplied by y. Maybe I can rewrite this equation to make it look more familiar. If I divide every term by cos x, that might help. Let me try that:(dy/dx) + y tan x = sec xOkay, so now it looks like a linear differential equation. I think the standard form for a linear equation is dy/dx + P(x) y = Q(x). Comparing that to what I have, P(x) must be tan x and Q(x) is sec x. I recall that to solve a linear differential equation, I need an integrating factor. The integrating factor, μ(x), is usually e raised to the integral of P(x) dx. So, let me compute that:μ(x) = e^{∫ tan x dx}Hmm, integrating tan x. I think the integral of tan x is -ln |cos x| + C, right? So, that would make μ(x) = e^{-ln |cos x|} which simplifies to 1/cos x or sec x. Wait, let me double-check that. If I have e^{∫ tan x dx} = e^{-ln |cos x|} = e^{ln |sec x|} = sec x. Yeah, that seems right. So, the integrating factor is sec x.Now, according to the method, I multiply the entire differential equation by the integrating factor:sec x * (dy/dx) + sec x * y tan x = sec x * sec xSimplifying each term:sec x dy/dx + y sec x tan x = sec² xI remember that the left side of the equation should now be the derivative of (μ(x) y). Let me check:d/dx (sec x * y) = sec x dy/dx + y * sec x tan xYes, that matches the left side of my equation. So, now I can write:d/dx (sec x * y) = sec² xNow, I need to integrate both sides with respect to x:∫ d/dx (sec x * y) dx = ∫ sec² x dxThe left side simplifies to sec x * y. The integral of sec² x dx is tan x + C, where C is the constant of integration. So, putting it all together:sec x * y = tan x + CNow, to solve for y, I need to multiply both sides by cos x (since sec x is 1/cos x):y = (tan x + C) * cos xSimplifying this expression:tan x is sin x / cos x, so when I multiply by cos x, it becomes sin x. Similarly, C * cos x remains as is. Therefore:y = sin x + C cos xSo, the general solution to the differential equation is y = sin x + C cos x, where C is an arbitrary constant.Wait a second, let me verify if this solution actually satisfies the original differential equation. Let's compute dy/dx:dy/dx = cos x - C sin xNow, plug y and dy/dx back into the original equation:(dy/dx) cos x + y sin x = (cos x - C sin x) cos x + (sin x + C cos x) sin xLet's expand this:= cos² x - C sin x cos x + sin² x + C sin x cos xNotice that the -C sin x cos x and +C sin x cos x cancel each other out:= cos² x + sin² xAnd since cos² x + sin² x = 1, the left side equals 1, which matches the right side of the original equation. So, yes, the solution satisfies the equation.Therefore, I'm confident that y = sin x + C cos x is the correct general solution.

answer：Okay, so I have this problem about selecting leadership for an ancient tribe. The tribe has 12 members, and the leadership structure is as follows: one chief, two supporting chiefs, and each supporting chief has two inferior officers. The tricky part is that the supporting chiefs are chosen simultaneously, meaning we don't distinguish between them, and the same goes for their respective inferior officers.Alright, let me break this down step by step. First, I need to choose the chief. There are 12 members, so there are 12 possible choices for the chief. Once the chief is selected, that leaves us with 11 members.Next, I need to choose the two supporting chiefs from these remaining 11 members. Since the supporting chiefs are indistinct, meaning it doesn't matter which one is chosen first or second, I should use combinations here. The number of ways to choose 2 supporting chiefs out of 11 is given by the combination formula C(11,2). Calculating that, C(11,2) = 55. So, there are 55 ways to choose the supporting chiefs.Now, after selecting the chief and the two supporting chiefs, we have 9 members left. Each supporting chief needs two inferior officers. Again, since the inferior officers for each supporting chief are indistinct, I need to be careful here. I think I need to choose 2 officers for the first supporting chief and then 2 officers for the second supporting chief from the remaining members.So, for the first supporting chief, the number of ways to choose 2 officers out of 9 is C(9,2). That equals 36. Then, for the second supporting chief, we have 7 members left, so the number of ways to choose 2 officers is C(7,2), which is 21.But wait, since the supporting chiefs are indistinct, does this mean that the order in which I choose their officers doesn't matter? That is, choosing officers A and B for the first chief and C and D for the second chief is the same as choosing C and D for the first chief and A and B for the second chief. If that's the case, I might be overcounting by a factor of 2! because there are two ways to assign the groups of officers to the chiefs.So, to correct for this overcounting, I should divide by 2!. That is, I need to multiply the number of ways to choose the officers by 1/2!.Putting it all together, the total number of ways to select the leadership should be:Number of ways = (Number of ways to choose chief) × (Number of ways to choose supporting chiefs) × (Number of ways to choose officers for each chief divided by 2!)Plugging in the numbers:Number of ways = 12 × 55 × (36 × 21 / 2)Let me calculate that step by step.First, 36 × 21 = 756.Then, 756 / 2 = 378.Next, 12 × 55 = 660.Finally, 660 × 378.Hmm, let me compute that. 660 × 378.Breaking it down:660 × 300 = 198,000660 × 78 = ?660 × 70 = 46,200660 × 8 = 5,280So, 46,200 + 5,280 = 51,480Adding that to 198,000: 198,000 + 51,480 = 249,480Wait, that doesn't match the previous answer of 248,040. Did I make a mistake somewhere?Let me double-check the calculations.First, 36 × 21 = 756. Correct.756 / 2 = 378. Correct.12 × 55 = 660. Correct.Now, 660 × 378.Let me try another way to compute 660 × 378.660 × 378 = 660 × (300 + 70 + 8) = 660×300 + 660×70 + 660×8660×300 = 198,000660×70 = 46,200660×8 = 5,280Adding them up: 198,000 + 46,200 = 244,200; 244,200 + 5,280 = 249,480Hmm, so I get 249,480, but the initial answer was 248,040. There's a discrepancy here.Wait, maybe I made a mistake in the initial step. Let me go back.When I chose the officers, I did C(9,2) × C(7,2) = 36 × 21 = 756, and then divided by 2! to account for the indistinctness of the supporting chiefs, getting 378.But perhaps I should have considered that the officers themselves are indistinct as well. That is, for each supporting chief, the two officers are indistinct, so I might need to divide by 2! for each chief's officers.Wait, no, because when I choose the officers, I'm already using combinations, which account for indistinctness. So C(9,2) is the number of ways to choose 2 indistinct officers, and similarly for C(7,2). So I think dividing by 2! is only necessary because the two supporting chiefs are indistinct, not because the officers are.But then why is there a discrepancy in the final number?Let me check the initial calculation again.12 × 55 × (36 × 21 / 2) = 12 × 55 × 378 = 12 × 20,790 = 249,480Wait, 55 × 378 = 20,79012 × 20,790 = 249,480But the initial answer was 248,040. So where is the mistake?Wait, perhaps I made a mistake in the initial step when choosing the supporting chiefs. Let me see.Is the number of ways to choose the supporting chiefs C(11,2) = 55 correct?Yes, because we have 11 members left after choosing the chief, and we need to choose 2 supporting chiefs without order.Then, for the officers, we have 9 members left. Choosing 2 for the first chief: C(9,2) = 36. Then 7 left, choosing 2 for the second chief: C(7,2) = 21. So 36 × 21 = 756.Since the supporting chiefs are indistinct, we need to divide by 2! to account for the fact that swapping the officers between the chiefs doesn't create a new arrangement.So 756 / 2 = 378.So, 12 × 55 × 378 = 249,480.But the initial answer was 248,040. So there's a difference of 1,440.Wait, maybe I made a mistake in the initial calculation. Let me see.Alternatively, perhaps the correct approach is to first choose the chief, then choose the two supporting chiefs, and then partition the remaining 9 members into two groups of 2 and one group of 5 (since 9 - 2 - 2 = 5). But wait, no, the remaining 5 are not part of the leadership, so they can be ignored.But actually, the leadership consists of the chief, the two supporting chiefs, and their four officers. So total leadership is 1 + 2 + 4 = 7 members. Wait, but the tribe has 12 members, so 12 - 7 = 5 are non-leadership members. So perhaps I don't need to worry about them.But back to the calculation.Alternatively, maybe I should think of it as:Total ways = C(12,1) × C(11,2) × [C(9,2) × C(7,2)] / 2!Which is what I did, and that gives 249,480.But the initial answer was 248,040. So perhaps the initial answer was incorrect.Alternatively, maybe I need to consider that the officers are also indistinct within each chief's group. But no, the problem says that the supporting chiefs are chosen simultaneously without distinguishing between them, and the same goes for their respective inferior officers.Wait, does that mean that the officers are also indistinct across the two chiefs? Or just within each chief's group?The problem says "the same goes for their respective inferior officers." So perhaps for each chief, the two officers are indistinct, but across chiefs, they are distinct because they belong to different chiefs.Wait, but the chiefs are indistinct, so their officers are also indistinct in terms of which chief they belong to.Wait, this is getting confusing.Let me try to clarify.If the supporting chiefs are indistinct, then swapping their officers doesn't create a new arrangement. Therefore, when choosing the officers, I need to account for that.So, the total number of ways to choose the officers is C(9,2) × C(7,2), but since the chiefs are indistinct, we need to divide by 2! to account for the fact that swapping the two groups of officers between the two chiefs doesn't create a new arrangement.Therefore, the calculation is:12 × C(11,2) × [C(9,2) × C(7,2)] / 2!Which is 12 × 55 × (36 × 21) / 2 = 12 × 55 × 756 / 2 = 12 × 55 × 378 = 12 × 20,790 = 249,480.So, I think the correct answer is 249,480, not 248,040.But wait, let me check the initial calculation again.In the initial answer, it was:12 × 55 × (36 × 21 / 2) = 12 × 55 × 18 × 21 = 248,040.Wait, 36 × 21 = 756, divided by 2 is 378, not 18 × 21.So, 12 × 55 × 378 = 249,480.But in the initial answer, it was written as 12 × 55 × 18 × 21 = 248,040.Wait, 18 × 21 = 378, so 12 × 55 × 378 = 249,480.So, the initial answer had a miscalculation in the multiplication steps, leading to 248,040 instead of 249,480.Therefore, the correct number of ways should be 249,480.