💡Okay, so I need to find the modulus of the complex number 7/4 + 3i. Hmm, I remember that the modulus of a complex number is like its distance from the origin in the complex plane. Right, it's calculated using the formula |a + bi| = sqrt(a² + b²). Alright, let's break this down. The real part here is 7/4 and the imaginary part is 3. So, I need to square both of these and then add them together before taking the square root. First, squaring the real part: (7/4)². Let me calculate that. 7 squared is 49, and 4 squared is 16, so that gives me 49/16. Okay, that seems right.Next, squaring the imaginary part: 3² is 9. Got that down. Now, I need to add these two results together. So, 49/16 plus 9. Hmm, to add these, they need a common denominator. 9 can be written as 144/16 because 16 times 9 is 144, right? So, 49/16 plus 144/16 equals 193/16.Now, I have to take the square root of 193/16. The square root of 193 is a bit tricky because it's not a perfect square. Let me see, 13 squared is 169 and 14 squared is 196, so 193 is between those. It doesn't simplify nicely, so I'll just leave it as sqrt(193). The square root of 16 is 4, so putting it all together, the modulus is sqrt(193)/4.Wait, let me double-check my steps to make sure I didn't make any mistakes. Squared the real part correctly: (7/4)² = 49/16. Squared the imaginary part: 3² = 9. Converted 9 to sixteenths: 144/16. Added them: 49 + 144 = 193, so 193/16. Square root of that is sqrt(193)/4. Yeah, that seems correct.I guess another way to think about it is plotting the point (7/4, 3) on a coordinate plane and finding the distance from the origin. Using the distance formula, which is essentially the same as the modulus formula for complex numbers. So, distance = sqrt((7/4)² + 3²) = sqrt(49/16 + 9) = sqrt(193/16) = sqrt(193)/4. Yep, that matches what I did earlier.I don't think I made any calculation errors. Maybe I should approximate sqrt(193) to see what kind of number we're dealing with. Sqrt(196) is 14, so sqrt(193) is a little less than 14, maybe around 13.89. So, sqrt(193)/4 would be approximately 13.89/4, which is roughly 3.47. But since the question just asks for the exact value, I don't need to approximate. Just to recap, modulus of a complex number a + bi is sqrt(a² + b²). Applied that here, squared both parts, added them, took the square root, and simplified. Everything seems to check out. I think I'm confident with this answer.**Final Answer**The modulus of the complex number is boxed{dfrac{sqrt{193}}{4}}.

💡Alright, so I came across this problem called the "General Drinking Horse" problem, and it's based on a poem by Li Gu from the Tang Dynasty. The problem involves finding the shortest path for a general to go from a starting point, drink water from a river, and then return to the military camp. It sounds like a classic optimization problem, maybe something to do with reflections or distances in geometry.Okay, let me parse the details given. The military camp is located at point A with coordinates (1,1). The general starts from point B at (4,4). The riverbank is represented by the line l with the equation x - y + 1 = 0. The goal is to find the shortest total distance the general can walk, which includes going from B to some point on the river, drinking, and then going back to A.Hmm, so it's like a mirror reflection problem, right? I remember that in problems where you have to find the shortest path that involves touching a line, reflecting one of the points over that line and then finding the straight line distance between the original point and the reflection gives the shortest path. Maybe that applies here.Let me try to recall. If I have two points and a line, the shortest path that touches the line is found by reflecting one point over the line and then drawing a straight line between the original point and the reflection. The point where this straight line intersects the original line is the optimal point to touch.So, in this case, I need to reflect either point A or point B over the riverbank line l. It might make sense to reflect point A over l and then find the distance from the reflection to point B, or reflect point B over l and find the distance from the reflection to point A. I think either way should work because reflection is symmetric.Let me try reflecting point B over line l. The coordinates of B are (4,4), and the equation of line l is x - y + 1 = 0. To find the reflection of a point over a line, I need to use some formula. I think the formula for reflecting a point (x,y) over the line ax + by + c = 0 is:x' = x - 2a(ax + by + c)/(a² + b²)y' = y - 2b(ax + by + c)/(a² + b²)Wait, let me make sure. I think it's:If the line is ax + by + c = 0, then the reflection of point (x,y) is:x' = x - 2a(ax + by + c)/(a² + b²)y' = y - 2b(ax + by + c)/(a² + b²)But in our case, the line is x - y + 1 = 0, so a = 1, b = -1, c = 1.So, plugging in point B(4,4):First, compute ax + by + c = 1*4 + (-1)*4 + 1 = 4 - 4 + 1 = 1.Then, compute the denominators: a² + b² = 1 + 1 = 2.So,x' = 4 - 2*1*(1)/2 = 4 - 1 = 3y' = 4 - 2*(-1)*(1)/2 = 4 + 1 = 5So, the reflection of point B over line l is point C(3,5).Now, the shortest path from B to l and then to A is equivalent to the straight line distance from C to A.So, let's compute the distance between A(1,1) and C(3,5).Using the distance formula: sqrt[(3-1)^2 + (5-1)^2] = sqrt[(2)^2 + (4)^2] = sqrt[4 + 16] = sqrt[20] = 2*sqrt(5).Wait, but let me double-check my reflection calculation because sometimes signs can be tricky.Given the line x - y + 1 = 0, which can be rewritten as y = x + 1. So, it's a line with a slope of 1, crossing the y-axis at (0,1).Reflecting point B(4,4) over this line. The formula I used earlier gave me (3,5). Let me see if that makes sense.Alternatively, I can think about the reflection geometrically. The line y = x + 1 is at 45 degrees. Reflecting a point over this line would swap the x and y coordinates and adjust for the intercept.But in this case, since the line is y = x + 1, not y = x, the reflection isn't just swapping coordinates. So, my earlier calculation seems more accurate.Alternatively, I can use another method to verify. Let's find the midpoint between B(4,4) and its reflection C(3,5). The midpoint should lie on the line l.Midpoint M = [(4+3)/2, (4+5)/2] = (3.5, 4.5)Now, check if M lies on l: x - y + 1 = 0.3.5 - 4.5 + 1 = 0. Yes, 3.5 - 4.5 = -1, -1 +1 = 0. So, it satisfies the equation. Good, so the reflection is correct.Therefore, the distance from A(1,1) to C(3,5) is indeed 2*sqrt(5), which is approximately 4.472.Looking at the options: A is 3*sqrt(6) ≈ 7.348, B is sqrt(34) ≈ 5.830, C is sqrt(5) ≈ 2.236, D is 2*sqrt(5) ≈ 4.472.So, D is the correct answer.But just to make sure I didn't make any mistakes, let me recap:1. Identified the problem as a shortest path involving reflection over a line.2. Chose to reflect point B over line l to get point C.3. Calculated the reflection using the formula, got C(3,5).4. Verified that the midpoint between B and C lies on line l.5. Calculated the distance from A to C, got 2*sqrt(5).6. Compared with the options, D matches.Yes, that seems solid. I don't see any errors in the reasoning or calculations.**Final Answer**boxed{D}

💡First, I need to determine the relative speed between the train and the man since they are moving in opposite directions. This means I will add their speeds to find the relative speed.Let’s denote the speed of the train as ( V_{text{train}} ) in km/h.The man is running at a speed of 6 km/h.The relative speed ( V_{text{relative}} ) in km/h is then:[V_{text{relative}} = V_{text{train}} + 6]Next, I know that the train passes the man in 6 seconds, and the length of the train is 140 meters. To find the relative speed in meters per second (m/s), I use the formula:[text{Speed} = frac{text{Distance}}{text{Time}}]The distance is the length of the train, which is 140 meters, and the time is 6 seconds.[V_{text{relative}} (text{m/s}) = frac{140 text{meters}}{6 text{seconds}} = frac{140}{6} approx 23.overline{3} text{m/s}]To convert this speed from m/s to km/h, I use the conversion factor:[1 text{m/s} = 3.6 text{km/h}]So,[V_{text{relative}} (text{km/h}) = 23.overline{3} times 3.6 approx 84 text{km/h}]Now that I have the relative speed in km/h, I can find the speed of the train by subtracting the speed of the man from the relative speed.[V_{text{train}} = V_{text{relative}} - 6][V_{text{train}} = 84 - 6 = 78 text{km/h}]Therefore, the speed of the train is (boxed{78}) km/h.

💡Mary has a total of 3 bedrooms, 1 kitchen, and 1 living room, which adds up to 5 rooms. It takes her 4 minutes to vacuum each room.So, the total time to vacuum all the rooms once is:5 rooms * 4 minutes/room = 20 minutesSince Mary needs to charge her vacuum cleaner 2 times to vacuum her whole house, this means she can vacuum a certain number of rooms before each charge.Let's assume the battery lasts for X minutes. She can vacuum the whole house in two charges, so she can vacuum for X minutes, charge, vacuum for another X minutes, and then she's done.Since it takes 20 minutes to vacuum the whole house and she needs 2 charges, we can set up the equation:2X = 20 minutesNow, we solve for X:X = 20 minutes / 2X = 10 minutesTherefore, the battery charge lasts for boxed{10} minutes.

💡Okay, so Mark got a new puppy, and I need to figure out how much everything cost him. Let's break it down step by step.First, the puppy itself cost 10. That's straightforward. Now, he also bought enough food for 3 weeks. I need to calculate how much food that is and then find out how much it cost.The puppy eats 1/3 cup of food each day. So, for one day, it's 1/3 cup. How much does it eat in a week? Well, there are 7 days in a week, so I can multiply 1/3 cup by 7 to get the weekly consumption.Let me do that calculation: 1/3 cup/day * 7 days = 7/3 cups per week. Hmm, 7 divided by 3 is approximately 2.333 cups per week. But since we're dealing with exact fractions, I'll keep it as 7/3 cups per week.Now, he needs enough food for 3 weeks. So, I need to multiply the weekly consumption by 3. That would be 7/3 cups/week * 3 weeks = 7 cups total. Okay, so Mark needs 7 cups of food.Next, I need to figure out how many bags of food Mark bought. Each bag contains 3.5 cups of food and costs 2. So, if he needs 7 cups, how many bags does he need? I can divide the total cups needed by the cups per bag.So, 7 cups / 3.5 cups per bag = 2 bags. That makes sense because 3.5 times 2 is exactly 7. So, he needs to buy 2 bags of food.Now, calculating the cost of the food: each bag is 2, and he bought 2 bags. So, 2 bags * 2 per bag = 4.Finally, to find the total cost, I need to add the cost of the puppy and the cost of the food. The puppy was 10, and the food was 4. So, 10 + 4 = 14.Wait, let me double-check everything to make sure I didn't make a mistake. The puppy is 10, the food consumption per day is 1/3 cup, over 21 days (3 weeks) that's 7 cups. Each bag is 3.5 cups, so 7 divided by 3.5 is 2 bags, costing 4. Adding that to the 10 for the puppy gives a total of 14. Yep, that seems right.I think I've covered all the steps and calculations correctly. So, Mark spent a total of 14 on his new puppy and the food.

💡First, I identify the given values: the principal amount is Rs. 780, the simple interest is Rs. 130, and the time period is 4 years.Next, I use the simple interest formula ( SI = frac{P times R times T}{100} ) to find the rate of interest ( R ).Plugging in the known values, I set up the equation ( 130 = frac{780 times R times 4}{100} ).To isolate ( R ), I multiply both sides of the equation by 100, resulting in ( 13000 = 3120 times R ).Finally, I solve for ( R ) by dividing both sides by 3120, which gives ( R approx 4.17% ).