💡Okay, so I have this problem where I need to find the value of |f(i) - 1| given that f(f(z)) equals (z times the conjugate of z minus z minus the conjugate of z) squared, and also that f(1) equals 0. Hmm, that seems a bit complex, but let me try to break it down step by step.First, let me write down what I know. The function f(z) satisfies the equation f(f(z)) = (z overline{z} - z - overline{z})². And I know that f(1) = 0. I need to find |f(i) - 1|.Let me start by understanding the expression (z overline{z} - z - overline{z})². Since z is a complex number, I can write z as x + yi, where x and y are real numbers. Then, the conjugate of z, which is overline{z}, would be x - yi. So, z overline{z} would be (x + yi)(x - yi) = x² + y². Then, subtracting z and its conjugate, we get z overline{z} - z - overline{z} = (x² + y²) - (x + yi) - (x - yi) = x² + y² - 2x. Therefore, (z overline{z} - z - overline{z})² becomes (x² + y² - 2x)². So, f(f(z)) is equal to (x² + y² - 2x)². Now, since z is a complex number, maybe I can represent it in polar form as well. Let me think. If z = re^{iθ}, then overline{z} = re^{-iθ}. Then, z overline{z} = r². So, z overline{z} - z - overline{z} = r² - re^{iθ} - re^{-iθ} = r² - 2r cosθ. So, f(f(z)) = (r² - 2r cosθ)². Hmm, that might be useful later.But let's get back to the problem. I need to find |f(i) - 1|. So, maybe I can find f(i) first. To do that, perhaps I can use the given condition f(f(z)) = (z overline{z} - z - overline{z})² and plug in z = i.Let me try that. If z = i, then overline{z} = -i. So, z overline{z} = i*(-i) = 1. Then, z overline{z} - z - overline{z} = 1 - i - (-i) = 1 - i + i = 1. So, (z overline{z} - z - overline{z})² = 1² = 1. Therefore, f(f(i)) = 1. So, f(f(i)) = 1. That means that f(i) is some complex number w such that f(w) = 1. But I also know that f(1) = 0. So, f(1) = 0. Maybe I can use this to find more information about f(w). Wait, if f(w) = 1, then f(f(w)) = f(1) = 0. But f(f(w)) is also equal to (w overline{w} - w - overline{w})². So, (w overline{w} - w - overline{w})² = 0. Therefore, w overline{w} - w - overline{w} = 0. Let me write that down: w overline{w} - w - overline{w} = 0. Let me denote w as a complex number, say w = a + bi, where a and b are real numbers. Then, overline{w} = a - bi. So, w overline{w} = a² + b². Substituting into the equation: a² + b² - (a + bi) - (a - bi) = 0. Simplifying, that's a² + b² - 2a = 0. So, a² + b² - 2a = 0. Let me rearrange that: a² - 2a + b² = 0. Completing the square for the a terms: (a - 1)² + b² = 1. So, this is the equation of a circle in the complex plane with center at (1, 0) and radius 1. Therefore, w must lie on this circle. But w is f(i), so f(i) lies on the circle centered at (1, 0) with radius 1. So, |f(i) - 1| = 1. Wait, that seems too straightforward. Let me check my steps again. Starting from f(f(z)) = (z overline{z} - z - overline{z})². For z = i, we found f(f(i)) = 1. So, f(i) is a point w such that f(w) = 1. Given that f(1) = 0, we can use the functional equation again: f(f(w)) = (w overline{w} - w - overline{w})². But f(w) = 1, so f(f(w)) = f(1) = 0. Therefore, (w overline{w} - w - overline{w})² = 0, which implies w overline{w} - w - overline{w} = 0. As I did before, this leads to (a - 1)² + b² = 1, meaning that |w - 1| = 1. Since w = f(i), this means |f(i) - 1| = 1. So, it seems that the answer is 1. But let me think if there's any possibility that I missed something. Is there any other condition or constraint on f(z) that I haven't considered? The problem only gives f(f(z)) = (z overline{z} - z - overline{z})² and f(1) = 0. So, unless there's more information about f(z), I think this is the only conclusion we can draw. Therefore, I think |f(i) - 1| is equal to 1.

💡First, determine the work rates of X and Y individually. X can complete the work in 15 days, so X's work rate is ( frac{1}{15} ) of the work per day. Y can complete the work in 10 days, so Y's work rate is ( frac{1}{10} ) of the work per day.Next, calculate the combined work rate of X and Y:[frac{1}{15} + frac{1}{10} = frac{2}{30} + frac{3}{30} = frac{5}{30} = frac{1}{6}]This means together, X and Y can complete ( frac{1}{6} ) of the work in one day.When Z joins them, the combined work rate of X, Y, and Z is:[frac{1}{5}]Let Z's work rate be ( frac{1}{z} ). Then:[frac{1}{15} + frac{1}{10} + frac{1}{z} = frac{1}{5}]Solving for ( frac{1}{z} ):[frac{1}{z} = frac{1}{5} - frac{1}{15} - frac{1}{10} = frac{3}{15} - frac{1}{15} - frac{1.5}{15} = frac{0.5}{15} = frac{1}{30}]Thus, Z can complete the work alone in 30 days.Now, determine the payment for Z. The total payment is Rs. 720, and the work was completed in 5 days with the help of Z. The payment should be divided based on the amount of work each person contributed.The ratio of their work rates is:[X : Y : Z = frac{1}{15} : frac{1}{10} : frac{1}{30} = 2 : 3 : 1]The sum of the ratios is ( 2 + 3 + 1 = 6 ).Z's share of the payment is:[frac{1}{6} times 720 = 120]

💡Alright, so I have this function f(x) = x² - 6x - 9, and I need to find its range when x is between 1 and 4. Hmm, okay. Let me think about how to approach this.First, I remember that quadratic functions have a parabola shape when graphed. Since the coefficient of x² is positive (which is 1 in this case), the parabola opens upwards. That means the function has a minimum point, called the vertex, and it goes to infinity as x increases or decreases beyond that point.So, to find the range of f(x) over the interval (1, 4), I need to figure out the minimum and maximum values that f(x) can take in that interval. Since the parabola opens upwards, the minimum value will be at the vertex, and the maximum will be at one of the endpoints of the interval.Let me write down the function again: f(x) = x² - 6x - 9. To find the vertex, I can complete the square or use the formula for the vertex of a parabola. The formula for the x-coordinate of the vertex is x = -b/(2a), where a and b are coefficients from the quadratic equation ax² + bx + c.In this case, a = 1 and b = -6. Plugging into the formula: x = -(-6)/(2*1) = 6/2 = 3. So, the x-coordinate of the vertex is 3. That means the vertex is at x = 3. Let me find the corresponding y-value by plugging x = 3 back into the function.f(3) = (3)² - 6*(3) - 9 = 9 - 18 - 9 = -18. So, the vertex is at (3, -18), which is the minimum point of the function.Now, since the interval we're considering is (1, 4), and the vertex is at x = 3, which is inside this interval, the minimum value of f(x) in this interval is -18. That's straightforward.Next, I need to find the maximum value. Since the parabola opens upwards, the maximum will occur at one of the endpoints of the interval. The interval is (1, 4), so the endpoints are x = 1 and x = 4. However, since the interval is open, x = 1 and x = 4 are not included. But wait, in terms of limits, as x approaches 1 from the right and as x approaches 4 from the left, the function will approach certain values.Let me compute f(1) and f(4) to see what those values are.First, f(1) = (1)² - 6*(1) - 9 = 1 - 6 - 9 = -14.Next, f(4) = (4)² - 6*(4) - 9 = 16 - 24 - 9 = -17.So, f(1) is -14 and f(4) is -17. Since the interval is open, x doesn't actually reach 1 or 4, but gets arbitrarily close to them. Therefore, the function values approach -14 and -17, but don't actually reach them.Now, I need to determine which of these is the maximum. Since -14 is greater than -17, the function approaches -14 as x approaches 1 from the right, and approaches -17 as x approaches 4 from the left. Therefore, the maximum value that the function approaches is -14, and the minimum is -18.But wait, since the interval is open, does that mean the function doesn't actually attain -14 or -17? Hmm. So, the range should include all values from the minimum up to, but not including, the maximum. But in this case, the function approaches -14 and -17, but doesn't reach them.Wait, but actually, since the function is continuous on the open interval (1, 4), it will take on all values between the minimum and the supremum and infimum of the endpoints. But since the function is decreasing from x = 1 to x = 3 and increasing from x = 3 to x = 4, the maximum value near x = 1 is approaching -14, and near x = 4 is approaching -17. But since -14 is greater than -17, the function's upper bound is approaching -14, and the lower bound is -18.Therefore, the range of f(x) over the interval (1, 4) is all real numbers from -18 up to, but not including, -14. So, in interval notation, that would be [-18, -14).Let me double-check my calculations to make sure I didn't make any mistakes.First, vertex at x = 3, f(3) = -18. Correct.f(1) = 1 - 6 - 9 = -14. Correct.f(4) = 16 - 24 - 9 = -17. Correct.Since the function is continuous on (1, 4), and it's decreasing from 1 to 3 and increasing from 3 to 4, the maximum value near the endpoints is -14 (since it's higher than -17). So, the function can get as low as -18 and as high as just below -14.Therefore, the range is [-18, -14). That seems right.I think I've got it. The range is from -18 inclusive to -14 exclusive.**Final Answer**The range of the function ( f(x) ) for ( x in (1, 4) ) is boxed{[-18, -14)}.

💡First, I'll determine the total amount of provisions available in the hostel. Since the provisions are enough for 250 men for 32 days, the total provisions can be calculated by multiplying the number of men by the number of days:[text{Total provisions} = 250 text{ men} times 32 text{ days} = 8000 text{ man-days}]Next, let ( x ) be the number of men who left the hostel. This means there are ( 250 - x ) men remaining. The provisions now need to last for 40 days for the remaining men. Therefore, the total provisions can also be expressed as:[text{Total provisions} = (250 - x) text{ men} times 40 text{ days} = (250 - x) times 40 text{ man-days}]Since the total provisions remain the same, I can set up the equation:[8000 = (250 - x) times 40]To solve for ( x ), I'll first divide both sides of the equation by 40:[frac{8000}{40} = 250 - x][200 = 250 - x]Next, I'll isolate ( x ) by subtracting 250 from both sides:[200 - 250 = -x][-50 = -x]Finally, I'll multiply both sides by -1 to find the value of ( x ):[x = 50]Therefore, 50 men left the hostel.

💡First, I recognize that arranging 12 people around a round table involves considering rotational symmetries. Unlike arranging people in a straight line, where each position is distinct, in a circular arrangement, rotating the entire group doesn't create a new arrangement.To account for these rotational symmetries, I start by calculating the total number of ways to arrange 12 people in a straight line, which is 12 factorial (12!). However, since the table is round, each unique arrangement can be rotated in 12 different ways that are considered identical. Therefore, I divide the total number of linear arrangements by 12 to eliminate the duplicate rotational arrangements.This simplifies the problem to calculating 11!, which represents the number of distinct seating arrangements around the round table. Finally, I compute 11! to find the exact number of ways.

💡Okay, so I have this problem where I'm given a function f(x) = ax + b, and I need to find a + b. The function is defined recursively: f₁(x) = f(x), and f_{n+1}(x) = f(fₙ(x)). They tell me that f₇(x) = 128x + 381, and I need to find a + b.Hmm, let me start by understanding what this recursive function does. So f₁(x) is just f(x), which is ax + b. Then f₂(x) would be f(f₁(x)), which is f(ax + b). Let me compute that: f(ax + b) = a*(ax + b) + b = a²x + ab + b.Similarly, f₃(x) would be f(f₂(x)) = f(a²x + ab + b) = a*(a²x + ab + b) + b = a³x + a²b + ab + b.Wait, I see a pattern here. Each time I apply f again, I'm multiplying by a and adding b. So for fₙ(x), it seems like the coefficient of x is aⁿ, and the constant term is b times the sum of a⁰ + a¹ + ... + a^{n-1}.Let me check that with n=1: f₁(x) = a¹x + b*(1) = ax + b, which is correct. For n=2: a²x + b*(1 + a), which matches what I got earlier. For n=3: a³x + b*(1 + a + a²). Yeah, that seems consistent.So in general, fₙ(x) = aⁿx + b*(1 + a + a² + ... + a^{n-1}). The sum 1 + a + a² + ... + a^{n-1} is a geometric series. The formula for the sum of a geometric series is (1 - aⁿ)/(1 - a) when a ≠ 1.So fₙ(x) can be written as fₙ(x) = aⁿx + b*(1 - aⁿ)/(1 - a).Given that f₇(x) = 128x + 381, I can set up the equation:a⁷x + b*(1 - a⁷)/(1 - a) = 128x + 381.This gives me two equations by comparing coefficients:1. a⁷ = 1282. b*(1 - a⁷)/(1 - a) = 381Let me solve the first equation: a⁷ = 128. I know that 2⁷ = 128, so a must be 2.Now plug a = 2 into the second equation:b*(1 - 2⁷)/(1 - 2) = 381Calculate 2⁷: that's 128. So 1 - 128 = -127. The denominator is 1 - 2 = -1.So the equation becomes:b*(-127)/(-1) = 381Simplify: (-127)/(-1) is 127, so 127b = 381.Divide both sides by 127: b = 381 / 127.Calculate that: 127 * 3 = 381, so b = 3.Therefore, a = 2 and b = 3. So a + b = 2 + 3 = 5.Wait, let me double-check my steps to make sure I didn't make a mistake. Starting from f₇(x) = 128x + 381, I used the formula for fₙ(x) which I derived as aⁿx + b*(1 - aⁿ)/(1 - a). Plugging in n=7, I got a⁷ = 128, which gives a=2. Then plugging a=2 into the constant term equation, I solved for b and got b=3. Seems correct.I think that's solid. So the answer should be 5.