💡Okay, so I have this complex number z, which is given as (i + 1)/i, and I need to find its modulus, |z|. Hmm, complex numbers can sometimes be tricky, especially when they're in fractions like this. Let me think about how to approach this step by step.First, I remember that the modulus of a complex number a + bi is calculated as the square root of (a squared plus b squared). So, |a + bi| = sqrt(a² + b²). But in this case, z is given as a fraction, so I might need to simplify it first before I can find the modulus.Alright, so z = (i + 1)/i. I think it would be easier to find the modulus if I can express z in the standard form a + bi. To do that, I need to eliminate the i in the denominator. I recall that multiplying the numerator and the denominator by the complex conjugate of the denominator can help simplify expressions like this.Wait, but the denominator here is just i, which is a purely imaginary number. The complex conjugate of i is -i, right? So, if I multiply both the numerator and the denominator by -i, that should help me get rid of the i in the denominator.Let me try that:z = (i + 1)/i * (-i)/(-i) = [(i + 1)(-i)] / (i * -i)Okay, let's compute the numerator first: (i + 1)(-i). I'll distribute the multiplication:= i*(-i) + 1*(-i)= -i² - iI know that i² = -1, so -i² = -(-1) = 1. So, substituting that in:= 1 - iNow, the denominator: i * -i = -i². Again, since i² = -1, this becomes:= -(-1) = 1So, putting it all together, z = (1 - i)/1 = 1 - i.Alright, so now z is simplified to 1 - i, which is in the standard form a + bi, where a = 1 and b = -1.Now, to find the modulus |z|, I use the formula |a + bi| = sqrt(a² + b²). Plugging in the values:|z| = sqrt(1² + (-1)²) = sqrt(1 + 1) = sqrt(2)So, the modulus of z is sqrt(2).Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from z = (i + 1)/i, I multiplied numerator and denominator by -i to get rid of the i in the denominator. That gave me (1 - i)/1, which simplifies to 1 - i. Then, using the modulus formula, I got sqrt(2). That seems correct.Alternatively, I could have used another method to find the modulus without simplifying z first. I remember that the modulus of a quotient is the quotient of the moduli. So, |z| = |(i + 1)/i| = |i + 1| / |i|.Let's try that approach to verify. First, find |i + 1|. Since i + 1 is 1 + i, which is a complex number with a = 1 and b = 1, so |1 + i| = sqrt(1² + 1²) = sqrt(2).Next, find |i|. The modulus of i is sqrt(0² + 1²) = 1.So, |z| = |i + 1| / |i| = sqrt(2)/1 = sqrt(2). That matches the result I got earlier.Okay, so both methods give me the same answer, which is reassuring. I think I've got it right.Just to make sure, I'll recall another property: multiplying a complex number by its conjugate gives the square of its modulus. But in this case, since I already have the modulus, I don't think that's necessary here. Still, it's good to remember these properties for future problems.In summary, whether I simplify z first or use the property of modulus for quotients, I end up with |z| = sqrt(2). I feel confident that this is the correct answer.

💡Alright, so I've got this problem about Yunmeng fish noodles, which sounds like a traditional delicacy from Hubei Province in China. The problem is about a fish noodle factory that sells these noodles online, and they want to figure out how to maximize their profit by adjusting the price. There are two parts to the problem: the first one asks about the profit when the price is reduced by 5 yuan, and the second one is about finding the optimal price reduction to achieve a specific profit while maximizing customer benefits.Okay, let's start with part (1). They tell us that in March, they sold 200 bags at 30 yuan per bag, and the cost per bag is 16 yuan. So, first, I need to understand what the profit is. Profit is usually calculated as (Selling Price - Cost Price) multiplied by the number of units sold. In March, the profit would be (30 - 16) * 200, which is 14 * 200 = 2800 yuan.But in April, they plan to reduce the price to increase sales. For every 1 yuan reduction, they can sell 20 more bags. So, if they reduce the price by 5 yuan, the new selling price would be 30 - 5 = 25 yuan per bag. The number of bags sold would increase by 20 * 5 = 100 bags, making the total sales 200 + 100 = 300 bags.Now, the profit per bag after the price reduction would be 25 - 16 = 9 yuan. So, the total profit would be 9 * 300 = 2700 yuan. That seems straightforward.Moving on to part (2), they want to know by how much the price should be reduced to maximize customer benefits while still making a profit of 2860 yuan. Hmm, this seems a bit more complex. I think I need to set up an equation where the profit is 2860 yuan and solve for the price reduction.Let's denote the price reduction as x yuan. So, the new selling price would be 30 - x yuan, and the number of bags sold would be 200 + 20x. The profit per bag would then be (30 - x - 16) = (14 - x) yuan. The total profit would be (14 - x) * (200 + 20x).We want this total profit to be 2860 yuan, so:(14 - x) * (200 + 20x) = 2860Let me expand this equation:14*200 + 14*20x - x*200 - x*20x = 28602800 + 280x - 200x - 20x² = 2860Combine like terms:2800 + 80x - 20x² = 2860Subtract 2860 from both sides:2800 + 80x - 20x² - 2860 = 0Simplify:-60 + 80x - 20x² = 0Let's rearrange it:-20x² + 80x - 60 = 0I can divide the entire equation by -20 to simplify:x² - 4x + 3 = 0Now, this is a quadratic equation. Let's factor it:(x - 1)(x - 3) = 0So, the solutions are x = 1 and x = 3.Now, I need to decide which one to choose. The problem mentions maximizing customer benefits. I think that means we want to reduce the price as much as possible to benefit the customers, but still make the desired profit. So, between reducing by 1 yuan or 3 yuan, reducing by 3 yuan would give a bigger price reduction, which is better for customers, but we need to check if it still gives the desired profit.Wait, but both solutions give the same profit of 2860 yuan. So, to maximize customer benefits, we should choose the larger reduction, which is 3 yuan. That way, customers get a bigger discount, and the manufacturer still makes the desired profit.But let me double-check my calculations to make sure I didn't make a mistake. Starting from the profit equation:(14 - x)(200 + 20x) = 2860Expanding:14*200 + 14*20x - x*200 - x*20x = 28602800 + 280x - 200x - 20x² = 28602800 + 80x - 20x² = 2860Subtract 2860:-60 + 80x - 20x² = 0Divide by -20:x² - 4x + 3 = 0Factor:(x - 1)(x - 3) = 0Solutions: x = 1 and x = 3Yes, that seems correct. So, the price should be reduced by 3 yuan to maximize customer benefits while achieving the desired profit.

💡Okay, so Denis has these cards numbered from 1 to 50, and I need to figure out how many ways he can choose two cards such that the difference between the numbers is 11, and their product is divisible by 5. Hmm, let's break this down step by step.First off, the product being divisible by 5 means that at least one of the numbers on the cards must be a multiple of 5. So, I should probably list out all the multiples of 5 between 1 and 50. Let me do that:5, 10, 15, 20, 25, 30, 35, 40, 45, 50.Alright, so there are 10 multiples of 5 in this range. Now, for each of these multiples, I need to see if adding or subtracting 11 gives me another number within the 1 to 50 range. Because the difference between the two numbers has to be exactly 11.Let me start with the first multiple, which is 5. If I add 11 to 5, I get 16. That's within 1 to 50, so that's a valid pair: (5, 16). If I subtract 11 from 5, I get -6, which is not valid because we can't have negative numbers on the cards. So, only one pair here.Next, 10. Adding 11 gives me 21, which is valid. Subtracting 11 gives me -1, which isn't valid. So, another pair: (10, 21).Moving on to 15. Adding 11 gives me 26, which is good. Subtracting 11 gives me 4, which is also within range. So, two pairs here: (15, 26) and (4, 15).For 20, adding 11 gives 31, and subtracting 11 gives 9. Both are valid, so two pairs: (20, 31) and (9, 20).25: Adding 11 is 36, subtracting 11 is 14. Both valid, so two pairs: (25, 36) and (14, 25).30: Adding 11 is 41, subtracting 11 is 19. Both valid, so two pairs: (30, 41) and (19, 30).35: Adding 11 is 46, subtracting 11 is 24. Both valid, so two pairs: (35, 46) and (24, 35).40: Adding 11 would be 51, which is over 50, so that's not valid. Subtracting 11 gives 29, which is valid. So, one pair: (40, 29).45: Adding 11 is 56, which is too big. Subtracting 11 gives 34, which is valid. So, one pair: (45, 34).50: Adding 11 is 61, which is way too big. Subtracting 11 gives 39, which is valid. So, one pair: (50, 39).Let me count how many pairs I have here. For 5, 10, 40, 45, and 50, each gave me one pair. That's 5 pairs. For 15, 20, 25, 30, and 35, each gave me two pairs. That's 5 times 2, which is 10 pairs. So, in total, 5 plus 10 equals 15 pairs.Wait, let me double-check to make sure I didn't miss anything. Each multiple of 5 was checked, and for each, I considered both adding and subtracting 11. I think that's comprehensive. Yeah, I think 15 is the right answer.**Final Answer**The number of ways is boxed{15}.

💡Okay, so I have this problem here: I need to find the distance between the center of the circle defined by the equation ( x^2 + y^2 = -6x + 8y - 18 ) and the point ( (3, 10) ). Hmm, let me think about how to approach this.First, I remember that the general equation of a circle is usually written in the form ( (x - h)^2 + (y - k)^2 = r^2 ), where ( (h, k) ) is the center and ( r ) is the radius. But the equation given here doesn't look like that. It's more like ( x^2 + y^2 ) equals some linear terms and a constant. So, I think I need to rearrange this equation to get it into the standard circle form.Let me start by moving all the terms to one side to see if that helps. The original equation is:( x^2 + y^2 = -6x + 8y - 18 )If I bring all the terms to the left side, it becomes:( x^2 + 6x + y^2 - 8y = -18 )Okay, so now I have ( x^2 + 6x + y^2 - 8y = -18 ). This looks like I can complete the square for both the x terms and the y terms. Completing the square is a method to convert a quadratic equation into a perfect square trinomial, which will help me get the equation into the standard circle form.Let me handle the x terms first. I have ( x^2 + 6x ). To complete the square, I take the coefficient of x, which is 6, divide it by 2 to get 3, and then square it to get 9. So, I can write:( x^2 + 6x = (x + 3)^2 - 9 )Similarly, for the y terms, I have ( y^2 - 8y ). The coefficient of y is -8. Dividing that by 2 gives -4, and squaring it gives 16. So, I can write:( y^2 - 8y = (y - 4)^2 - 16 )Now, substituting these back into the equation:( (x + 3)^2 - 9 + (y - 4)^2 - 16 = -18 )Let me simplify the constants on the left side:( (x + 3)^2 + (y - 4)^2 - 25 = -18 )Now, I'll move the constant term to the right side:( (x + 3)^2 + (y - 4)^2 = -18 + 25 )Calculating the right side:( (x + 3)^2 + (y - 4)^2 = 7 )Okay, now the equation is in the standard circle form. From this, I can see that the center of the circle is at ( (-3, 4) ) because the equation is ( (x - (-3))^2 + (y - 4)^2 = 7 ). The radius is ( sqrt{7} ), but I don't think I need the radius for this problem.Now, I need to find the distance between this center ( (-3, 4) ) and the point ( (3, 10) ). To find the distance between two points, I can use the distance formula, which is:( d = sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} )Let me label the points to avoid confusion. Let me call ( (x_1, y_1) = (-3, 4) ) and ( (x_2, y_2) = (3, 10) ).Plugging these into the distance formula:( d = sqrt{(3 - (-3))^2 + (10 - 4)^2} )Simplifying inside the parentheses:( d = sqrt{(3 + 3)^2 + (10 - 4)^2} )Which becomes:( d = sqrt{6^2 + 6^2} )Calculating the squares:( d = sqrt{36 + 36} )Adding them together:( d = sqrt{72} )Hmm, ( sqrt{72} ) can be simplified. I know that 72 is 36 times 2, and 36 is a perfect square. So,( sqrt{72} = sqrt{36 times 2} = sqrt{36} times sqrt{2} = 6sqrt{2} )So, the distance between the center of the circle and the point ( (3, 10) ) is ( 6sqrt{2} ).Let me just double-check my steps to make sure I didn't make any mistakes. I rearranged the original equation correctly, completed the square for both x and y terms, simplified the constants, and then applied the distance formula. Everything seems to add up. I think I'm confident with this answer.

💡Okay, so I have this math problem here: I need to calculate the value of this expression:( frac {9}{4})^{ frac {1}{2}} times ( frac {27}{8})^{- frac {1}{3}} - (lg2)^{2} - (lg5)^{2} - 2lg2 cdot lg5.Hmm, that's a bit intimidating at first glance, but let me break it down step by step. I think I can handle it if I take it one part at a time.First, let's look at the expression: it's two main parts. The first part is ( frac {9}{4})^{ frac {1}{2}} times ( frac {27}{8})^{- frac {1}{3}}, and the second part is (lg2)^{2} + (lg5)^{2} + 2lg2 cdot lg5. Then, we subtract the second part from the first part.Alright, let's tackle the first part first: ( frac {9}{4})^{ frac {1}{2}} times ( frac {27}{8})^{- frac {1}{3}}.I remember that when you have an exponent like frac{1}{2}, it's the same as taking the square root. Similarly, an exponent of -frac{1}{3} is the same as taking the reciprocal and then the cube root. So, let me rewrite these parts to make it clearer.Starting with ( frac {9}{4})^{ frac {1}{2}}, that should be the square root of frac{9}{4}. The square root of 9 is 3, and the square root of 4 is 2, so this simplifies to frac{3}{2}.Next, let's look at ( frac {27}{8})^{- frac {1}{3}}. The negative exponent means I need to take the reciprocal first. So, the reciprocal of frac{27}{8} is frac{8}{27}. Then, the exponent frac{1}{3} is the cube root. The cube root of 8 is 2, and the cube root of 27 is 3. So, this simplifies to frac{2}{3}.Now, multiplying these two results together: frac{3}{2} times frac{2}{3}. Let me compute that. The 3 in the numerator and the 3 in the denominator cancel out, and the 2 in the numerator and the 2 in the denominator also cancel out. So, that leaves me with 1. Okay, so the first part simplifies nicely to 1.Alright, moving on to the second part: (lg2)^{2} + (lg5)^{2} + 2lg2 cdot lg5. This looks familiar. It reminds me of the formula for (a + b)^2, which is a^2 + 2ab + b^2. If I let a = lg2 and b = lg5, then this expression is exactly (lg2 + lg5)^2.So, the second part simplifies to (lg2 + lg5)^2. Now, I know that lg2 + lg5 is equal to lg(2 times 5) because of the logarithm property that lg a + lg b = lg(ab). So, lg2 + lg5 = lg10. And since lg10 is the logarithm of 10 with base 10, that's equal to 1 because 10^1 = 10.Therefore, (lg2 + lg5)^2 = (1)^2 = 1. So, the second part simplifies to 1 as well.Now, putting it all back into the original expression: the first part was 1, and we subtract the second part, which is also 1. So, 1 - 1 equals 0.Wait, that seems too straightforward. Did I do that right? Let me double-check each step to make sure I didn't make a mistake.First part: ( frac {9}{4})^{ frac {1}{2}} is indeed frac{3}{2} because square root of 9 is 3 and square root of 4 is 2. Then, ( frac {27}{8})^{- frac {1}{3}} is the reciprocal of the cube root of frac{27}{8}, which is frac{8}{27}, and the cube root of that is frac{2}{3}. Multiplying frac{3}{2} and frac{2}{3} gives 1. That seems correct.Second part: (lg2)^2 + (lg5)^2 + 2lg2 cdot lg5 is indeed (lg2 + lg5)^2, which is (lg10)^2 = 1^2 = 1. So, subtracting 1 from 1 gives 0. Hmm, that seems right.I guess the problem was testing my knowledge of exponent rules and logarithm properties. It's interesting how breaking it down step by step makes it manageable. I was initially worried because the expression looked complicated, but simplifying each part made it much easier.Just to recap:1. Simplify each exponent part: - ( frac {9}{4})^{ frac {1}{2}} = frac{3}{2} - ( frac {27}{8})^{- frac {1}{3}} = frac{2}{3} - Multiplying them gives 1.2. Simplify the logarithmic part: - Recognize the square of a binomial: (lg2 + lg5)^2 - Use logarithm properties: lg2 + lg5 = lg10 = 1 - So, the square is 1.3. Subtract the second part from the first: 1 - 1 = 0.Yep, that all checks out. I think I'm confident with this answer now.**Final Answer**boxed{0}

💡Okay, so I have this problem about a function f(x) that's defined on all real numbers. It has two properties: first, that f(x + 1) equals negative f(x), and second, that when x is between 0 and 1, f(x) is equal to 2^x minus 1. I need to find the value of f(log base 2 of 20). The options are A: 1/4, B: 4/5, C: 5/4, D: -1/5.Hmm, let me start by understanding the function f(x). The first property says that f(x + 1) = -f(x). That seems like a recursive relationship. So, if I know f at some point x, I can find f at x + 1 by just taking the negative. Maybe this implies some periodicity? Let me check.If I apply the property twice, f(x + 2) = f((x + 1) + 1) = -f(x + 1) = -(-f(x)) = f(x). So, f is periodic with period 2. That means f(x + 2) = f(x) for all x. So, every 2 units, the function repeats its values. That's useful because if I can find an equivalent point within one period, I can use the given definition of f(x) on (0,1).Now, I need to compute f(log₂20). Let me figure out where log₂20 is. I know that 2^4 is 16 and 2^5 is 32. Since 20 is between 16 and 32, log₂20 is between 4 and 5. So, log₂20 is approximately 4.something. Specifically, log₂20 is log₂(16 * 1.25) = log₂16 + log₂1.25 = 4 + log₂(5/4). So, log₂20 = 4 + log₂(5/4). Therefore, log₂(5/4) is between 0 and 1 because 5/4 is between 1 and 2. So, log₂(5/4) is in (0,1).Since f is periodic with period 2, f(log₂20) = f(4 + log₂(5/4)) = f(log₂(5/4)). Because adding 4, which is a multiple of the period 2, doesn't change the value of the function.Now, log₂(5/4) is in (0,1), so we can use the given definition of f(x) on that interval. The function f(x) = 2^x - 1 for x in (0,1). So, f(log₂(5/4)) = 2^{log₂(5/4)} - 1.But wait, 2^{log₂(5/4)} is just 5/4 because 2 and log base 2 are inverse functions. So, 2^{log₂(5/4)} = 5/4. Therefore, f(log₂(5/4)) = 5/4 - 1 = 1/4.So, putting it all together, f(log₂20) = f(log₂(5/4)) = 1/4. Therefore, the answer should be A: 1/4.But let me double-check to make sure I didn't make any mistakes. So, f(x + 1) = -f(x), which leads to f(x + 2) = f(x). So, period 2. log₂20 is between 4 and 5, so subtract 4 to get it into the interval (0,1). Then, apply f(x) = 2^x - 1, which gives 5/4 - 1 = 1/4. Yeah, that seems right.Wait, another thought: since f(x + 1) = -f(x), maybe I should check if subtracting 4 is the right approach. Let me think. If I have f(log₂20) = f(4 + log₂(5/4)). Since the period is 2, subtracting 2 twice would bring it down by 4, which is still within the periodicity. So, f(4 + log₂(5/4)) = f(log₂(5/4)). So, that seems correct.Alternatively, I could have considered f(log₂20) = f(log₂20 - 4) because 4 is a multiple of the period 2. So, subtracting 4 is the same as subtracting two periods, which brings it into the interval (0,1). So, yes, that's correct.Therefore, I think my answer is correct. The value of f(log₂20) is 1/4, which is option A.**Final Answer**boxed{A}