answer：First, I recognize that the given solution ({x = a, y = b}) satisfies the equation (2x - 3y - 5 = 0). This means that when I substitute (x) with (a) and (y) with (b), the equation holds true.Substituting the values, the equation becomes:[2a - 3b - 5 = 0]From this, I can isolate (2a - 3b) by adding 5 to both sides:[2a - 3b = 5]Now, the problem asks for the value of (2a - 3b + 3). Using the equation I derived, I substitute (2a - 3b) with 5:[2a - 3b + 3 = 5 + 3 = 8]Therefore, the value of (2a - 3b + 3) is 8.

answer：Alright, so I have this problem here where I need to use the digits 1, 2, 5, 6, and 9 to form five-digit even numbers without repeating any digits. Then, I have to find the difference between the largest and smallest of these numbers. The options given are (A) 83916, (B) 79524, (C) 83952, (D) 79236, and (E) 83016. Hmm, okay, let's break this down step by step.First off, since we're dealing with five-digit numbers, each number must use all five digits exactly once. Also, the number has to be even, which means the last digit has to be an even number. Looking at the digits we have: 1, 2, 5, 6, 9. The even digits here are 2 and 6. So, the last digit of our five-digit number must be either 2 or 6.Alright, so to find the largest possible number, I want the digits to be as large as possible from left to right. That means the first digit should be the largest available, then the next, and so on. Similarly, for the smallest number, I want the digits to be as small as possible from left to right.Let's tackle the largest number first. Since the number has to be even, the last digit must be 6 or 2. But to make the largest number, I should choose the largest possible digit for the last place. So, between 6 and 2, 6 is larger. Therefore, the last digit should be 6.Now, with 6 fixed in the last position, I need to arrange the remaining digits (1, 2, 5, 9) in descending order for the first four positions. Let's see: the largest digit is 9, then 5, then 2, then 1. So, putting that together, the largest number would be 9 5 2 1 6, which is 95216.Wait, hold on. Let me double-check that. If I fix 6 at the end, the remaining digits are 1, 2, 5, 9. To make the largest number, I should arrange them in descending order: 9, 5, 2, 1. So yes, that gives 95216. That seems correct.Now, moving on to the smallest number. Again, since it's even, the last digit has to be 2 or 6. But this time, to make the smallest number, I should choose the smallest possible even digit for the last place. Between 2 and 6, 2 is smaller. So, the last digit should be 2.With 2 fixed at the end, I need to arrange the remaining digits (1, 5, 6, 9) in ascending order for the first four positions. The smallest digit is 1, followed by 5, then 6, then 9. So, the smallest number would be 1 5 6 9 2, which is 15692.Wait, hold on again. Let me verify. If I fix 2 at the end, the remaining digits are 1, 5, 6, 9. To make the smallest number, the first digit should be the smallest available, which is 1. Then, the next digits should be the next smallest, so 5, then 6, then 9. So yes, 15692 is correct.Now, I need to find the difference between the largest number and the smallest number. That would be 95216 minus 15692. Let me calculate that.First, subtract the units place: 6 - 2 = 4.Then, the tens place: 1 - 9. Hmm, can't do that, so I need to borrow. So, 1 becomes 11, and the 2 in the hundreds place becomes 1. Now, 11 - 9 = 2.Moving to the hundreds place: 1 (after borrowing) - 6. Again, can't do that, so borrow from the thousands place. The 5 becomes 4, and the hundreds place becomes 11. 11 - 6 = 5.Thousands place: 4 (after borrowing) - 5. Hmm, can't do that either. So, borrow from the ten-thousands place. The 9 becomes 8, and the thousands place becomes 14. 14 - 5 = 9.Ten-thousands place: 8 (after borrowing) - 1 = 7.Putting it all together, the difference is 79524. Hmm, let me write that down: 95216 - 15692 = 79524.Looking back at the options, (B) is 79524. So, that should be the answer. But wait, let me double-check my subtraction to make sure I didn't make a mistake.Let me do the subtraction again:95216-15692--------?Starting from the right:6 - 2 = 4.1 - 9: Can't do, borrow. 11 - 9 = 2. The next digit becomes 1 (since we borrowed 1).2 - 6: Can't do, borrow. 12 - 6 = 6. The next digit becomes 4 (since we borrowed 1).5 - 5: Wait, hold on, the thousands place. Wait, in the minuend, it's 5, but we had borrowed 1 earlier, so it's 4. 4 - 5: Can't do, borrow. 14 - 5 = 9. The ten-thousands place becomes 8 (since we borrowed 1).9 - 1 = 8.So, writing it out: 79524. Yep, same result. So, that seems correct.But just to be thorough, let me add 15692 and 79524 to see if I get 95216.15692 + 79524:2 + 4 = 6.9 + 2 = 11, carryover 1.6 + 5 + 1 = 12, carryover 1.5 + 9 + 1 = 15, carryover 1.1 + 7 + 1 = 9.So, adding them together gives 95216, which matches the largest number. So, that confirms that the difference is indeed 79524.Therefore, the answer should be option (B) 79524.But hold on, let me just make sure that I didn't make a mistake in determining the largest and smallest numbers.For the largest number, I fixed 6 at the end and arranged the rest in descending order: 9, 5, 2, 1. So, 95216. Is that the largest possible?Alternatively, if I fix 2 at the end, would that give a larger number? Let's see: 9, 6, 5, 1, 2. That would be 96512. Wait, that's larger than 95216. Hmm, so maybe I made a mistake there.Wait, hold on, that can't be. Because if I fix 2 at the end, the remaining digits are 1, 5, 6, 9. To make the largest number, I should arrange them in descending order: 9, 6, 5, 1. So, 96512. That's actually larger than 95216.Wait, so why did I fix 6 at the end earlier? Because I thought 6 is larger than 2, so to make the number as large as possible, the last digit should be as large as possible. But actually, the last digit is the units place, so it's the least significant digit. So, making the last digit larger doesn't contribute as much as making the higher place digits larger.Therefore, perhaps I was wrong earlier. Maybe 96512 is actually the larger number. Let me think.Wait, let's compare 95216 and 96512.95216 vs. 96512.Looking at the first digit: both are 9.Second digit: 5 vs. 6. 6 is larger, so 96512 is larger than 95216.So, actually, 96512 is the larger number. So, I made a mistake earlier by fixing 6 at the end. Instead, I should fix the smaller even digit at the end if that allows the higher place digits to be larger.Wait, no, actually, to maximize the number, you want the higher place digits to be as large as possible, but the last digit also affects the overall number. So, both the last digit and the other digits are important.So, perhaps, to get the largest number, you need to choose the last digit as large as possible, but also arrange the other digits in descending order.Wait, but in this case, if I fix 6 at the end, the remaining digits are 1, 2, 5, 9, which can be arranged as 9,5,2,1, giving 95216.Alternatively, if I fix 2 at the end, the remaining digits are 1,5,6,9, which can be arranged as 9,6,5,1, giving 96512.Comparing 95216 and 96512, which is larger? 96512 is larger because the second digit is 6 vs. 5.So, 96512 is actually larger. So, my initial approach was wrong. I thought that fixing the larger even digit at the end would give the larger number, but actually, fixing the smaller even digit allows the other digits to be arranged in a way that makes the overall number larger.Therefore, I need to reconsider how I determine the largest number.So, perhaps, the way to do it is to fix the even digit at the end, and then arrange the remaining digits in descending order. But depending on which even digit you fix, the remaining digits may allow for a larger number.So, let's try both possibilities.Case 1: Last digit is 6.Remaining digits: 1,2,5,9. Arrange them in descending order: 9,5,2,1. So, the number is 95216.Case 2: Last digit is 2.Remaining digits: 1,5,6,9. Arrange them in descending order: 9,6,5,1. So, the number is 96512.Comparing both, 96512 is larger. So, 96512 is the largest number.Similarly, for the smallest number, I fixed 2 at the end and arranged the remaining digits in ascending order: 1,5,6,9, giving 15692.But let me check if fixing 6 at the end gives a smaller number.Case 1: Last digit is 2.Remaining digits: 1,5,6,9. Arrange them in ascending order: 1,5,6,9. So, the number is 15692.Case 2: Last digit is 6.Remaining digits: 1,2,5,9. Arrange them in ascending order: 1,2,5,9. So, the number is 12596.Comparing both, 12596 is smaller than 15692. So, 12596 is the smaller number.Wait, so that changes things. So, actually, the smallest number is 12596, not 15692. So, I made a mistake earlier.So, let me correct that.To find the smallest number, we need to fix the smallest even digit at the end, which is 2, but also arrange the remaining digits in ascending order. However, if we fix 6 at the end, we might get an even smaller number.Wait, let's see:If last digit is 2, remaining digits are 1,5,6,9. The smallest number is 15692.If last digit is 6, remaining digits are 1,2,5,9. The smallest number is 12596.Comparing 12596 and 15692, 12596 is smaller. So, 12596 is the smallest number.Therefore, my initial conclusion was wrong because I didn't consider that fixing 6 at the end could lead to a smaller number.So, now, the largest number is 96512, and the smallest number is 12596.Therefore, the difference is 96512 - 12596.Let me calculate that.96512-12596--------?Starting from the right:2 - 6: Can't do, borrow. 12 - 6 = 6. Next digit becomes 0 (since we borrowed 1).1 (after borrowing) - 9: Can't do, borrow. 11 - 9 = 2. Next digit becomes 5 (since we borrowed 1).5 (after borrowing) - 5 = 0.6 - 2 = 4.9 - 1 = 8.So, putting it together: 83916.Wait, that's 83916. Hmm, which is option (A). But earlier, when I subtracted 15692 from 95216, I got 79524, which is option (B). But now, with the corrected numbers, 96512 - 12596 = 83916, which is option (A). So, which one is correct?Wait, let me verify the subtraction again.96512-12596--------?Starting from the units place:2 - 6: Can't do, borrow 1 from the tens place. So, 12 - 6 = 6. The tens digit becomes 0.0 (after borrowing) - 9: Can't do, borrow 1 from the hundreds place. So, 10 - 9 = 1. The hundreds digit becomes 4 (since we borrowed 1).4 (after borrowing) - 5: Can't do, borrow 1 from the thousands place. So, 14 - 5 = 9. The thousands digit becomes 5 (since we borrowed 1).5 (after borrowing) - 2 = 3.9 - 1 = 8.So, writing it out: 83916. So, that seems correct.But wait, let me add 12596 and 83916 to see if I get 96512.12596 + 83916:6 + 6 = 12, carryover 1.9 + 1 + 1 = 11, carryover 1.5 + 9 + 1 = 15, carryover 1.2 + 3 + 1 = 6.1 + 8 = 9.So, adding them together gives 96512, which is correct.Therefore, the difference is indeed 83916, which is option (A).But wait, earlier, I thought the largest number was 95216 and the smallest was 15692, giving a difference of 79524. But upon re-evaluating, I found that the largest number is actually 96512 and the smallest is 12596, giving a difference of 83916.So, why did I get confused earlier? Because I didn't consider that fixing a different even digit at the end could lead to a larger or smaller number.Therefore, the correct approach is:To find the largest number:- Choose the largest possible digit for the last place (even digit). But wait, actually, no. To maximize the overall number, you need to consider both the last digit and the other digits.Wait, perhaps a better approach is:To form the largest number:1. The last digit must be even, so choose between 2 and 6. To maximize the number, you need to consider which choice allows the remaining digits to form the largest possible number.So, if you choose 6 as the last digit, the remaining digits are 1,2,5,9. The largest number you can form with these is 9521, so the number is 95216.If you choose 2 as the last digit, the remaining digits are 1,5,6,9. The largest number you can form with these is 9651, so the number is 96512.Comparing 95216 and 96512, 96512 is larger because the second digit is 6 vs. 5.Therefore, the largest number is 96512.Similarly, for the smallest number:1. The last digit must be even, so choose between 2 and 6. To minimize the number, you need to consider which choice allows the remaining digits to form the smallest possible number.If you choose 2 as the last digit, the remaining digits are 1,5,6,9. The smallest number you can form with these is 1569, so the number is 15692.If you choose 6 as the last digit, the remaining digits are 1,2,5,9. The smallest number you can form with these is 1259, so the number is 12596.Comparing 15692 and 12596, 12596 is smaller.Therefore, the smallest number is 12596.Thus, the difference is 96512 - 12596 = 83916.So, the correct answer is (A) 83916.But wait, let me just make sure that 96512 is indeed the largest possible number.Is there any way to get a larger number than 96512?Let's see. The digits are 1,2,5,6,9. To make the largest number, the first digit should be 9. Then, the next digit should be the next largest available, which is 6. Then, the next largest is 5, then 2, and the last digit is 1. Wait, but the last digit has to be even, so it can't be 1. So, that approach doesn't work.Alternatively, if I fix 6 at the end, the remaining digits are 1,2,5,9. So, arranging them in descending order: 9,5,2,1. So, 95216.But as we saw earlier, 96512 is larger because the second digit is 6 instead of 5.So, yeah, 96512 is indeed larger.Similarly, for the smallest number, fixing 6 at the end allows the remaining digits to be arranged as 1,2,5,9, giving 12596, which is smaller than 15692.Therefore, the difference is 83916.So, the correct answer is (A) 83916.But wait, looking back at the options, (A) is 83916, which is what I got. However, initially, I thought the answer was (B) 79524, but that was based on incorrect largest and smallest numbers.Therefore, the correct answer should be (A) 83916.But just to make sure, let me list all possible five-digit even numbers and see if 96512 is indeed the largest and 12596 is the smallest.Wait, listing all possible numbers would be time-consuming, but let's try to see.Possible even numbers must end with 2 or 6.Let's consider numbers ending with 6:The remaining digits are 1,2,5,9. The largest number is 95216, as we saw.Numbers ending with 2:The remaining digits are 1,5,6,9. The largest number is 96512.So, 96512 is larger than 95216.Similarly, for the smallest numbers:Numbers ending with 2: smallest is 15692.Numbers ending with 6: smallest is 12596.So, 12596 is smaller.Therefore, the difference is 96512 - 12596 = 83916.Yes, that seems correct.Therefore, the answer is (A) 83916.**Final Answer**boxed{A}

answer：Okay, so I have this problem here: Given a complex number Z with a real part of 1 and an imaginary part of -2, I need to find the imaginary part of (1 + 3i)/Z. Hmm, let me think about how to approach this.First, let me write down what Z is. Since the real part is 1 and the imaginary part is -2, Z must be 1 - 2i. So, Z = 1 - 2i. Got that down.Now, the expression I need to evaluate is (1 + 3i)/Z, which is (1 + 3i)/(1 - 2i). Hmm, complex numbers in the denominator can be tricky. I remember that to simplify expressions like this, we multiply the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of 1 - 2i is 1 + 2i. So, I'll do that.Let me write that out:(1 + 3i)/(1 - 2i) * (1 + 2i)/(1 + 2i)Multiplying the numerators together and the denominators together. Let me compute the numerator first.Numerator: (1 + 3i)(1 + 2i)Let me expand this using the distributive property, also known as the FOIL method for binomials.First: 1*1 = 1Outer: 1*2i = 2iInner: 3i*1 = 3iLast: 3i*2i = 6i²So, adding all these together: 1 + 2i + 3i + 6i².Combine like terms: 1 + (2i + 3i) + 6i² = 1 + 5i + 6i².Wait, I remember that i² is equal to -1. So, 6i² is 6*(-1) = -6.So, substituting that back in: 1 + 5i - 6.Combine the real parts: 1 - 6 = -5.So, the numerator simplifies to -5 + 5i.Now, let's compute the denominator: (1 - 2i)(1 + 2i)Again, using the distributive property.First: 1*1 = 1Outer: 1*2i = 2iInner: -2i*1 = -2iLast: -2i*2i = -4i²Adding these together: 1 + 2i - 2i - 4i².Combine like terms: 1 + (2i - 2i) - 4i² = 1 + 0 - 4i².Again, i² is -1, so -4i² is -4*(-1) = 4.So, substituting back in: 1 + 4 = 5.Therefore, the denominator simplifies to 5.Putting it all together, the expression (1 + 3i)/(1 - 2i) simplifies to (-5 + 5i)/5.Now, let's divide both terms in the numerator by 5:-5/5 + (5i)/5 = -1 + i.So, the simplified form is -1 + i.The problem asks for the imaginary part of this expression. In the complex number -1 + i, the real part is -1 and the imaginary part is 1 (since it's the coefficient of i).Therefore, the imaginary part is 1.Wait, let me double-check my steps to make sure I didn't make a mistake.First, Z = 1 - 2i. Correct.Expression: (1 + 3i)/(1 - 2i). Correct.Multiply numerator and denominator by the conjugate, 1 + 2i. Correct.Numerator: (1 + 3i)(1 + 2i) = 1 + 2i + 3i + 6i² = 1 + 5i - 6 = -5 + 5i. Correct.Denominator: (1 - 2i)(1 + 2i) = 1 - (2i)^2 = 1 - (-4) = 5. Wait, hold on, is that another way to compute it? Because (a - b)(a + b) = a² - b². So, here, a = 1, b = 2i. So, it's 1² - (2i)² = 1 - 4i² = 1 - 4*(-1) = 1 + 4 = 5. Yep, same result. So that's correct.Then, (-5 + 5i)/5 = -1 + i. Correct.Imaginary part is 1. So, yeah, that seems right.Alternatively, maybe I can use another method to verify. Let's represent Z in polar form and then compute 1/Z, then multiply by (1 + 3i). Hmm, that might be more complicated, but let's try.First, Z = 1 - 2i. Its modulus is sqrt(1² + (-2)²) = sqrt(1 + 4) = sqrt(5). Its argument is arctangent of (-2)/1, which is arctangent(-2). Since it's in the fourth quadrant, the argument is -arctan(2).So, Z can be written as sqrt(5) * (cos(-arctan(2)) + i sin(-arctan(2))).Then, 1/Z is 1/sqrt(5) * (cos(arctan(2)) - i sin(arctan(2))). Because 1/Z has modulus 1/|Z| and argument -arg(Z).Now, let's compute (1 + 3i) * (1/Z). So, that would be (1 + 3i) * [1/sqrt(5) * (cos(arctan(2)) - i sin(arctan(2)))].Hmm, this seems more involved. Maybe I can compute it step by step.First, let's compute cos(arctan(2)) and sin(arctan(2)).If theta = arctan(2), then tan(theta) = 2. So, we can imagine a right triangle where the opposite side is 2 and the adjacent side is 1, so the hypotenuse is sqrt(1 + 4) = sqrt(5). Therefore, cos(theta) = adjacent/hypotenuse = 1/sqrt(5), and sin(theta) = opposite/hypotenuse = 2/sqrt(5).Therefore, cos(arctan(2)) = 1/sqrt(5), and sin(arctan(2)) = 2/sqrt(5).So, 1/Z = (1/sqrt(5)) * [1/sqrt(5) - i*(2/sqrt(5))] = (1/sqrt(5))*(1/sqrt(5)) - i*(1/sqrt(5))*(2/sqrt(5)).Compute each term:First term: (1/sqrt(5))*(1/sqrt(5)) = 1/5.Second term: -i*(1/sqrt(5))*(2/sqrt(5)) = -i*(2/5).Therefore, 1/Z = 1/5 - (2/5)i.Now, multiply this by (1 + 3i):(1 + 3i)*(1/5 - 2i/5).Let's distribute this multiplication:First, multiply 1 by (1/5 - 2i/5): 1*(1/5) + 1*(-2i/5) = 1/5 - 2i/5.Then, multiply 3i by (1/5 - 2i/5): 3i*(1/5) + 3i*(-2i/5) = 3i/5 - 6i²/5.Combine these two results:1/5 - 2i/5 + 3i/5 - 6i²/5.Simplify term by term:Real parts: 1/5 and -6i²/5. Since i² = -1, -6i²/5 = -6*(-1)/5 = 6/5. So, real parts: 1/5 + 6/5 = 7/5.Imaginary parts: -2i/5 + 3i/5 = ( -2 + 3 )i/5 = 1i/5.So, the result is 7/5 + (1/5)i.Wait, hold on, that's different from what I got earlier. Earlier, I had -1 + i, which is -5/5 + 5i/5. But here, I have 7/5 + i/5. That's conflicting. So, which one is correct?Hmm, that means I must have made a mistake somewhere in one of the methods.Let me check the first method again.First method:(1 + 3i)/(1 - 2i) * (1 + 2i)/(1 + 2i) = [ (1 + 3i)(1 + 2i) ] / [ (1 - 2i)(1 + 2i) ]Numerator: 1*1 + 1*2i + 3i*1 + 3i*2i = 1 + 2i + 3i + 6i² = 1 + 5i - 6 = -5 + 5i.Denominator: 1 - (2i)^2 = 1 - (-4) = 5.So, (-5 + 5i)/5 = -1 + i. So, that's correct.Second method:Expressed Z in polar form, found 1/Z as 1/5 - 2i/5, then multiplied by (1 + 3i):(1 + 3i)*(1/5 - 2i/5) = 1*(1/5) + 1*(-2i/5) + 3i*(1/5) + 3i*(-2i/5) = 1/5 - 2i/5 + 3i/5 - 6i²/5.Simplify: 1/5 + ( -2i + 3i )/5 + ( -6i² )/5.Which is 1/5 + i/5 + 6/5, since i² = -1.So, 1/5 + 6/5 = 7/5, and i/5.So, 7/5 + i/5.Wait, so why is this different from the first method?Hmm, perhaps I made a mistake in the second method. Let me check.Wait, in the second method, I said that 1/Z is 1/sqrt(5) * (cos(theta) - i sin(theta)), where theta is arctan(2). Then, I computed cos(theta) as 1/sqrt(5) and sin(theta) as 2/sqrt(5). So, 1/Z = (1/sqrt(5))*(1/sqrt(5) - i*2/sqrt(5)) = (1/5 - 2i/5). That seems correct.Then, multiplying (1 + 3i) by (1/5 - 2i/5):1*(1/5) = 1/51*(-2i/5) = -2i/53i*(1/5) = 3i/53i*(-2i/5) = -6i²/5 = 6/5So, adding all together: 1/5 + 6/5 + (-2i/5 + 3i/5) = 7/5 + i/5.But in the first method, I got -1 + i, which is -5/5 + 5i/5.Wait, so which one is correct?Wait, perhaps I made a mistake in the second method in the multiplication step.Wait, let me recompute (1 + 3i)*(1/5 - 2i/5):First term: 1*(1/5) = 1/5Second term: 1*(-2i/5) = -2i/5Third term: 3i*(1/5) = 3i/5Fourth term: 3i*(-2i/5) = -6i²/5 = 6/5So, adding all together:Real parts: 1/5 + 6/5 = 7/5Imaginary parts: (-2i/5 + 3i/5) = (1i)/5So, total: 7/5 + i/5.But in the first method, I had -1 + i, which is -5/5 + 5i/5.Wait, so that's conflicting. So, which one is correct?Wait, maybe I messed up the multiplication in the first method.Wait, first method:(1 + 3i)/(1 - 2i) * (1 + 2i)/(1 + 2i) = [ (1 + 3i)(1 + 2i) ] / [ (1 - 2i)(1 + 2i) ]Compute numerator:1*1 = 11*2i = 2i3i*1 = 3i3i*2i = 6i² = -6So, numerator: 1 + 2i + 3i -6 = (1 -6) + (2i + 3i) = -5 + 5iDenominator: (1)^2 - (2i)^2 = 1 - (-4) = 5So, (-5 + 5i)/5 = -1 + i.Wait, that seems correct.But in the second method, I have 7/5 + i/5.Wait, so which one is correct?Wait, perhaps I made a mistake in the second method when I expressed 1/Z.Wait, let me re-examine that.Z is 1 - 2i, so its modulus is sqrt(1 + 4) = sqrt(5). Its argument is arctangent(-2/1) = -arctan(2). So, in polar form, Z = sqrt(5) [ cos(-arctan(2)) + i sin(-arctan(2)) ].Therefore, 1/Z is 1/sqrt(5) [ cos(arctan(2)) - i sin(arctan(2)) ].Wait, cos(-theta) = cos(theta), and sin(-theta) = -sin(theta). So, 1/Z should be 1/sqrt(5) [ cos(theta) - i sin(theta) ], where theta = arctan(2). So, that part is correct.Then, cos(theta) = 1/sqrt(5), sin(theta) = 2/sqrt(5). So, 1/Z is 1/sqrt(5)*(1/sqrt(5) - i*2/sqrt(5)) = (1/5 - 2i/5). So, that is correct.Then, multiplying (1 + 3i)*(1/5 - 2i/5):1*(1/5) = 1/51*(-2i/5) = -2i/53i*(1/5) = 3i/53i*(-2i/5) = -6i²/5 = 6/5Adding up: 1/5 + 6/5 = 7/5, and -2i/5 + 3i/5 = i/5.So, 7/5 + i/5.But in the first method, I have -1 + i.Wait, so that's a problem. Which one is correct?Wait, let me compute (1 + 3i)/(1 - 2i) numerically.Let me compute it as fractions.Let me compute (1 + 3i)/(1 - 2i):Multiply numerator and denominator by (1 + 2i):Numerator: (1 + 3i)(1 + 2i) = 1 + 2i + 3i + 6i² = 1 + 5i -6 = -5 + 5i.Denominator: (1)^2 - (2i)^2 = 1 - (-4) = 5.So, (-5 + 5i)/5 = -1 + i.So, that's correct.But in the second method, I get 7/5 + i/5.Wait, so that must mean I made a mistake in the second method.Wait, perhaps I messed up the multiplication step.Wait, let me recompute (1 + 3i)*(1/5 - 2i/5):First, 1*(1/5) = 1/51*(-2i/5) = -2i/53i*(1/5) = 3i/53i*(-2i/5) = -6i²/5 = 6/5So, adding all together:Real parts: 1/5 + 6/5 = 7/5Imaginary parts: (-2i/5 + 3i/5) = (1i)/5So, total: 7/5 + i/5.Wait, so that's conflicting with the first method.Wait, perhaps in the second method, I made a mistake in the expression of 1/Z.Wait, let me compute 1/Z directly.Z = 1 - 2i.So, 1/Z = (1 + 2i)/(1^2 + (2)^2) = (1 + 2i)/5.Wait, that's different from what I had before.Wait, hold on, that's a different approach.Wait, 1/Z is the conjugate of Z divided by |Z| squared.So, 1/Z = conjugate(Z)/|Z|².Since Z = 1 - 2i, conjugate(Z) = 1 + 2i.|Z|² = (1)^2 + (-2)^2 = 1 + 4 = 5.Therefore, 1/Z = (1 + 2i)/5.So, 1/Z = 1/5 + 2i/5.Wait, so that's different from what I had earlier.Wait, in the second method, I had 1/Z = 1/5 - 2i/5. That was incorrect.Wait, so that was my mistake.Because in the second method, I thought that 1/Z is (1/sqrt(5))*(cos(theta) - i sin(theta)), but actually, since theta is the argument of Z, which is negative, so 1/Z should have the conjugate of Z over |Z|².Wait, so 1/Z is (1 + 2i)/5, not (1 - 2i)/5.Wait, so that was my mistake in the second method. I incorrectly took the conjugate as (1 - 2i), but actually, the conjugate of Z = 1 - 2i is 1 + 2i.So, 1/Z is (1 + 2i)/5.Therefore, in the second method, I should have:(1 + 3i)*(1 + 2i)/5.Let me compute that.(1 + 3i)*(1 + 2i) = 1*1 + 1*2i + 3i*1 + 3i*2i = 1 + 2i + 3i + 6i² = 1 + 5i -6 = -5 + 5i.Then, divide by 5: (-5 + 5i)/5 = -1 + i.Which matches the first method.So, my mistake in the second method was incorrectly computing 1/Z as (1 - 2i)/5 instead of (1 + 2i)/5. That was the error.Therefore, the correct result is -1 + i, so the imaginary part is 1.So, yeah, that's consistent.Alternatively, another way to compute the imaginary part without simplifying the entire expression is to use the formula for the imaginary part of a quotient.Given two complex numbers, A = a + bi and B = c + di, then the imaginary part of A/B is (ad - bc)/(c² + d²).Wait, let me recall the formula.The imaginary part of (A/B) is equal to (Im(A)Re(B) - Re(A)Im(B)) / |B|².Yes, that's correct.So, in this case, A = 1 + 3i, so Re(A) = 1, Im(A) = 3.B = Z = 1 - 2i, so Re(B) = 1, Im(B) = -2.Therefore, Im(A/B) = (Im(A)Re(B) - Re(A)Im(B)) / |B|².Compute that:Im(A/B) = (3*1 - 1*(-2)) / (1² + (-2)²) = (3 + 2)/ (1 + 4) = 5/5 = 1.So, that's another way to get the imaginary part directly without computing the entire expression.So, that confirms that the imaginary part is indeed 1.Therefore, the answer is 1.**Final Answer**The imaginary part is boxed{1}.

answer：To find the real part of the product of the complex numbers z₁ = 1 + 2i and z₂ = 1 - i, I will start by multiplying the two complex numbers using the distributive property.First, I'll distribute each term in z₁ by each term in z₂:(1)(1) + (1)(-i) + (2i)(1) + (2i)(-i).Next, I'll simplify each term:1 - i + 2i - 2i².Since i² equals -1, the term -2i² becomes +2. Combining like terms, the expression becomes:1 + i + 2.Finally, adding the real parts together, the real part of the product is 3.

answer：First, Amanda starts with 7 candy bars and gives 3 to her sister. This leaves her with 4 candy bars.The next day, Amanda wins prize money and buys an additional 30 candy bars, bringing her total to 34 candy bars.She then gives her sister 4 times the number she gave the first time, which is 4 multiplied by 3, equaling 12 candy bars.Finally, to find out how many candy bars Amanda kept for herself, subtract the 12 she gave her sister from the 34 she had after buying more. This results in 22 candy bars.

answer：Okay, so I have this problem here: given that ( x geq 0 ) and ( y geq 0 ), and ( x^2 + y^2 = 4 ), I need to find the minimum value of the expression ( xy - 4(x + y) - 2 ). Hmm, alright, let me try to figure this out step by step.First, let me write down what I know. The variables ( x ) and ( y ) are both non-negative, and they satisfy the equation ( x^2 + y^2 = 4 ). So, this is the equation of a circle with radius 2 centered at the origin, but since ( x ) and ( y ) are both non-negative, we're only looking at the first quadrant part of this circle. That makes sense.Now, the expression I need to minimize is ( xy - 4(x + y) - 2 ). Let me denote this expression as ( f(x, y) = xy - 4x - 4y - 2 ). So, I need to find the minimum value of ( f(x, y) ) subject to the constraint ( x^2 + y^2 = 4 ) with ( x, y geq 0 ).Since this is a constrained optimization problem, I think I can use the method of Lagrange multipliers. But wait, I remember that sometimes for problems with circles, it might be easier to parameterize ( x ) and ( y ) using trigonometric functions. Let me consider both approaches and see which one is more straightforward.Starting with the Lagrange multipliers method. The idea is to find the points where the gradient of ( f ) is proportional to the gradient of the constraint function ( g(x, y) = x^2 + y^2 - 4 ). So, let's compute the gradients.The gradient of ( f ) is:[nabla f = left( frac{partial f}{partial x}, frac{partial f}{partial y} right) = (y - 4, x - 4)]And the gradient of ( g ) is:[nabla g = left( frac{partial g}{partial x}, frac{partial g}{partial y} right) = (2x, 2y)]According to the method, there exists a scalar ( lambda ) such that:[nabla f = lambda nabla g]Which gives us the system of equations:[y - 4 = 2lambda x quad (1)][x - 4 = 2lambda y quad (2)]And of course, the constraint:[x^2 + y^2 = 4 quad (3)]So, now I have three equations with three variables: ( x ), ( y ), and ( lambda ). Let me try to solve this system.From equation (1): ( y = 2lambda x + 4 )From equation (2): ( x = 2lambda y + 4 )Hmm, so I can substitute ( y ) from equation (1) into equation (2). Let's do that.Substituting ( y = 2lambda x + 4 ) into equation (2):[x = 2lambda (2lambda x + 4) + 4]Let me expand this:[x = 4lambda^2 x + 8lambda + 4]Now, let's bring all terms to one side:[x - 4lambda^2 x - 8lambda - 4 = 0]Factor out ( x ):[x(1 - 4lambda^2) - 8lambda - 4 = 0]So, solving for ( x ):[x = frac{8lambda + 4}{1 - 4lambda^2}]Similarly, from equation (1), ( y = 2lambda x + 4 ), so substituting the expression for ( x ):[y = 2lambda left( frac{8lambda + 4}{1 - 4lambda^2} right) + 4]Let me compute this:[y = frac{16lambda^2 + 8lambda}{1 - 4lambda^2} + 4]To combine the terms, I'll write 4 as ( frac{4(1 - 4lambda^2)}{1 - 4lambda^2} ):[y = frac{16lambda^2 + 8lambda + 4 - 16lambda^2}{1 - 4lambda^2}]Simplify numerator:[16lambda^2 - 16lambda^2 + 8lambda + 4 = 8lambda + 4]So, ( y = frac{8lambda + 4}{1 - 4lambda^2} )Wait a second, that's interesting. So, both ( x ) and ( y ) have the same expression:[x = y = frac{8lambda + 4}{1 - 4lambda^2}]Hmm, does this mean that ( x = y ) at the extremum points? Let me check.If ( x = y ), then from the constraint equation ( x^2 + y^2 = 4 ), we have ( 2x^2 = 4 ), so ( x^2 = 2 ), which gives ( x = sqrt{2} ) since ( x geq 0 ). So, ( x = y = sqrt{2} ).But wait, let's see if this is consistent with our earlier expressions. If ( x = y ), then from equation (1):[y - 4 = 2lambda x implies x - 4 = 2lambda x]Similarly, from equation (2):[x - 4 = 2lambda y implies x - 4 = 2lambda x]So, both equations reduce to the same thing, which is ( x - 4 = 2lambda x ). So, solving for ( lambda ):[x - 4 = 2lambda x implies lambda = frac{x - 4}{2x}]But if ( x = sqrt{2} ), then:[lambda = frac{sqrt{2} - 4}{2sqrt{2}} = frac{sqrt{2} - 4}{2sqrt{2}} = frac{1 - 2sqrt{2}}{2}]Hmm, okay. So, that gives a specific value for ( lambda ). But let me see if this is the only solution or if there are other possibilities.Wait, earlier when I substituted, I got that both ( x ) and ( y ) have the same expression in terms of ( lambda ), which suggests that ( x = y ). So, that seems like the only critical point. But I should also check the boundaries, right? Because sometimes the extrema can occur on the boundary of the domain.In this case, the domain is the quarter-circle in the first quadrant. The boundaries would be where either ( x = 0 ) or ( y = 0 ). So, let me check those cases as well.First, let's consider ( x = 0 ). Then, from the constraint ( x^2 + y^2 = 4 ), we get ( y = 2 ). So, the point is ( (0, 2) ). Let's compute ( f(0, 2) ):[f(0, 2) = (0)(2) - 4(0 + 2) - 2 = 0 - 8 - 2 = -10]Okay, that's a value of -10.Next, consider ( y = 0 ). Similarly, ( x = 2 ), so the point is ( (2, 0) ). Compute ( f(2, 0) ):[f(2, 0) = (2)(0) - 4(2 + 0) - 2 = 0 - 8 - 2 = -10]Same value, -10.So, both boundary points give me -10. Now, let's check the critical point we found earlier where ( x = y = sqrt{2} ). Compute ( f(sqrt{2}, sqrt{2}) ):[f(sqrt{2}, sqrt{2}) = (sqrt{2})(sqrt{2}) - 4(sqrt{2} + sqrt{2}) - 2 = 2 - 8sqrt{2} - 2 = -8sqrt{2}]Hmm, so that's approximately -11.31, which is less than -10. So, that's actually a lower value.Wait, but hold on. The problem is asking for the minimum value. So, is -8√2 the minimum? But wait, let me make sure that this critical point is indeed a minimum.Alternatively, maybe I should also check other points on the circle to see if there are any lower values. But before that, perhaps I can parameterize the circle and express ( f(x, y) ) in terms of a single variable.Let me try that approach. Since ( x ) and ( y ) lie on the circle ( x^2 + y^2 = 4 ), I can write ( x = 2costheta ) and ( y = 2sintheta ) where ( 0 leq theta leq frac{pi}{2} ) because ( x ) and ( y ) are non-negative.So, substituting into ( f(x, y) ):[f(theta) = (2costheta)(2sintheta) - 4(2costheta + 2sintheta) - 2]Simplify each term:- ( (2costheta)(2sintheta) = 4costhetasintheta = 2sin(2theta) )- ( 4(2costheta + 2sintheta) = 8costheta + 8sintheta )So, putting it all together:[f(theta) = 2sin(2theta) - 8costheta - 8sintheta - 2]Hmm, that seems a bit complicated, but maybe I can simplify it further or take its derivative with respect to ( theta ) to find the minima.Let me compute the derivative ( f'(theta) ):[f'(theta) = 4cos(2theta) + 8sintheta - 8costheta]Set this equal to zero to find critical points:[4cos(2theta) + 8sintheta - 8costheta = 0]Let me simplify this equation. First, recall that ( cos(2theta) = 1 - 2sin^2theta ), but that might complicate things. Alternatively, ( cos(2theta) = 2cos^2theta - 1 ). Let's try that.Substitute ( cos(2theta) = 2cos^2theta - 1 ):[4(2cos^2theta - 1) + 8sintheta - 8costheta = 0]Multiply out:[8cos^2theta - 4 + 8sintheta - 8costheta = 0]Bring all terms to one side:[8cos^2theta - 8costheta + 8sintheta - 4 = 0]Hmm, this still looks complicated. Maybe I can factor out some terms. Let's see:Factor out 8 from the first two terms:[8(cos^2theta - costheta) + 8sintheta - 4 = 0]Hmm, not sure if that helps. Alternatively, maybe I can write everything in terms of sine or cosine.Alternatively, let me try to express ( costheta ) and ( sintheta ) in terms of a single variable. Let me denote ( t = theta ), so:[8cos^2 t - 8cos t + 8sin t - 4 = 0]Hmm, perhaps I can write ( cos^2 t ) as ( 1 - sin^2 t ), so:[8(1 - sin^2 t) - 8cos t + 8sin t - 4 = 0]Simplify:[8 - 8sin^2 t - 8cos t + 8sin t - 4 = 0]Which simplifies to:[4 - 8sin^2 t - 8cos t + 8sin t = 0]Divide both sides by 4:[1 - 2sin^2 t - 2cos t + 2sin t = 0]Hmm, still complicated. Maybe I can rearrange terms:[-2sin^2 t + 2sin t - 2cos t + 1 = 0]Multiply both sides by -1:[2sin^2 t - 2sin t + 2cos t - 1 = 0]Still not so helpful. Maybe I need a different approach.Alternatively, perhaps I can consider using substitution or some trigonometric identities. Let me think.Wait, another approach: since I have both ( sin t ) and ( cos t ), maybe I can write this equation in terms of ( sin t ) only or ( cos t ) only.Alternatively, perhaps I can use substitution ( u = sin t ) and ( v = cos t ), but I don't know if that will help.Alternatively, maybe I can use the method of auxiliary angles or something like that.Wait, let me try to write the equation as:( 8cos^2 t - 8cos t + 8sin t - 4 = 0 )Let me group terms:( 8cos^2 t - 8cos t - 4 + 8sin t = 0 )Hmm, maybe factor out 8 from the first two terms:( 8(cos^2 t - cos t) - 4 + 8sin t = 0 )Not sure. Alternatively, maybe I can write ( cos^2 t ) as ( (1 + cos 2t)/2 ):So, ( 8*(1 + cos 2t)/2 - 8cos t - 4 + 8sin t = 0 )Simplify:( 4(1 + cos 2t) - 8cos t - 4 + 8sin t = 0 )Which is:( 4 + 4cos 2t - 8cos t - 4 + 8sin t = 0 )Simplify further:( 4cos 2t - 8cos t + 8sin t = 0 )Divide both sides by 4:( cos 2t - 2cos t + 2sin t = 0 )Hmm, still not straightforward.Wait, maybe I can write ( cos 2t ) as ( cos^2 t - sin^2 t ), but that might not help. Alternatively, perhaps I can express ( cos 2t ) as ( 1 - 2sin^2 t ):So, substituting:( 1 - 2sin^2 t - 2cos t + 2sin t = 0 )Which is similar to what I had earlier.Alternatively, maybe I can set ( s = sin t ) and ( c = cos t ), with ( s^2 + c^2 = 1 ). Then, the equation becomes:( 1 - 2s^2 - 2c + 2s = 0 )But since ( s^2 = 1 - c^2 ), substitute that in:( 1 - 2(1 - c^2) - 2c + 2s = 0 )Simplify:( 1 - 2 + 2c^2 - 2c + 2s = 0 )Which is:( -1 + 2c^2 - 2c + 2s = 0 )Hmm, still complicated.Wait, maybe I can express ( s ) in terms of ( c ) or vice versa.Alternatively, perhaps I can consider that this equation might have a solution where ( t = 45^circ ) or ( pi/4 ), since in the Lagrange multipliers method, we found that ( x = y ), which corresponds to ( theta = pi/4 ).Let me test ( theta = pi/4 ):Compute ( f'(pi/4) ):First, ( cos(2*pi/4) = cos(pi/2) = 0 )( sin(pi/4) = sqrt{2}/2 )( cos(pi/4) = sqrt{2}/2 )So,( f'(pi/4) = 4*0 + 8*(sqrt{2}/2) - 8*(sqrt{2}/2) = 0 + 4sqrt{2} - 4sqrt{2} = 0 )So, yes, ( theta = pi/4 ) is a critical point.So, that's consistent with our earlier result from the Lagrange multipliers method. So, that gives me confidence that ( x = y = sqrt{2} ) is indeed a critical point.But I also need to check if there are other critical points. Let me see.Suppose ( theta ) is such that ( f'(theta) = 0 ). So, we have:[4cos(2theta) + 8sintheta - 8costheta = 0]Let me try to see if there are other solutions besides ( theta = pi/4 ).Let me consider ( theta = 0 ):( cos(0) = 1 ), ( sin(0) = 0 )So, ( f'(0) = 4*1 + 0 - 8*1 = 4 - 8 = -4 neq 0 )Similarly, ( theta = pi/2 ):( cos(pi) = -1 ), ( sin(pi/2) = 1 )So, ( f'(pi/2) = 4*(-1) + 8*1 - 8*0 = -4 + 8 = 4 neq 0 )So, at the endpoints ( theta = 0 ) and ( theta = pi/2 ), the derivative is not zero, which makes sense because we already considered those boundary points.What about ( theta = pi/6 ) (30 degrees):( cos(pi/3) = 0.5 ), ( sin(pi/6) = 0.5 )So, ( f'(pi/6) = 4*0.5 + 8*0.5 - 8*(sqrt{3}/2) )Wait, hold on. Let me compute it correctly.Wait, ( f'(theta) = 4cos(2theta) + 8sintheta - 8costheta )So, for ( theta = pi/6 ):( cos(2*pi/6) = cos(pi/3) = 0.5 )( sin(pi/6) = 0.5 )( cos(pi/6) = sqrt{3}/2 approx 0.866 )So,( f'(pi/6) = 4*0.5 + 8*0.5 - 8*0.866 )Compute each term:- ( 4*0.5 = 2 )- ( 8*0.5 = 4 )- ( 8*0.866 approx 6.928 )So, total:( 2 + 4 - 6.928 approx -0.928 neq 0 )So, not zero.What about ( theta = pi/3 ) (60 degrees):( cos(2*pi/3) = cos(2pi/3) = -0.5 )( sin(pi/3) = sqrt{3}/2 approx 0.866 )( cos(pi/3) = 0.5 )So,( f'(pi/3) = 4*(-0.5) + 8*(0.866) - 8*(0.5) )Compute each term:- ( 4*(-0.5) = -2 )- ( 8*(0.866) approx 6.928 )- ( 8*(0.5) = 4 )Total:( -2 + 6.928 - 4 approx 0.928 neq 0 )Still not zero.Hmm, so it seems that besides ( theta = pi/4 ), there are no other critical points in the interval ( [0, pi/2] ). Therefore, the only critical point is at ( theta = pi/4 ), which corresponds to ( x = y = sqrt{2} ).So, now let me compute the value of ( f ) at this critical point and compare it with the boundary values.Earlier, I found that ( f(sqrt{2}, sqrt{2}) = -8sqrt{2} approx -11.31 ), and the boundary points gave me ( f = -10 ). So, clearly, ( -8sqrt{2} ) is less than -10, so it's the minimum.But wait, let me make sure that this is indeed the global minimum on the domain. Since the function ( f(x, y) ) is continuous on a compact set (the quarter-circle is compact), by Extreme Value Theorem, it must attain its minimum and maximum. So, we have found all critical points and boundary points, and the minimum occurs at ( (sqrt{2}, sqrt{2}) ).Therefore, the minimum value is ( -8sqrt{2} ).But just to be thorough, let me check another point on the circle to see if the function can get any lower. Let's pick ( theta = pi/8 ) (22.5 degrees):Compute ( x = 2cos(pi/8) approx 2*0.9239 = 1.8478 )Compute ( y = 2sin(pi/8) approx 2*0.3827 = 0.7654 )Compute ( f(x, y) = (1.8478)(0.7654) - 4(1.8478 + 0.7654) - 2 )Calculate each term:- ( 1.8478 * 0.7654 approx 1.4142 )- ( 4*(1.8478 + 0.7654) = 4*(2.6132) = 10.4528 )So, ( f approx 1.4142 - 10.4528 - 2 = 1.4142 - 12.4528 approx -11.0386 )Which is approximately -11.04, which is higher than ( -8sqrt{2} approx -11.31 ). So, indeed, the minimum is at ( theta = pi/4 ).Alternatively, let me pick ( theta = pi/12 ) (15 degrees):( x = 2cos(pi/12) approx 2*0.9659 = 1.9319 )( y = 2sin(pi/12) approx 2*0.2588 = 0.5176 )Compute ( f(x, y) = (1.9319)(0.5176) - 4(1.9319 + 0.5176) - 2 )Calculate each term:- ( 1.9319 * 0.5176 approx 1.000 )- ( 4*(1.9319 + 0.5176) = 4*(2.4495) = 9.798 )So, ( f approx 1.000 - 9.798 - 2 = 1.000 - 11.798 approx -10.798 )Which is approximately -10.8, still higher than ( -8sqrt{2} ).So, it seems that as we move away from ( theta = pi/4 ), the value of ( f ) increases, confirming that the minimum is indeed at ( theta = pi/4 ).Therefore, the minimum value of ( xy - 4(x + y) - 2 ) is ( -8sqrt{2} ).But just to make sure I didn't make any calculation mistakes earlier, let me recompute ( f(sqrt{2}, sqrt{2}) ):( xy = sqrt{2}*sqrt{2} = 2 )( 4(x + y) = 4(sqrt{2} + sqrt{2}) = 4*(2sqrt{2}) = 8sqrt{2} )So, ( f = 2 - 8sqrt{2} - 2 = -8sqrt{2} ). Yep, that's correct.Also, checking the derivative approach again, since ( f'(pi/4) = 0 ) and the second derivative test might help confirm if it's a minimum.Compute the second derivative ( f''(theta) ):Starting from ( f'(theta) = 4cos(2theta) + 8sintheta - 8costheta )So, ( f''(theta) = -8sin(2theta) + 8costheta + 8sintheta )At ( theta = pi/4 ):( sin(2*pi/4) = sin(pi/2) = 1 )( cos(pi/4) = sqrt{2}/2 )( sin(pi/4) = sqrt{2}/2 )So,( f''(pi/4) = -8*1 + 8*(sqrt{2}/2) + 8*(sqrt{2}/2) = -8 + 4sqrt{2} + 4sqrt{2} = -8 + 8sqrt{2} )Since ( sqrt{2} approx 1.414 ), so ( 8sqrt{2} approx 11.31 ), so ( f''(pi/4) approx -8 + 11.31 = 3.31 > 0 ). Therefore, the function is concave upward at this point, which means it's a local minimum. Since we've checked the boundaries and found higher values, this must be the global minimum.Therefore, I'm confident that the minimum value is ( -8sqrt{2} ).**Final Answer**The minimum value is boxed{-8sqrt{2}}.