answer：Okay, so I have this complex number z, which is equal to (3 + i) divided by (1 - 2i). I need to find the absolute value of its conjugate, which is written as | conjugate(z) |. Hmm, complex numbers can be a bit tricky, but let me try to break this down step by step.First, I remember that for any complex number z, the absolute value of its conjugate is the same as the absolute value of z itself. So, | conjugate(z) | is equal to |z|. That's a helpful property because it means I don't actually need to find the conjugate of z; instead, I can just find the absolute value of z directly. That should save me some time and effort.Alright, so z is given as (3 + i)/(1 - 2i). To find |z|, I can use the formula for the absolute value of a complex number. If I have a complex number in the form a + bi, its absolute value is sqrt(a² + b²). But since z is a fraction, I need to handle that carefully.I recall that when you have a complex number in the denominator, you can simplify it by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi. So, in this case, the denominator is (1 - 2i), and its conjugate would be (1 + 2i).Let me write that out:z = (3 + i)/(1 - 2i) * (1 + 2i)/(1 + 2i)Multiplying both the numerator and the denominator by (1 + 2i) should help me eliminate the imaginary unit from the denominator.Now, let's multiply out the numerators and denominators.First, the numerator: (3 + i)(1 + 2i)Using the distributive property (FOIL method):3*1 + 3*2i + i*1 + i*2i= 3 + 6i + i + 2i²I know that i² is equal to -1, so 2i² is 2*(-1) = -2.So, combining like terms:3 + 6i + i - 2= (3 - 2) + (6i + i)= 1 + 7iOkay, so the numerator simplifies to 1 + 7i.Now, the denominator: (1 - 2i)(1 + 2i)Again, using the distributive property:1*1 + 1*2i - 2i*1 - 2i*2i= 1 + 2i - 2i - 4i²Simplify the terms:1 + 2i - 2i - 4i²The 2i and -2i cancel each other out, so we're left with:1 - 4i²Again, since i² = -1, this becomes:1 - 4*(-1)= 1 + 4= 5So, the denominator simplifies to 5.Putting it all together, z = (1 + 7i)/5, which can also be written as (1/5) + (7/5)i.Now, to find |z|, which is the absolute value of z, I can use the formula for the absolute value of a complex number a + bi, which is sqrt(a² + b²).In this case, a is 1/5 and b is 7/5.So, |z| = sqrt( (1/5)² + (7/5)² )Let's compute each part:(1/5)² = 1/25(7/5)² = 49/25Adding them together:1/25 + 49/25 = 50/25 = 2So, |z| = sqrt(2)Since | conjugate(z) | is equal to |z|, that means | conjugate(z) | is also sqrt(2).Wait, let me double-check my steps to make sure I didn't make any mistakes.1. I started with z = (3 + i)/(1 - 2i).2. I multiplied numerator and denominator by the conjugate of the denominator, which is (1 + 2i).3. Expanded the numerator: (3 + i)(1 + 2i) = 3 + 6i + i + 2i² = 3 + 7i - 2 = 1 + 7i.4. Expanded the denominator: (1 - 2i)(1 + 2i) = 1 + 2i - 2i - 4i² = 1 + 4 = 5.5. So, z = (1 + 7i)/5.6. Calculated |z| = sqrt( (1/5)² + (7/5)² ) = sqrt(1/25 + 49/25) = sqrt(50/25) = sqrt(2).Everything seems to check out. I don't see any errors in my calculations. So, I'm confident that | conjugate(z) | is indeed sqrt(2).Just to recap, the key steps were recognizing that | conjugate(z) | equals |z|, simplifying the complex fraction by multiplying numerator and denominator by the conjugate of the denominator, expanding the products, simplifying, and then applying the absolute value formula. It all comes together nicely to give sqrt(2) as the final answer.I think that's it. I don't have any lingering doubts about this solution. It makes sense, and each step logically follows from the previous one. I feel good about this!

answer：Okay, so I have this problem where I need to evaluate three statements about three positive numbers a, b, and c, which are not all equal. Let me try to understand each statement one by one.First, statement ① says: (a - b)² + (b - c)² + (c - a)² ≠ 0. Hmm, I know that squares of real numbers are always non-negative, right? So each term here is either zero or positive. If all three numbers a, b, and c are equal, then each squared term would be zero, and the whole sum would be zero. But the problem says that a, b, and c are not all equal. So, at least one of the terms must be positive, which means the sum can't be zero. So, statement ① should be correct. That makes sense.Moving on to statement ②: Among a > b, a < b, and a = b, at least one is true. Hmm, this sounds familiar. I think this is related to the law of trichotomy in mathematics, which states that for any two real numbers, exactly one of the following is true: the first is greater than the second, the first is less than the second, or they are equal. Since a and b are real numbers, this should apply here. So, at least one of these three must be true. Therefore, statement ② is also correct.Now, statement ③: It is impossible for a ≠ c, b ≠ c, and a ≠ b to all be true simultaneously. Wait, so this is saying that it's impossible for all three pairs to be unequal. But hold on, if a, b, and c are all different, then a ≠ c, b ≠ c, and a ≠ b would all be true. For example, if a = 1, b = 2, c = 3, then all three inequalities hold. So, actually, it is possible for all three to be true at the same time. That means statement ③ is incorrect.Let me just recap:- Statement ①: Correct, because since not all are equal, the sum of squares can't be zero.- Statement ②: Correct, based on the law of trichotomy.- Statement ③: Incorrect, because we can have all three numbers different, making all three inequalities true.So, the correct judgments are statements ① and ②.**Final Answer**The correct judgments are boxed{①②}.

answer：Okay, so I have this problem here: Given that b is a multiple of 345, I need to find the greatest common divisor (GCD) of the polynomial 5b³ + 2b² + 7b + 69 and b. Hmm, that sounds a bit tricky, but let me try to break it down step by step.First, I remember that the GCD of two numbers is the largest number that divides both of them without leaving a remainder. In this case, the two numbers are the polynomial expression and b. Since b is a multiple of 345, that means b can be written as 345 times some integer, right? So, b = 345k, where k is an integer. That might come in handy later.Now, to find the GCD of 5b³ + 2b² + 7b + 69 and b, I think the Euclidean algorithm could be useful here. The Euclidean algorithm is a method to find the GCD of two numbers by repeatedly applying division and taking remainders. I remember that for polynomials, the same idea applies. So, let me try to apply that.Let me denote the polynomial as P(b) = 5b³ + 2b² + 7b + 69. So, I need to find GCD(P(b), b). According to the Euclidean algorithm, GCD(P(b), b) is the same as GCD(b, P(b) mod b). That makes sense because when you divide P(b) by b, the remainder will be the constant term, right?Wait, let me think about that. If I divide P(b) by b, what's the remainder? Well, when you divide a polynomial by b, the remainder is the constant term. So, in this case, the remainder should be 69. Let me check that.If I write P(b) as b*(5b² + 2b + 7) + 69, then when I divide P(b) by b, the quotient is 5b² + 2b + 7, and the remainder is 69. So, yes, that's correct. Therefore, P(b) mod b is 69. So, GCD(P(b), b) is the same as GCD(b, 69).Alright, so now I need to find GCD(b, 69). But I know that b is a multiple of 345, so b = 345k. Therefore, GCD(345k, 69). Hmm, I need to find the GCD of 345k and 69. I remember that GCD(mk, n) is equal to GCD(m, n) times k if k is a multiple of GCD(m, n). Wait, no, that's not quite right. Let me think again. Actually, GCD(mk, n) is equal to GCD(m, n/k) if k divides n. Hmm, maybe I'm complicating things.Alternatively, since 345 is a multiple of 69, because 345 divided by 69 is 5. Let me check that: 69 times 5 is 345. Yes, that's correct. So, 345 is 5 times 69. Therefore, b = 345k = 5*69*k. So, b is 5*69*k, which means that 69 is a factor of b.Therefore, GCD(b, 69) is 69 because 69 divides b. So, the GCD of P(b) and b is 69. Wait, let me make sure I didn't skip any steps. So, starting from GCD(P(b), b), we reduced it to GCD(b, 69). Since b is a multiple of 345, and 345 is a multiple of 69, then 69 divides b. Therefore, the GCD is 69. That seems solid.But just to double-check, let me consider an example. Let's pick b = 345, which is the smallest multiple of 345. Then, P(b) = 5*(345)^3 + 2*(345)^2 + 7*(345) + 69. That's a huge number, but I don't need to compute it exactly. Instead, I can compute P(b) mod 345 to see if it's 69, which it should be.Wait, actually, since we already established that P(b) mod b is 69, and b is 345, then P(b) mod 345 is 69. So, GCD(P(b), 345) is GCD(69, 345). Since 69 divides 345, as we saw earlier, GCD(69, 345) is 69. So, that checks out.What if I pick another multiple, say b = 690, which is 2*345. Then, P(b) = 5*(690)^3 + 2*(690)^2 + 7*(690) + 69. Again, P(b) mod 690 is 69, so GCD(P(b), 690) is GCD(69, 690). Since 69 divides 690 (690 = 10*69), GCD(69, 690) is 69. So, that also works.Therefore, regardless of the multiple k, as long as b is a multiple of 345, GCD(P(b), b) is 69. I think that solidifies the conclusion.Just to recap, the steps were:1. Recognize that GCD(P(b), b) can be simplified using the Euclidean algorithm to GCD(b, 69).2. Since b is a multiple of 345, and 345 is a multiple of 69, GCD(b, 69) is 69.3. Verified with specific examples to ensure the reasoning holds.So, I feel confident that the greatest common divisor is 69.

answer：Okay, so I need to find a monic quadratic polynomial f(x) such that f(0) equals 6 and f(1) equals 12. Hmm, let me think about how to approach this.First, I remember that a quadratic polynomial generally looks like f(x) = ax² + bx + c. Since it's monic, that means the coefficient of x² is 1, right? So, a = 1. That simplifies things a bit. So now, the polynomial is f(x) = x² + bx + c.Next, I need to use the given conditions to find the values of b and c. The first condition is f(0) = 6. Let me substitute x = 0 into the polynomial:f(0) = (0)² + b*(0) + c = 0 + 0 + c = c.So, f(0) = c, and we know that equals 6. Therefore, c = 6. That was straightforward.Now, the second condition is f(1) = 12. Let me substitute x = 1 into the polynomial:f(1) = (1)² + b*(1) + c = 1 + b + c.We know that f(1) is 12, so:1 + b + c = 12.But we already found that c = 6, so I can substitute that in:1 + b + 6 = 12.Simplifying that:7 + b = 12.Subtracting 7 from both sides:b = 12 - 7 = 5.So, b equals 5. Now, putting it all together, the polynomial is:f(x) = x² + 5x + 6.Let me double-check to make sure I didn't make any mistakes. When x = 0, f(0) should be 6, and indeed, plugging in 0 gives 0 + 0 + 6 = 6. When x = 1, f(1) should be 12, and plugging in 1 gives 1 + 5 + 6 = 12. That checks out.I think that's it. The polynomial is x² + 5x + 6.

answer：Okay, so I have this problem where a line intersects a circle at two points, A and B, and the triangle formed by these two points and the center of the circle is an isosceles right triangle. I need to find the value of 'a'. Hmm, let me try to break this down step by step.First, let me write down the given equations to have a clear picture.The line is given by:[ ax + y - 1 = 0 ]And the circle is:[ (x - 1)^2 + (y + a)^2 = 1 ]So, the circle has its center at (1, -a) and a radius of 1 because the equation is in the standard form ((x - h)^2 + (y - k)^2 = r^2).Now, the line intersects the circle at points A and B. The triangle ABC, where C is the center of the circle, is an isosceles right triangle. That means two sides are equal, and one angle is 90 degrees. Since it's an isosceles right triangle, the two legs are equal, and the hypotenuse is ( sqrt{2} ) times longer than each leg.So, in triangle ABC, either AC = BC, or AB = AC, or AB = BC. But since C is the center of the circle, AC and BC are both radii of the circle. Wait, that's right! The radius of the circle is 1, so AC = BC = 1. Therefore, triangle ABC has two sides of length 1, and the third side AB is the hypotenuse.But wait, if AC and BC are both 1, then triangle ABC is an isosceles right triangle with legs of length 1. That would mean that the hypotenuse AB should be ( sqrt{2} ). So, the distance between points A and B should be ( sqrt{2} ).But how do I find the distance between A and B? Well, since A and B lie on both the line and the circle, I can find their coordinates by solving the system of equations given by the line and the circle. Then, I can compute the distance between A and B and set it equal to ( sqrt{2} ) to solve for 'a'.Alternatively, maybe there's a smarter way using the distance from the center to the line. Let me recall that the length of the chord (which is AB in this case) can be found using the formula:[ text{Length of chord} = 2 sqrt{r^2 - d^2} ]where 'r' is the radius of the circle and 'd' is the distance from the center to the line.Given that the length of AB is ( sqrt{2} ), we can set up the equation:[ sqrt{2} = 2 sqrt{1 - d^2} ]Let me solve this for 'd'.Divide both sides by 2:[ frac{sqrt{2}}{2} = sqrt{1 - d^2} ]Square both sides:[ left( frac{sqrt{2}}{2} right)^2 = 1 - d^2 ][ frac{2}{4} = 1 - d^2 ][ frac{1}{2} = 1 - d^2 ]Subtract 1 from both sides:[ -frac{1}{2} = -d^2 ]Multiply both sides by -1:[ frac{1}{2} = d^2 ]Take square roots:[ d = frac{sqrt{2}}{2} ]So, the distance from the center of the circle to the line is ( frac{sqrt{2}}{2} ).Now, let's find the distance from the center (1, -a) to the line ( ax + y - 1 = 0 ). The formula for the distance from a point (x₀, y₀) to the line ( Ax + By + C = 0 ) is:[ d = frac{|Ax₀ + By₀ + C|}{sqrt{A^2 + B^2}} ]In our case, the line is ( ax + y - 1 = 0 ), so A = a, B = 1, and C = -1. The point is (1, -a). Plugging into the formula:[ d = frac{|a(1) + 1(-a) - 1|}{sqrt{a^2 + 1^2}} ]Simplify the numerator:[ |a - a - 1| = |-1| = 1 ]So, the distance is:[ d = frac{1}{sqrt{a^2 + 1}} ]But we know from earlier that this distance should be ( frac{sqrt{2}}{2} ). So, set them equal:[ frac{1}{sqrt{a^2 + 1}} = frac{sqrt{2}}{2} ]Let me solve for 'a'. First, take reciprocals on both sides:[ sqrt{a^2 + 1} = frac{2}{sqrt{2}} ]Simplify ( frac{2}{sqrt{2}} ):[ frac{2}{sqrt{2}} = sqrt{2} ]So,[ sqrt{a^2 + 1} = sqrt{2} ]Square both sides:[ a^2 + 1 = 2 ]Subtract 1:[ a^2 = 1 ]Take square roots:[ a = pm 1 ]So, the possible values for 'a' are 1 and -1. Let me check the options given:A: 1/7 or -1B: -1C: 1 or -1D: 1So, the correct answer is option C: 1 or -1.Wait, just to make sure I didn't make any mistakes. Let me recap:1. The triangle is isosceles right-angled, so sides AC and BC are equal (both radii of the circle, so length 1).2. Therefore, AB must be the hypotenuse, length ( sqrt{2} ).3. The length of chord AB is related to the distance from the center to the line by the formula ( 2sqrt{r^2 - d^2} ).4. Plugging in, we found that the distance 'd' must be ( frac{sqrt{2}}{2} ).5. Calculated the distance from center (1, -a) to the line, got ( frac{1}{sqrt{a^2 + 1}} ).6. Set equal to ( frac{sqrt{2}}{2} ), solved for 'a' and got ( a = pm1 ).Everything seems to check out. So, yeah, the answer is C.**Final Answer**boxed{C}

answer：Let's denote the total value of the estate as ( E ).1. **Sons' Share:** - The two sons together receive ( frac{3}{5} ) of the estate. - Let ( b ) be the amount received by the younger son. - The elder son receives twice as much as the younger son, so the elder son gets ( 2b ). - Therefore, ( 2b + b = frac{3}{5}E ) which simplifies to ( 3b = frac{3}{5}E ) and hence ( b = frac{1}{5}E ).2. **Husband's Share:** - The husband receives three times the amount that the younger son gets, so the husband's share is ( 3b = 3 cdot frac{1}{5}E = frac{3}{5}E ).3. **Total Estate Calculation:** - The total estate is the sum of the shares of the two sons, the husband, and the charity. - Therefore, ( E = frac{2}{5}E + frac{1}{5}E + frac{3}{5}E + 4000 ). - Simplifying, ( E = frac{6}{5}E + 4000 ). - Rearranging, ( frac{-1}{5}E = -4000 ) which leads to ( E = 20000 ).Thus, the total value of the estate is ( 20000 ).