💡Alright, so I have this problem here: I need to find the coefficient of ( x^3 y^3 ) in the expansion of ( left(x + frac{y^2}{x}right)(x + y)^5 ). The options are A: 5, B: 10, C: 15, D: 20. Hmm, okay, let's try to figure this out step by step.First, I remember that when expanding expressions like ( (x + y)^n ), we can use the binomial theorem. The binomial theorem says that ( (x + y)^n = sum_{k=0}^{n} binom{n}{k} x^{n - k} y^k ). So, in this case, ( (x + y)^5 ) would expand to ( sum_{k=0}^{5} binom{5}{k} x^{5 - k} y^k ).Now, the given expression is ( left(x + frac{y^2}{x}right)(x + y)^5 ). I think I should first expand ( (x + y)^5 ) and then multiply it by ( x + frac{y^2}{x} ). Alternatively, maybe I can distribute ( x + frac{y^2}{x} ) over ( (x + y)^5 ) without expanding the entire thing. Let me think about that.If I distribute, it would be ( x cdot (x + y)^5 + frac{y^2}{x} cdot (x + y)^5 ). So, I have two separate terms to consider: one is ( x cdot (x + y)^5 ) and the other is ( frac{y^2}{x} cdot (x + y)^5 ). Maybe I can find the coefficient of ( x^3 y^3 ) in each of these terms separately and then add them together.Let me start with the first term: ( x cdot (x + y)^5 ). If I expand ( (x + y)^5 ), as I mentioned earlier, it's ( sum_{k=0}^{5} binom{5}{k} x^{5 - k} y^k ). Then, multiplying by ( x ) would give me ( sum_{k=0}^{5} binom{5}{k} x^{6 - k} y^k ). So, in this expansion, each term has an exponent of ( x ) increased by 1 compared to the original expansion.Now, I need the term where the exponent of ( x ) is 3 and the exponent of ( y ) is 3. So, let's set up the equation: ( 6 - k = 3 ) for the exponent of ( x ), and ( k = 3 ) for the exponent of ( y ). Solving ( 6 - k = 3 ) gives ( k = 3 ). So, the coefficient for this term is ( binom{5}{3} ).Calculating ( binom{5}{3} ), that's ( frac{5!}{3!2!} = frac{120}{6 times 2} = 10 ). So, the coefficient from the first term is 10.Now, moving on to the second term: ( frac{y^2}{x} cdot (x + y)^5 ). Similarly, expanding ( (x + y)^5 ) gives ( sum_{k=0}^{5} binom{5}{k} x^{5 - k} y^k ). Multiplying this by ( frac{y^2}{x} ) would give ( sum_{k=0}^{5} binom{5}{k} x^{4 - k} y^{k + 2} ).Again, I need the term where the exponent of ( x ) is 3 and the exponent of ( y ) is 3. So, setting up the equations: ( 4 - k = 3 ) for ( x ) and ( k + 2 = 3 ) for ( y ). Solving ( 4 - k = 3 ) gives ( k = 1 ). Checking ( k + 2 = 3 ) also gives ( k = 1 ). So, the coefficient here is ( binom{5}{1} ).Calculating ( binom{5}{1} ), that's simply 5. So, the coefficient from the second term is 5.Now, to find the total coefficient of ( x^3 y^3 ) in the original expression, I need to add the coefficients from both terms. That would be 10 (from the first term) plus 5 (from the second term), which equals 15.Therefore, the coefficient of ( x^3 y^3 ) is 15, which corresponds to option C.Wait, let me double-check to make sure I didn't make a mistake. For the first term, I had ( x cdot (x + y)^5 ), which led to ( x^{6 - k} y^k ). Setting ( 6 - k = 3 ) gives ( k = 3 ), and ( binom{5}{3} = 10 ). That seems correct.For the second term, ( frac{y^2}{x} cdot (x + y)^5 ), which gave ( x^{4 - k} y^{k + 2} ). Setting ( 4 - k = 3 ) gives ( k = 1 ), and ( binom{5}{1} = 5 ). That also seems correct.Adding them together: 10 + 5 = 15. Yeah, that seems right. So, I think the answer is C: 15.**Final Answer**boxed{C}

💡Okay, so I have this function f(x) defined piecewise. For x less than or equal to a, it's a quadratic function: -(x - a)^2 - k - a. And for x greater than a, it's an exponential function divided by (a - x): e^x / (a - x). The question is asking about the range of the real number k such that there exists some x0 in (-∞, a] where for all x1 in (a, ∞), f(x1) is less than or equal to f(x0). The options are A: (-∞,1), B: [1, ∞), C: (-∞,2], D: [2, ∞).Alright, let me try to break this down. So, we need to find the range of k such that the maximum value of f(x1) for x1 > a is less than or equal to some value f(x0) where x0 is in (-∞, a]. So, essentially, the maximum of the right piece of the function (x > a) should be less than or equal to the maximum of the left piece (x ≤ a).First, let me analyze both pieces of the function separately.Starting with the left piece: f(x) = -(x - a)^2 - k - a for x ≤ a. This is a downward-opening parabola because of the negative sign. The vertex of this parabola is at x = a, and the value at the vertex is f(a) = -(0)^2 - k - a = -k - a. Since it's a downward-opening parabola, this vertex is the maximum point on the left side. So, the maximum value of f(x) on (-∞, a] is -k - a.Now, moving on to the right piece: f(x) = e^x / (a - x) for x > a. Hmm, this looks a bit more complicated. Let me see if I can find its maximum. To find the maximum, I should take the derivative and set it equal to zero.So, let's compute f'(x). The function is e^x divided by (a - x). Using the quotient rule: if f(x) = u/v, then f'(x) = (u'v - uv') / v^2.Here, u = e^x, so u' = e^x. v = a - x, so v' = -1.Therefore, f'(x) = [e^x*(a - x) - e^x*(-1)] / (a - x)^2.Simplify the numerator: e^x*(a - x) + e^x = e^x*(a - x + 1) = e^x*(a + 1 - x).So, f'(x) = e^x*(a + 1 - x) / (a - x)^2.To find critical points, set f'(x) = 0. Since e^x is always positive and (a - x)^2 is always positive (as it's squared), the only way for f'(x) to be zero is if the numerator is zero. So, set a + 1 - x = 0, which gives x = a + 1.So, the critical point is at x = a + 1. Now, we should check if this is a maximum or a minimum. Let's test the sign of f'(x) around x = a + 1.For x < a + 1, say x = a, then f'(x) = e^a*(a + 1 - a) / (a - a)^2. Wait, but x can't be equal to a because the function is defined differently there. Let me pick a point slightly less than a + 1, say x = a + 0.5. Then, a + 1 - x = a + 1 - (a + 0.5) = 0.5, which is positive. So, f'(x) is positive here, meaning the function is increasing.For x > a + 1, say x = a + 2. Then, a + 1 - x = a + 1 - (a + 2) = -1, which is negative. So, f'(x) is negative here, meaning the function is decreasing.Therefore, x = a + 1 is a local maximum. So, the maximum value of f(x) on (a, ∞) is f(a + 1).Let's compute f(a + 1): e^(a + 1) / (a - (a + 1)) = e^(a + 1) / (-1) = -e^(a + 1).So, the maximum value of the right piece is -e^(a + 1).Now, according to the problem, there exists an x0 in (-∞, a] such that for all x1 in (a, ∞), f(x1) ≤ f(x0). Since the maximum of f(x1) is -e^(a + 1), we need this maximum to be less than or equal to some f(x0). But the maximum of f(x0) on (-∞, a] is -k - a, as we found earlier.Therefore, we need:-e^(a + 1) ≤ -k - aLet me rewrite this inequality:-e^(a + 1) ≤ -k - aMultiply both sides by -1, which reverses the inequality:e^(a + 1) ≥ k + aSo,k + a ≤ e^(a + 1)Therefore,k ≤ e^(a + 1) - aSo, k must be less than or equal to e^(a + 1) - a.Now, the question is, what is the range of k? But k is a real number, so we need to find the possible values of k such that this inequality holds for some a. Wait, but a is a parameter here, right? Or is a a variable? Wait, in the function f(x), a is a constant, right? Because the function is defined in terms of a. So, is a given? Or is a a variable as well?Wait, the problem says "the range of the real number k", so I think a is a parameter, and we need to find k such that for some a, the inequality holds. Wait, but the problem doesn't specify a. Hmm, maybe I need to consider a as a variable as well.Wait, let me reread the problem."The function f(x) is defined as... if there exists x0 in (-∞, a] such that for all x1 in (a, ∞), f(x1) ≤ f(x0), then the range of the real number k is..."So, it's saying that for this function f(x), which depends on a and k, there exists an x0 in (-∞, a] such that for all x1 in (a, ∞), f(x1) ≤ f(x0). So, we need to find the range of k such that this condition holds for some a.Wait, but the function is defined for a specific a, right? So, is a given? Or is a a variable? Hmm.Wait, the problem is asking for the range of k, so maybe a is given? Or perhaps, since a is in the function, it's a parameter, and we have to find k such that for some a, the condition holds. Hmm, I'm a bit confused.Wait, maybe the problem is that a is a fixed constant, and we have to find k such that for this a, the condition holds. But the problem doesn't specify a, so maybe we need to find k such that for all a, the condition holds? Or for some a?Wait, the problem says "if there exists x0 in (-∞, a] such that for all x1 in (a, ∞), f(x1) ≤ f(x0)", so it's for a specific a, right? So, for a given a, we can find x0 such that f(x1) ≤ f(x0) for all x1 > a. So, the condition is for a specific a, and we need to find the range of k.But the problem is asking for the range of k, so maybe a is a fixed constant, but since it's not given, perhaps we have to find k such that for all a, this condition holds? Or maybe for some a?Wait, the wording is a bit ambiguous. Let me see."If there exists x0 in (-∞, a] such that for all x1 in (a, ∞), f(x1) ≤ f(x0), then the range of the real number k is..."So, it's conditional: if such an x0 exists, then k is in which range. So, it's saying that given that such an x0 exists, what can we say about k.So, in other words, for the function f(x), if there exists an x0 in (-∞, a] such that f(x1) ≤ f(x0) for all x1 > a, then find the range of k.So, in this case, a is fixed because it's part of the function definition. So, a is a constant, and we need to find k such that the condition holds.But the problem is, a isn't given, so maybe we have to find k such that for all a, the condition holds? Or for some a?Wait, the problem is a bit unclear. Let me think.Wait, the function is defined as f(x) with a specific a, so a is a parameter. So, the problem is saying that for this function f(x), which depends on a and k, if there exists an x0 in (-∞, a] such that for all x1 in (a, ∞), f(x1) ≤ f(x0), then find the range of k.So, for a given a, find k such that the maximum of f(x1) is less than or equal to the maximum of f(x0). But since a is part of the function, it's a parameter, so maybe we need to find k in terms of a, but the problem is asking for the range of k without any reference to a, so perhaps we need to find k such that for all a, the condition holds? Or maybe for some a.Wait, the problem is a bit confusing because it doesn't specify whether a is given or not. Maybe I need to assume that a is a fixed constant, and we have to find k in terms of a, but the answer choices are numerical ranges, not in terms of a. So, perhaps a is a specific value?Wait, no, the answer choices are numerical, so maybe a is a specific value that we can determine.Wait, let me think again.We have the condition that -e^(a + 1) ≤ -k - a, which simplifies to k ≤ e^(a + 1) - a.So, k must be less than or equal to e^(a + 1) - a.But the problem is asking for the range of k, so we need to find the possible values of k such that this inequality holds for some a.Wait, but if a can be any real number, then e^(a + 1) - a can take various values. So, perhaps we need to find the minimum value of e^(a + 1) - a over all real numbers a, and then k must be less than or equal to that minimum value.Because if k is less than or equal to the minimum of e^(a + 1) - a, then for some a, k will satisfy the inequality.Wait, let me see.Let me define h(a) = e^(a + 1) - a. We need to find the minimum value of h(a) over all real numbers a.To find the minimum, take the derivative of h(a) with respect to a:h'(a) = d/da [e^(a + 1) - a] = e^(a + 1) - 1.Set h'(a) = 0:e^(a + 1) - 1 = 0 => e^(a + 1) = 1 => a + 1 = ln(1) => a + 1 = 0 => a = -1.So, the critical point is at a = -1.Now, check the second derivative to confirm if it's a minimum.h''(a) = d/da [e^(a + 1) - 1] = e^(a + 1), which is always positive because e^(anything) is positive. Therefore, the function h(a) has a minimum at a = -1.Compute h(-1):h(-1) = e^(-1 + 1) - (-1) = e^0 + 1 = 1 + 1 = 2.So, the minimum value of h(a) is 2.Therefore, h(a) ≥ 2 for all real numbers a.So, since k ≤ h(a) for some a, and h(a) can be as small as 2, the maximum lower bound for k is 2. Therefore, k must be less than or equal to 2.Wait, but hold on. The condition is k ≤ h(a) for some a. Since h(a) can be as small as 2, the maximum value that k can take is 2. Because if k is greater than 2, then h(a) is always greater than or equal to 2, so k cannot be greater than 2.Wait, no. Let me clarify.We have k ≤ h(a) for some a. Since h(a) can be as small as 2, k must be less than or equal to 2. Because if k is greater than 2, then h(a) is always at least 2, so k would not be less than or equal to h(a) for any a.Wait, actually, no. If k is less than or equal to 2, then since h(a) is always at least 2, k ≤ h(a) is always true. But the problem says "if there exists x0 in (-∞, a] such that for all x1 in (a, ∞), f(x1) ≤ f(x0)", which translates to k ≤ h(a) for some a.But wait, actually, for the condition to hold, we need that for some a, k ≤ h(a). But h(a) can be as small as 2, so k must be less than or equal to 2 to satisfy k ≤ h(a) for some a.Wait, I'm getting confused.Let me think differently. The condition is that for some a, k ≤ h(a). Since h(a) can be as small as 2, the maximum possible k that satisfies k ≤ h(a) for some a is k ≤ 2. Because if k is greater than 2, then h(a) is always at least 2, so k cannot be less than or equal to h(a) for any a.Wait, no, that's not correct. If k is greater than 2, then h(a) is always at least 2, so k would not be less than or equal to h(a) for any a. Therefore, k must be less than or equal to 2.But wait, the problem is asking for the range of k such that there exists an x0 in (-∞, a] such that for all x1 in (a, ∞), f(x1) ≤ f(x0). So, the condition is k ≤ h(a) for some a. Since h(a) can be as small as 2, the maximum k can be is 2. Therefore, k must be less than or equal to 2.But wait, let me think again. If k is less than or equal to 2, then for a = -1, h(a) = 2, so k ≤ 2 satisfies k ≤ h(a). For other values of a, h(a) is larger than 2, so k ≤ h(a) is automatically satisfied. Therefore, the range of k is (-∞, 2].Wait, but the options are A: (-∞,1), B: [1, ∞), C: (-∞,2], D: [2, ∞).So, the correct answer should be C: (-∞,2].But wait, let me make sure.Wait, in the initial analysis, I concluded that k must be less than or equal to h(a) for some a, and h(a) has a minimum of 2, so k must be less than or equal to 2. Therefore, the range of k is (-∞,2], which is option C.Yes, that makes sense.So, the answer is C.

💡Okay, so I have this problem about finding the equation of a parabola given some conditions. Let me try to understand it step by step.First, the problem gives me two lines: ( l_1 ) is ( 4x - 3y + 6 = 0 ) and ( l_2 ) is ( x = -frac{p}{2} ). It also mentions a parabola ( C ) with the equation ( y^2 = 2px ) where ( p > 0 ). The key condition is that the minimum sum of distances from any point on this parabola to the two lines ( l_1 ) and ( l_2 ) is 2. I need to find the equation of parabola ( C ) from the given options.Alright, let's break this down. I know that for a parabola ( y^2 = 4ax ), the standard form is similar, but here it's ( y^2 = 2px ). So, comparing to the standard form, ( 4a = 2p ) which means ( a = frac{p}{2} ). So, the focus of this parabola would be at ( (a, 0) = (frac{p}{2}, 0) ) and the directrix is ( x = -frac{p}{2} ). Wait, that's interesting because ( l_2 ) is given as ( x = -frac{p}{2} ), which is exactly the directrix of the parabola. So, ( l_2 ) is the directrix of the parabola ( C ).Now, the other line ( l_1 ) is ( 4x - 3y + 6 = 0 ). I need to find the sum of distances from a point on the parabola to these two lines and then find the minimum value of this sum, which is given as 2.Let me recall that for any point on a parabola, the distance to the focus is equal to the distance to the directrix. But in this problem, we are dealing with distances to two different lines: one is the directrix ( l_2 ), and the other is an arbitrary line ( l_1 ). So, it's not directly the definition of a parabola, but we can use the distance formulas.Let me denote a general point on the parabola ( C ) as ( (x, y) ). Since ( y^2 = 2px ), I can express ( x ) in terms of ( y ): ( x = frac{y^2}{2p} ). Alternatively, I can parametrize the parabola. Maybe using parametric equations would be easier here.For a parabola ( y^2 = 4ax ), the parametric equations are ( x = at^2 ) and ( y = 2at ). In our case, comparing to ( y^2 = 2px ), we have ( 4a = 2p ) so ( a = frac{p}{2} ). Therefore, the parametric equations would be ( x = frac{p}{2} t^2 ) and ( y = 2 cdot frac{p}{2} t = pt ). So, a general point on the parabola can be written as ( left( frac{p}{2} t^2, pt right) ).Alternatively, sometimes people use ( x = frac{y^2}{2p} ) as the parametric form, but using a parameter ( t ) might make differentiation easier when finding the minimum.So, let's stick with the parametric form: ( x = frac{p}{2} t^2 ), ( y = pt ).Now, I need to find the distance from this point ( ( frac{p}{2} t^2, pt ) ) to both lines ( l_1 ) and ( l_2 ), sum them, and then find the minimum of this sum.First, let's find the distance to ( l_2 ): ( x = -frac{p}{2} ). The distance from a point ( (x, y) ) to the vertical line ( x = c ) is ( |x - c| ). So, in this case, the distance ( d_2 ) is:( d_2 = left| frac{p}{2} t^2 - left( -frac{p}{2} right) right| = left| frac{p}{2} t^2 + frac{p}{2} right| = frac{p}{2} t^2 + frac{p}{2} ).Since ( p > 0 ) and ( t^2 ) is always non-negative, this simplifies to ( d_2 = frac{p}{2} t^2 + frac{p}{2} ).Next, let's find the distance to ( l_1 ): ( 4x - 3y + 6 = 0 ). The formula for the distance from a point ( (x, y) ) to the line ( ax + by + c = 0 ) is ( frac{|ax + by + c|}{sqrt{a^2 + b^2}} ).So, substituting our point into this formula:( d_1 = frac{ |4 cdot frac{p}{2} t^2 - 3 cdot pt + 6| }{ sqrt{4^2 + (-3)^2} } = frac{ |2p t^2 - 3pt + 6| }{5} ).So, now, the sum of the distances ( d_1 + d_2 ) is:( d_1 + d_2 = frac{ |2p t^2 - 3pt + 6| }{5} + frac{p}{2} t^2 + frac{p}{2} ).Our goal is to find the minimum value of this expression with respect to ( t ), and set that minimum equal to 2, then solve for ( p ).Hmm, dealing with absolute values can complicate things. Let me see if I can figure out when the expression inside the absolute value is positive or negative.Let me denote the numerator inside the absolute value as ( N(t) = 2p t^2 - 3pt + 6 ).I need to analyze the sign of ( N(t) ). Let's compute its discriminant:Discriminant ( D = (-3p)^2 - 4 cdot 2p cdot 6 = 9p^2 - 48p ).So, ( D = 9p^2 - 48p = 3p(3p - 16) ).Since ( p > 0 ), the discriminant is positive when ( 3p - 16 > 0 ) i.e., ( p > frac{16}{3} approx 5.33 ). So, if ( p > frac{16}{3} ), ( N(t) ) has two real roots, meaning it can be positive or negative depending on ( t ). If ( p leq frac{16}{3} ), the quadratic ( N(t) ) is always positive because the discriminant is negative or zero, so the quadratic doesn't cross the t-axis.Given that the options for ( p ) are such that ( y^2 = 2px ) corresponds to ( p = frac{1}{6} ) (option A), ( p = frac{1}{4} ) (option B), ( p = 1 ) (option C), and ( p = 2 ) (option D). So, all these ( p ) values are less than ( frac{16}{3} ), which is approximately 5.33. Therefore, for all these cases, ( N(t) ) is always positive because the discriminant is negative. So, the absolute value can be removed without changing the sign:( |2p t^2 - 3pt + 6| = 2p t^2 - 3pt + 6 ).Therefore, the sum ( d_1 + d_2 ) simplifies to:( d_1 + d_2 = frac{2p t^2 - 3pt + 6}{5} + frac{p}{2} t^2 + frac{p}{2} ).Let me combine these terms:First, expand the first term:( frac{2p t^2}{5} - frac{3pt}{5} + frac{6}{5} ).Then, add the second term:( + frac{p}{2} t^2 + frac{p}{2} ).So, combining like terms:For ( t^2 ):( frac{2p}{5} t^2 + frac{p}{2} t^2 = left( frac{4p}{10} + frac{5p}{10} right) t^2 = frac{9p}{10} t^2 ).For ( t ):( - frac{3p}{5} t ).Constants:( frac{6}{5} + frac{p}{2} ).So, putting it all together:( d_1 + d_2 = frac{9p}{10} t^2 - frac{3p}{5} t + frac{6}{5} + frac{p}{2} ).Now, this is a quadratic function in terms of ( t ). To find its minimum, we can use calculus or complete the square. Since it's quadratic, the minimum occurs at ( t = -frac{b}{2a} ), where ( a = frac{9p}{10} ) and ( b = -frac{3p}{5} ).Calculating the value of ( t ) where the minimum occurs:( t = -frac{ -frac{3p}{5} }{ 2 cdot frac{9p}{10} } = frac{ frac{3p}{5} }{ frac{18p}{10} } = frac{3p}{5} cdot frac{10}{18p} = frac{30p}{90p} = frac{1}{3} ).So, the minimum occurs at ( t = frac{1}{3} ).Now, let's plug ( t = frac{1}{3} ) back into the expression for ( d_1 + d_2 ):First, compute each term:1. ( frac{9p}{10} t^2 = frac{9p}{10} cdot left( frac{1}{3} right)^2 = frac{9p}{10} cdot frac{1}{9} = frac{p}{10} ).2. ( -frac{3p}{5} t = -frac{3p}{5} cdot frac{1}{3} = -frac{p}{5} ).3. Constants: ( frac{6}{5} + frac{p}{2} ).So, adding them up:( frac{p}{10} - frac{p}{5} + frac{6}{5} + frac{p}{2} ).Let me combine the ( p ) terms:( frac{p}{10} - frac{2p}{10} + frac{5p}{10} = left( frac{1 - 2 + 5}{10} right) p = frac{4p}{10} = frac{2p}{5} ).So, the total sum is:( frac{2p}{5} + frac{6}{5} ).According to the problem, this minimum sum is equal to 2. So, we set up the equation:( frac{2p}{5} + frac{6}{5} = 2 ).Let's solve for ( p ):Multiply both sides by 5:( 2p + 6 = 10 ).Subtract 6:( 2p = 4 ).Divide by 2:( p = 2 ).So, ( p = 2 ). Therefore, the equation of the parabola ( C ) is ( y^2 = 2px = 2 cdot 2x = 4x ). So, the equation is ( y^2 = 4x ), which corresponds to option D.Let me double-check my steps to make sure I didn't make any mistakes.1. I parametrized the parabola correctly as ( x = frac{p}{2} t^2 ), ( y = pt ). That seems right.2. Calculated the distance to ( l_2 ): Correct, since ( l_2 ) is the directrix, the distance is ( frac{p}{2} t^2 + frac{p}{2} ).3. Calculated the distance to ( l_1 ): Applied the distance formula correctly, substituted the parametric coordinates.4. Removed the absolute value because the discriminant was negative for the given options, which is correct since ( p ) in the options is less than ( frac{16}{3} ).5. Combined the terms correctly: Yes, the coefficients for ( t^2 ), ( t ), and constants were correctly calculated.6. Found the minimum by taking ( t = -b/(2a) ): Correct, and the calculation led to ( t = 1/3 ).7. Plugged back ( t = 1/3 ) into the expression: Correct, each term was computed accurately.8. Solved for ( p ): Correct, got ( p = 2 ).So, all steps seem correct, and the final equation is ( y^2 = 4x ), which is option D.**Final Answer**boxed{D}

💡To determine which of the given options is a pure imaginary number, let's analyze each one:**Option A: ( i^2 )**- ( i^2 = -1 ), which is a real number.**Option B: ( pi )**- ( pi ) is a well-known real number.**Option C: ( 1 + sqrt{3}i )**- This is a complex number with both real and imaginary parts.**Option D: ( (1 + sqrt{3})i )**- This can be rewritten as ( 0 + (1 + sqrt{3})i ), which fits the definition of a pure imaginary number since it has no real part.Therefore, the correct answer is **D**.

💡Okay, so I have this problem here: "sinA = sinB" is a condition for "A = B" to be (　　). The options are A: Sufficient but not necessary, B: Necessary but not sufficient, C: Both necessary and sufficient, D: Neither sufficient nor necessary.Hmm, let me try to understand what this is asking. It's about the relationship between the equality of two angles, A and B, and the equality of their sines. So, if sinA equals sinB, does that mean A must equal B? Or is there more to it?First, I know that the sine function is periodic. That means sinA = sinB doesn't necessarily imply that A and B are the same angle. For example, sin(0) = sin(π) = 0, but 0 and π are not equal. So, just because their sines are equal doesn't mean the angles themselves are equal. That suggests that sinA = sinB is not a sufficient condition for A = B because it doesn't guarantee that A and B are the same.But wait, if A does equal B, then certainly sinA would equal sinB. So, if A = B, then sinA = sinB must hold true. That means that sinA = sinB is a necessary condition for A = B. Because without sinA = sinB, A and B can't be equal. But as I thought earlier, sinA = sinB alone isn't enough to conclude that A = B.So, putting it together: sinA = sinB is necessary for A = B because if A and B are equal, their sines must be equal. But it's not sufficient because there are cases where sinA = sinB but A ≠ B.Let me think of another example to confirm. If A is 30 degrees, then sinA is 0.5. If B is 150 degrees, sinB is also 0.5. So, sinA = sinB, but A ≠ B. This shows that even though their sines are equal, the angles themselves aren't. So, sinA = sinB doesn't guarantee A = B, which reinforces that it's not a sufficient condition.On the flip side, if A = B, then sinA must equal sinB. So, it's a necessary condition because equality of angles ensures equality of their sines. But as we saw, the reverse isn't true.Therefore, sinA = sinB is a necessary condition for A = B, but it's not sufficient. So, the correct answer should be option B: Necessary but not sufficient.I think that makes sense. Let me just recap to make sure I didn't miss anything. If two angles are equal, their sines are equal—that's necessity. But equal sines don't necessarily mean the angles are equal because sine is periodic and has the same value at multiple angles. So, it's necessary but not sufficient. Yeah, that seems right.I don't think I made any mistakes here. I considered both directions: if A = B, then sinA = sinB, which is necessity. And if sinA = sinB, A doesn't have to equal B, which shows it's not sufficient. So, yeah, option B is the one.**Final Answer**boxed{B}

💡Alright, so I've got this problem here about four numbers on a board, none of which are zero. The condition is that when each number is multiplied by the sum of the other three, all four results are the same. I need to prove that the squares of these numbers are equal. Hmm, okay, let me try to unpack this step by step.First, let's denote the four numbers as ( a, b, c, ) and ( d ). The problem states that multiplying each number by the sum of the other three gives the same result. So, mathematically, that would mean:[a(b + c + d) = b(a + c + d) = c(a + b + d) = d(a + b + c)]Alright, so all four expressions are equal. Let's pick the first two and set them equal to each other:[a(b + c + d) = b(a + c + d)]Expanding both sides:[ab + ac + ad = ba + bc + bd]Wait, since ( ab = ba ), those terms cancel out on both sides. So, subtracting ( ab ) from both sides:[ac + ad = bc + bd]Hmm, factor out ( a ) from the left side and ( b ) from the right side:[a(c + d) = b(c + d)]Now, assuming ( c + d neq 0 ) (since none of the numbers are zero, their sum can't be zero either, right?), we can divide both sides by ( c + d ):[a = b]Okay, so that tells me that ( a = b ). Interesting. Let me try another pair to see if I can find a similar relationship. Let's take the third and fourth expressions:[c(a + b + d) = d(a + b + c)]Expanding both sides:[ca + cb + cd = da + db + dc]Again, ( ca = ac ) and ( da = ad ), so let's rearrange:[ac + cb + cd = ad + db + dc]Subtract ( cd ) from both sides:[ac + cb = ad + db]Factor out ( c ) from the left and ( d ) from the right:[c(a + b) = d(a + b)]Assuming ( a + b neq 0 ) (since none of the numbers are zero), we can divide both sides by ( a + b ):[c = d]So now I know that ( c = d ). So far, so good. So from the first pair, ( a = b ), and from the second pair, ( c = d ). But wait, the problem says that all four results are equal, not just the first two and the last two. So maybe I need to consider more relationships here. Let me see. Let's take another pair, say the first and the third:[a(b + c + d) = c(a + b + d)]Substituting ( a = b ) and ( c = d ) into this equation:[a(a + c + c) = c(a + a + c)]Simplify:[a(a + 2c) = c(2a + c)]Expanding both sides:[a^2 + 2ac = 2ac + c^2]Subtract ( 2ac ) from both sides:[a^2 = c^2]So, ( a^2 = c^2 ). Which means ( a = c ) or ( a = -c ). But since ( a = b ) and ( c = d ), this would imply that all four numbers have the same square. Wait, hold on. If ( a^2 = c^2 ), then ( a = c ) or ( a = -c ). But if ( a = c ), then since ( a = b ) and ( c = d ), all four numbers are equal. If ( a = -c ), then ( b = a = -c = -d ), so ( b = -c ) and ( d = -c ). So in either case, the squares of all four numbers are equal because ( a^2 = b^2 = c^2 = d^2 ).Let me just verify this with another pair to make sure I'm not missing something. Let's take the second and the fourth expressions:[b(a + c + d) = d(a + b + c)]Again, substituting ( a = b ) and ( c = d ):[a(a + c + c) = c(a + a + c)]Which simplifies to the same equation as before:[a(a + 2c) = c(2a + c)]And as before, this leads to ( a^2 = c^2 ). So regardless of which pair I choose, I end up with the conclusion that ( a^2 = c^2 ), and by extension, all four squares are equal.Just to be thorough, let's consider the case where ( a = -c ). If ( a = -c ), then ( b = a = -c ) and ( d = c = -a ). So, ( b = -c ) and ( d = -a ). Therefore, ( a^2 = (-c)^2 = c^2 ), and similarly ( b^2 = (-c)^2 = c^2 ), ( d^2 = (-a)^2 = a^2 ). So all squares are equal.Therefore, regardless of whether the numbers are equal or negatives of each other, their squares are equal. Hence, the squares of the numbers written on the board are equal.I think that covers all the cases. I started by setting pairs equal and found relationships between the variables, leading me to the conclusion that their squares must be equal. It seems consistent across all pairs, so I feel confident about this proof.**Final Answer**boxed{a^2 = b^2 = c^2 = d^2}