💡Okay, so I have this function f(x) = (x - 1)e^{-x}, and I need to do two things with it. First, I need to find the monotonic intervals, which means figuring out where the function is increasing or decreasing. Second, I need to find the range of values for a real number c such that for all x in [0, +∞), f(x) is less than or equal to 1/c². Hmm, okay, let's tackle the first part first.Starting with part (I): Finding the monotonic intervals. I remember that to find where a function is increasing or decreasing, I need to look at its first derivative. If the derivative is positive, the function is increasing; if it's negative, the function is decreasing. So, I should compute f'(x).Given f(x) = (x - 1)e^{-x}, I can use the product rule for differentiation. The product rule states that if you have two functions multiplied together, their derivative is the derivative of the first times the second plus the first times the derivative of the second. So, let me denote u = (x - 1) and v = e^{-x}. Then, f(x) = u*v.First, find u': the derivative of (x - 1) with respect to x is 1. Then, find v': the derivative of e^{-x} with respect to x is -e^{-x}. So, putting it together, f'(x) = u'v + uv' = (1)(e^{-x}) + (x - 1)(-e^{-x}).Simplify that: f'(x) = e^{-x} - (x - 1)e^{-x}. Let's factor out e^{-x} since it's common to both terms. So, f'(x) = e^{-x}[1 - (x - 1)]. Simplify inside the brackets: 1 - x + 1 = 2 - x. Therefore, f'(x) = (2 - x)e^{-x}.Alright, so f'(x) = (2 - x)e^{-x}. Now, to find where the function is increasing or decreasing, we need to determine where f'(x) is positive or negative. Since e^{-x} is always positive for all real x (because the exponential function is always positive), the sign of f'(x) depends solely on the numerator, which is (2 - x).So, when is (2 - x) positive? When 2 - x > 0, which implies x < 2. Similarly, when is (2 - x) negative? When 2 - x < 0, which implies x > 2. Therefore, f'(x) is positive when x < 2 and negative when x > 2.This tells me that the function f(x) is increasing on the interval (-∞, 2) and decreasing on the interval (2, +∞). So, that's part (I) done.Moving on to part (II): We need to find the range of c such that for all x in [0, +∞), f(x) ≤ 1/c². Hmm, okay. So, essentially, we need to find the maximum value of f(x) on the interval [0, +∞) and set that less than or equal to 1/c². Then, solve for c.From part (I), we know that f(x) is increasing on [0, 2) and decreasing on (2, +∞). Therefore, the maximum value of f(x) on [0, +∞) occurs at x = 2. So, let's compute f(2).f(2) = (2 - 1)e^{-2} = (1)e^{-2} = 1/e².So, the maximum value of f(x) on [0, +∞) is 1/e². Therefore, we have 1/e² ≤ 1/c².We need to solve this inequality for c. Let's write it down:1/e² ≤ 1/c².To solve for c, let's take reciprocals on both sides. But remember, when taking reciprocals in an inequality, the direction of the inequality flips if both sides are positive. Since e² and c² are both positive (as squares are always non-negative and e² is positive), we can safely take reciprocals and reverse the inequality:e² ≥ c².Which is the same as c² ≤ e².So, c² ≤ e² implies that |c| ≤ e, which means that c is between -e and e. However, we have to consider the original inequality 1/c². Since c is in the denominator, c cannot be zero because division by zero is undefined. Therefore, c ≠ 0.So, combining these results, c must satisfy -e ≤ c ≤ e, but c cannot be zero. Therefore, the range of c is all real numbers from -e to e, excluding zero.Let me just double-check my steps to make sure I didn't make any mistakes. Starting with the derivative: yes, product rule, correct. Simplified correctly to (2 - x)e^{-x}, that seems right. Then, since e^{-x} is always positive, the sign of f'(x) depends on (2 - x). So, increasing when x < 2, decreasing when x > 2. That seems correct.For part (II), finding the maximum on [0, +∞). Since the function increases up to x=2 and then decreases, the maximum is indeed at x=2. Calculated f(2) as 1/e², that's correct.Then, setting 1/e² ≤ 1/c², solving for c. Taking reciprocals and flipping the inequality, getting c² ≤ e², which leads to |c| ≤ e, so c is between -e and e. But since c is in the denominator, c cannot be zero. So, c ∈ [-e, e] {0}. That seems right.I think that's solid. I don't see any errors in my reasoning.**Final Answer**(I) The function is increasing on boxed{(-infty, 2)} and decreasing on boxed{(2, +infty)}.(II) The range of values for c is boxed{[-e, e] setminus {0}}.

💡Alright, so I need to figure out the range of the function ( g(x) = frac{1}{(x-1)^2} ). Hmm, okay. Let me start by understanding what the function looks like and how it behaves.First, I know that the function is defined for all real numbers except where the denominator is zero. So, ( (x-1)^2 ) equals zero when ( x = 1 ). That means the function is undefined at ( x = 1 ). So, the domain of ( g(x) ) is all real numbers except 1. Got that.Now, to find the range, I need to determine all possible output values ( g(x) ) can take. Since ( g(x) ) is a reciprocal of a squared term, it's always positive because any real number squared is non-negative, and the reciprocal of a positive number is positive. So, ( g(x) ) is always greater than zero. That means the range is at least ( (0, infty) ), but I need to confirm if it actually covers all positive real numbers.Let me think about how ( g(x) ) behaves as ( x ) approaches 1 and as ( x ) moves away from 1. When ( x ) is close to 1, say ( x = 1.1 ), then ( (x-1)^2 = (0.1)^2 = 0.01 ), so ( g(x) = 1/0.01 = 100 ). If ( x ) is even closer to 1, like ( x = 1.01 ), then ( (x-1)^2 = (0.01)^2 = 0.0001 ), and ( g(x) = 1/0.0001 = 10000 ). So, as ( x ) approaches 1, ( g(x) ) becomes very large, approaching infinity.On the other hand, when ( x ) is far from 1, say ( x = 10 ), then ( (x-1)^2 = (9)^2 = 81 ), so ( g(x) = 1/81 approx 0.0123 ). If ( x ) is even larger, like ( x = 100 ), then ( (x-1)^2 = (99)^2 = 9801 ), and ( g(x) = 1/9801 approx 0.000102 ). So, as ( x ) moves away from 1, ( g(x) ) approaches zero but never actually reaches zero because the denominator can't be infinite.Putting this together, ( g(x) ) can take on any positive value greater than zero. It can get as large as we want when ( x ) is near 1 and as small as we want (approaching zero) when ( x ) is far from 1. Therefore, the range of ( g(x) ) should be all positive real numbers.Wait, let me double-check if there are any restrictions or special cases I might have missed. Since ( (x-1)^2 ) is always positive, ( g(x) ) is always positive, so negative numbers are definitely out of the question. Also, ( g(x) ) can't be zero because the reciprocal of a positive number can't be zero. So, zero isn't included either.Is there any value between zero and infinity that ( g(x) ) can't reach? Suppose I pick an arbitrary positive number ( y ). Can I find an ( x ) such that ( g(x) = y )? Let's solve for ( x ):( y = frac{1}{(x-1)^2} )Multiply both sides by ( (x-1)^2 ):( y(x-1)^2 = 1 )Divide both sides by ( y ):( (x-1)^2 = frac{1}{y} )Take the square root of both sides:( x - 1 = pm sqrt{frac{1}{y}} )So,( x = 1 pm sqrt{frac{1}{y}} )Since ( y ) is positive, ( sqrt{frac{1}{y}} ) is a real number, meaning there are two real solutions for ( x ) for any positive ( y ). Therefore, every positive ( y ) is attainable by ( g(x) ), confirming that the range is indeed all positive real numbers.I think that covers it. I've considered the behavior near the asymptote at ( x = 1 ), the behavior as ( x ) approaches infinity, and checked if every positive number is achievable. So, the range should be ( (0, infty) ).

💡Okay, so I have this problem where I need to hang a painting using four nails in such a way that if any one of the nails is removed, the painting falls. Hmm, that sounds tricky but interesting. I guess I need to figure out a way to loop the string around the nails so that each nail is essential for holding the painting up. First, I should probably visualize how the string is wrapped around the nails. Maybe if I use some kind of knot or loop that depends on all four nails, then removing any one would cause the whole thing to unravel. But I'm not sure how to design such a configuration. Maybe I can start by thinking about simpler cases with fewer nails and see if I can find a pattern or principle that applies here.Let's say I start with two nails. If I loop the string around both nails in a certain way, removing either nail would make the painting fall. How is that done? I think it involves wrapping the string around one nail, then the other, and then back in a way that creates a dependency between them. Maybe something like a figure-eight pattern? Okay, so for two nails, a figure-eight loop ensures that removing either nail will cause the painting to fall. Now, how do I extend this idea to four nails? Maybe I can create two separate figure-eight loops, each involving two nails, and then somehow combine them so that all four nails are interconnected. Wait, but if I have two separate figure-eight loops, removing one nail from one loop might not affect the other loop. So the painting might still stay up if only one nail is removed from one loop. That's not what I want. I need a configuration where all four nails are interdependent in such a way that removing any one nail disrupts the entire structure.Perhaps I need to create a more complex loop that involves all four nails simultaneously. Maybe something like a chain where each nail supports the next one in the loop. If I can make the string pass through all four nails in a way that each nail's removal would cause the string to lose its grip on the others, that would work.Let me try to sketch this out mentally. Imagine the string starts at the painting, goes around nail 1, then nail 2, then nail 3, then nail 4, and back to the painting. But that seems too straightforward, and removing any one nail wouldn't necessarily make the painting fall—it might just loosen the string a bit.Maybe I need to involve some kind of overhand knots or more intricate weaving around the nails. What if I create loops around pairs of nails and then connect those loops in a way that they depend on each other? For example, loop around nail 1 and 2, then loop around nail 3 and 4, and then connect these two loops so that they are mutually dependent.But I'm not sure how to connect them without making the whole thing too complicated. Maybe there's a specific mathematical way to represent this problem. I recall something about group theory and commutators being used in such problems. Maybe I can represent each nail as a generator and the loops as commutators.Let me think about that. If I denote each nail as a generator ( a_1, a_2, a_3, a_4 ), then the way the string is wrapped around them can be represented as a sequence of these generators and their inverses. The goal is to create a sequence where each generator is essential, meaning that removing any one of them would make the sequence trivial or cause the painting to fall.So, perhaps the solution involves creating a commutator of commutators. A commutator is something like ( [a, b] = aba^{-1}b^{-1} ). If I create a commutator involving all four nails, maybe that would ensure that each nail is essential.But how exactly? Maybe I can create two separate commutators, each involving two nails, and then combine them in a way that they are interdependent. For example, ( [[a_1, a_2], [a_3, a_4]] ). This would mean that the overall structure depends on both commutators, and removing any nail from either commutator would disrupt the whole structure.Wait, but I'm not sure if that's the right approach. Maybe I need to think about it differently. What if I create a loop that goes around all four nails in a specific order, making sure that each nail is necessary for the loop to hold?I think I need to look up some examples or similar problems to get a better idea. Maybe there's a standard way to solve this kind of problem with four nails. I recall seeing something about using a figure-eight pattern with multiple nails, but I'm not sure how to extend it to four nails.Alternatively, maybe I can use a more mathematical approach. If I consider the fundamental group of the space with four nails removed, the solution would correspond to a non-trivial element in that group. The key is to find a word in the generators ( a_1, a_2, a_3, a_4 ) such that removing any generator would make the word trivial.This seems abstract, but maybe it can help me construct the sequence. I need to create a word where each generator appears in both positive and negative exponents, ensuring that their removal would simplify the word to something trivial.Perhaps I can use nested commutators. For example, start with ( [a_1, a_2] ), then take the commutator of that with ( [a_3, a_4] ). This would give me ( [[a_1, a_2], [a_3, a_4]] ), which might be a sufficiently complex word that depends on all four nails.But I'm not entirely sure if this is the minimal or optimal solution. Maybe there's a shorter sequence that achieves the same effect. I need to think about what makes a solution optimal. Is it the number of crossings, the number of nails used, or the complexity of the loop?In this case, since the problem specifies four nails, I think optimality might refer to the minimal number of crossings or the simplicity of the loop. So, I need to find the simplest loop that involves all four nails in such a way that each is essential.Going back to the commutator idea, maybe I can write out the sequence explicitly. Let's see:1. Start with ( a_1 ), go around nail 1.2. Then ( a_2 ), go around nail 2.3. Then ( a_1^{-1} ), go back around nail 1 in the opposite direction.4. Then ( a_2^{-1} ), go back around nail 2 in the opposite direction.5. Now, repeat this process for nails 3 and 4: ( a_3 ), ( a_4 ), ( a_3^{-1} ), ( a_4^{-1} ).6. Then, to connect them, maybe go back through some of the nails again: ( a_2 ), ( a_1 ), ( a_2^{-1} ), ( a_1^{-1} ), ( a_4 ), ( a_3 ), ( a_4^{-1} ), ( a_3^{-1} ).Putting it all together, the sequence would be:( a_1 a_2 a_1^{-1} a_2^{-1} a_3 a_4 a_3^{-1} a_4^{-1} a_2 a_1 a_2^{-1} a_1^{-1} a_4 a_3 a_4^{-1} a_3^{-1} )This seems to involve all four nails in a way that each is essential. If I remove any one nail, the corresponding generator would be missing, and the sequence would simplify, causing the painting to fall.But is this the optimal solution? It uses 16 symbols, which seems efficient. I don't think I can make it shorter without losing the essential dependency on all four nails. So, I think this might be the optimal solution.However, I'm still a bit unsure. Maybe there's a different configuration that achieves the same result with fewer steps or a simpler loop. I should probably try drawing this out or looking for similar problems to confirm.In summary, my approach is to use nested commutators involving all four nails, ensuring that each nail is essential for the overall structure. This should make the painting fall if any one nail is removed, and the sequence I've constructed seems to be optimal in terms of the number of symbols used.

💡Okay, so I have this triangle ABC, and I need to find the measure of angle B. The problem gives me two equations: ( b^2 + c^2 - a^2 = bc ) and ( sin^2 A + sin^2 B = sin^2 C ). Hmm, let me try to figure this out step by step.First, I remember the Law of Cosines, which says that ( a^2 = b^2 + c^2 - 2bc cos A ). The first equation given is ( b^2 + c^2 - a^2 = bc ). If I rearrange the Law of Cosines, I get ( b^2 + c^2 - a^2 = 2bc cos A ). Comparing this to the given equation ( b^2 + c^2 - a^2 = bc ), it looks like ( 2bc cos A = bc ). So, if I divide both sides by bc (assuming bc isn't zero, which it can't be in a triangle), I get ( 2 cos A = 1 ), which simplifies to ( cos A = frac{1}{2} ). I know that ( cos frac{pi}{3} = frac{1}{2} ), so angle A must be ( frac{pi}{3} ) radians, or 60 degrees. Okay, that's angle A figured out.Now, moving on to the second equation: ( sin^2 A + sin^2 B = sin^2 C ). I remember that in any triangle, the sum of the angles is ( pi ) radians, so ( A + B + C = pi ). Since I already know A is ( frac{pi}{3} ), I can write ( B + C = frac{2pi}{3} ). That might come in handy later.Looking at the equation ( sin^2 A + sin^2 B = sin^2 C ), I wonder if there's a trigonometric identity that can help me here. I recall that ( sin^2 theta = frac{1 - cos 2theta}{2} ), but I'm not sure if that will help directly. Maybe I can express ( sin^2 C ) in terms of A and B since ( C = pi - A - B ).So, ( sin^2 C = sin^2 (pi - A - B) ). I remember that ( sin (pi - x) = sin x ), so ( sin^2 (pi - A - B) = sin^2 (A + B) ). Therefore, the equation becomes ( sin^2 A + sin^2 B = sin^2 (A + B) ).Let me expand ( sin^2 (A + B) ) using the identity ( sin (A + B) = sin A cos B + cos A sin B ). So, squaring both sides:( sin^2 (A + B) = (sin A cos B + cos A sin B)^2 )= ( sin^2 A cos^2 B + 2 sin A cos A sin B cos B + cos^2 A sin^2 B ).So, substituting back into the equation:( sin^2 A + sin^2 B = sin^2 A cos^2 B + 2 sin A cos A sin B cos B + cos^2 A sin^2 B ).Hmm, this looks complicated. Maybe I can subtract ( sin^2 A + sin^2 B ) from both sides to see if something cancels out:0 = ( sin^2 A cos^2 B + 2 sin A cos A sin B cos B + cos^2 A sin^2 B - sin^2 A - sin^2 B ).Let me factor terms:= ( sin^2 A (cos^2 B - 1) + sin^2 B (cos^2 A - 1) + 2 sin A cos A sin B cos B ).Since ( cos^2 B - 1 = -sin^2 B ) and ( cos^2 A - 1 = -sin^2 A ), substituting these in:= ( -sin^4 A - sin^4 B + 2 sin A cos A sin B cos B ).So, the equation becomes:( -sin^4 A - sin^4 B + 2 sin A cos A sin B cos B = 0 ).Hmm, this seems a bit messy. Maybe there's a simpler approach. Let me think about the Law of Sines. It states that ( frac{a}{sin A} = frac{b}{sin B} = frac{c}{sin C} = 2R ), where R is the radius of the circumscribed circle.Since I know angle A is ( frac{pi}{3} ), maybe I can express sides b and c in terms of a. Let me denote ( 2R = k ), so ( a = k sin A = k sin frac{pi}{3} = k frac{sqrt{3}}{2} ). Similarly, ( b = k sin B ) and ( c = k sin C ).Given that ( C = pi - A - B = pi - frac{pi}{3} - B = frac{2pi}{3} - B ), so ( c = k sin (frac{2pi}{3} - B) ).I can use the sine of a difference identity: ( sin (frac{2pi}{3} - B) = sin frac{2pi}{3} cos B - cos frac{2pi}{3} sin B ).We know that ( sin frac{2pi}{3} = frac{sqrt{3}}{2} ) and ( cos frac{2pi}{3} = -frac{1}{2} ), so:( c = k left( frac{sqrt{3}}{2} cos B - (-frac{1}{2}) sin B right ) = k left( frac{sqrt{3}}{2} cos B + frac{1}{2} sin B right ) ).So, ( c = k left( frac{sqrt{3}}{2} cos B + frac{1}{2} sin B right ) ).Now, going back to the first equation given: ( b^2 + c^2 - a^2 = bc ).Substituting the expressions for a, b, c in terms of k and angles:( (k sin B)^2 + left( k left( frac{sqrt{3}}{2} cos B + frac{1}{2} sin B right ) right )^2 - left( k frac{sqrt{3}}{2} right )^2 = (k sin B) left( k left( frac{sqrt{3}}{2} cos B + frac{1}{2} sin B right ) right ) ).Let me factor out ( k^2 ) from all terms:( k^2 [ sin^2 B + left( frac{sqrt{3}}{2} cos B + frac{1}{2} sin B right )^2 - left( frac{3}{4} right ) ] = k^2 [ sin B left( frac{sqrt{3}}{2} cos B + frac{1}{2} sin B right ) ] ).Since ( k^2 ) is non-zero, I can divide both sides by ( k^2 ):( sin^2 B + left( frac{sqrt{3}}{2} cos B + frac{1}{2} sin B right )^2 - frac{3}{4} = sin B left( frac{sqrt{3}}{2} cos B + frac{1}{2} sin B right ) ).Let me expand the left side:First, expand ( left( frac{sqrt{3}}{2} cos B + frac{1}{2} sin B right )^2 ):= ( left( frac{sqrt{3}}{2} cos B right )^2 + 2 cdot frac{sqrt{3}}{2} cos B cdot frac{1}{2} sin B + left( frac{1}{2} sin B right )^2 )= ( frac{3}{4} cos^2 B + frac{sqrt{3}}{2} sin B cos B + frac{1}{4} sin^2 B ).So, the left side becomes:( sin^2 B + frac{3}{4} cos^2 B + frac{sqrt{3}}{2} sin B cos B + frac{1}{4} sin^2 B - frac{3}{4} ).Combine like terms:= ( sin^2 B + frac{1}{4} sin^2 B + frac{3}{4} cos^2 B + frac{sqrt{3}}{2} sin B cos B - frac{3}{4} )= ( frac{5}{4} sin^2 B + frac{3}{4} cos^2 B + frac{sqrt{3}}{2} sin B cos B - frac{3}{4} ).Now, the right side is:( sin B left( frac{sqrt{3}}{2} cos B + frac{1}{2} sin B right ) )= ( frac{sqrt{3}}{2} sin B cos B + frac{1}{2} sin^2 B ).So, putting it all together, the equation is:( frac{5}{4} sin^2 B + frac{3}{4} cos^2 B + frac{sqrt{3}}{2} sin B cos B - frac{3}{4} = frac{sqrt{3}}{2} sin B cos B + frac{1}{2} sin^2 B ).Let me subtract the right side from both sides to bring everything to the left:( frac{5}{4} sin^2 B + frac{3}{4} cos^2 B + frac{sqrt{3}}{2} sin B cos B - frac{3}{4} - frac{sqrt{3}}{2} sin B cos B - frac{1}{2} sin^2 B = 0 ).Simplify term by term:- ( frac{5}{4} sin^2 B - frac{1}{2} sin^2 B = frac{5}{4} - frac{2}{4} = frac{3}{4} sin^2 B ).- ( frac{3}{4} cos^2 B ) remains.- ( frac{sqrt{3}}{2} sin B cos B - frac{sqrt{3}}{2} sin B cos B = 0 ).- ( - frac{3}{4} ) remains.So, the equation simplifies to:( frac{3}{4} sin^2 B + frac{3}{4} cos^2 B - frac{3}{4} = 0 ).Factor out ( frac{3}{4} ):( frac{3}{4} ( sin^2 B + cos^2 B - 1 ) = 0 ).But ( sin^2 B + cos^2 B = 1 ), so this becomes:( frac{3}{4} (1 - 1) = 0 ), which is ( 0 = 0 ).Hmm, that's interesting. It seems like the equation simplifies to an identity, which doesn't give me new information. Maybe I need to approach this differently.Let me go back to the second given equation: ( sin^2 A + sin^2 B = sin^2 C ). I know that ( A = frac{pi}{3} ), so ( sin A = frac{sqrt{3}}{2} ), so ( sin^2 A = frac{3}{4} ).Thus, the equation becomes:( frac{3}{4} + sin^2 B = sin^2 C ).But since ( C = frac{2pi}{3} - B ), ( sin C = sin (frac{2pi}{3} - B) ). Using the sine of a difference identity again:( sin (frac{2pi}{3} - B) = sin frac{2pi}{3} cos B - cos frac{2pi}{3} sin B )= ( frac{sqrt{3}}{2} cos B - (-frac{1}{2}) sin B )= ( frac{sqrt{3}}{2} cos B + frac{1}{2} sin B ).Therefore, ( sin^2 C = left( frac{sqrt{3}}{2} cos B + frac{1}{2} sin B right )^2 ).Expanding this:= ( left( frac{sqrt{3}}{2} cos B right )^2 + 2 cdot frac{sqrt{3}}{2} cos B cdot frac{1}{2} sin B + left( frac{1}{2} sin B right )^2 )= ( frac{3}{4} cos^2 B + frac{sqrt{3}}{2} sin B cos B + frac{1}{4} sin^2 B ).So, substituting back into the equation ( frac{3}{4} + sin^2 B = sin^2 C ):( frac{3}{4} + sin^2 B = frac{3}{4} cos^2 B + frac{sqrt{3}}{2} sin B cos B + frac{1}{4} sin^2 B ).Let me move all terms to the left side:( frac{3}{4} + sin^2 B - frac{3}{4} cos^2 B - frac{sqrt{3}}{2} sin B cos B - frac{1}{4} sin^2 B = 0 ).Combine like terms:- ( sin^2 B - frac{1}{4} sin^2 B = frac{3}{4} sin^2 B ).- ( - frac{3}{4} cos^2 B ).- ( - frac{sqrt{3}}{2} sin B cos B ).- ( frac{3}{4} ).So, the equation becomes:( frac{3}{4} sin^2 B - frac{3}{4} cos^2 B - frac{sqrt{3}}{2} sin B cos B + frac{3}{4} = 0 ).Let me factor out ( frac{3}{4} ) from the first and last terms:( frac{3}{4} ( sin^2 B - cos^2 B + 1 ) - frac{sqrt{3}}{2} sin B cos B = 0 ).Hmm, ( sin^2 B - cos^2 B = -cos 2B ), and ( 1 = sin^2 B + cos^2 B ), so:= ( frac{3}{4} ( -cos 2B + sin^2 B + cos^2 B ) - frac{sqrt{3}}{2} sin B cos B )= ( frac{3}{4} ( -cos 2B + 1 ) - frac{sqrt{3}}{2} sin B cos B ).But ( 1 - cos 2B = 2 sin^2 B ), so:= ( frac{3}{4} cdot 2 sin^2 B - frac{sqrt{3}}{2} sin B cos B )= ( frac{3}{2} sin^2 B - frac{sqrt{3}}{2} sin B cos B ).So, the equation is:( frac{3}{2} sin^2 B - frac{sqrt{3}}{2} sin B cos B = 0 ).Factor out ( frac{sqrt{3}}{2} sin B ):= ( frac{sqrt{3}}{2} sin B ( sqrt{3} sin B - cos B ) = 0 ).So, either ( frac{sqrt{3}}{2} sin B = 0 ) or ( sqrt{3} sin B - cos B = 0 ).Case 1: ( frac{sqrt{3}}{2} sin B = 0 ).This implies ( sin B = 0 ), but in a triangle, angles are between 0 and ( pi ), so ( sin B = 0 ) would mean B = 0 or ( pi ), which isn't possible. So, this case is invalid.Case 2: ( sqrt{3} sin B - cos B = 0 ).Let me solve for B:( sqrt{3} sin B = cos B ).Divide both sides by ( cos B ) (assuming ( cos B neq 0 )):( sqrt{3} tan B = 1 ).So, ( tan B = frac{1}{sqrt{3}} ).I know that ( tan frac{pi}{6} = frac{1}{sqrt{3}} ), so ( B = frac{pi}{6} ) radians, or 30 degrees.Let me verify if this makes sense. If B is 30 degrees, then angle C would be ( 180 - 60 - 30 = 90 ) degrees. So, angle C is 90 degrees.Let me check the second equation ( sin^2 A + sin^2 B = sin^2 C ):( sin^2 60^circ + sin^2 30^circ = sin^2 90^circ )= ( left( frac{sqrt{3}}{2} right )^2 + left( frac{1}{2} right )^2 = 1^2 )= ( frac{3}{4} + frac{1}{4} = 1 )= ( 1 = 1 ).Yes, that checks out. Also, let's verify the first equation ( b^2 + c^2 - a^2 = bc ).If angle C is 90 degrees, then by Pythagoras' theorem, ( c^2 = a^2 + b^2 ). Wait, but the given equation is ( b^2 + c^2 - a^2 = bc ). If ( c^2 = a^2 + b^2 ), then substituting:( b^2 + (a^2 + b^2) - a^2 = bc )= ( 2b^2 = bc )= ( 2b = c ).So, in this case, side c is twice the length of side b. Let me see if that's consistent with the Law of Sines.From the Law of Sines, ( frac{a}{sin 60^circ} = frac{b}{sin 30^circ} = frac{c}{sin 90^circ} ).Let me denote ( frac{a}{frac{sqrt{3}}{2}} = frac{b}{frac{1}{2}} = frac{c}{1} = 2R ).So, ( a = 2R cdot frac{sqrt{3}}{2} = R sqrt{3} ),( b = 2R cdot frac{1}{2} = R ),( c = 2R cdot 1 = 2R ).So, indeed, c = 2b, which matches the earlier result from the first equation. So, everything checks out.Therefore, angle B is ( frac{pi}{6} ) radians or 30 degrees.**Final Answer**The measure of angle ( B ) is boxed{dfrac{pi}{6}}.

💡Okay, so I have this problem with two complex numbers, z₁ = 3 - bi and z₂ = 1 - 2i. The question is asking me to find the value of the real number b such that the ratio z₁/z₂ is a pure imaginary number. The options are A: 0, B: 8/15, C: 1/2, and D: -3/2. Hmm, okay, let me think about how to approach this.First, I remember that a pure imaginary number is a complex number with no real part. That means if I divide z₁ by z₂, the result should have a real part equal to zero. So, my goal is to perform the division z₁/z₂ and then set the real part to zero and solve for b.To divide complex numbers, I know that I can multiply the numerator and the denominator by the conjugate of the denominator to simplify the expression. The conjugate of z₂, which is 1 - 2i, is 1 + 2i. So, I'll multiply both the numerator and the denominator by 1 + 2i.Let me write that down:z₁/z₂ = (3 - bi)/(1 - 2i) Multiply numerator and denominator by (1 + 2i):= [(3 - bi)(1 + 2i)] / [(1 - 2i)(1 + 2i)]Now, I'll expand both the numerator and the denominator.Starting with the numerator:(3 - bi)(1 + 2i) = 3*1 + 3*2i - bi*1 - bi*2i = 3 + 6i - bi - 2b i²Wait, I remember that i² = -1, so -2b i² becomes +2b. So, substituting that in:= 3 + 6i - bi + 2b Now, combine like terms:Real parts: 3 + 2b Imaginary parts: (6i - bi) = (6 - b)iSo, the numerator simplifies to (3 + 2b) + (6 - b)i.Now, the denominator:(1 - 2i)(1 + 2i) = 1*1 + 1*2i - 2i*1 - 2i*2i = 1 + 2i - 2i - 4i²Again, i² = -1, so -4i² becomes +4. Therefore:= 1 + 0i + 4 = 5So, the denominator simplifies to 5.Putting it all together, the division becomes:[(3 + 2b) + (6 - b)i] / 5 Which can be written as:(3 + 2b)/5 + (6 - b)/5 iSince the problem states that this ratio is a pure imaginary number, the real part must be zero. Therefore, I can set the real part equal to zero and solve for b.So, set (3 + 2b)/5 = 0.Multiplying both sides by 5:3 + 2b = 0 Subtract 3 from both sides:2b = -3 Divide both sides by 2:b = -3/2Hmm, that gives me b = -3/2, which is option D. Let me just double-check my steps to make sure I didn't make any mistakes.First, I multiplied the numerator and denominator by the conjugate of the denominator, which is correct. Then I expanded both the numerator and the denominator, remembering to replace i² with -1. That seems right. Then I combined like terms in the numerator, separating real and imaginary parts. The denominator simplified to 5, which makes sense because (1 - 2i)(1 + 2i) is a difference of squares, 1² - (2i)² = 1 - 4i² = 1 + 4 = 5.Then, I set the real part of the result equal to zero because the ratio is supposed to be purely imaginary. That gives me the equation (3 + 2b)/5 = 0, leading to b = -3/2. That seems consistent.Let me also check if the imaginary part is non-zero because if the imaginary part were zero as well, then the ratio would be zero, which is technically a real number, not purely imaginary. So, I should ensure that (6 - b)/5 ≠ 0.Plugging b = -3/2 into (6 - b):6 - (-3/2) = 6 + 3/2 = 7.5, which is 15/2. So, (15/2)/5 = 15/10 = 3/2, which is not zero. Therefore, the imaginary part is indeed non-zero, confirming that the ratio is purely imaginary.So, I think my answer is correct. b is -3/2, which is option D.**Final Answer**boxed{D}

💡Okay, so I have this problem where I'm given integers (a, b, c) and a natural number (n) such that (a + b + c = 1) and (a^2 + b^2 + c^2 = 2n + 1). I need to prove that (a^3 + b^2 - a^2 - b^3) is divisible by (n). Hmm, that sounds a bit tricky, but let me try to break it down step by step.First, I know that (a + b + c = 1). Maybe I can express (c) in terms of (a) and (b). So, (c = 1 - a - b). That might help simplify things later on.Next, I have the equation (a^2 + b^2 + c^2 = 2n + 1). Since I already expressed (c) in terms of (a) and (b), I can substitute that into this equation. Let me do that:(a^2 + b^2 + (1 - a - b)^2 = 2n + 1).Now, I need to expand that squared term. Let's see:((1 - a - b)^2 = 1 - 2a - 2b + a^2 + 2ab + b^2).So, substituting back into the equation:(a^2 + b^2 + 1 - 2a - 2b + a^2 + 2ab + b^2 = 2n + 1).Now, let's combine like terms:- (a^2 + a^2 = 2a^2)- (b^2 + b^2 = 2b^2)- The constants: just 1- The linear terms: (-2a - 2b)- The cross term: (2ab)So, putting it all together:(2a^2 + 2b^2 - 2a - 2b + 2ab + 1 = 2n + 1).Hmm, I notice that both sides have a "+1", so maybe I can subtract 1 from both sides to simplify:(2a^2 + 2b^2 - 2a - 2b + 2ab = 2n).Now, I can factor out a 2 from the left side:(2(a^2 + b^2 + ab - a - b) = 2n).Dividing both sides by 2:(a^2 + b^2 + ab - a - b = n).Okay, so now I have an expression for (n) in terms of (a) and (b). That might be useful later.Now, the expression I need to prove is divisible by (n) is (a^3 + b^2 - a^2 - b^3). Let me write that down:(a^3 + b^2 - a^2 - b^3).Hmm, maybe I can factor this expression somehow. Let me see if I can group terms or factor by grouping.Looking at (a^3 - b^3), I remember that (a^3 - b^3 = (a - b)(a^2 + ab + b^2)). So, maybe I can use that identity here.Similarly, (b^2 - a^2 = (b - a)(b + a)). So, that's another identity I can use.Let me try to rewrite the expression using these identities:(a^3 + b^2 - a^2 - b^3 = (a^3 - b^3) + (b^2 - a^2)).Substituting the identities:((a - b)(a^2 + ab + b^2) + (b - a)(b + a)).Wait, I notice that ((b - a)) is the same as (-(a - b)). So, I can factor out ((a - b)):((a - b)(a^2 + ab + b^2) - (a - b)(b + a)).Now, factor out ((a - b)):((a - b)[(a^2 + ab + b^2) - (a + b)]).Simplify inside the brackets:(a^2 + ab + b^2 - a - b).Wait a minute, that looks familiar! Earlier, I found that (a^2 + b^2 + ab - a - b = n). So, that entire expression inside the brackets is equal to (n).So, substituting back in:((a - b)(n)).Therefore, (a^3 + b^2 - a^2 - b^3 = (a - b)n).Since (n) is a natural number, and ((a - b)) is an integer (because (a) and (b) are integers), the entire expression is a multiple of (n). Therefore, (a^3 + b^2 - a^2 - b^3) is divisible by (n).I think that makes sense. Let me just recap to make sure I didn't miss anything:1. Expressed (c) in terms of (a) and (b).2. Substituted into the second equation and simplified to find an expression for (n).3. Factored the target expression using known identities.4. Recognized that the factored form includes (n), thus proving divisibility.Yeah, that seems solid. I don't see any mistakes in my steps, so I'm confident this is correct.**Final Answer**boxed{a^3 + b^2 - a^2 - b^3} text{ is divisible by } n.