💡Alright, so I have this problem about sets A and B, and I need to find A ∪ B and (A⁰ᵤ) ∩ B when m = -1, and then find the range of m if A ∩ B = A. Let me try to break this down step by step.First, let's understand what set A is. It's defined as {x | 0 < log₃x < 1}. Okay, so that's all real numbers x such that the logarithm base 3 of x is between 0 and 1. I remember that log₃x = y means that 3^y = x. So, if log₃x is between 0 and 1, that means x is between 3^0 and 3^1. Since 3^0 is 1 and 3^1 is 3, set A is the interval (1, 3). Got that.Now, set B is {x | 2m < x < 1 - m}. This is an interval that depends on the value of m. When m = -1, let's substitute that in. So, 2m becomes 2*(-1) = -2, and 1 - m becomes 1 - (-1) = 2. So, set B when m = -1 is (-2, 2). Okay, that makes sense.For the first part, I need to find A ∪ B and (A⁰ᵤ) ∩ B when m = -1. Let's start with A ∪ B. A is (1, 3) and B is (-2, 2). The union of these two intervals would cover all numbers from -2 up to 3, because A starts at 1 and goes to 3, and B starts at -2 and goes to 2. So, overlapping between 1 and 2, and combining the non-overlapping parts, the union should be (-2, 3). That seems straightforward.Next, I need to find (A⁰ᵤ) ∩ B. A⁰ᵤ is the complement of A in the universal set U, which is all real numbers. Since A is (1, 3), its complement would be everything less than or equal to 1 and everything greater than or equal to 3. So, A⁰ᵤ is (-∞, 1] ∪ [3, ∞). Now, intersecting this with B, which is (-2, 2), we're looking for the overlap between (-∞, 1] ∪ [3, ∞) and (-2, 2). The overlap with (-∞, 1] is (-2, 1], and the overlap with [3, ∞) is empty because B only goes up to 2. So, (A⁰ᵤ) ∩ B is (-2, 1]. That makes sense.Moving on to the second part, I need to find the range of real number m if A ∩ B = A. This means that A is a subset of B. So, every element of A must also be in B. Since A is (1, 3), B must cover the entire interval (1, 3). B is defined as (2m, 1 - m). So, for B to cover (1, 3), the lower bound of B must be less than or equal to 1, and the upper bound must be greater than or equal to 3.Let's write that down as inequalities:1. 2m ≤ 12. 1 - m ≥ 3Solving the first inequality: 2m ≤ 1 ⇒ m ≤ 1/2.Solving the second inequality: 1 - m ≥ 3 ⇒ -m ≥ 2 ⇒ m ≤ -2.Wait, so m has to satisfy both m ≤ 1/2 and m ≤ -2. The more restrictive condition is m ≤ -2. So, m must be less than or equal to -2.But I should also check if B is a valid interval. For B to be valid, the lower bound must be less than the upper bound. So, 2m < 1 - m. Let's solve that:2m < 1 - m ⇒ 2m + m < 1 ⇒ 3m < 1 ⇒ m < 1/3.Since we already have m ≤ -2, which is more restrictive than m < 1/3, this condition is automatically satisfied. So, the range of m is all real numbers less than or equal to -2.Let me double-check my work. If m = -2, then B would be (2*(-2), 1 - (-2)) = (-4, 3). A is (1, 3), so A ∩ B would be (1, 3), which is equal to A. If m is less than -2, say m = -3, then B would be (-6, 4), and A ∩ B would still be (1, 3), which is A. If m is greater than -2, say m = -1, then B is (-2, 2), and A ∩ B would be (1, 2), which is not equal to A. So, my conclusion seems correct.I think I've covered all the necessary steps and checked my work. Hopefully, I didn't make any mistakes.

💡Alright, so I've got this problem here about finding the area of a quadrilateral ( ABCD ). It says that the area of ( ABCD ) is four times the sum of the areas of two other shapes, ( OKSL ) and ( ONAM ). Specifically, it mentions that ( S_{square ABCD} = 4(S_{text{OKSL}} + S_{text{ONAM}}) = 4(6 + 12) = 72 ). Then, it asks me to determine the area ( S_{text{OMVK}} ).Okay, first things first, let's make sure I understand the given information correctly. The area of quadrilateral ( ABCD ) is given as 72, which is four times the sum of the areas of ( OKSL ) and ( ONAM ). So, ( OKSL ) has an area of 6, and ( ONAM ) has an area of 12. Adding those together gives 18, and multiplying by 4 gives 72, which checks out.Now, I need to find the area of ( OMKV ). Hmm, ( OMKV ) isn't mentioned in the initial problem statement, so I wonder if it's a typo or if it refers to another quadrilateral within ( ABCD ). Maybe ( OMKV ) is a part of ( ABCD ) that's not covered by ( OKSL ) and ( ONAM ). Let me visualize this. If ( ABCD ) is a quadrilateral, and within it, there are two smaller shapes, ( OKSL ) and ( ONAM ), each with their own areas. Perhaps ( OMKV ) is another region within ( ABCD ) that, when combined with ( OKSL ) and ( ONAM ), makes up the entire area of ( ABCD ).If that's the case, then the area of ( OMKV ) would be the remaining area after subtracting the areas of ( OKSL ) and ( ONAM ) from the total area of ( ABCD ). So, mathematically, that would be:[S_{text{OMKV}} = S_{square ABCD} - S_{text{OKSL}} - S_{text{ONAM}}]Plugging in the numbers:[S_{text{OMKV}} = 72 - 6 - 12 = 72 - 18 = 54]Wait, that doesn't seem right. The problem mentions ( S_{text{OMVK}} ), not ( OMKV ). Maybe I misread it. Let me check again.Oh, it's ( OMKV ). So, perhaps there's a different configuration. Maybe ( OMKV ) is overlapping with ( OKSL ) and ( ONAM ), or maybe it's a separate region. I need to think about how these shapes are positioned within ( ABCD ).If ( ABCD ) is divided into four regions: ( OKSL ), ( ONAM ), and two others, one of which is ( OMKV ), then the total area would be the sum of all four regions. But the problem only mentions two regions, ( OKSL ) and ( ONAM ), whose combined area is 18, and the total area is 72, which is four times that sum. So, perhaps each of the four regions has an area equal to ( OKSL ) and ( ONAM ), but that doesn't quite make sense.Alternatively, maybe ( OKSL ) and ( ONAM ) are each part of the quadrilateral, and ( OMKV ) is another part. If ( ABCD ) is divided into four equal parts, each part would have an area of 18, since 72 divided by 4 is 18. But ( OKSL ) is 6 and ( ONAM ) is 12, which don't add up to 18. So, that might not be the case.Perhaps ( OMKV ) is a combination of parts of ( OKSL ) and ( ONAM ). Or maybe it's a separate region altogether. I'm getting a bit confused here.Let me try another approach. If the total area of ( ABCD ) is 72, and the areas of ( OKSL ) and ( ONAM ) are 6 and 12 respectively, then the combined area of these two is 18. If ( OMKV ) is another region within ( ABCD ), then its area would be 72 minus 18, which is 54. But that seems too large, considering the other areas are much smaller.Wait, maybe ( OMKV ) is a specific region that's part of ( ABCD ) but not overlapping with ( OKSL ) and ( ONAM ). If that's the case, then the area of ( OMKV ) would indeed be 72 minus 6 minus 12, which is 54. But I'm not sure if that's the correct interpretation.Alternatively, perhaps ( OMKV ) is a triangle or another shape within ( ABCD ) that's formed by connecting certain points. Without a diagram, it's hard to be precise. Maybe I need to consider that ( OMKV ) is one of the four regions that make up ( ABCD ), each scaled by a factor.Given that the total area is four times the sum of ( OKSL ) and ( ONAM ), it's possible that ( ABCD ) is composed of four smaller quadrilaterals, each similar to ( OKSL ) and ( ONAM ). If ( OKSL ) is 6 and ( ONAM ) is 12, then perhaps ( OMKV ) is another such region with a different area.But I'm not sure. Maybe I need to think about the ratios. If ( S_{square ABCD} = 4(S_{text{OKSL}} + S_{text{ONAM}}) ), then each of the four regions could have areas proportional to ( OKSL ) and ( ONAM ). But without more information, it's hard to determine.Wait, perhaps ( OMKV ) is a specific region whose area can be found by subtracting the areas of ( OKSL ) and ( ONAM ) from the total area. So, ( S_{text{OMKV}} = 72 - 6 - 12 = 54 ). But earlier, I thought that might be too large. Maybe it's correct.Alternatively, if ( OMKV ) is a part of ( ABCD ) that's not covered by ( OKSL ) and ( ONAM ), then yes, its area would be 54. But I'm not entirely certain without more context.I think I need to go with the straightforward approach: subtract the areas of ( OKSL ) and ( ONAM ) from the total area of ( ABCD ) to find the area of ( OMKV ).So, ( S_{text{OMKV}} = 72 - 6 - 12 = 54 ).But wait, the problem mentions ( S_{text{OMVK}} ), not ( OMKV ). Maybe it's a typo, and it's supposed to be ( OMKV ). If that's the case, then 54 would be the answer. But I'm not sure.Alternatively, maybe ( OMKV ) is a different region that overlaps with ( OKSL ) and ( ONAM ), so its area would be calculated differently. But without more information, it's hard to say.I think the most logical approach is to assume that ( OMKV ) is another region within ( ABCD ) that, when combined with ( OKSL ) and ( ONAM ), makes up the entire area. Therefore, its area would be 72 minus 6 minus 12, which is 54.But I'm still a bit uncertain. Maybe I should double-check my calculations.Total area of ( ABCD ) is 72.Areas of ( OKSL ) and ( ONAM ) are 6 and 12, respectively.So, 72 - 6 - 12 = 54.Yes, that seems correct.Therefore, the area of ( OMKV ) is 54.

💡First, I need to calculate the product of ( x ) and ( y ). Given ( x = sqrt{3} + 1 ) and ( y = sqrt{3} - 1 ), I can use the difference of squares formula to simplify the multiplication.Next, I'll find the difference between ( x ) and ( y ). This involves subtracting ( y ) from ( x ) and simplifying the expression.Once I have both the product ( xy ) and the difference ( x - y ), I can substitute these values into the expression ( x^2 y - x y^2 ). By factoring out ( xy ), the expression simplifies to ( xy(x - y) ).Finally, I'll multiply the values of ( xy ) and ( x - y ) to obtain the final result.

💡Alright, so I have this problem with four propositions, and I need to figure out which ones are true. Let me take it step by step.First, let's understand each proposition one by one.**Proposition (1):** The converse of "If ( xy = 1 ), then ( x ) and ( y ) are reciprocals of each other."Okay, so the original statement is: If the product of ( x ) and ( y ) is 1, then ( x ) and ( y ) are reciprocals. The converse of this statement would switch the hypothesis and the conclusion. So, the converse would be: If ( x ) and ( y ) are reciprocals, then ( xy = 1 ).Hmm, does that make sense? Well, if ( x ) and ( y ) are reciprocals, by definition, ( y = frac{1}{x} ) or ( x = frac{1}{y} ). So, multiplying them together, ( xy = x times frac{1}{x} = 1 ). Yeah, that seems true. So, proposition (1) is true.**Proposition (2):** The negation of "Triangles with equal areas are congruent."Alright, the original statement is: If two triangles have equal areas, then they are congruent. The negation of this would be: There exist triangles with equal areas that are not congruent.Is that true? Well, I know that two triangles can have the same area but different shapes. For example, imagine two triangles with the same base and height but different side lengths. Their areas would be the same because area is ( frac{1}{2} times text{base} times text{height} ), but they wouldn't necessarily be congruent because their side lengths could differ. So, yes, the negation is true. Therefore, proposition (2) is true.**Proposition (3):** The contrapositive of "If ( m leqslant 1 ), then ( x^2 - 2x + m = 0 ) has real solutions."First, let's recall that the contrapositive of "If P, then Q" is "If not Q, then not P." So, the contrapositive here would be: If ( x^2 - 2x + m = 0 ) does not have real solutions, then ( m > 1 ).To check if this is true, let's analyze the quadratic equation ( x^2 - 2x + m = 0 ). The discriminant ( D ) is given by ( D = b^2 - 4ac ). Here, ( a = 1 ), ( b = -2 ), and ( c = m ), so ( D = (-2)^2 - 4(1)(m) = 4 - 4m ).For the equation to have real solutions, the discriminant must be non-negative: ( 4 - 4m geq 0 ) which simplifies to ( m leq 1 ). So, if ( m > 1 ), the discriminant is negative, meaning there are no real solutions. Therefore, the contrapositive statement is true. So, proposition (3) is true.**Proposition (4):** The contrapositive of "If ( A cap B = B ), then ( A subseteq B )."Let's break this down. The original statement is: If the intersection of sets ( A ) and ( B ) is equal to ( B ), then ( A ) is a subset of ( B ). The contrapositive would be: If ( A ) is not a subset of ( B ), then ( A cap B neq B ).Is the original statement true? Let's see. If ( A cap B = B ), that means every element of ( B ) is also in ( A ), which is the definition of ( B subseteq A ). Wait, but the conclusion is ( A subseteq B ). That doesn't seem right. If ( A cap B = B ), it actually implies ( B subseteq A ), not ( A subseteq B ). So, the original statement is false.Since the original statement is false, its contrapositive must also be false because a statement and its contrapositive are logically equivalent. Therefore, proposition (4) is false.**Summary:**- Proposition (1): True- Proposition (2): True- Proposition (3): True- Proposition (4): FalseSo, the true propositions are (1), (2), and (3). Looking back at the options:A: (1)(2)B: (2)(3)C: (4)D: (1)(2)(3)Therefore, the correct answer is D.**Final Answer**boxed{D}

💡Okay, so Dexter wants to buy an iPhone 12 and an iWatch, and both are on sale with different discounts. The iPhone has a 15% discount, and the iWatch has a 10% discount. After applying those discounts, there's also a 2% cashback at checkout. I need to figure out the total cost Dexter will pay for both items after all these discounts and cashback.First, I should probably find out how much each item costs after their respective discounts. Let's start with the iPhone 12. The original price is 800, and there's a 15% discount. To find the discount amount, I can calculate 15% of 800. So, 15% of 800 is 0.15 multiplied by 800, which equals 120. That means the iPhone will cost 800 minus 120, which is 680 after the discount.Next, the iWatch costs 300 with a 10% discount. To find the discount amount, I'll calculate 10% of 300. That's 0.10 multiplied by 300, which is 30. So, the iWatch will cost 300 minus 30, which is 270 after the discount.Now, I need to add the discounted prices of both items to get the total before cashback. That would be 680 for the iPhone plus 270 for the iWatch, totaling 950.Then, there's a 2% cashback on the total amount. To find out how much cashback Dexter gets, I'll calculate 2% of 950. That's 0.02 multiplied by 950, which equals 19.Finally, to find the total cost after applying the cashback, I'll subtract the cashback amount from the total before cashback. So, 950 minus 19 equals 931.Therefore, Dexter will pay a total of 931 for both the iPhone 12 and the iWatch after all discounts and cashback.

💡First, I need to determine the initial walking speed of the person. I'll denote this speed as ( v ) km/h.When the person walks at 5 km/h, they reach the bus stand 5 minutes early. This means the time taken to walk 5 km at 5 km/h is 5 minutes less than the bus's travel time. I'll calculate the time taken at 5 km/h:[text{Time} = frac{text{Distance}}{text{Speed}} = frac{5 text{ km}}{5 text{ km/h}} = 1 text{ hour} = 60 text{ minutes}]Since they reach 5 minutes early, the bus's travel time is:[60 text{ minutes} + 5 text{ minutes} = 65 text{ minutes}]Now, when the person walks at their initial speed ( v ), they miss the bus by 10 minutes. This means the time taken to walk 5 km at speed ( v ) is 10 minutes more than the bus's travel time:[text{Time at speed } v = 65 text{ minutes} + 10 text{ minutes} = 75 text{ minutes}]Converting 75 minutes to hours:[75 text{ minutes} = frac{75}{60} text{ hours} = 1.25 text{ hours}]Using the formula for speed:[text{Speed} = frac{text{Distance}}{text{Time}} = frac{5 text{ km}}{1.25 text{ hours}} = 4 text{ km/h}]Therefore, the initial walking speed of the person is 4 km/h.