💡Okay, so I have this question about Avogadro's constant, and I need to figure out which statement is correct among the four options: A, B, C, and D. Let me go through each one step by step.Starting with option A: "The number of D atoms in 18 g of D₂O is 2Nₐ." Hmm, D₂O is deuterium oxide, right? So, the molar mass of D₂O would be different from regular water because deuterium has a higher atomic mass. Let me calculate the molar mass of D₂O. Deuterium has an atomic mass of about 2 g/mol, and oxygen is about 16 g/mol. So, D₂O would be 2*2 + 16 = 20 g/mol. Now, if I have 18 g of D₂O, the number of moles would be 18 g divided by 20 g/mol, which is 0.9 mol. Each mole of D₂O contains 2 moles of D atoms, so 0.9 mol * 2 = 1.8 mol of D atoms. Since Avogadro's number is Nₐ, the number of D atoms would be 1.8Nₐ. But option A says it's 2Nₐ, which is higher than what I calculated. So, A is incorrect.Moving on to option B: "The number of anions in 1 L of 0.1 mol/L Na₂CO₃ solution is less than 0.1Nₐ." Sodium carbonate dissociates in water into 2 Na⁺ ions and one CO₃²⁻ ion. So, for every mole of Na₂CO₃, we get one mole of CO₃²⁻ anions. Given that the concentration is 0.1 mol/L and the volume is 1 L, the number of moles of CO₃²⁻ would be 0.1 mol. So, the number of anions should be 0.1Nₐ. But the statement says it's less than 0.1Nₐ. Wait, why would that be? Oh, because carbonate ions can undergo hydrolysis in water, right? CO₃²⁻ + H₂O ↔ HCO₃⁻ + OH⁻. This hydrolysis would produce more anions, like HCO₃⁻ and OH⁻, so the total number of anions would actually be more than 0.1Nₐ. Therefore, option B is incorrect.Next is option C: "The total number of electrons in 14 g of a mixed gas composed of N₂ and CO is 7Nₐ." Both N₂ and CO have a molar mass of 28 g/mol. So, 14 g of this mixture would be 0.5 mol. Now, each molecule of N₂ has 14 electrons (7 from each N atom), and each molecule of CO also has 14 electrons (6 from C and 8 from O). So, regardless of the ratio of N₂ to CO in the mixture, each molecule contributes 14 electrons. Therefore, 0.5 mol of the mixture would have 0.5 mol * 14Nₐ = 7Nₐ electrons. That matches what option C says, so C seems correct.Finally, option D: "The number of electrons transferred in the complete reaction of 2.24 L of chlorine gas with an excess of sodium hydroxide solution is 0.1Nₐ." Chlorine gas reacting with sodium hydroxide typically follows the reaction: Cl₂ + 2OH⁻ → Cl⁻ + ClO⁻ + H₂O. In this reaction, each Cl₂ molecule transfers one electron per Cl atom, so two electrons per Cl₂ molecule.But wait, the volume of Cl₂ is given as 2.24 L. However, the problem doesn't specify the temperature and pressure conditions. If it's at standard temperature and pressure (STP), 1 mole of gas is 22.4 L, so 2.24 L would be 0.1 mol. But if it's not at STP, we can't assume that. Since the conditions aren't specified, we can't accurately determine the number of moles of Cl₂, and thus can't determine the number of electrons transferred. Therefore, option D is incorrect.So, after going through all the options, it looks like option C is the correct one.

💡Okay, so I need to find the 1000th term in the sequence of positive integers that are coprime to 105, arranged in ascending order. Hmm, let me start by understanding what it means for two numbers to be coprime. Two numbers are coprime if their greatest common divisor (GCD) is 1. So, I'm looking for numbers that don't share any common factors with 105 except for 1.First, let me factorize 105 to understand its prime components. 105 can be broken down into prime factors: 105 = 3 × 5 × 7. So, any number that doesn't have 3, 5, or 7 as a factor will be coprime to 105.Now, to find how many numbers less than or equal to 105 are coprime to it, I can use Euler's Totient Function, φ(n). The formula for φ(n) when n is the product of distinct primes is:φ(n) = n × (1 - 1/p1) × (1 - 1/p2) × ... × (1 - 1/pk)Where p1, p2, ..., pk are the prime factors of n.Applying this to 105:φ(105) = 105 × (1 - 1/3) × (1 - 1/5) × (1 - 1/7)Calculating each part:1 - 1/3 = 2/31 - 1/5 = 4/51 - 1/7 = 6/7So,φ(105) = 105 × (2/3) × (4/5) × (6/7)Let me compute this step by step:First, 105 × (2/3) = 70Then, 70 × (4/5) = 56Next, 56 × (6/7) = 48So, φ(105) = 48. That means there are 48 numbers less than or equal to 105 that are coprime to 105.Now, the sequence of numbers coprime to 105 repeats every 105 numbers. So, every block of 105 numbers contains 48 numbers that are coprime to 105.To find the 1000th term, I can think of how many complete blocks of 105 fit into 1000 and then find the position within the next block.Let me calculate how many complete blocks there are:Number of complete blocks = 1000 ÷ 48Calculating this:48 × 20 = 960So, 20 complete blocks account for 960 numbers. That leaves 1000 - 960 = 40 numbers remaining.So, the 1000th term is the 40th number in the next block of 105 numbers.Now, each block is 105 numbers, so the starting number of the 21st block is 20 × 105 + 1 = 2101.Wait, actually, if each block is 105 numbers, the first block is 1-105, the second is 106-210, and so on. So, the 21st block would be from 20 × 105 + 1 = 2101 to 21 × 105 = 2205.But I need the 40th number in this block. So, I need to list the numbers from 2101 to 2205 and pick the 40th one that is coprime to 105.But listing all these numbers would be tedious. Maybe there's a smarter way.Since the coprimality is periodic with period 105, the pattern of coprime numbers repeats every 105 numbers. So, the 40th coprime number in the 21st block is the same as the 40th coprime number in the first block, but shifted by 20 × 105.Therefore, if I can find the 40th number in the first block (1-105) that is coprime to 105, then add 20 × 105 to it, I will get the 1000th term.So, let me list the numbers from 1 to 105 and pick the 40th one that is coprime to 105.But listing all 105 numbers is time-consuming. Maybe I can find a pattern or a formula.Alternatively, I can note that the coprime numbers to 105 are those not divisible by 3, 5, or 7.So, in each block of 105 numbers, the coprime numbers are the ones not divisible by 3, 5, or 7.To find the 40th such number, I can use the inclusion-exclusion principle.Wait, but maybe it's easier to list them.Alternatively, I can note that the coprime numbers to 105 are the numbers that are not multiples of 3, 5, or 7.So, in the first block (1-105), the coprime numbers are:1, 2, 4, 8, 11, 13, 16, 17, 19, 22, 23, 26, 29, 31, 32, 34, 37, 38, 41, 43, 44, 46, 47, 52, 53, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 82, 83, 86, 88, 89, 92, 94, 97, 101, 103, 104.Wait, that seems like 48 numbers, which matches φ(105)=48.Now, I need the 40th number in this list.Let me count them:1. 12. 23. 44. 85. 116. 137. 168. 179. 1910. 2211. 2312. 2613. 2914. 3115. 3216. 3417. 3718. 3819. 4120. 4321. 4422. 4623. 4724. 5225. 5326. 5827. 5928. 6129. 6230. 6431. 6732. 6833. 7134. 7335. 7436. 7637. 7938. 8239. 8340. 86So, the 40th number is 86.Therefore, the 1000th term is 86 + 20 × 105 = 86 + 2100 = 2186.Wait, let me double-check that.Each block of 105 numbers contributes 48 coprime numbers. So, 20 blocks give 960 numbers. The 1000th number is the 40th number in the 21st block.The 21st block starts at 2101, so the 40th coprime number in this block is 2100 + 86 = 2186.Yes, that seems correct.So, the 1000th term is 2186.

💡Okay, so I have this problem about a set A defined by the quadratic equation ax² + 2x + 1 = 0, where a is a real number. There are three parts to this problem, and I need to solve each one step by step. Let me try to understand each part and figure out how to approach them.Starting with part 1: If 1 is an element of A, represent A using the roster method. Hmm, okay. So, if 1 is in A, that means when x = 1, the equation ax² + 2x + 1 = 0 should hold true. So, I can substitute x = 1 into the equation and solve for a.Let me write that down:a*(1)² + 2*(1) + 1 = 0 a + 2 + 1 = 0 a + 3 = 0 So, a = -3.Alright, so when a is -3, the equation becomes -3x² + 2x + 1 = 0. Now, I need to find all real solutions x to this equation to represent set A using the roster method. That means I need to solve the quadratic equation.Let me write the equation again:-3x² + 2x + 1 = 0I can use the quadratic formula to find the roots. The quadratic formula is x = [-b ± sqrt(b² - 4ac)] / (2a). Here, a is -3, b is 2, and c is 1.Calculating the discriminant first:b² - 4ac = (2)² - 4*(-3)*(1) = 4 + 12 = 16Since the discriminant is positive, there are two real roots.Now, applying the quadratic formula:x = [-2 ± sqrt(16)] / (2*(-3)) x = [-2 ± 4] / (-6)So, let's compute both roots:First root: [-2 + 4]/(-6) = (2)/(-6) = -1/3 Second root: [-2 - 4]/(-6) = (-6)/(-6) = 1So, the solutions are x = -1/3 and x = 1. Therefore, set A is {-1/3, 1}.Wait, that seems straightforward. So, part 1 is done. I think I got that right.Moving on to part 2: If A contains exactly one element, find the set B composed of the real number a. Hmm, okay. So, A has exactly one element, which means the quadratic equation has exactly one real root. That happens when the discriminant is zero.But wait, I need to be careful here. If a is zero, the equation becomes linear, not quadratic. So, in that case, it's a linear equation, which has exactly one solution. So, I need to consider both cases: when a is zero and when a is not zero.Let me break it down.Case 1: a = 0 Then the equation becomes 0x² + 2x + 1 = 0, which simplifies to 2x + 1 = 0. Solving for x, we get x = -1/2. So, A = {-1/2}, which has exactly one element.Case 2: a ≠ 0 In this case, the equation is quadratic. For a quadratic equation to have exactly one real root, the discriminant must be zero.The discriminant D is given by D = b² - 4ac. Here, a is the coefficient of x², which is a, b is 2, and c is 1.So, D = (2)² - 4*a*1 = 4 - 4a.Setting D = 0 for exactly one real root:4 - 4a = 0 4a = 4 a = 1So, when a = 1, the quadratic equation becomes x² + 2x + 1 = 0, which factors to (x + 1)² = 0, giving a repeated root at x = -1. So, A = {-1}, which has exactly one element.Therefore, the set B, which consists of all real numbers a for which A has exactly one element, is {0, 1}.Wait, that makes sense. So, part 2 is also done.Now, part 3: If A contains at most one element, determine the range of values for a. Hmm, "at most one element" means that A can have either one element or no elements. So, I need to find all a such that A has either one element or is empty.From part 2, we know that A has exactly one element when a = 0 or a = 1.Now, when does A have no elements? That would be when the quadratic equation has no real roots. For a quadratic equation, that happens when the discriminant is negative.But again, I need to consider whether a is zero or not.Case 1: a = 0 As before, the equation is linear, 2x + 1 = 0, which has exactly one solution. So, A cannot be empty when a = 0.Case 2: a ≠ 0 Then, the equation is quadratic. For it to have no real roots, the discriminant must be negative.So, discriminant D = 4 - 4a < 0 4 - 4a < 0 -4a < -4 Dividing both sides by -4 (remembering to reverse the inequality sign when dividing by a negative number):a > 1So, when a > 1, the quadratic equation has no real roots, meaning A is empty.Therefore, combining both cases:- A has exactly one element when a = 0 or a = 1.- A has no elements when a > 1.So, the values of a for which A contains at most one element are a = 0, a = 1, and a > 1.Expressed as a range, that would be a ≥ 1 or a = 0. So, in interval notation, it's a ∈ {0} ∪ [1, ∞).Wait, let me make sure I didn't miss anything. When a = 1, A has exactly one element. When a > 1, A has no elements. When a = 0, A has exactly one element. For a between 0 and 1, excluding 1, the quadratic equation would have two real roots, so A would have two elements, which is more than one. So, those a's are excluded.Therefore, the range of a is all real numbers a such that a is either 0 or greater than or equal to 1.So, in set-builder notation, that would be {a | a = 0 or a ≥ 1}.Alternatively, using interval notation, it's {0} ∪ [1, ∞).I think that's correct.Let me recap:1. When x = 1 is in A, we found a = -3, leading to A = {-1/3, 1}.2. For A to have exactly one element, a must be 0 or 1.3. For A to have at most one element, a must be 0 or a ≥ 1.Yes, that seems consistent.I don't think I made any mistakes here, but let me double-check part 3.If a = 1, discriminant is zero, so one element. If a > 1, discriminant negative, so no elements. If a = 0, linear equation, one element. For a < 1 and a ≠ 0, discriminant positive, two elements. So, yes, at most one element when a = 0 or a ≥ 1.Alright, I think I've got all parts figured out.**Final Answer**1. boxed{left{-frac{1}{3}, 1right}} 2. boxed{{0, 1}} 3. boxed{{a mid a geq 1 text{ or } a = 0}}

💡Okay, so I have this problem where a cubic function intersects a line at three points, and I need to find the area between them. Let me try to break this down step by step.First, the cubic function is given by ( y = ax^3 + bx^2 + cx + d ) where ( a neq 0 ). The line is ( y = px + q ). They intersect at ( x = alpha, beta, gamma ) with ( alpha < beta < gamma ). So, these are the x-coordinates where the two graphs meet.To find the area between the two graphs, I know I need to integrate the difference between the cubic function and the line from ( x = alpha ) to ( x = gamma ). But wait, actually, since they intersect at three points, the area might be split into regions where one function is above the other. Hmm, but the problem just says "the region bounded by these graphs," so maybe it's the area between all three intersection points? Or is it just between two of them? Hmm, I need to clarify that.Wait, the problem says "the region bounded by these graphs," so I think it's the entire area between all three intersection points. But actually, since a cubic and a line can intersect at up to three points, the area between them would consist of two separate regions: one between ( alpha ) and ( beta ), and another between ( beta ) and ( gamma ). But depending on which function is on top in each interval, the area might alternate.But maybe the problem is simplified, and it's considering the total area between ( alpha ) and ( gamma ), regardless of which function is on top. Hmm, the problem says "the region bounded by these graphs," so perhaps it's considering the union of all regions where they enclose an area. So, that would be the sum of the areas between ( alpha ) and ( beta ) and between ( beta ) and ( gamma ).But wait, actually, since the cubic is a continuous function and the line is also continuous, the area between them from ( alpha ) to ( gamma ) would involve integrating the absolute difference between the two functions over that interval. However, integrating the absolute value can be tricky because we don't know which function is on top in each sub-interval.But maybe there's a smarter way to approach this. Since the cubic intersects the line at ( alpha, beta, gamma ), the difference between the cubic and the line, which is ( ax^3 + bx^2 + cx + d - (px + q) ), will have roots at ( x = alpha, beta, gamma ). So, this difference can be factored as ( a(x - alpha)(x - beta)(x - gamma) ).So, the difference between the cubic and the line is ( a(x - alpha)(x - beta)(x - gamma) ). Therefore, the area between the two graphs from ( alpha ) to ( gamma ) is the integral of the absolute value of this difference. But integrating the absolute value is complicated because we don't know the sign of the cubic in each interval.But maybe, since we have the roots, we can figure out the sign changes. Let's think about the behavior of the cubic. Since the leading coefficient is ( a ), as ( x ) approaches positive infinity, the cubic will go to positive infinity if ( a > 0 ) and negative infinity if ( a < 0 ). Similarly, as ( x ) approaches negative infinity, it will go to negative infinity if ( a > 0 ) and positive infinity if ( a < 0 ).But in our case, the roots are at ( alpha, beta, gamma ). So, the cubic crosses the line at these points. Between ( alpha ) and ( beta ), the cubic might be above or below the line, and similarly between ( beta ) and ( gamma ). The sign of the cubic difference will alternate between these intervals.But since we have the factorization ( a(x - alpha)(x - beta)(x - gamma) ), we can analyze the sign of this expression in each interval.Let me consider the sign of ( (x - alpha)(x - beta)(x - gamma) ) in each interval.1. For ( x < alpha ): All three factors are negative, so the product is negative.2. For ( alpha < x < beta ): ( (x - alpha) ) is positive, ( (x - beta) ) and ( (x - gamma) ) are negative, so the product is positive.3. For ( beta < x < gamma ): ( (x - alpha) ) and ( (x - beta) ) are positive, ( (x - gamma) ) is negative, so the product is negative.4. For ( x > gamma ): All three factors are positive, so the product is positive.But since we are integrating from ( alpha ) to ( gamma ), we are concerned with the intervals ( alpha ) to ( beta ) and ( beta ) to ( gamma ).In ( alpha ) to ( beta ), the product ( (x - alpha)(x - beta)(x - gamma) ) is positive, as we saw. In ( beta ) to ( gamma ), it's negative.Therefore, the difference ( ax^3 + bx^2 + cx + d - (px + q) ) will be positive in ( alpha ) to ( beta ) and negative in ( beta ) to ( gamma ) if ( a > 0 ). If ( a < 0 ), the signs would be reversed.But since we are taking the absolute value for the area, we can write the area as the sum of the integrals where the cubic is above the line and vice versa.So, the total area ( A ) is:[A = int_{alpha}^{beta} |f(x) - g(x)| dx + int_{beta}^{gamma} |f(x) - g(x)| dx]Where ( f(x) ) is the cubic and ( g(x) ) is the line.But since we know the sign of ( f(x) - g(x) ) in each interval, we can drop the absolute value by considering the sign.If ( a > 0 ):- From ( alpha ) to ( beta ), ( f(x) - g(x) ) is positive, so ( |f(x) - g(x)| = f(x) - g(x) ).- From ( beta ) to ( gamma ), ( f(x) - g(x) ) is negative, so ( |f(x) - g(x)| = g(x) - f(x) ).If ( a < 0 ), it would be the opposite.But since the area is a positive quantity, we can write:[A = int_{alpha}^{beta} (f(x) - g(x)) dx + int_{beta}^{gamma} (g(x) - f(x)) dx]But since ( f(x) - g(x) = a(x - alpha)(x - beta)(x - gamma) ), we can write:[A = int_{alpha}^{beta} a(x - alpha)(x - beta)(x - gamma) dx + int_{beta}^{gamma} -a(x - alpha)(x - beta)(x - gamma) dx]But this seems a bit complicated. Maybe there's a better way.Wait, actually, since ( f(x) - g(x) = a(x - alpha)(x - beta)(x - gamma) ), we can express the area as:[A = int_{alpha}^{gamma} |a(x - alpha)(x - beta)(x - gamma)| dx]But integrating the absolute value is still tricky. Alternatively, since we know the sign changes, we can split the integral into two parts where the function is positive and negative.But perhaps there's a clever substitution or symmetry we can exploit here.Let me consider substituting ( t = x - alpha ). Then, when ( x = alpha ), ( t = 0 ), and when ( x = gamma ), ( t = gamma - alpha ). But I'm not sure if that helps directly.Alternatively, maybe we can use substitution to make the integral symmetric. Let me think about the roots.The roots are ( alpha, beta, gamma ). Let me denote ( beta = frac{alpha + gamma}{2} + h ), but I'm not sure.Wait, perhaps we can use the fact that the integral of ( (x - alpha)(x - beta)(x - gamma) ) can be expressed in terms of symmetric functions.Alternatively, maybe we can perform a substitution to make the integral symmetric around zero.Let me try to shift the variable so that the interval ( [alpha, gamma] ) is centered at zero. Let me set ( t = x - frac{alpha + gamma}{2} ). Then, ( x = t + frac{alpha + gamma}{2} ), and when ( x = alpha ), ( t = -frac{gamma - alpha}{2} ), and when ( x = gamma ), ( t = frac{gamma - alpha}{2} ).So, the integral becomes:[A = int_{-frac{gamma - alpha}{2}}^{frac{gamma - alpha}{2}} |a(t + frac{alpha + gamma}{2} - alpha)(t + frac{alpha + gamma}{2} - beta)(t + frac{alpha + gamma}{2} - gamma)| dt]Simplify each term:1. ( t + frac{alpha + gamma}{2} - alpha = t - frac{alpha - gamma}{2} = t + frac{gamma - alpha}{2} )2. ( t + frac{alpha + gamma}{2} - beta = t + frac{alpha + gamma}{2} - beta )3. ( t + frac{alpha + gamma}{2} - gamma = t - frac{gamma - alpha}{2} )So, the integral becomes:[A = |a| int_{-frac{gamma - alpha}{2}}^{frac{gamma - alpha}{2}} |(t + frac{gamma - alpha}{2})(t + frac{alpha + gamma}{2} - beta)(t - frac{gamma - alpha}{2})| dt]Hmm, this seems more complicated. Maybe I should instead consider expanding the polynomial and integrating term by term.Let me recall that ( (x - alpha)(x - beta)(x - gamma) ) is a cubic polynomial. Let me expand it:[(x - alpha)(x - beta)(x - gamma) = x^3 - (alpha + beta + gamma)x^2 + (alphabeta + betagamma + gammaalpha)x - alphabetagamma]So, the difference ( f(x) - g(x) = a(x^3 - (alpha + beta + gamma)x^2 + (alphabeta + betagamma + gammaalpha)x - alphabetagamma) )Therefore, the integral becomes:[A = int_{alpha}^{gamma} |a(x^3 - (alpha + beta + gamma)x^2 + (alphabeta + betagamma + gammaalpha)x - alphabetagamma)| dx]But integrating this absolute value is still complicated. Maybe instead of integrating the absolute value, I can consider the definite integral without the absolute value and then take the absolute value at the end, but that might not work because the function changes sign.Wait, but earlier I thought that the function is positive between ( alpha ) and ( beta ) and negative between ( beta ) and ( gamma ) if ( a > 0 ). So, the integral from ( alpha ) to ( beta ) is positive, and from ( beta ) to ( gamma ) is negative. So, the total integral without absolute value would be the area from ( alpha ) to ( beta ) minus the area from ( beta ) to ( gamma ). But since area is positive, we need to take the absolute value of each part.But maybe instead of splitting the integral, I can find a formula for the integral of ( (x - alpha)(x - beta)(x - gamma) ) from ( alpha ) to ( gamma ).Let me compute the integral:[int_{alpha}^{gamma} (x - alpha)(x - beta)(x - gamma) dx]Let me make a substitution to simplify this integral. Let me set ( t = x - alpha ), so ( x = t + alpha ), and when ( x = alpha ), ( t = 0 ), and when ( x = gamma ), ( t = gamma - alpha ).So, the integral becomes:[int_{0}^{gamma - alpha} (t)(t + alpha - beta)(t + alpha - gamma) dt]Simplify each term:1. ( t ) remains ( t )2. ( t + alpha - beta = t - (beta - alpha) )3. ( t + alpha - gamma = t - (gamma - alpha) )So, the integral is:[int_{0}^{gamma - alpha} t(t - (beta - alpha))(t - (gamma - alpha)) dt]Let me denote ( L = gamma - alpha ), which is the length of the interval. Then, the integral becomes:[int_{0}^{L} t(t - (beta - alpha))(t - L) dt]Let me expand the integrand:First, compute ( (t - (beta - alpha))(t - L) ):[(t - (beta - alpha))(t - L) = t^2 - (L + beta - alpha)t + (beta - alpha)L]So, the integrand becomes:[t cdot [t^2 - (L + beta - alpha)t + (beta - alpha)L] = t^3 - (L + beta - alpha)t^2 + (beta - alpha)L t]Therefore, the integral is:[int_{0}^{L} [t^3 - (L + beta - alpha)t^2 + (beta - alpha)L t] dt]Now, integrate term by term:1. ( int t^3 dt = frac{t^4}{4} )2. ( int (L + beta - alpha)t^2 dt = (L + beta - alpha) cdot frac{t^3}{3} )3. ( int (beta - alpha)L t dt = (beta - alpha)L cdot frac{t^2}{2} )Evaluate each from 0 to ( L ):1. ( frac{L^4}{4} - 0 = frac{L^4}{4} )2. ( (L + beta - alpha) cdot frac{L^3}{3} - 0 = frac{(L + beta - alpha)L^3}{3} )3. ( (beta - alpha)L cdot frac{L^2}{2} - 0 = frac{(beta - alpha)L^3}{2} )So, putting it all together:[int_{0}^{L} [t^3 - (L + beta - alpha)t^2 + (beta - alpha)L t] dt = frac{L^4}{4} - frac{(L + beta - alpha)L^3}{3} + frac{(beta - alpha)L^3}{2}]Simplify each term:First, note that ( L = gamma - alpha ), so ( beta - alpha = beta - alpha ), and ( L + beta - alpha = (gamma - alpha) + (beta - alpha) = gamma + beta - 2alpha ).But let's see:Let me factor ( L^3 ) from the second and third terms:[- frac{(L + beta - alpha)L^3}{3} + frac{(beta - alpha)L^3}{2} = L^3 left( -frac{L + beta - alpha}{3} + frac{beta - alpha}{2} right )]Compute the expression inside the parentheses:[- frac{L + beta - alpha}{3} + frac{beta - alpha}{2} = - frac{L}{3} - frac{beta - alpha}{3} + frac{beta - alpha}{2}]Combine the terms with ( beta - alpha ):[- frac{L}{3} + left( - frac{beta - alpha}{3} + frac{beta - alpha}{2} right ) = - frac{L}{3} + left( frac{-2(beta - alpha) + 3(beta - alpha)}{6} right ) = - frac{L}{3} + frac{(beta - alpha)}{6}]So, the integral becomes:[frac{L^4}{4} + L^3 left( - frac{L}{3} + frac{beta - alpha}{6} right ) = frac{L^4}{4} - frac{L^4}{3} + frac{L^3(beta - alpha)}{6}]Combine the ( L^4 ) terms:[frac{L^4}{4} - frac{L^4}{3} = - frac{L^4}{12}]So, the integral is:[- frac{L^4}{12} + frac{L^3(beta - alpha)}{6}]Factor ( frac{L^3}{12} ):[frac{L^3}{12} left( -L + 2(beta - alpha) right ) = frac{L^3}{12} (2(beta - alpha) - L)]But ( L = gamma - alpha ), so:[2(beta - alpha) - (gamma - alpha) = 2beta - 2alpha - gamma + alpha = 2beta - gamma - alpha]Therefore, the integral is:[frac{(gamma - alpha)^3}{12} (2beta - gamma - alpha)]So, going back to the original integral:[int_{alpha}^{gamma} (x - alpha)(x - beta)(x - gamma) dx = frac{(gamma - alpha)^3}{12} (2beta - gamma - alpha)]But remember, the difference ( f(x) - g(x) = a(x - alpha)(x - beta)(x - gamma) ). So, the integral of the difference is:[a cdot frac{(gamma - alpha)^3}{12} (2beta - gamma - alpha)]But this is the integral without taking absolute value. However, since we are interested in the area, which is the absolute value of this integral, we need to take the absolute value.But wait, actually, the integral we computed is the integral of ( (x - alpha)(x - beta)(x - gamma) ), which is the same as ( frac{f(x) - g(x)}{a} ). So, the integral of ( f(x) - g(x) ) is ( a ) times that integral.But since the area is the integral of the absolute difference, and we know the sign changes, we can write the area as the absolute value of the integral from ( alpha ) to ( beta ) plus the absolute value of the integral from ( beta ) to ( gamma ). However, since we have the entire integral from ( alpha ) to ( gamma ), which is ( a cdot frac{(gamma - alpha)^3}{12} (2beta - gamma - alpha) ), the area would be the absolute value of this.But wait, actually, the integral from ( alpha ) to ( gamma ) is the net area, which is the area where the cubic is above the line minus the area where it's below. But since we need the total area, regardless of which is on top, we need to take the absolute value of each part.But given that the integral we computed is ( a cdot frac{(gamma - alpha)^3}{12} (2beta - gamma - alpha) ), and since the area is the sum of the absolute integrals over each interval, we can express the area as:[text{Area} = left| a cdot frac{(gamma - alpha)^3}{12} (2beta - gamma - alpha) right|]But let's verify this. Suppose ( a > 0 ). Then, between ( alpha ) and ( beta ), the cubic is above the line, so the integral is positive. Between ( beta ) and ( gamma ), the cubic is below the line, so the integral is negative. Therefore, the total integral from ( alpha ) to ( gamma ) is positive area minus negative area, which is the net area. But we need the total area, which is the sum of the absolute values.However, in our case, we have expressed the integral as ( a cdot frac{(gamma - alpha)^3}{12} (2beta - gamma - alpha) ). The term ( 2beta - gamma - alpha ) can be rewritten as ( 2beta - (gamma + alpha) ). If ( beta ) is the middle root, then ( beta = frac{alpha + gamma}{2} ) only if the roots are symmetric, which they are not necessarily. So, ( 2beta - gamma - alpha ) is just a constant based on the roots.But regardless, the key point is that the integral we computed is proportional to ( (gamma - alpha)^3 ) and the term ( 2beta - gamma - alpha ). Therefore, the area is the absolute value of this integral.So, putting it all together, the area is:[text{Area} = frac{|a|}{12} (gamma - alpha)^3 |2beta - gamma - alpha|]Wait, but let me check the units. The integral of a cubic function over an interval has units of area, which is consistent. The term ( (gamma - alpha)^3 ) is a volume, but multiplied by ( a ), which has units of 1/volume (since it's a cubic term), so it cancels out, leaving area. Hmm, actually, no, because ( a ) is the coefficient of ( x^3 ), so it has units of 1/(length)^2 if y has units of length. Wait, maybe I'm overcomplicating.But regardless, the formula seems dimensionally consistent because ( a ) is multiplied by ( (gamma - alpha)^3 ), which is a length cubed, and the term ( 2beta - gamma - alpha ) is a length, so overall, it's ( a times text{length}^4 ), but since ( a ) is 1/(length)^2, it becomes 1/(length)^2 * length^4 = length^2, which is area. So, that checks out.Therefore, the final answer is:[boxed{frac{|a|}{12} (gamma - alpha)^3 |2beta - gamma - alpha|}]

💡Alright, so I have this problem here that asks me to figure out which of the given square roots is the simplest. The options are:A: √3B: √0.5C: √(2/5)D: √8Okay, let's break this down step by step. First, I need to understand what it means for a square root to be "simplest." From what I remember, a square root is considered simplest when it doesn't have any perfect square factors inside the radical, and if it's a fraction, it shouldn't have a radical in the denominator. So, my goal is to simplify each of these square roots as much as possible and see which one is already in its simplest form or becomes the simplest after simplification.Starting with Option A: √3. Hmm, 3 is a prime number, right? So, it doesn't have any perfect square factors other than 1. That means √3 can't be simplified any further. So, √3 is already in its simplest form. That seems straightforward.Moving on to Option B: √0.5. Okay, 0.5 is the same as 1/2. So, √0.5 is the same as √(1/2). Now, I remember that √(a/b) is equal to √a / √b. So, √(1/2) is the same as √1 / √2, which simplifies to 1/√2. But wait, having a radical in the denominator isn't considered simplest form. To rationalize the denominator, I need to multiply both the numerator and the denominator by √2. That gives me (√2)/(√2 * √2) = √2/2. So, √0.5 simplifies to √2/2. That's better, but it's still not as simple as √3 because it involves a fraction.Next up is Option C: √(2/5). Similar to Option B, this is a square root of a fraction. Using the same property, √(a/b) = √a / √b, so √(2/5) is √2 / √5. Again, we have a radical in the denominator, so we need to rationalize it. Multiply numerator and denominator by √5: (√2 * √5)/(√5 * √5) = √10/5. So, √(2/5) simplifies to √10/5. That's simpler than √2/2, but still not as simple as √3.Finally, Option D: √8. Let's see, 8 can be factored into 4 and 2, and 4 is a perfect square. So, √8 = √(4*2) = √4 * √2 = 2√2. So, √8 simplifies to 2√2. That's simpler than both √2/2 and √10/5, but still, it's not as simple as √3 because it has a coefficient in front of the radical.So, summarizing my findings:- A: √3 (already simplest)- B: √0.5 simplifies to √2/2- C: √(2/5) simplifies to √10/5- D: √8 simplifies to 2√2Comparing all these, √3 is the simplest because it doesn't require any further simplification. The other options, while they can be simplified, still involve fractions or coefficients, making them more complex than √3.I think I've covered all the steps and made sure to simplify each option properly. I don't see any mistakes in my reasoning, but just to double-check:For Option B, √0.5 = √(1/2) = 1/√2 = √2/2 after rationalizing. That seems correct.For Option C, √(2/5) = √2/√5 = √10/5 after rationalizing. That also looks right.For Option D, √8 = √(4*2) = 2√2. Yep, that's correct.And Option A, √3, is already in its simplest form because 3 is prime. So, no issues there.Therefore, I'm confident that Option A: √3 is the simplest square root among the given options.

💡Okay, so I have these four propositions, and I need to figure out which ones are correct. Let me go through each one step by step.**Proposition 1:** If ( m > 0 ), the function ( f(x) = x^2 + x - m ) has zero points.Hmm, okay. So, a zero point means the function crosses the x-axis, right? That happens when the quadratic equation ( x^2 + x - m = 0 ) has real roots. To find out if it has real roots, I can check the discriminant. The discriminant ( D ) of a quadratic ( ax^2 + bx + c ) is ( D = b^2 - 4ac ). In this case, ( a = 1 ), ( b = 1 ), and ( c = -m ). So, the discriminant is ( D = 1^2 - 4(1)(-m) = 1 + 4m ). Since ( m > 0 ), ( 4m ) is positive, so ( D = 1 + 4m > 0 ). A positive discriminant means there are two distinct real roots. Therefore, the function does have zero points. So, Proposition 1 is correct.**Proposition 2:** Given four points ( E ), ( F ), ( G ), ( H ) in space, proposition A: points ( E ), ( F ), ( G ), ( H ) are not coplanar, and proposition B: lines ( EF ) and ( GH ) do not intersect. Then, A is a necessary but not sufficient condition for B.Alright, so A is about the four points not being coplanar, and B is about the lines ( EF ) and ( GH ) not intersecting. The statement is saying that A is necessary for B but not sufficient.First, let me recall that if four points are not coplanar, then any two lines connecting them (like ( EF ) and ( GH )) cannot intersect because they are skew lines. So, if A is true, then B must be true. That means A is a sufficient condition for B. But the proposition says A is necessary but not sufficient. Wait, that seems contradictory. If A is sufficient, then it's not just necessary. Maybe I'm misunderstanding. Let me think again.If four points are coplanar, then lines ( EF ) and ( GH ) could intersect or not. So, if B is true (lines don't intersect), does that necessarily mean A is true (points are not coplanar)? No, because even if the points are coplanar, the lines might still not intersect if they are parallel. So, B can be true without A being true. Therefore, A is not a necessary condition for B. Hmm, but the proposition says A is necessary but not sufficient. That doesn't seem right because A is actually sufficient, not necessary.Wait, maybe I got it backwards. Let me rephrase:- If A is true (points not coplanar), then B is true (lines don't intersect). So, A implies B. Therefore, A is a sufficient condition for B.- However, B can be true even if A is false. For example, if the points are coplanar, but the lines are parallel, so they don't intersect. Therefore, B doesn't necessarily imply A.So, A is sufficient for B, but not necessary. The proposition says A is necessary but not sufficient, which is incorrect. Therefore, Proposition 2 is incorrect.**Proposition 3:** "( a < 2 )" is a necessary and sufficient condition for "( |x + 1| + |x - 1| geqslant a ) holds for any real number ( x )".Okay, so we need to find the condition on ( a ) such that the inequality ( |x + 1| + |x - 1| geq a ) is true for all real ( x ).First, let me analyze the expression ( |x + 1| + |x - 1| ). This is the sum of distances from ( x ) to -1 and 1 on the real line. The minimum value of this expression occurs when ( x ) is between -1 and 1. Let me compute the minimum value. If ( x ) is between -1 and 1, then ( |x + 1| + |x - 1| = (x + 1) + (1 - x) = 2 ). If ( x ) is outside this interval, say ( x > 1 ), then ( |x + 1| + |x - 1| = (x + 1) + (x - 1) = 2x ), which is greater than 2. Similarly, if ( x < -1 ), it's ( -(x + 1) - (x - 1) = -2x ), which is also greater than 2. So, the minimum value of ( |x + 1| + |x - 1| ) is 2.Therefore, for the inequality ( |x + 1| + |x - 1| geq a ) to hold for all real ( x ), ( a ) must be less than or equal to 2. But the proposition says "( a < 2 )" is a necessary and sufficient condition. Wait, if ( a = 2 ), the inequality still holds because the minimum is 2. So, the condition should be ( a leq 2 ), not just ( a < 2 ). Therefore, "( a < 2 )" is not a necessary condition because ( a = 2 ) also works. It is sufficient, but not necessary. So, Proposition 3 is incorrect.**Proposition 4:** "( 0 < m < 1 )" is a necessary and sufficient condition for "the equation ( mx^2 + (m - 1)y^2 = 1 ) represents a hyperbola".Alright, so I need to determine when the equation ( mx^2 + (m - 1)y^2 = 1 ) represents a hyperbola.First, recall that the general form of a conic section is ( Ax^2 + By^2 + Cx + Dy + E = 0 ). For it to be a hyperbola, the coefficients of ( x^2 ) and ( y^2 ) must have opposite signs. In our equation, ( A = m ) and ( B = m - 1 ). So, for the equation to represent a hyperbola, we need ( m ) and ( m - 1 ) to have opposite signs.Let me solve the inequality ( m(m - 1) < 0 ). This product is negative when one factor is positive and the other is negative.Case 1: ( m > 0 ) and ( m - 1 < 0 ). This implies ( 0 < m < 1 ).Case 2: ( m < 0 ) and ( m - 1 > 0 ). This implies ( m < 0 ) and ( m > 1 ), which is impossible.Therefore, the only solution is ( 0 < m < 1 ). So, "( 0 < m < 1 )" is indeed a necessary and sufficient condition for the equation to represent a hyperbola. Therefore, Proposition 4 is correct.**Summary:**1. Correct2. Incorrect3. Incorrect4. CorrectSo, out of the four propositions, two are correct (Propositions 1 and 4). Therefore, the number of correct propositions is 2.**Final Answer**boxed{B}