🧠:[<think>Okay, so I have this problem here about defining an operation called "∗" for natural numbers n. The problem gives two properties: first, that 1∗1 equals 1, and second, that (n+1)∗1 equals n∗1 plus 1. Then they ask what n∗1 equals, with options A through D. Let me try to figure this out step by step.First, let me restate the problem to make sure I understand it. We need to define this operation "∗" such that when both operands are 1, the result is 1. Then, for any natural number n, if we take (n+1)∗1, it should be equal to n∗1 plus 1. The question is then asking what the general expression for n∗1 is, given these two rules. The options are A: n, B: n+1, C: n-1, D: n². So we need to determine which of these expressions satisfies the given properties.Let me start by analyzing the given properties. The first one is straightforward: when both numbers are 1, the operation gives 1. The second one is a recursive definition. It tells us how to compute (n+1)∗1 if we know n∗1. This seems similar to how addition is defined recursively. For example, in Peano arithmetic, the successor function is used to define addition. Let me see if this operation "∗" is actually just addition.Wait, but if we look at the second property: (n+1)∗1 = n∗1 + 1. If we think of "∗" as a function f(n) = n∗1, then the equation becomes f(n+1) = f(n) + 1. This is a simple recurrence relation. Let me write that down:f(n+1) = f(n) + 1, with the initial condition f(1) = 1.Solving this recurrence relation should give us the general formula for f(n) = n∗1.Recurrence relations of the form f(n+1) = f(n) + c, where c is a constant, have linear solutions. Specifically, if each term is the previous term plus a constant, the sequence is arithmetic. The general solution for such a recurrence is f(n) = f(1) + c*(n - 1). In this case, c is 1 because each time we add 1. Let's check that.Given f(1) = 1.Then f(2) = f(1) + 1 = 1 + 1 = 2.f(3) = f(2) + 1 = 2 + 1 = 3.Continuing this pattern, f(n) = 1 + 1*(n - 1) = 1 + n - 1 = n.Wait, so according to this, f(n) = n. Therefore, n∗1 would equal n. So the answer is A: n.But let me verify this with the given properties. Let's test the base case: 1∗1 = 1, which matches the given condition (i). Now, let's check the recursive step. Suppose n∗1 = n. Then (n+1)∗1 should be n∗1 + 1 = n + 1. But according to our solution, (n+1)∗1 = n + 1. Which is exactly what we get if we follow the operation. Wait a second, but according to this, (n+1)∗1 = n + 1, which would be the same as the result of our function. But according to our recurrence, f(n+1) = f(n) + 1, which gives us f(n+1) = n + 1. So everything checks out. Therefore, n∗1 = n.But let me double-check to make sure I haven't made a mistake. Let's compute a few values manually.For n=1: 1∗1=1, which is correct.For n=2: 2∗1 should be (1+1)∗1 = 1∗1 + 1 = 1 + 1 = 2. So 2∗1=2. That's consistent with option A: n=2 gives 2.For n=3: 3∗1 = (2+1)∗1 = 2∗1 + 1 = 2 + 1 = 3. So again, 3∗1=3. Similarly, n=4 would be 4, etc. So each time, n∗1 equals n. Therefore, the answer must be A: n.But wait, let me check the other options to make sure none of them could also satisfy the conditions. For example, if someone thought that maybe the operation was addition, then n∗1 = n + 1, which is option B. Let's test that.If n∗1 were equal to n + 1, then let's check the base case. 1∗1 would be 1 + 1 = 2, but according to the problem statement, 1∗1=1. So that's a contradiction. Therefore, option B is invalid.Option C: n - 1. Testing n=1: 1 - 1 = 0, which contradicts 1∗1=1. So that's out.Option D: n². For n=1, 1²=1, which is okay. For n=2, (1+1)∗1 should equal 1∗1 + 1 = 1 + 1 = 2. But if n=2, 2²=4, which is not 2. So D is invalid.Therefore, only option A satisfies both conditions. So the answer is A: n.But wait, let me consider another angle. Maybe the operation "∗" isn't defined as a standard operation, so maybe there's a trick here. Let me re-examine the problem statement.The problem says: "Define an operation '∗' for natural numbers n with the following properties: (i) 1∗1=1, (ii) (n+1)∗1 = n∗1 + 1. Then, n∗1 equals to ( )".Wait, the operation is defined for natural numbers n. But how is the operation defined? The operation "∗" takes two arguments, right? Because we have expressions like 1∗1 and (n+1)∗1. So it's a binary operation. But the problem says "for natural numbers n", which might be slightly confusing. Wait, perhaps the operation is only defined for pairs where the second argument is 1? Because both properties only mention operations where the second operand is 1. That is, 1∗1 and (n+1)∗1. So maybe the operation is only being defined in cases where the second operand is 1, and we are to find n∗1 for any n.In that case, the operation is being defined recursively on the first argument, keeping the second argument fixed at 1. So, starting with 1∗1=1, then each subsequent (n+1)∗1 is equal to n∗1 + 1. Therefore, as I thought before, this is defining n∗1 as a function f(n) where f(1)=1 and f(n+1)=f(n)+1, leading to f(n)=n.Alternatively, if we model this as a recursive function:f(1) = 1f(n+1) = f(n) + 1This is exactly the definition of the successor function in Peano arithmetic, where each natural number is defined as the successor of the previous one. So f(n) = n for all natural numbers n. Therefore, n∗1 = n, which is option A.Therefore, the answer is A.But let me think again if there's any possible way this could be different. For example, is there a different function that could satisfy these properties? Suppose someone thought that the operation might involve multiplication or another operation. Let's test that.Suppose someone thought that "∗" was multiplication. Then, 1∗1=1, which holds. Then, (n+1)∗1 would be (n+1)*1 = n + 1. On the other hand, n∗1 + 1 would be n*1 + 1 = n + 1. So in this case, it would hold. Wait, that's interesting. If "∗" was multiplication, then n∗1 would be n*1 = n, which is option A. So multiplication does satisfy both properties. Therefore, the operation "∗" is just multiplication, and n∗1 = n. So answer is A.Alternatively, if the operation was addition, then n∗1 would be n + 1, but as we saw earlier, that doesn't satisfy the base case. If the operation was subtraction, then n∗1 would be n - 1, which fails the base case. If it was exponentiation, 1∗1=1^1=1, which works, but (n+1)∗1 = (n+1)^1 = n + 1, and n∗1 + 1 would be n^1 + 1 = n + 1, so that also works. Wait, so exponentiation would also satisfy these properties?Wait, let me check:If "∗" is exponentiation, so a∗b = a^b.Then 1∗1 = 1^1 = 1, which is okay.(n+1)∗1 = (n+1)^1 = n + 1.On the other hand, n∗1 + 1 = n^1 + 1 = n + 1.So (n+1)∗1 equals n∗1 + 1. Therefore, exponentiation also satisfies these properties. Then, according to this, n∗1 would be n^1 = n, which is again option A.So both multiplication and exponentiation, when restricted to the case where the second operand is 1, satisfy the given properties, and both result in n∗1 = n. Therefore, regardless of whether the operation is multiplication or exponentiation, the answer would still be A.Therefore, the answer must be A: n.Another way to see it is that the given properties define the operation n∗1 as a function that starts at 1 and increments by 1 each time n increases by 1. So it's essentially the identity function on natural numbers. Therefore, n∗1 = n.Yes, this all makes sense. The recursive definition given is a standard way to define the natural numbers themselves, where each number is built by adding 1 successively. Therefore, the operation here is essentially just counting, which results in n∗1 being equal to n.So after going through various possibilities and checking each one, the conclusion is that the correct answer is A: n.Final Answerboxed{A}</think>Given the operation "ast" for natural numbers n with the properties:1. 1 ast 1 = 12. (n + 1) ast 1 = n ast 1 + 1We need to determine the value of n ast 1.First, we recognize that the second property defines a recurrence relation. Let f(n) = n ast 1. Then the recurrence relation is:[ f(n + 1) = f(n) + 1 ]with the initial condition:[ f(1) = 1 ]This recurrence relation is a simple arithmetic sequence where each term is the previous term plus 1. Solving this recurrence relation, we find:[ f(n) = 1 + 1 cdot (n - 1) = n ]Thus, n ast 1 = n.To verify, we check the base case and a few values:- For n = 1, 1 ast 1 = 1 which satisfies the first property.- For n = 2, (1 + 1) ast 1 = 1 ast 1 + 1 = 1 + 1 = 2.- For n = 3, (2 + 1) ast 1 = 2 ast 1 + 1 = 2 + 1 = 3.All these values confirm that n ast 1 = n. Checking other options (B, C, D) shows they do not satisfy the given properties.Therefore, the answer is:[boxed{A}],<think>Okay, let me try to figure out this problem. So, we have this operation "∗" defined for natural numbers n, and there are two properties given. The first one is that 1∗1 equals 1. The second property is that (n+1)∗1 equals n∗1 plus 1. Then, we need to find what n∗1 equals, with the options given. Let me start by writing down the given information to clarify things. The operation "∗" has two rules:1. When both numbers are 1, the result is 1. So, 1∗1 = 1.2. For any natural number n, if we take (n+1)∗1, it's equal to n∗1 + 1. The question is asking what n∗1 equals in general. The options are A: n, B: n+1, C: n-1, D: n². So, we need to figure out which one of these is consistent with the given properties.Hmm, this seems like a recursive definition. The first condition gives the base case, and the second condition gives the recursive step. Maybe this operation is similar to addition or something else? Let's see.Let me try to compute some small cases to see if I can spot a pattern. Let's start with n=1 since we know 1∗1=1. Then, let's compute 2∗1 using the second rule. If n=1, then (1+1)∗1 = 2∗1 = 1∗1 + 1. According to the first property, 1∗1 is 1, so 2∗1 = 1 + 1 = 2. Okay, so 2∗1 is 2. Let's do the next one. For n=2, (2+1)∗1 = 3∗1 = 2∗1 + 1. We just found that 2∗1 is 2, so 3∗1 = 2 + 1 = 3. Hmm, so 3∗1 is 3. Let's check n=3: (3+1)∗1 = 4∗1 = 3∗1 + 1 = 3 + 1 = 4. Wait a second, this seems like each time, n∗1 is just equal to n. Let me verify for another value. Let's do n=4: 4∗1 = 3∗1 + 1 = 3 + 1 = 4. Yep, same pattern. So, if this pattern continues, n∗1 = n. But let me make sure that this holds for all natural numbers. The first case, n=1, works because 1∗1=1. Then, assuming that for some natural number k, k∗1 = k, then (k+1)∗1 = k∗1 + 1 = k + 1. So, by induction, this holds for all natural numbers. Therefore, n∗1 is equal to n. So, the answer should be option A: n. Wait, but let me check the problem statement again to ensure I didn't misinterpret anything. The operation is defined for natural numbers n with properties (i) 1∗1=1, and (ii) (n+1)∗1 = n∗1 + 1. Then, n∗1 equals which of the options. The options are A: n, B: n+1, etc. Hmm, so according to the calculations, n∗1 is n. That would be option A. Let me check if the options are correctly labeled. Yes, A is n. So, that's the answer. But just to be thorough, let's check if any other options could fit. Suppose, for instance, if someone thought maybe it's n+1. But when n=1, 1∗1 would have to be 1+1=2, but according to (i), it's 1. So, that's not possible. Similarly, option C: n-1. If n=1, 1-1=0, but 1∗1=1, so that doesn't work. Option D: n squared. For n=1, 1²=1, which works, but for n=2, 2²=4, but we already calculated 2∗1=2, which is different. So D is out. Therefore, only option A works. So, the answer is A: n.</think>The operation "ast" is defined recursively with two properties: 1. Base case: 1 ast 1 = 1. 2. Recursive step: (n + 1) ast 1 = n ast 1 + 1. To determine n ast 1, we compute values for small n: - For n = 1, 1 ast 1 = 1 (given). - For n = 2, (1 + 1) ast 1 = 2 ast 1 = 1 ast 1 + 1 = 1 + 1 = 2. - For n = 3, (2 + 1) ast 1 = 3 ast 1 = 2 ast 1 + 1 = 2 + 1 = 3. - This pattern continues: n ast 1 = n for all tested n. Proof by induction: - Base case: 1 ast 1 = 1 holds. - Inductive step: Assume k ast 1 = k. Then, (k + 1) ast 1 = k ast 1 + 1 = k + 1. Thus, n ast 1 = n for all natural numbers n. Answer: boxed{A}]

🔑:Given 1ast1=1 and (n+1)ast1=nast1+1, we can deduce that:(n+1)ast1=nast1+1=(n-1)ast1+1+1=(n-2)ast1+3=dots=[n-(n-1)]ast1+n=1+n.Therefore, nast1=n.Hence, the correct choice is boxed{A}.

🧠:[<think>Okay, so I need to find the minimum distance between a point P on the curve y = x² + 2 and a point Q on the curve y = √(x - 2). Hmm, let's start by understanding what's given here. First, both curves are defined in the coordinate plane. The point P is somewhere on the parabola y = x² + 2, which opens upward and has its vertex at (0, 2). The point Q is on the curve y = √(x - 2), which is a square root function shifted to the right by 2 units. This function starts at x = 2, y = 0 and increases slowly as x increases. The distance between two points P and Q in the plane can be found using the distance formula. If P has coordinates (a, a² + 2) and Q has coordinates (b, √(b - 2)), then the distance squared between them (since minimizing distance squared is equivalent to minimizing distance) would be:Distance squared = (a - b)² + [(a² + 2) - √(b - 2)]².To find the minimum distance, I can minimize this expression with respect to both a and b. However, this seems a bit complicated because there are two variables involved. Maybe there's a way to relate a and b? Let me think.Alternatively, perhaps the minimal distance occurs when the line segment PQ is perpendicular to both curves at those points. That is, the tangent lines to each curve at points P and Q are both perpendicular to PQ. This is similar to the idea that the shortest distance from a point to a curve is along the line perpendicular to the curve's tangent at that point. But since both points are moving, maybe the direction of PQ has to be perpendicular to both tangents? I might need to use calculus here.Let me try to set up the problem using calculus. Let’s denote the coordinates of P as (x1, y1) where y1 = x1² + 2, and Q as (x2, y2) where y2 = √(x2 - 2). The distance squared between P and Q is:D = (x1 - x2)² + (y1 - y2)²= (x1 - x2)² + (x1² + 2 - √(x2 - 2))².To find the minimum, we can take the partial derivatives of D with respect to x1 and x2, set them equal to zero, and solve the resulting system of equations.First, compute ∂D/∂x1:The derivative of (x1 - x2)² with respect to x1 is 2(x1 - x2).The derivative of (x1² + 2 - √(x2 - 2))² with respect to x1 is 2(x1² + 2 - √(x2 - 2)) * 2x1.So, ∂D/∂x1 = 2(x1 - x2) + 4x1(x1² + 2 - √(x2 - 2)).Similarly, compute ∂D/∂x2:The derivative of (x1 - x2)² with respect to x2 is -2(x1 - x2).The derivative of (x1² + 2 - √(x2 - 2))² with respect to x2 is 2(x1² + 2 - √(x2 - 2)) * (-1/(2√(x2 - 2))).So, ∂D/∂x2 = -2(x1 - x2) - (x1² + 2 - √(x2 - 2))/√(x2 - 2).Set both partial derivatives equal to zero:1. 2(x1 - x2) + 4x1(x1² + 2 - √(x2 - 2)) = 02. -2(x1 - x2) - (x1² + 2 - √(x2 - 2))/√(x2 - 2) = 0This system of equations looks pretty complicated. Let me see if I can simplify or relate them.Notice that both equations have terms involving (x1 - x2) and (x1² + 2 - √(x2 - 2)). Maybe I can express one equation in terms of the other.Let’s denote A = x1 - x2 and B = x1² + 2 - √(x2 - 2). Then equation 1 becomes:2A + 4x1B = 0 → 2A = -4x1B → A = -2x1BEquation 2 becomes:-2A - B / √(x2 - 2) = 0 → -2A = B / √(x2 - 2)But from equation 1, A = -2x1B. Substitute that into equation 2:-2*(-2x1B) = B / √(x2 - 2) → 4x1B = B / √(x2 - 2)Assuming B ≠ 0, we can divide both sides by B:4x1 = 1 / √(x2 - 2)So, 4x1 = 1 / √(x2 - 2) → √(x2 - 2) = 1 / (4x1) → x2 - 2 = 1 / (16x1²) → x2 = 2 + 1/(16x1²)So, we have a relation between x2 and x1. Let’s keep this in mind.Also, recall that A = x1 - x2 = -2x1B. Let’s substitute x2 from the above equation into this.A = x1 - x2 = x1 - [2 + 1/(16x1²)] = x1 - 2 - 1/(16x1²)But A = -2x1B. Let’s also express B in terms of x1.B = x1² + 2 - √(x2 - 2) = x1² + 2 - [1/(4x1)] (since √(x2 - 2) = 1/(4x1))Therefore, B = x1² + 2 - 1/(4x1)So, substituting into A = -2x1B:x1 - 2 - 1/(16x1²) = -2x1[x1² + 2 - 1/(4x1)]Let’s compute the right-hand side:-2x1(x1²) - 2x1(2) + 2x1*(1/(4x1)) = -2x1³ - 4x1 + 2*(1/4) = -2x1³ - 4x1 + 0.5So, the equation becomes:x1 - 2 - 1/(16x1²) = -2x1³ - 4x1 + 0.5Let’s bring all terms to the left-hand side:x1 - 2 - 1/(16x1²) + 2x1³ + 4x1 - 0.5 = 0Combine like terms:2x1³ + (x1 + 4x1) + (-2 - 0.5) - 1/(16x1²) = 0Which simplifies to:2x1³ + 5x1 - 2.5 - 1/(16x1²) = 0Hmm, this is a non-linear equation involving x1³ and 1/x1². Solving this algebraically might be challenging. Maybe we can multiply through by 16x1² to eliminate denominators.Multiplying each term by 16x1²:16x1²*(2x1³) + 16x1²*(5x1) - 16x1²*(2.5) - 16x1²*(1/(16x1²)) = 0Calculates to:32x1^5 + 80x1^3 - 40x1² - 1 = 0So, we have a quintic equation: 32x1^5 + 80x1^3 - 40x1² - 1 = 0Quintic equations don't have general solutions in radicals, so we might need to approximate the root numerically. Let's see if we can find an approximate solution.First, check if there are any rational roots using the Rational Root Theorem. The possible rational roots are ±1, ±1/2, ±1/4, ±1/8, ±1/16, ±1/32, etc., considering the factors of the constant term and leading coefficient.Test x = 1: 32 + 80 - 40 - 1 = 71 ≠ 0x = 0.5: 32*(0.5)^5 + 80*(0.5)^3 - 40*(0.5)^2 -1= 32*(1/32) + 80*(1/8) - 40*(1/4) -1= 1 + 10 - 10 -1 = 0Oh! Wait, x = 0.5 is a root. Let me verify:32*(0.5)^5 = 32*(1/32) = 180*(0.5)^3 = 80*(1/8) = 10-40*(0.5)^2 = -40*(1/4) = -10-1 = -1Adding them up: 1 + 10 -10 -1 = 0. Perfect. So x1 = 0.5 is a root.Therefore, we can factor out (x - 0.5) from the quintic.Let me perform polynomial division or use synthetic division.Dividing 32x^5 + 80x^3 - 40x² - 1 by (x - 0.5). Wait, but the quintic is 32x^5 + 0x^4 +80x^3 -40x² +0x -1. Let's write all coefficients:32, 0, 80, -40, 0, -1Divide by (x - 0.5). Using synthetic division with root 0.5:We bring down 32.Multiply by 0.5: 16Add to next coefficient: 0 +16 =16Multiply by 0.5: 8Add to next coefficient: 80 +8=88Multiply by 0.5:44Add to next coefficient: -40 +44=4Multiply by 0.5:2Add to next coefficient: 0 +2=2Multiply by 0.5:1Add to last coefficient: -1 +1=0. Perfect, remainder zero.So the quintic factors as (x - 0.5)(32x^4 +16x^3 +88x² +4x +2).Now, we need to check if the quartic 32x^4 +16x^3 +88x² +4x +2 has real roots. Let's check possible rational roots again. Potential roots are ±1, ±2, ±1/2, etc. Let's test x = -1:32(-1)^4 +16(-1)^3 +88(-1)^2 +4(-1) +2 = 32 -16 +88 -4 +2 = 102 ≠0x= -0.5:32*(0.0625) +16*(-0.125) +88*(0.25) +4*(-0.5) +2= 2 -2 +22 -2 +2= 22 ≠0x=0.5:32*(0.5)^4 +16*(0.5)^3 +88*(0.5)^2 +4*(0.5) +2=32*(1/16) +16*(1/8) +88*(1/4) +2 +2=2 +2 +22 +2 +2=30 ≠0So no rational roots. Also, for real roots, let's check the quartic's behavior. Since all coefficients except the leading term are positive for x positive, but even for x negative, likely the quartic is always positive. Let's check for minima.Compute derivative of quartic: 128x³ +48x² +176x +4. Set to zero. This cubic equation might have real roots, but solving it would be complicated. Alternatively, since the quartic evaluated at x=0 is 2, and as x approaches infinity, it goes to positive infinity, and x approaches negative infinity, it also goes to positive infinity (since even degree with positive leading coefficient). So, if it has a minimum above zero, then no real roots. Alternatively, maybe the quartic is always positive. Let me check discriminant or use the sum of squares. Not straightforward. Alternatively, since all coefficients are positive except the cubic term which can be positive or negative, but perhaps the quartic is always positive. For real x, 32x^4 +16x^3 +88x² +4x +2. Let me group terms:32x^4 +16x^3 + 88x² +4x +2.Perhaps write as 32x^4 +16x^3 + ... Maybe factor:Not obvious. Alternatively, complete squares. Hmm. Alternatively, note that 32x^4 +16x^3 = 16x^3(2x +1). Not helpful. Alternatively, write as:32x^4 + 16x^3 + 88x² +4x +2 = 32x^4 +16x^3 + (let's see, 88x²) +4x +2.Alternatively, check if it's always positive. For x >=0, all terms are positive, so quartic is positive. For x <0, let’s substitute x = -y where y >0:32y^4 -16y^3 +88y² -4y +2. Let's check if this is positive for all y>0.Check if 32y^4 -16y^3 +88y² -4y +2 >0. Let’s group terms:(32y^4 -16y^3) + (88y² -4y) +2.Factor:16y^3(2y -1) + 4y(22y -1) +2.Hmm, for y >0.5, 2y -1 >0, so first term positive. Second term 22y -1 >0 when y>1/22, which is true for y>0.5. So for y>0.5, all terms positive. For y between 0 and 0.5, let's check.Let's take y=0.25:32*(0.25)^4 -16*(0.25)^3 +88*(0.25)^2 -4*(0.25) +2=32*(1/256) -16*(1/64) +88*(1/16) -1 +2= (32/256) - (16/64) + (88/16) -1 +2= (1/8) - (1/4) + 5.5 -1 +2= (-1/8) +5.5 +1= 6.5 -0.125=6.375>0At y=0.1:32*(0.1)^4 -16*(0.1)^3 +88*(0.1)^2 -4*(0.1) +2=32*0.0001 -16*0.001 +88*0.01 -0.4 +2=0.0032 -0.016 +0.88 -0.4 +2≈0.0032 -0.016= -0.0128; -0.0128 +0.88=0.8672; 0.8672 -0.4=0.4672; 0.4672 +2≈2.4672>0So even for y>0, the expression is positive. Therefore, the quartic 32x^4 +16x^3 +88x² +4x +2 is always positive. Thus, the only real root of the quintic equation is x1=0.5.Therefore, x1 = 0.5. Then, from earlier relation:x2 = 2 + 1/(16x1²) = 2 + 1/(16*(0.5)^2) = 2 + 1/(16*0.25) = 2 + 1/4 = 2.25So x2 = 2.25 = 9/4. Now, compute y1 and y2:y1 = x1² + 2 = (0.5)^2 + 2 = 0.25 + 2 = 2.25y2 = √(x2 - 2) = √(2.25 - 2) = √(0.25) = 0.5So point P is (0.5, 2.25) and point Q is (2.25, 0.5). Let me verify if these points satisfy the conditions.Wait, let me check the distance between these two points. The distance squared is (0.5 - 2.25)^2 + (2.25 - 0.5)^2 = (-1.75)^2 + (1.75)^2 = 3.0625 + 3.0625 = 6.125. So distance is √6.125. But 6.125 is 49/8, since 6 = 48/8 + 1/8 = 49/8. Therefore, √(49/8) = 7/(2√2) ≈ 2.474.But let me verify if this is indeed the minimal distance. Let me check if the line connecting P and Q is perpendicular to both curves at those points.First, compute the derivative of y = x² + 2 at x = 0.5. The derivative is 2x, so slope of tangent at P is 2*(0.5) = 1. The slope of PQ is (0.5 - 2.25)/(2.25 - 0.5) = (-1.75)/1.75 = -1. The product of the slopes is 1*(-1) = -1, which means they are perpendicular. Good.Now for the curve y = √(x - 2) at Q (x=9/4=2.25). The derivative is (1/(2√(x - 2))). At x=2.25, derivative is 1/(2√(0.25)) = 1/(2*0.5) = 1/1 =1. The slope of PQ is -1, so the product is 1*(-1)= -1, which also means they are perpendicular. Perfect! So both tangent lines are perpendicular to PQ, confirming that this is indeed the minimal distance.Therefore, the minimal distance is 7/(2√2) which simplifies to (7√2)/4. To check, 7/(2√2) rationalized is 7√2/(2*2)=7√2/4. So the minimal distance is 7√2/4.Wait, but let me confirm the distance calculation again. The coordinates are P(0.5, 2.25) and Q(2.25, 0.5). The differences are Δx = 2.25 - 0.5 = 1.75, Δy = 0.5 - 2.25 = -1.75. So distance squared is (1.75)^2 + (-1.75)^2 = 2*(1.75)^2. 1.75 is 7/4, so (7/4)^2 = 49/16. Thus, distance squared is 2*(49/16) = 49/8. Therefore, distance is √(49/8) = 7/√8 = 7/(2√2) = 7√2/4. Yes, that's correct.So the minimal distance is 7√2/4. Let me just check if there could be another minimal distance. Since we found the only real solution to the quintic equation, and the other roots are complex or not real, this should be the only critical point. Also, given the geometry of the curves, the minimal distance should occur at this point. The parabola y = x² + 2 is above the square root curve y = √(x - 2), which starts at (2,0). The closest points seem to be where the line connecting them is perpendicular to both curves, as we found. Therefore, this should be the minimal distance.Final AnswerThe minimum value of the distance (|PQ|) is boxed{dfrac{7sqrt{2}}{4}}.</think>Given a point ( P ) on the curve ( y = x^2 + 2 ) and a point ( Q ) on the curve ( y = sqrt{x - 2} ), we need to find the minimum value of the distance ( |PQ| ).1. Coordinates of Points: - Point ( P ) on ( y = x^2 + 2 ) has coordinates ( (x_1, x_1^2 + 2) ). - Point ( Q ) on ( y = sqrt{x - 2} ) has coordinates ( (x_2, sqrt{x_2 - 2}) ).2. Distance Formula: - The distance squared between ( P ) and ( Q ) is given by: [ D = (x_1 - x_2)^2 + left( x_1^2 + 2 - sqrt{x_2 - 2} right)^2 ]3. Partial Derivatives: - Compute partial derivatives of ( D ) with respect to ( x_1 ) and ( x_2 ), set them to zero, and solve the resulting system of equations.4. Critical Points: - Solving the system of equations leads to the relationship ( x_2 = 2 + frac{1}{16x_1^2} ). - Substituting this back into the equations, we find ( x_1 = 0.5 ) and ( x_2 = 2.25 ).5. Verification: - Coordinates of ( P ) are ( (0.5, 2.25) ) and coordinates of ( Q ) are ( (2.25, 0.5) ). - The distance between ( P ) and ( Q ) is calculated as: [ sqrt{(0.5 - 2.25)^2 + (2.25 - 0.5)^2} = sqrt{(-1.75)^2 + (1.75)^2} = sqrt{2 times (1.75)^2} = sqrt{2 times left(frac{7}{4}right)^2} = frac{7sqrt{2}}{4} ]6. Perpendicular Check: - The slopes of the tangents at ( P ) and ( Q ) are both 1, and the slope of ( PQ ) is -1, confirming perpendicularity.The minimum value of the distance ( |PQ| ) is (boxed{dfrac{7sqrt{2}}{4}}).,<think>Okay, so I need to find the minimum distance between a point P on the curve y = x² + 2 and a point Q on the curve y = √(x - 2). Hmm, let me think. First, I remember that the distance between two points (x₁, y₁) and (x₂, y₂) is given by the formula sqrt[(x₂ - x₁)² + (y₂ - y₁)²]. But since we want to minimize this distance, maybe I can work with the square of the distance to make the calculations easier. Because the square root function is increasing, the minimum of the distance squared will also give the minimum distance. So, let's denote point P as (a, a² + 2) since it's on y = x² + 2. Similarly, point Q will be (b, sqrt(b - 2)) because it's on y = sqrt(x - 2). I need to be careful with the domain here; for Q, the square root is defined when x - 2 ≥ 0, so b ≥ 2. Now, the distance squared between P and Q is [(b - a)² + (sqrt(b - 2) - (a² + 2))²]. Let me write that out as a function D(a, b):D(a, b) = (b - a)² + (sqrt(b - 2) - a² - 2)².To find the minimum, I should take the partial derivatives of D with respect to both a and b, set them equal to zero, and solve the resulting system of equations. That should give me the critical points, which could be minima. Then I can check which one gives the smallest distance.First, let's compute the partial derivative with respect to a:∂D/∂a = 2(b - a)(-1) + 2(sqrt(b - 2) - a² - 2)(-2a).Simplifying:∂D/∂a = -2(b - a) - 4a(sqrt(b - 2) - a² - 2).Similarly, the partial derivative with respect to b:∂D/∂b = 2(b - a)(1) + 2(sqrt(b - 2) - a² - 2)( (1)/(2sqrt(b - 2)) ).Simplifying:∂D/∂b = 2(b - a) + (sqrt(b - 2) - a² - 2)/sqrt(b - 2).So now, we have two equations:1) -2(b - a) - 4a(sqrt(b - 2) - a² - 2) = 02) 2(b - a) + (sqrt(b - 2) - a² - 2)/sqrt(b - 2) = 0Hmm, these look a bit complicated. Maybe there's a substitution or a relationship between a and b that can simplify things. Let me see.Alternatively, maybe instead of parameterizing both points with a and b, I can express one variable in terms of the other. For example, since Q is on y = sqrt(x - 2), maybe we can let Q = (q² + 2, q), but wait, no. Let me check:Wait, if y = sqrt(x - 2), then x = y² + 2. So, if I let the y-coordinate of Q be q, then the x-coordinate is q² + 2. So point Q would be (q² + 2, q). That might make things easier, perhaps? Let me try that.Similarly, point P is on y = x² + 2, so if I let the x-coordinate of P be p, then the coordinates are (p, p² + 2).So then, the distance squared between P(p, p² + 2) and Q(q² + 2, q) is:D(p, q) = (q² + 2 - p)² + (q - p² - 2)².Now, this seems symmetric in a way. Maybe if we can find a relationship between p and q? Let's see.Alternatively, maybe there's a point where the line PQ is perpendicular to both curves at those points? Because the minimal distance should occur when the line connecting P and Q is perpendicular to both curves at those points. Is that a valid approach?I remember that for the shortest distance between two curves, the line connecting the points must be perpendicular to both tangents of the curves at those points. So, if I can find points P and Q where the vector PQ is perpendicular to both tangent lines at P and Q, then that should be the minimal distance.Let's try this approach.First, find the tangent vector to each curve at points P and Q.For curve y = x² + 2 (point P), the derivative dy/dx = 2x, so the tangent vector at P can be written as (1, 2p), since the direction is in the direction of (1, dy/dx).For curve y = sqrt(x - 2) (point Q), the derivative dy/dx = 1/(2sqrt(x - 2)). If Q is (q² + 2, q), then x = q² + 2, so dy/dx = 1/(2q). So the tangent vector at Q can be written as (1, 1/(2q)).Now, the vector PQ is (q² + 2 - p, q - (p² + 2)).For this vector to be perpendicular to both tangent vectors at P and Q, the dot product of PQ with the tangent vectors should be zero.So, first, the dot product with tangent at P:(q² + 2 - p)(1) + (q - p² - 2)(2p) = 0.Second, the dot product with tangent at Q:(q² + 2 - p)(1) + (q - p² - 2)(1/(2q)) = 0.So now, we have two equations:1) (q² + 2 - p) + 2p(q - p² - 2) = 02) (q² + 2 - p) + (q - p² - 2)/(2q) = 0Hmm, these equations might be complex, but perhaps we can solve them.Let me write them out again:Equation 1: (q² + 2 - p) + 2p(q - p² - 2) = 0Equation 2: (q² + 2 - p) + (q - p² - 2)/(2q) = 0Let me denote equation 1 and equation 2. Let's try to subtract equation 2 from equation 1 to eliminate the (q² + 2 - p) term.Equation 1 - Equation 2:[2p(q - p² - 2) - (q - p² - 2)/(2q)] = 0Factor out (q - p² - 2):(q - p² - 2)[2p - 1/(2q)] = 0So either (q - p² - 2) = 0 or [2p - 1/(2q)] = 0Case 1: q - p² - 2 = 0 => q = p² + 2Case 2: 2p - 1/(2q) = 0 => 2p = 1/(2q) => p = 1/(4q)Let me check Case 1 first.Case 1: q = p² + 2Substitute this into equation 2:(q² + 2 - p) + (q - p² - 2)/(2q) = 0But since q = p² + 2, the term (q - p² - 2) becomes 0. So equation 2 becomes:(q² + 2 - p) + 0 = 0 => q² + 2 - p = 0But from q = p² + 2, substitute into this:(p² + 2)^2 + 2 - p = 0Expand (p² + 2)^2: p^4 + 4p² + 4So:p^4 + 4p² + 4 + 2 - p = 0 => p^4 + 4p² + 6 - p = 0That's a quartic equation: p^4 + 4p² - p + 6 = 0This seems difficult to solve. Maybe there are real roots? Let's check.Try p = -2: (-2)^4 +4*(-2)^2 - (-2) +6 = 16 + 16 +2 +6=40≠0p = -1: 1 + 4 - (-1) +6=1+4+1+6=12≠0p=0: 0 +0 -0 +6=6≠0p=1:1 +4 -1 +6=10≠0p=2:16 +16 -2 +6=36≠0All these trial values don't give zero. So maybe there are no real roots here. Therefore, Case 1 does not yield a solution.So move to Case 2: p = 1/(4q)Substitute p = 1/(4q) into equation 1:(q² + 2 - p) + 2p(q - p² - 2) = 0Replace p with 1/(4q):(q² + 2 - 1/(4q)) + 2*(1/(4q))*(q - (1/(4q))² - 2) = 0Let me compute each part step by step.First part: q² + 2 - 1/(4q)Second part: 2*(1/(4q))*(q - (1/(16q²)) - 2) = (1/(2q))*(q - 1/(16q²) - 2)So, let's compute the second part:(1/(2q))*(q - 2 - 1/(16q²)) = (1/(2q))*(q - 2) - (1/(2q))*(1/(16q²)) = (1/2 - 1/q) - 1/(32q³)Wait, maybe better to expand directly:Multiply each term inside the parentheses by 1/(2q):= (1/(2q))*q - (1/(2q))*2 - (1/(2q))*(1/(16q²))= 1/2 - 1/q - 1/(32q³)So the entire equation becomes:(q² + 2 - 1/(4q)) + (1/2 - 1/q - 1/(32q³)) = 0Combine like terms:q² + 2 + 1/2 - 1/(4q) -1/q -1/(32q³) = 0Simplify constants: 2 + 1/2 = 5/2Combine -1/(4q) -1/q = -(1/(4q) + 4/(4q)) = -5/(4q)So:q² + 5/2 - 5/(4q) -1/(32q³) = 0Multiply both sides by 32q³ to eliminate denominators:32q³*q² + 32q³*(5/2) -32q³*(5/(4q)) -32q³*(1/(32q³)) =0Simplify each term:32q^5 + 16*5 q^3 - 8*5 q² -1 =0Which is:32q^5 +80q³ -40q² -1 =0Hmm, this is a fifth-degree equation. Solving this algebraically might be impossible. Maybe we can try to find a rational root. Let's apply the Rational Root Theorem. The possible rational roots are ±1, ±1/2, ±1/4, ±1/8, ±1/16, ±1/32.Test q=1:32(1)^5 +80(1)^3 -40(1)^2 -1=32+80-40-1=71≠0q= -1:32(-1)^5 +80(-1)^3 -40(-1)^2 -1= -32 -80 -40 -1= -153≠0q=1/2:32*(1/32) +80*(1/8) -40*(1/4) -1=1 +10 -10 -1=0Wait, q=1/2:32*(1/2)^5 +80*(1/2)^3 -40*(1/2)^2 -1Calculate each term:32*(1/32)=180*(1/8)=10-40*(1/4)=-10-1So total:1 +10 -10 -1=0Oh! So q=1/2 is a root.Therefore, we can factor out (q - 1/2) from the polynomial.Use polynomial division or synthetic division.Let's use synthetic division with root q=1/2:Coefficients: 32, 0, 80, -40, 0, -1Wait, the polynomial is 32q^5 +0q^4 +80q^3 -40q^2 +0q -1.So coefficients:32,0,80,-40,0,-1Divide by (q -1/2):Bring down 32. Multiply by 1/2:16. Add to next term:0+16=16.Multiply by1/2:8. Add to 80:88.Multiply by1/2:44. Add to -40:4.Multiply by1/2:2. Add to 0:2.Multiply by1/2:1. Add to -1:0. Perfect.So the polynomial factors as (q -1/2)(32q^4 +16q³ +88q² +4q +2).Now, check if the quartic has real roots. Let's see:32q^4 +16q³ +88q² +4q +2=0This quartic equation. Let me check for possible rational roots. Possible roots are ±1, ±2, ±1/2, etc., but testing q=1:32 +16 +88 +4 +2=142≠0q=-1:32 -16 +88 -4 +2=102≠0q=1/2:32*(1/16) +16*(1/8) +88*(1/4) +4*(1/2) +2=2 +2 +22 +2 +2=30≠0q=-1/2: 32*(1/16) -16*(1/8) +88*(1/4) -4*(1/2) +2=2 -2 +22 -2 +2=22≠0So no rational roots. Let's check discriminant or see if it can be factored.Alternatively, note that all coefficients are positive, so for q>0, the quartic is positive. Since we know from the original problem that q must be ≥0 because Q is on y=√(x - 2), which requires x ≥2, but q is the y-coordinate, which is sqrt(x-2), so q ≥0. Wait, actually, q is the y-coordinate, which is sqrt(x -2), so q ≥0, but since x= q² +2, x ≥2 as needed.But if q is real, but the quartic is positive for all q, since all terms are positive when q>0. So the quartic 32q^4 +16q³ +88q² +4q +2 is always positive for real q. Therefore, the only real root of the fifth-degree equation is q=1/2.Therefore, q=1/2 is the solution in Case 2. Then p=1/(4q)=1/(4*(1/2))=1/2.So p=1/2 and q=1/2.Now, let's find the points P and Q:Point P is (p, p² + 2) = (1/2, (1/2)^2 + 2) = (1/2, 1/4 + 2) = (1/2, 9/4)Point Q is (q² + 2, q) = ((1/2)^2 + 2, 1/2) = (1/4 + 2, 1/2) = (9/4, 1/2)So the distance between P(1/2, 9/4) and Q(9/4, 1/2) is sqrt[(9/4 -1/2)^2 + (1/2 -9/4)^2]Compute the differences:9/4 -1/2 =9/4 -2/4=7/41/2 -9/4=2/4 -9/4= -7/4So distance squared is (7/4)^2 + (-7/4)^2=2*(49/16)=49/8Thus, distance is sqrt(49/8)=7/(2√2)=7√2/4≈2.474Wait, but let me check if this is indeed the minimal distance. Because in Case 1, we didn't find any real solutions, so the only critical point is from Case 2. Also, since the distance function is continuous and the curves are both closed in some sense (parabola opening upwards and a square root curve), the minimal distance should be attained here.But just to be thorough, let's check another point to see if the distance is indeed minimal. For example, take Q at (2, 0), which is when x=2, y=0. Then, what's the closest point P on y=x² +2?The point Q=(2,0). Let's find the closest P on y=x² +2. Let P=(p, p² +2). The distance squared is (2 - p)^2 + (0 - p² -2)^2. Taking derivative with respect to p:d/dp [ (2 - p)^2 + (-p² -2)^2 ] = -2(2 - p) + 2(-p² -2)(-2p) = -4 + 2p +4p(p² +2)Set equal to zero: -4 + 2p +4p(p² +2)=0Simplify: -4 +2p +4p^3 +8p=0 =>4p^3 +10p -4=0This cubic equation. Let's test p=0.5:4*(0.125) +10*(0.5) -4=0.5 +5 -4=1.5≠0p=1:4 +10 -4=10≠0p≈0.4:4*(0.064) +10*0.4 -4≈0.256 +4 -4≈0.256≠0Not sure, but maybe there's a real root here. But even if there is, the distance would be sqrt[(2 - p)^2 + (p² +2)^2], which for p=0.5, gives sqrt[(1.5)^2 + (0.25 +2)^2]=sqrt[2.25 +5.0625]=sqrt[7.3125]≈2.706, which is larger than 7√2/4≈2.474. So this seems larger. Similarly, if we take Q=(3,1), then P would need to be found, but again, likely the distance would be larger. Therefore, the critical point found is likely the minimum.Therefore, the minimal distance is 7√2/4.Wait, but let me double-check my calculation for the distance between P(1/2, 9/4) and Q(9/4,1/2):x-coordinate difference:9/4 -1/2=9/4 -2/4=7/4y-coordinate difference:1/2 -9/4=2/4 -9/4=-7/4Distance squared: (7/4)^2 + (-7/4)^2=49/16 +49/16=98/16=49/8Distance: sqrt(49/8)=7/(2√2)=7√2/4≈2.474Yes, that's correct.So the minimal distance is 7√2/4.But just to be thorough, let's check if there might be a case where a = b? That is, if the closest points are vertically or horizontally aligned. But given the curves, it's unlikely. The curve y=x² +2 is a parabola opening upwards, and y=sqrt(x -2) is a square root curve starting at (2,0) and increasing. The closest points are probably not with a = b.Alternatively, suppose we set a = b and see what happens. Let a = b. Then point P is (a, a² +2) and Q is (a, sqrt(a -2)). Wait, but Q must have x ≥2, so a must be ≥2. Then, the distance squared is (a - a)^2 + (sqrt(a -2) - a² -2)^2=0 + (sqrt(a -2) -a² -2)^2. To minimize this, we need sqrt(a -2) -a² -2=0. But sqrt(a -2)=a² +2. Since a ≥2, the right-hand side is at least 2² +2=6, while the left-hand side sqrt(a -2) when a=2 is 0, and increases as a increases. But sqrt(a -2) grows much slower than a² +2. So there's no solution here, since sqrt(a -2) can't catch up to a² +2. So no solution in this case.Therefore, the minimal distance is indeed 7√2/4.Final AnswerThe minimum value of the distance (|PQ|) is (boxed{dfrac{7sqrt{2}}{4}}).</think>Given a point ( P ) on the curve ( y = x^2 + 2 ) and a point ( Q ) on the curve ( y = sqrt{x - 2} ), we need to find the minimum distance ( |PQ| ).1. Parameterize Points: - Point ( P ) on ( y = x^2 + 2 ) is ((a, a^2 + 2)). - Point ( Q ) on ( y = sqrt{x - 2} ) is ((b, sqrt{b - 2})) where ( b geq 2 ).2. Distance Squared Function: The distance squared between ( P ) and ( Q ) is given by: [ D(a, b) = (b - a)^2 + left( sqrt{b - 2} - (a^2 + 2) right)^2 ]3. Partial Derivatives: - Partial derivative with respect to ( a ): [ frac{partial D}{partial a} = -2(b - a) - 4a left( sqrt{b - 2} - a^2 - 2 right) ] - Partial derivative with respect to ( b ): [ frac{partial D}{partial b} = 2(b - a) + frac{sqrt{b - 2} - a^2 - 2}{sqrt{b - 2}} ]4. Critical Points: Solving the system of equations from setting the partial derivatives to zero, we find that the critical points occur when the line segment ( PQ ) is perpendicular to the tangents of both curves at those points.5. Tangent Vectors: - For ( P ), the tangent vector is ((1, 2a)). - For ( Q ), the tangent vector is ((1, frac{1}{2sqrt{b - 2}})).6. Perpendicular Condition: The vector ( PQ ) must be perpendicular to both tangent vectors. This leads to solving the system of equations derived from the dot products being zero.7. Solving the System: - Through substitution and solving the resulting polynomial equations, we find the critical points ( a = frac{1}{2} ) and ( b = frac{1}{2} ).8. Points Calculation: - Point ( P ) is (left( frac{1}{2}, frac{9}{4} right)). - Point ( Q ) is (left( frac{9}{4}, frac{1}{2} right)).9. Distance Calculation: The distance between ( P ) and ( Q ) is: [ sqrt{left( frac{9}{4} - frac{1}{2} right)^2 + left( frac{1}{2} - frac{9}{4} right)^2} = sqrt{left( frac{7}{4} right)^2 + left( -frac{7}{4} right)^2} = sqrt{frac{49}{16} + frac{49}{16}} = sqrt{frac{98}{16}} = frac{7sqrt{2}}{4} ]Thus, the minimum value of the distance ( |PQ| ) is (boxed{dfrac{7sqrt{2}}{4}}).]

🔑:Let the coordinates of P be (x_P, y_P) on the curve y = x^2 + 2, we have:y_P = x_P^2 + 2 quad text{(1)}And for the coordinates of Q (x_Q, y_Q) on the curve y = sqrt{x - 2}, we have:y_Q = sqrt{x_Q - 2} quad text{(2)}Since the two curves are inverses of each other (for x geq 0) with respect to the line y = x, the minimum distance between P and Q will be twice the minimum distance from either point to the line y = x.Without loss of generality, let's find the shortest distance from Q to the line y = x. The distance d from point Q to the line is given by the perpendicular distance formula:d = frac{|x_Q - y_Q|}{sqrt{1^2 + (-1)^2}}Substituting y_Q from equation (2),d = frac{|sqrt{x_Q - 2} - x_Q|}{sqrt{2}}Squares are always positive, so we have:d = frac{|x_Q - sqrt{x_Q - 2}|}{sqrt{2}} = frac{|sqrt{(x_Q - 2)^2} + 2 - sqrt{x_Q - 2}|}{sqrt{2}}Observing the expression carefully, notice that this distance achieves its minimum when the quantity inside the absolute value is minimized. The minimum possible value of this expression, without the absolute value, is when sqrt{x_Q - 2} - frac{1}{2} is zero, making the minimum distance solely dependent on the constant term frac{7}{4}:d_{min} = frac{frac{7}{4}}{sqrt{2}} = frac{7sqrt{2}}{8}Thus, the minimum distance between P and Q is twice this distance:|PQ|_{min} = 2 times frac{7sqrt{2}}{8} = frac{7sqrt{2}}{4}Thus, the minimum value of |PQ| is boxed{frac{7sqrt{2}}{4}}.

🧠:[<think>Okay, so I need to find the domain of the function f(x) = (3x²)/√(1 - x) + lg(3x + 1). Hmm, the domain is all the real numbers x for which the function is defined. Let me break this down into parts because there are two main components here: a rational expression with a square root in the denominator and a logarithmic function. Both of these have their own domain restrictions, so the domain of f(x) will be the intersection of the domains of these two parts.First, let's look at the square root in the denominator: √(1 - x). For the square root to be defined, the expression inside it must be non-negative. So, 1 - x ≥ 0. Solving that inequality: 1 - x ≥ 0 ⇒ -x ≥ -1 ⇒ x ≤ 1. But wait, since this is in the denominator, the expression under the square root can't be zero because division by zero is undefined. Therefore, 1 - x must be strictly greater than zero. So, 1 - x > 0 ⇒ x < 1. That gives me the first restriction: x must be less than 1.Next, the logarithmic function: lg(3x + 1). The logarithm is defined only when its argument is positive. So, 3x + 1 > 0. Solving that: 3x + 1 > 0 ⇒ 3x > -1 ⇒ x > -1/3. That gives me the second restriction: x must be greater than -1/3.Now, combining both conditions: x has to be greater than -1/3 and less than 1. So, the domain should be (-1/3, 1). Let me check the answer choices.Option A: [-1/3, 1). Hmm, that's a closed interval on the left. But the logarithm requires x > -1/3, not greater than or equal. If x were equal to -1/3, then 3x + 1 would be 3*(-1/3) + 1 = -1 + 1 = 0, and log of zero is undefined. So, x cannot be -1/3. Therefore, the left endpoint should be open. So, A is incorrect.Option B: (-1/3, 1). This matches my initial conclusion. Both endpoints are open where necessary. Let me verify once again.For the square root denominator: x < 1, which is open at 1. For the logarithm: x > -1/3, which is open at -1/3. So combining these, yes, the domain is all x such that -1/3 < x < 1. So, interval notation is (-1/3, 1). That's option B.Just to make sure I didn't miss anything else. The numerator in the first term is 3x², which is a polynomial and defined for all real numbers. The denominator is √(1 - x), which we already handled. The logarithm's argument is 3x + 1, which is handled as well. So, there are no other components that would restrict the domain further.So, yes, the domain should be (-1/3, 1), which is option B. Let me just check the other options to be thorough.Option C: (-1/3, +∞). Well, that's too broad on the right side. Because when x approaches 1 from the left, the denominator √(1 - x) becomes zero, making the function undefined. So, x can't be greater than or equal to 1. Hence, C is wrong.Option D: (-∞, 1). Similarly, this includes values less than or equal to -1/3, but when x is less than or equal to -1/3, the argument of the logarithm becomes 3x + 1 ≤ 0, which is invalid. So, D is also incorrect.Therefore, confirming again, the correct answer is B.Final Answerboxed{B}</think>To find the domain of the function ( f(x) = dfrac{3x^2}{sqrt{1 - x}} + lg(3x + 1) ), we need to consider the restrictions imposed by both the square root in the denominator and the logarithmic function.1. Square Root in the Denominator: The expression inside the square root, (1 - x), must be positive because the square root of a non-positive number is undefined and the denominator cannot be zero. This gives the inequality: [ 1 - x > 0 implies x < 1 ]2. Logarithmic Function: The argument of the logarithm, (3x + 1), must be positive. This gives the inequality: [ 3x + 1 > 0 implies x > -dfrac{1}{3} ]Combining these two conditions, we get:[-dfrac{1}{3} < x < 1]Thus, the domain of the function is the interval (left( -dfrac{1}{3}, 1 right)), which corresponds to option B.[boxed{B}],<think>Okay, let's see. I need to find the domain of the function f(x) = (3x²)/(√(1 - x)) + lg(3x + 1). Hmm, the domain is all the real numbers x for which the function is defined. So, I need to check where each part of this function is defined and then find the intersection of those domains because both parts need to be defined for the whole function to be defined.First, let's break down the function into two parts: the rational expression with a square root in the denominator and the logarithmic function. Let me handle each part separately.Starting with the first part: (3x²)/(√(1 - x)). For this expression to be defined, the denominator cannot be zero because division by zero is undefined. Also, since there's a square root, the expression inside the square root must be positive. So, the square root √(1 - x) requires that 1 - x > 0. Solving that inequality: 1 - x > 0 ⇒ -x > -1 ⇒ x < 1. So, x has to be less than 1. Additionally, the denominator √(1 - x) cannot be zero, so 1 - x ≠ 0 ⇒ x ≠ 1. But since we already have x < 1 from the square root, that's already covered. So the domain for the first part is all real numbers x where x < 1.Now, moving to the second part: the logarithmic function lg(3x + 1). I remember that logarithmic functions are defined only when their argument is positive. So, 3x + 1 must be greater than 0. Solving that: 3x + 1 > 0 ⇒ 3x > -1 ⇒ x > -1/3. So, the domain for the logarithmic part is all real numbers x where x > -1/3.Now, the overall domain of f(x) will be the intersection of the domains of the two parts. That is, x must satisfy both x < 1 and x > -1/3. So, combining these two inequalities, the domain is (-1/3, 1).But wait, let me double-check. The first part had x < 1 and the second part had x > -1/3. So, the intersection is indeed all x such that -1/3 < x < 1. Therefore, the domain should be the open interval (-1/3, 1). Let me check the answer options.Option A is [-1/3, 1). Option B is (-1/3, 1). Option C is (-1/3, +∞). Option D is (-∞, 1). So, according to my reasoning, the correct answer is B: (-1/3, 1).Wait, but hold on. Let me make sure there aren't any other restrictions. For the first part, the denominator is √(1 - x), which is okay as long as 1 - x > 0, so x < 1. The numerator 3x² is a polynomial, which is defined for all real numbers, so no problem there. For the logarithm, lg(3x + 1) is defined when 3x + 1 > 0, which is x > -1/3. So combining these, x must be greater than -1/3 and less than 1. Therefore, open interval at both ends? Wait, but the original function has √(1 - x) in the denominator. So, at x approaching 1 from below, 1 - x approaches zero, making the denominator approach zero. But since the denominator is under a square root, it's approaching zero from the positive side, so √(1 - x) is approaching zero, so the entire fraction (3x²)/√(1 - x) would approach infinity. But in terms of domain, x=1 is not included because 1 - x = 0 is not allowed. Similarly, for x approaching -1/3 from above, 3x +1 approaches 0 from the positive side, so log of a number approaching zero is defined but approaches negative infinity. However, x cannot be equal to -1/3 because 3x +1 would be zero, and logarithm of zero is undefined. So, x must be greater than -1/3. Therefore, the domain is open interval at both ends: (-1/3, 1). So answer B.But let me check the options again. Option A is [-1/3,1). Why would someone think that? Maybe if there was a mistake in the logarithm part. Wait, if x = -1/3, then 3x +1 = 0, so log(0) is undefined. Therefore, x cannot be -1/3. So, the left endpoint is excluded. Similarly, x =1 is excluded. So the interval is open on both ends. Therefore, the answer is B.Yes, so the answer should be option B. Let me check again. For the square root denominator, x must be less than 1. For the logarithm, x must be greater than -1/3. So, the domain is the overlap: x must be greater than -1/3 and less than 1. Since both endpoints make the function undefined, they are excluded. Hence, open interval. So, B is correct.Final Answerboxed{B}</think>To find the domain of the function ( f(x) = dfrac{3x^2}{sqrt{1 - x}} + lg(3x + 1) ), we need to consider the domains of each part separately and then find their intersection.1. For the rational expression (dfrac{3x^2}{sqrt{1 - x}}): - The denominator (sqrt{1 - x}) must be positive, so (1 - x > 0). Solving this gives (x < 1). - Additionally, the denominator cannot be zero, so (1 - x neq 0), which is already covered by (x < 1).2. For the logarithmic function (lg(3x + 1)): - The argument of the logarithm must be positive, so (3x + 1 > 0). Solving this gives (x > -dfrac{1}{3}).Combining these two conditions, we need (x) to satisfy both (x < 1) and (x > -dfrac{1}{3}). Therefore, the domain of (f(x)) is the intersection of these intervals, which is ((- dfrac{1}{3}, 1)).Thus, the correct answer is (boxed{B}).]

🔑:From the question, we have: begin{cases} 1-x > 0 3x+1 > 0 end{cases}Solving these inequalities gives us: (- dfrac {1}{3} < x < 1).Hence, the domain of the function is (- dfrac {1}{3},1).Therefore, the answer is boxed{B}.The domain of the function can be found by using the properties of the square root and logarithmic functions. This problem tests the understanding of these properties and is a basic question.

🧠:[<think>Okay, let's see. The problem is to express 8,000,000,000 in scientific notation. Hmm, I need to remember how scientific notation works. From what I recall, scientific notation is a way to write very large or very small numbers more compactly by using powers of ten. The general form is a number between 1 and 10 multiplied by 10 raised to some exponent. So, for example, 300 can be written as 3 × 10² because 3 times 100 is 300.Alright, so applying that to 8,000,000,000. First, I need to find a number between 1 and 10 that, when multiplied by a power of 10, gives 8,000,000,000. Let's count the digits in 8,000,000,000. Starting from 8, there are 9 zeros after it. So that's 8 followed by 9 zeros, which is 10 digits total. Wait, actually, 8 is the first digit, then 9 zeros, making it 10 digits. So, 8,000,000,000 is 8 times 10 to the power of how much?Let me think. Each zero adds a power of ten. So, 8 × 10^1 is 80, 8 × 10^2 is 800, and so on. Since there are 9 zeros after the 8, that should be 8 × 10^9. Let me check that. 10^9 is 1,000,000,000. Multiply that by 8, and you get 8,000,000,000. Yep, that seems right.But wait, just to make sure I didn't make a mistake. Let's count the places again. Starting from 8, moving the decimal point to the right once gives 80 (10^1), twice gives 800 (10^2), three times 8,000 (10^3), four times 80,000 (10^4), five times 800,000 (10^5), six times 8,000,000 (10^6), seven times 80,000,000 (10^7), eight times 800,000,000 (10^8), nine times 8,000,000,000 (10^9). Exactly. So moving the decimal nine places to the right from 8.0 gives 8,000,000,000. Therefore, the exponent is 9.Alternatively, another way to figure this out is to start with the number 8,000,000,000 and convert it to a number between 1 and 10. So, we take the decimal point, which is implicitly after the last zero, and move it to the left until we get a number between 1 and 10. Each move to the left increases the exponent by 1. Starting at 8,000,000,000.0, moving the decimal nine places to the left gives 8.0. So, that's 9 moves, which would be 10^9. Hence, 8.0 × 10^9. But in scientific notation, we usually don't write the decimal if it's a whole number, so it's just 8 × 10^9.Wait, but hold on. Let me verify with another example. Let's take a smaller number, like 500. That's 5 × 10^2. Right, two zeros, exponent is 2. Similarly, 7,000 is 7 × 10^3. Three zeros, exponent 3. So following that logic, 8,000,000,000 has nine zeros, so exponent should be 9. Therefore, 8 × 10^9. That matches. So I think that's correct.Is there a chance I miscounted the zeros? Let me count again. 8,000,000,000. Breaking it down: 8, followed by a comma, then three zeros (for the first billion), then another comma, three more zeros (second billion), another comma, three more zeros (third billion). Wait, no, actually, in numbers, commas are used every three digits. So starting from the right, each comma represents a thousand, million, billion, etc.Wait, let's parse 8,000,000,000 properly. In the US numbering system, the first comma separates thousands, the next separates millions, then billions. So 8,000,000,000 is 8 followed by three commas. Each comma represents three zeros. So first comma: 8,000 is eight thousand. Then 8,000,000 is eight million. Then 8,000,000,000 is eight billion. Each comma adds three zeros. So three commas after the 8 would be 3 sets of three zeros, which is 9 zeros total. Therefore, 8 followed by 9 zeros is 8 × 10^9. So that's correct.Another way to think about it: 1 billion is 10^9. So 8 billion is 8 × 10^9. Since the UN announced the population as 8 billion, converting that to scientific notation is straightforward. 1 billion in scientific notation is 1 × 10^9, so 8 billion is 8 × 10^9. Therefore, that's the answer.Wait, just to double-check, if I write 8 × 10^9 out, that's 8 multiplied by 1,000,000,000. 8 times 1,000,000,000 is 8,000,000,000. Exactly the number we started with. So that's correct. There doesn't seem to be any error here.So, putting it all together, the number 8,000,000,000 in scientific notation is 8 × 10^9. Therefore, the answer should be boxed{8 times 10^{9}}.Final AnswerThe scientific notation of 8,000,000,000 is boxed{8 times 10^{9}}.</think>To express 8,000,000,000 in scientific notation, we need to convert it into a number between 1 and 10 multiplied by a power of 10. 1. The number 8,000,000,000 can be written as 8 followed by 9 zeros.2. In scientific notation, this is achieved by moving the decimal point 9 places to the left from the original number, which gives us 8.0.3. Moving the decimal point 9 places to the left corresponds to multiplying by (10^9).Thus, 8,000,000,000 in scientific notation is (8 times 10^9).[boxed{8 times 10^{9}}],<think>Okay, let's see. The problem is to express 8,000,000,000 in scientific notation. Hmm, I need to remember how scientific notation works. From what I recall, scientific notation is a way to write very large or very small numbers more compactly by using powers of 10. The general form is a number between 1 and 10 multiplied by 10 raised to some exponent. So, for example, 300 would be written as 3 × 10² in scientific notation because 3 is between 1 and 10, and 10² is 100, so 3 times 100 is 300.Alright, so applying that to 8,000,000,000. First, I need to get a number between 1 and 10. Let's look at the number 8,000,000,000. The digits here are 8 followed by 9 zeros. So if I want a number between 1 and 10, the obvious choice is 8 because 8 is already between 1 and 10. Then, how many times do I have to multiply 8 by 10 to get back to 8,000,000,000?Well, starting with 8, if I multiply by 10 once, I get 80. Multiply by 10 again, I get 800, and so on. Each multiplication by 10 adds a zero. Since there are 9 zeros after the 8 in 8,000,000,000, that suggests that the exponent should be 9. So, 8 multiplied by 10 to the 9th power would give 8,000,000,000. Let me check that.10 to the 1st power is 10, so 8 × 10¹ = 80. 10² is 100, so 8 × 10² = 800. Continuing this, 10³ is 1,000, so 8 × 10³ = 8,000. Each exponent increases the number of zeros by one. Therefore, 10⁹ should be 1,000,000,000 (that's a billion), so 8 × 10⁹ is 8,000,000,000. That seems right.Wait, but just to make sure I haven't messed up the number of zeros. Let's count them in 8,000,000,000. Starting after the 8: first comma after three digits is 8,000, then next three digits is 000, so that's 8,000,000, and then another three digits 000, making it 8,000,000,000. So each comma represents three zeros. There are three commas: the first after 8, then after three more zeros, then another three, then another three. Wait, actually, in numbers, commas are placed every three digits. So starting from the right, but here we have 8 followed by nine zeros. Let's see:8,000,000,000. Breaking it down: 8 (billion), 000 (million), 000 (thousand), 000. So that's 8 billion, which is 8 followed by 9 zeros. So yes, nine zeros. Therefore, 10⁹ is correct.Alternatively, maybe another way to think about it is moving the decimal point. In scientific notation, you move the decimal point so that there's only one non-zero digit to the left of it. In 8,000,000,000, the decimal is implicitly after the last zero. To get to 8.0, we need to move the decimal point 9 places to the left. Each move to the left increases the exponent by 1. So moving it 9 places gives 10⁹. Therefore, 8,000,000,000 becomes 8 × 10⁹.Yes, that seems consistent. Let me confirm with another example. Let's take 5,000. That's 5 × 10³ because there are three zeros. Similarly, 7,000,000 is 7 × 10⁶. So following this pattern, 8,000,000,000 with nine zeros is 8 × 10⁹.Another check: 10⁹ is 1 billion. So 8 × 1 billion is 8 billion, which is exactly 8,000,000,000. That makes sense.Therefore, the scientific notation for 8,000,000,000 is 8 × 10⁹.Final AnswerThe number 8,000,000,000 in scientific notation is boxed{8 times 10^{9}}.</think>To express 8,000,000,000 in scientific notation, we need to write it as a number between 1 and 10 multiplied by a power of 10. 1. The number 8,000,000,000 has 8 followed by 9 zeros.2. To convert this into scientific notation, we identify that 8 is the number between 1 and 10.3. The decimal point is moved 9 places to the left from its original position, which corresponds to multiplying 8 by (10^9).Thus, 8,000,000,000 in scientific notation is (boxed{8 times 10^{9}}).]

🔑:To express the number 8,000,000,000 in scientific notation, we follow the standard procedure of moving the decimal point to the right of the first non-zero digit and counting the number of places moved. In this case, the decimal point is moved 9 places to the left:[8000000000 = 8 times 10^{9}]Thus, the number 8,000,000,000 expressed in scientific notation is:[boxed{8 times 10^{9}}]

🧠:[<think>Alright, so I need to figure out the range of 'a' such that for any x₁ in [1, 3] and x₂ in [1, 3], the inequality f(x₁) ≥ g(x₂) always holds. The functions are given as f(x) = (x² - x - 1)/(x + 1) and g(x) = -e^(x-1) - ln x + a. The answer choices are A through D, with various inequalities involving a and 1/2. Let me start by understanding what exactly I need to do here.First, the problem states that for any x₁ and x₂ in the interval [1, 3], f(x₁) is always greater than or equal to g(x₂). So, this means that the minimum value of f(x) on [1, 3] has to be greater than or equal to the maximum value of g(x) on [1, 3]. Because if the smallest f(x) is still bigger than the largest g(x), then all other combinations will automatically satisfy the inequality. That makes sense. So, my goal is to find the minimum of f(x) on [1, 3] and the maximum of g(x) on [1, 3], then set the minimum f(x) ≥ maximum g(x), and solve for a.Alright, so step 1: Find the minimum of f(x) on [1, 3].Given f(x) = (x² - x - 1)/(x + 1). Let me first simplify this function if possible. Let's try polynomial division or factorization. Let's see:Numerator: x² - x - 1Denominator: x + 1Divide x² - x -1 by x + 1. Let's do polynomial long division.x + 1 ) x² - x -1First term: x² divided by x is x. Multiply (x + 1) by x: x² + x. Subtract from the numerator:(x² - x -1) - (x² + x) = -2x -1.Now, the remainder is -2x -1. So, f(x) can be written as x + (-2x -1)/(x + 1). Wait, that seems a bit messy. Alternatively, maybe we can write f(x) as x - 2 + (something). Let me check again.Wait, maybe I made a mistake in the division. Let me try again:Divide x² - x -1 by x + 1.First term: x² / x = x. Multiply (x + 1) by x: x² + x. Subtract that from the numerator:(x² - x -1) - (x² + x) = -2x -1. So, the division gives x with a remainder of -2x -1. So, f(x) = x + (-2x -1)/(x + 1). Alternatively, perhaps simplify the remainder:(-2x -1)/(x + 1) = - (2x + 1)/(x + 1) = - [2x +1]/[x +1]. Maybe factor numerator: 2x +1 = 2(x + 0.5). But denominator is x +1, so not sure if that helps. Alternatively, write 2x +1 = 2(x +1) -1. So:2x +1 = 2(x +1) -1, so:[2x +1]/(x +1) = 2 - 1/(x +1). Therefore, f(x) = x - [2 - 1/(x +1)] = x -2 + 1/(x +1). Ah, that's better. So f(x) simplifies to x - 2 + 1/(x +1). That might be easier to work with.So f(x) = x - 2 + 1/(x +1). Now, to find the minimum of f(x) on [1, 3], I need to find critical points by taking the derivative and checking endpoints.First, compute the derivative f’(x):f’(x) = derivative of x -2 + 1/(x +1). The derivative of x is 1, derivative of -2 is 0, derivative of 1/(x +1) is -1/(x +1)^2. So:f’(x) = 1 - 1/(x +1)^2.Set derivative equal to zero to find critical points:1 - 1/(x +1)^2 = 0So, 1 = 1/(x +1)^2 => (x +1)^2 = 1 => x +1 = ±1. Since x is in [1,3], x +1 is always positive, so x +1 =1 => x=0. But x=0 is not in [1,3], so the only critical point would be at x +1 =1 => x=0, which is outside our interval. Wait, but (x +1)^2 =1 => x +1 =1 or x +1 =-1. Since x +1 >=2 in [1,3], as x is at least 1, x +1 is at least 2, so (x +1)^2 =1 has no solutions in the interval [1,3]. Therefore, f(x) has no critical points in [1,3], so its extrema must occur at the endpoints.Therefore, the minimum of f(x) on [1,3] is either at x=1 or x=3.Compute f(1):f(1) = (1)^2 -1 -1)/(1 +1) = (1 -1 -1)/2 = (-1)/2 = -0.5Wait, but using the simplified version f(x) = x -2 + 1/(x +1):f(1) = 1 -2 +1/(2) = -1 + 0.5 = -0.5. Same result.Compute f(3):f(3) = (9 -3 -1)/(3 +1) = (5)/4 = 1.25. Alternatively, simplified:3 -2 +1/(4) = 1 + 0.25 = 1.25. Correct.So since f(x) is increasing or decreasing? Since the derivative f’(x) =1 -1/(x +1)^2. Let's check the sign of the derivative on [1,3].For x in [1,3], x +1 ranges from 2 to 4, so (x +1)^2 ranges from 4 to 16. Then 1/(x +1)^2 ranges from 1/16 to 1/4. So f’(x) =1 - [1/16 to 1/4] = [15/16 to 3/4]. Since 15/16 ≈0.9375 and 3/4=0.75. Both are positive. So derivative is positive throughout [1,3]. Therefore, f(x) is increasing on [1,3]. Therefore, the minimum is at x=1, which is -0.5, and maximum at x=3, which is 1.25.Okay, so the minimum value of f(x) on [1,3] is -1/2. So f(x₁) ≥ -1/2 for all x₁ in [1,3]. Therefore, in order for f(x₁) ≥ g(x₂) for all x₁, x₂ in [1,3], we need that the minimum f(x₁) which is -1/2 is ≥ the maximum g(x₂). So -1/2 ≥ max(g(x₂)). Therefore, the maximum of g(x) on [1,3] must be less than or equal to -1/2. Then, since g(x) = -e^(x-1) - ln x + a, so the maximum of g(x) is when the expression -e^(x-1) - ln x is maximized, because a is a constant. Therefore, max(g(x)) = [max(-e^(x-1) - ln x)] + a. Therefore, if we can compute the maximum of (-e^(x-1) - ln x) over [1,3], then add 'a' to it, set that ≤ -1/2, then solve for 'a'.So step 2: Find the maximum of g(x) = -e^(x-1) - ln x + a on [1,3]. Which is equivalent to finding the maximum of h(x) = -e^(x-1) - ln x, then adding 'a'. So we need to find max h(x) on [1,3], then set h_max + a ≤ -1/2, so a ≤ -1/2 - h_max.Wait, but let me confirm. Since g(x) = h(x) + a, then the maximum of g(x) is h_max + a. Therefore, to have h_max + a ≤ -1/2, then a ≤ -1/2 - h_max. So we need to compute h_max first.Let me compute h(x) = -e^(x-1) - ln x. Find its maximum on [1,3].To find the maximum, we can take the derivative of h(x), set to zero, and check critical points as well as endpoints.Compute h'(x):h'(x) = derivative of -e^(x-1) - ln x. The derivative of -e^(x-1) is -e^(x-1) (chain rule: derivative of x-1 is 1). The derivative of -ln x is -1/x. So:h'(x) = -e^(x-1) - 1/x.Set h'(x) = 0:-e^(x-1) -1/x =0 => e^(x-1) = -1/x. However, e^(x-1) is always positive for all real x, and -1/x is negative for x>0. So there is no solution to h'(x)=0 in x>0. Therefore, h(x) has no critical points in [1,3]. Therefore, its maximum must occur at one of the endpoints, x=1 or x=3.Compute h(1):h(1) = -e^(0) - ln 1 = -1 -0 = -1.Compute h(3):h(3) = -e^(2) - ln 3 ≈ -7.389 -1.0986 ≈ -8.4876.Therefore, h(x) is decreasing on [1,3], since its derivative is always negative (as h'(x) = -e^(x-1) -1/x, both terms are negative in [1,3]). Therefore, the maximum of h(x) is at x=1, which is -1.Hence, the maximum value of g(x) on [1,3] is h_max + a = -1 + a.So, we need that -1 + a ≤ -1/2. Solving for a:-1 + a ≤ -1/2 => a ≤ -1/2 +1 => a ≤ 1/2.Therefore, the range of a is all real numbers less than or equal to 1/2. So the answer should be A: a ≤ 1/2.Wait, but let me double-check my logic. The minimum of f(x) is -1/2, and the maximum of g(x) is -1 + a. So to ensure that f(x₁) ≥ g(x₂) for all x₁, x₂ in [1,3], the minimum f(x) must be ≥ maximum g(x). Therefore, -1/2 ≥ -1 + a ⇒ a ≤ 1/2. Yes, that seems correct.Let me verify the endpoints. For example, if a=1/2, then the maximum g(x) is -1 +1/2= -1/2, so f(x₁) ≥ -1/2 and g(x₂) ≤ -1/2. Therefore, equality holds when f(x₁) is at its minimum and g(x₂) is at its maximum. If a is larger than 1/2, then the maximum g(x) would be greater than -1/2, so there would exist some x₂ where g(x₂) > -1/2, but since f(x₁) can be as low as -1/2, then f(x₁) might not be ≥ g(x₂) in that case. If a is smaller than 1/2, then the maximum g(x) is less than -1/2, so f(x₁) which is always ≥ -1/2 would certainly be ≥ g(x₂). Therefore, the correct range is a ≤1/2. So answer A.But let me check the answer choices again. The options are:A: a ≤ 1/2B: a ≥ 1/2C: 0 < a ≤1/2D: -1/2 ≤a ≤1/2So according to my reasoning, A is correct. But let me check if maybe I confused minimum and maximum somewhere.Wait, the function g(x) = -e^(x-1) - ln x + a. So as x increases from 1 to 3, -e^(x-1) becomes more negative, -ln x becomes more negative (since ln x increases), so overall h(x) = -e^(x-1) -ln x is decreasing, hence maximum at x=1. So h(1) = -1 -0 = -1. Then g(x) = h(x) +a, so maximum g(x) is -1 +a. Then set the minimum f(x) = -1/2 ≥ maximum g(x) = -1 +a. Therefore, -1/2 ≥ -1 +a ⇒ a ≤1/2. So that's correct.Therefore, answer A.Final Answerboxed{A}</think>To determine the range of ( a ) such that ( f(x_1) geq g(x_2) ) for all ( x_1, x_2 in [1, 3] ), we need to ensure that the minimum value of ( f(x) ) on ([1, 3]) is greater than or equal to the maximum value of ( g(x) ) on ([1, 3]).1. Finding the minimum value of ( f(x) ): - The function ( f(x) = frac{x^2 - x - 1}{x + 1} ) simplifies to ( f(x) = x - 2 + frac{1}{x + 1} ). - The derivative ( f'(x) = 1 - frac{1}{(x + 1)^2} ) is positive for all ( x in [1, 3] ), indicating ( f(x) ) is increasing on this interval. - Evaluating ( f(x) ) at the endpoints: - ( f(1) = -0.5 ) - ( f(3) = 1.25 ) - The minimum value of ( f(x) ) on ([1, 3]) is ( -0.5 ).2. Finding the maximum value of ( g(x) ): - The function ( g(x) = -e^{x-1} - ln x + a ). - The derivative ( g'(x) = -e^{x-1} - frac{1}{x} ) is negative for all ( x in [1, 3] ), indicating ( g(x) ) is decreasing on this interval. - Evaluating ( g(x) ) at the endpoints: - ( g(1) = -1 + a ) - ( g(3) approx -8.4876 + a ) - The maximum value of ( g(x) ) on ([1, 3]) is ( -1 + a ).3. Setting the inequality: - We need ( -0.5 geq -1 + a ). - Solving for ( a ): [ -0.5 geq -1 + a implies a leq frac{1}{2} ]Thus, the range of ( a ) is ( a leq frac{1}{2} ), which corresponds to option A.[boxed{A}],<think>Alright, let me try to figure this problem out. So, we have two functions, f(x) and g(x). The problem says that for any x₁ in [1, 3] and x₂ in [1, 3], the inequality f(x₁) ≥ g(x₂) always holds. We need to find the range of a given the options. Hmm, okay.First, let me parse the functions. f(x) is (x² - x - 1)/(x + 1). And g(x) is -e^(x - 1) - ln x + a. So, the inequality f(x₁) ≥ g(x₂) must be true for all x₁ and x₂ in [1, 3]. That means the minimum value of f(x) on [1, 3] has to be greater than or equal to the maximum value of g(x) on [1, 3]. Because if even the smallest f(x) is still bigger than the largest g(x), then all other combinations will satisfy the inequality too. So, essentially, we need to find min f(x) and max g(x) over [1, 3], then set min f(x) ≥ max g(x), and solve for a. That seems like the right approach.Let me start with f(x). Let's simplify f(x) first. The function is (x² - x - 1)/(x + 1). Maybe I can perform polynomial division or factor the numerator to see if it simplifies. Let's try dividing x² - x -1 by x + 1.Dividing x² - x -1 by x + 1. How many times does x + 1 go into x²? That's x times. Multiply x by x + 1: x² + x. Subtract that from the original numerator: (x² - x -1) - (x² + x) = -2x -1. Now, how many times does x + 1 go into -2x -1? That's -2 times. Multiply -2 by x + 1: -2x -2. Subtract that from -2x -1: (-2x -1) - (-2x -2) = (-2x -1 + 2x + 2) = 1. So, the division gives x - 2 with a remainder of 1. Therefore, f(x) can be written as x - 2 + 1/(x + 1). Let me check that:x - 2 + 1/(x +1) = [ (x - 2)(x + 1) + 1 ] / (x +1) = [x² + x - 2x -2 + 1]/(x +1) = [x² -x -1]/(x +1). Yes, that's correct. So f(x) simplifies to x - 2 + 1/(x +1). That might be easier to work with for finding the minimum.Now, to find the minimum of f(x) on [1, 3], we can take the derivative and find critical points. Let's compute f'(x). Since f(x) = x - 2 + 1/(x +1), the derivative is f'(x) = 1 - 1/(x +1)^2. Let me verify:d/dx [x - 2] = 1, and d/dx [1/(x +1)] = -1/(x +1)^2. So yes, f'(x) = 1 - 1/(x +1)^2.Set derivative equal to zero to find critical points:1 - 1/(x +1)^2 = 0 => 1 = 1/(x +1)^2 => (x +1)^2 = 1 => x +1 = ±1. Since x is in [1, 3], x +1 is always positive (minimum x=1, x+1=2), so x +1 = 1 => x = 0. But x=0 is outside of [1, 3], so there are no critical points inside the interval [1, 3]. Therefore, the extrema of f(x) on [1, 3] must occur at the endpoints.So we just need to evaluate f(x) at x=1 and x=3.Compute f(1):f(1) = (1² -1 -1)/(1 +1) = (1 -1 -1)/2 = (-1)/2 = -0.5.Compute f(3):f(3) = (9 -3 -1)/(3 +1) = (5)/4 = 1.25.But since there are no critical points in between, the function is increasing or decreasing throughout the interval? Let's check the derivative. Take x=1, f'(1) = 1 - 1/(2)^2 = 1 - 1/4 = 3/4 >0. So derivative is positive at x=1. Similarly, at x=3, f'(3) =1 -1/(4)^2=1 -1/16=15/16>0. So the derivative is positive throughout the interval, meaning f(x) is increasing on [1, 3]. Therefore, the minimum value is at x=1, which is -0.5, and the maximum is at x=3, 1.25.Wait, but the problem says for any x₁ and x₂ in [1,3], f(x₁) ≥ g(x₂). So the minimum of f(x) must be greater than or equal to the maximum of g(x). So min f(x) = -0.5, and we need -0.5 ≥ max g(x). Therefore, the maximum of g(x) on [1,3] must be ≤ -0.5. Then, solving for a.Now, let's work on g(x). g(x) = -e^(x -1) - ln x + a. We need to find the maximum of g(x) on [1,3], then set that maximum ≤ -0.5, and solve for a.First, find the maximum of g(x). Let's compute its derivative:g'(x) = derivative of -e^(x -1) is -e^(x -1), derivative of -ln x is -1/x, derivative of a is 0. So g'(x) = -e^(x -1) -1/x.To find critical points, set g'(x)=0:- e^(x -1) - 1/x =0 => e^(x -1) + 1/x =0. But e^(x -1) is always positive for any real x, and 1/x is positive for x>0. Therefore, their sum is always positive. Therefore, there's no solution to g'(x)=0. So there are no critical points. Therefore, the maximum of g(x) on [1,3] occurs at one of the endpoints.Therefore, compute g(1) and g(3):Compute g(1):g(1) = -e^(1 -1) - ln 1 + a = -e^0 -0 +a = -1 +a.Compute g(3):g(3) = -e^(3 -1) - ln3 +a = -e^2 - ln3 +a.So between x=1 and x=3, which one is larger? Let's see:g(1) = -1 +ag(3) = -e² - ln3 +a ≈ -7.389 -1.0986 +a ≈ -8.4876 +aSo clearly, g(1) is much larger than g(3) because -1 +a is greater than -8.4876 +a. Therefore, the maximum of g(x) on [1,3] is at x=1, which is -1 +a.Therefore, to ensure that min f(x) ≥ max g(x):-0.5 ≥ -1 +aSolve for a:Add 1 to both sides:0.5 ≥ aWhich is equivalent to a ≤ 0.5.Therefore, the range of a is all real numbers less than or equal to 1/2. So option A: a ≤ 1/2.But wait, let me check again. The problem says for any x₁ and x₂ in [1,3], f(x₁) ≥ g(x₂). So that the minimal f(x₁) is greater than or equal to the maximal g(x₂). So the minimal f is -0.5, and the maximal g is -1 +a. Therefore, -0.5 ≥ -1 +a => a ≤ 0.5. So the answer is A.But let me double-check if there are any mistakes in reasoning. For f(x), we found that it's increasing on [1,3] with minimum at x=1, -0.5, and maximum at x=3, 1.25. That seems correct. For g(x), since the derivative is always negative (since g'(x) = -e^(x -1) -1/x, which is negative for all x>0), so g(x) is decreasing on [1,3]. Therefore, its maximum is at x=1, which is -1 +a. Therefore, the maximum of g is -1 +a, and we need that to be less than or equal to the minimum of f, which is -0.5. So yes, -1 +a ≤ -0.5 => a ≤ 0.5. So answer A.But wait, let's check the options again. The options are A: a ≤ 1/2, B: a ≥1/2, C: 0 < a ≤1/2, D: -1/2 ≤a ≤1/2. So answer is A.But wait, the problem statement says g(x) = -e^{x-1} - lnx +a. So when x=1, e^{0}=1, ln1=0, so g(1)=-1 -0 +a= a -1. Similarly, when x=3, e^{2}, ln3, so g(3)= -e^2 - ln3 +a, which is a very negative number. So as x increases, g(x) decreases. So maximum of g is at x=1: a -1, and minimum at x=3. Therefore, the maximum is a -1, which must be less than or equal to -0.5. So a -1 ≤ -0.5 => a ≤ 0.5. So yes, answer is A. So the correct answer is A.But just to be thorough, let's check if there's any possibility that somewhere else in the interval [1,3], g(x) could be larger? For example, could there be a critical point where derivative is zero? Wait, we already concluded that g'(x) = -e^{x -1} -1/x, which is always negative. Since e^{x -1} is positive and increasing, so -e^{x -1} is negative and decreasing, and -1/x is also negative and increasing (since 1/x decreases as x increases). But regardless, both terms are negative for all x in [1,3], so their sum is negative. Therefore, g(x) is strictly decreasing on [1,3], so indeed the maximum is at x=1.Similarly, f(x) was found to be increasing on [1,3], so the minimum is at x=1. Therefore, combining these, yes, the minimal f(x) is -0.5 and the maximal g(x) is a -1. So inequality is -0.5 ≥ a -1 => a ≤ 0.5. So answer A.I think that's solid. So the answer is A.Final Answerboxed{A}</think>Given the functions ( f(x) = frac{x^2 - x - 1}{x + 1} ) and ( g(x) = -e^{x-1} - ln x + a ), we need to determine the range of ( a ) such that for any ( x_1 in [1, 3] ) and ( x_2 in [1, 3] ), the inequality ( f(x_1) geq g(x_2) ) always holds.First, we simplify ( f(x) ):[f(x) = frac{x^2 - x - 1}{x + 1} = x - 2 + frac{1}{x + 1}]To find the minimum value of ( f(x) ) on the interval ([1, 3]), we compute the derivative:[f'(x) = 1 - frac{1}{(x + 1)^2}]Setting ( f'(x) = 0 ) gives no critical points within ([1, 3]). Evaluating ( f(x) ) at the endpoints:[f(1) = frac{1^2 - 1 - 1}{1 + 1} = -frac{1}{2}, quad f(3) = frac{3^2 - 3 - 1}{3 + 1} = frac{5}{4}]Since ( f(x) ) is increasing on ([1, 3]), the minimum value is ( -frac{1}{2} ).Next, we analyze ( g(x) ):[g(x) = -e^{x-1} - ln x + a]The derivative is:[g'(x) = -e^{x-1} - frac{1}{x}]Since ( g'(x) ) is always negative, ( g(x) ) is strictly decreasing on ([1, 3]). Evaluating ( g(x) ) at the endpoints:[g(1) = -1 + a, quad g(3) = -e^2 - ln 3 + a]The maximum value of ( g(x) ) on ([1, 3]) is ( -1 + a ).To ensure ( f(x_1) geq g(x_2) ) for all ( x_1, x_2 in [1, 3] ), we need:[-frac{1}{2} geq -1 + a implies a leq frac{1}{2}]Thus, the range of ( a ) is (boxed{A}).]

🔑:The function f(x) = frac{x^2 - x - 1}{x + 1} can be simplified to x + 1 + frac{1}{x + 1} - 3, which is always greater than or equal to -1, with equality holding only when x = 0. However, since x₁ ∈ [1, 3], the minimum value of the function is f(1) = -frac{1}{2}.The function g(x) = -ex-1 - lnx + a is a decreasing function for any x₂ ∈ [1, 3]. Therefore, its maximum value is g(1) = -1 + a.Given that f(x₁) ≥ g(x₂) always holds for any x₁ ∈ [1, 3] and x₂ ∈ [1, 3], we can derive that -frac{1}{2} ≥ -1 + a, which simplifies to a ≤ frac{1}{2}.Therefore, the correct answer is: boxed{text{A}}This problem requires understanding the monotonicity of functions and their application, as well as the ability to convert the always-true conditions into inequalities to solve for the range of a.

🧠:[<think>Alright, let me try to figure out this probability problem. So, the question is about identifying which pair of events are non-exclusive when two balls are randomly selected from a pocket containing two red balls and three black balls. The options are A, B, C, and D. Each option pairs two events, and we need to determine which pair is non-exclusive. First, I need to recall what non-exclusive events mean. If two events are non-exclusive, it means that they can happen at the same time. In other words, there's an overlap between them; their intersection is not empty. So, if there's a scenario where both events occur simultaneously, then they are non-exclusive. Conversely, if they can't happen together, they are mutually exclusive.Let me list out the given options again:A: "At least one black ball" and "Both are black balls"B: "At least one black ball" and "At least one red ball"C: "Exactly one black ball" and "Exactly two black balls"D: "At least one black ball" and "Both are red balls"I need to check each pair to see if they can occur together.Starting with option A: "At least one black ball" and "Both are black balls". Let's parse these. The first event is getting at least one black ball when selecting two balls. The second event is getting both black balls. Now, if both are black balls, that certainly satisfies "at least one black ball". So, the event "Both are black balls" is a subset of "At least one black ball". Therefore, these two events can definitely occur together. In fact, whenever both are black, both events happen. Hence, they are non-exclusive. So, A is a possible answer.But let's check the other options to be thorough.Option B: "At least one black ball" and "At least one red ball". So, if we pick two balls, can we have at least one black and at least one red? Yes, that's possible. For example, if one is black and one is red. So, in that case, both events would occur. Therefore, these two events can happen together, making them non-exclusive. Hmm, so B might also be a correct answer. Wait, but let me think again. Wait, "at least one black" and "at least one red" – if we have both, that is, one black and one red, then both events are satisfied. So, yes, they can co-occur. Therefore, B is also non-exclusive. But the question is giving options, and we need to pick the correct one. Wait, but maybe I made a mistake here. Let me verify.Wait, the problem states "non-exclusive events", which is another term for events that are not mutually exclusive. So, mutually exclusive events cannot happen at the same time, while non-exclusive events can. So, in the case of B: "At least one black" and "At least one red", can these both happen? Yes, if you have one black and one red. So, their intersection is not empty. Therefore, they are non-exclusive. So, both A and B seem to be non-exclusive. But let me check the answer options again. Wait, the problem might be designed to have only one correct answer. Hmm.Wait, let me check the options again. Let's go through each carefully.First, let's model the sample space. There are 2 red balls and 3 black balls. So, total of 5 balls. When selecting two balls, the possible outcomes are combinations of these. Let's calculate all possible outcomes. The total number of ways to choose 2 balls from 5 is C(5,2) = 10. Let's list them:Let me denote the red balls as R1, R2 and the black balls as B1, B2, B3.So, the possible pairs are:1. R1, R22. R1, B13. R1, B24. R1, B35. R2, B16. R2, B27. R2, B38. B1, B29. B1, B310. B2, B3So, 10 outcomes.Now, let's analyze each option:Option A: "At least one black ball" and "Both are black balls"Event A1: At least one black ball. This includes all outcomes where there is one or two black balls. So, outcomes 2-10. That's 9 outcomes.Event A2: Both are black balls. This is outcomes 8,9,10. 3 outcomes.So, the intersection of A1 and A2 is A2 itself. Since A2 is a subset of A1, their intersection is A2. So, there are 3 outcomes where both occur. Therefore, they are non-exclusive. So, A is correct.Option B: "At least one black ball" and "At least one red ball"Event B1: At least one black ball. Same as A1: outcomes 2-10.Event B2: At least one red ball. This includes all outcomes with one or two red balls. Outcomes 1-7.Intersection of B1 and B2: outcomes that have at least one black and at least one red. These are the outcomes where there is one red and one black. So, outcomes 2-7. That's 6 outcomes. Therefore, their intersection is not empty, so they are non-exclusive. Therefore, B is also non-exclusive.Option C: "Exactly one black ball" and "Exactly two black balls"Event C1: Exactly one black ball. This is outcomes 2-7. Each of these has one black and one red. So, 6 outcomes.Event C2: Exactly two black balls. This is outcomes 8-10. 3 outcomes.Intersection of C1 and C2: There's no overlap because a pair can't have exactly one and exactly two black balls at the same time. Therefore, they are mutually exclusive. So, C is exclusive, not non-exclusive.Option D: "At least one black ball" and "Both are red balls"Event D1: At least one black ball. Outcomes 2-10.Event D2: Both are red balls. Outcome 1.Intersection of D1 and D2: Outcome 1 is both red, which is not in D1. Therefore, their intersection is empty. So, they are mutually exclusive. Hence, D is exclusive, not non-exclusive.So, based on this analysis, both A and B are non-exclusive. But the options are given as A, B, C, D, and the problem states "the non-exclusive events are", but it might be a translation issue. Wait, the original problem says "the non-exclusive events are:", and then lists options A to D. So, maybe more than one could be correct, but in the problem statement, in the original Chinese maybe it's different. However, in the problem presented here, in English, the user is giving four options, each with two events, and the question is which pair is non-exclusive.But looking at the options, in the original problem statement, the way it's phrased might be expecting a single answer. Let me check again.Wait, but let me check the definition. In probability, non-exclusive events are events that can occur together, i.e., their intersection is not empty. So, in the options given, both A and B are non-exclusive. However, let me check the problem again. The problem says "the non-exclusive events are", with options A to D. So, perhaps in the original problem, multiple options could be correct, but the way it's presented here, maybe only one is intended. Wait, but the user hasn't specified multiple answers; the instruction says "put your final answer within boxed{}", which usually indicates a single answer.Hmm. Let me re-examine my analysis for any mistakes.Option A: "At least one black" and "Both are black". As I saw, "Both are black" is a subset of "At least one black", so their intersection is the event "Both are black", which is non-empty. So, they are non-exclusive.Option B: "At least one black" and "At least one red". Their intersection is the event "Exactly one black and one red", which is non-empty. So, they are non-exclusive.Option C: "Exactly one black" and "Exactly two black". These cannot happen at the same time, so they are exclusive.Option D: "At least one black" and "Both are red". These can't happen together, since "Both are red" means zero black balls, which contradicts "At least one black". So, they are exclusive.Therefore, both A and B are non-exclusive. But the options are given as A, B, C, D. If the problem is from a multiple-choice question where only one option is correct, there might be an error. But according to the analysis, both A and B are correct. However, maybe the problem is translated from Chinese, and perhaps in the original, the options are different. Alternatively, maybe I made a mistake.Wait, let me check the problem again. It says "the non-exclusive events are", followed by the four options. In Chinese exams, sometimes they use "non-exclusive" to mean "not mutually exclusive", so events that can co-occur. So, in that case, both A and B are correct. But if the answer requires only one, perhaps there is a trick.Wait, let me check the exact wording. The problem says: "When two balls are randomly selected from a pocket containing two red balls and three black balls, the non-exclusive events are: A, B, C, D". So, in the original problem, maybe the answer expects to select all that apply, but in the context here, since the user is asking for the answer to be boxed, maybe there's a mistake in the problem, or maybe the answer is multiple. But in standard multiple-choice questions, sometimes only one is correct.Wait, let me double-check the analysis for B. If you have "at least one black" and "at least one red", the overlap is the cases where there is one black and one red. So, they can co-occur. Hence, non-exclusive. So, B is non-exclusive. But maybe the problem considers "non-exclusive" differently?Wait, no, the standard definition is that two events are non-exclusive if they can happen together. So, both A and B are non-exclusive. However, perhaps in the problem's context, there is a different intended answer. Let me check the options again.Option C: "Exactly one black" and "Exactly two black". These are clearly mutually exclusive. So, C is exclusive.Option D: "At least one black" and "Both are red". Also mutually exclusive.Thus, only A and B are non-exclusive. However, in the problem given, the user is to present an answer as boxed{} which is typically one box. But maybe the original problem had a different setup. Wait, in the problem statement as given here, the user wrote:"When two balls are randomly selected from a pocket containing two red balls and three black balls, the non-exclusive events are:A: ... B: ... C: ... D: ..."So, the user is likely expecting to choose one of the options A-D as the correct answer. However, according to the analysis, both A and B are correct. But perhaps in the original problem, there's a mistake, or maybe I'm missing something.Wait, let me check the definition once again. Non-exclusive events are those that can occur simultaneously, i.e., P(A ∩ B) ≠ 0. So, for A, as we saw, if both are black, then both events A occur. So, A is non-exclusive.For B, if you have one black and one red, then both events B occur. So, B is also non-exclusive.But maybe in the problem's options, the answer is multiple, but in the formatting here, the user might expect a single answer. Wait, looking at the original problem statement again: "the non-exclusive events are: A:..., B:..., C:..., D:...". So, in Chinese tests, sometimes the wording might be "which of the following are non-exclusive events", and the options are pairs. If the answer is multiple, but the user here expects a single boxed answer, perhaps there's a discrepancy.Alternatively, maybe the problem was mistranslated, and actually, it's asking for "mutually exclusive" events, in which case C and D are exclusive, but A and B are non-exclusive. But the problem says "non-exclusive events are", so the answer should be A and B. But since the user is expecting a single answer, maybe the correct answer is both A and B. However, in the options given, A and B are separate options.Wait, in the problem as presented, each option is a pair of events. So, the question is asking "which pair of events are non-exclusive". Therefore, the answer is the options (A, B, C, D) where the two events in the pair are non-exclusive. So, in this case, both A and B are pairs where the two events are non-exclusive, while C and D are pairs where the events are exclusive.But in the original problem, maybe only one of A or B is considered correct. Wait, let's check again:Option A: "At least one black" and "Both are black". The intersection is "Both are black", which is non-empty. Therefore, they are non-exclusive.Option B: "At least one black" and "At least one red". The intersection is "one black and one red", which is non-empty. Therefore, non-exclusive.So, both A and B are correct. But if the problem expects only one answer, perhaps there's a mistake. Alternatively, maybe the problem is from a source where the answer is different.Wait, maybe the term "non-exclusive" is being used differently. Wait, in some contexts, "non-exclusive" might mean that neither is a subset of the other, but I don't think that's the standard definition. The standard definition is that their intersection is non-empty.Alternatively, maybe the problem is asking for events that are not independent? But no, the term used is "non-exclusive".Alternatively, maybe in Chinese terminology, "non-exclusive" has a different meaning. But generally, in probability, "exclusive" means mutually exclusive, i.e., cannot happen together, so "non-exclusive" means can happen together. So, in that case, both A and B are non-exclusive. However, given that the answer requires boxing a single option, perhaps there's a mistake in the problem's options or in my analysis.Wait, let me verify once again:For Option B: "At least one black" and "At least one red". If we have two balls, can we have at least one black and at least one red? Yes, if one is black and one is red. So, yes, their intersection is not empty. Hence, they are non-exclusive.But maybe the problem is considering that "At least one black" and "At least one red" are complementary? Wait, no. The complement of "At least one black" is "No black balls", i.e., both red. Similarly, the complement of "At least one red" is "No red balls", i.e., both black. So, they are not complementary. Their intersection is the case where one is black and one is red.Therefore, my conclusion is that both A and B are non-exclusive. But since the options are labeled A to D, and the user expects one answer boxed, perhaps the intended answer is both A and B, but in the original problem, maybe the answer is different.Wait, looking back at the problem statement: "the non-exclusive events are: A:..., B:..., C:..., D:...", so perhaps the answer is multiple, but the way the user has presented the problem here, they might expect multiple answers. However, given the final instruction says "put your final answer within boxed{}", which usually is for a single answer.Alternatively, maybe the problem is from a source where the correct answer is C, but that contradicts our analysis. Wait, no, C is "Exactly one black" and "Exactly two black" which are mutually exclusive.Alternatively, maybe a mistranslation occurred. The term "non-exclusive" might have been intended as "not mutually exclusive", which is the same as non-exclusive. So, in that case, A and B are correct. However, given the options, and if forced to choose one, perhaps the answer is A and B. But the system here expects a single boxed letter.Wait, perhaps in the original problem, the answer is C, but that's not possible. Alternatively, maybe I need to check the problem again.Wait, let me think differently. Maybe the problem is referring to events that are not exclusive in the sense of forming a partition, but that seems unlikely.Alternatively, maybe the term "non-exclusive" is being used in the sense of overlapping in the context of probability formulas. Wait, but I think the standard term is that two events are mutually exclusive if they can't occur together, and non-exclusive otherwise. So, A and B are non-exclusive, C and D are exclusive.But if the answer options are A, B, C, D, then both A and B are correct. However, in Chinese textbooks, sometimes they might have multiple correct answers, but the way the user presented the problem here, with instructions to box one answer, maybe there's confusion.Alternatively, maybe I miscounted the intersections. Let me check again.For option A: "At least one black" (outcomes 2-10) and "Both are black" (outcomes 8-10). The intersection is outcomes 8-10, so yes, non-empty. So, non-exclusive.For option B: "At least one black" (2-10) and "At least one red" (1-7). The intersection is outcomes 2-7, which is non-empty. So, non-exclusive.So, indeed, both A and B are non-exclusive. Since the problem is presented with options A to D, and if it's a multiple-select question, the answer would be A and B. But given that the user is asking for a single boxed answer, perhaps there's a mistake in the problem's options or the original intended answer.Alternatively, perhaps the answer is C, but no, C is mutually exclusive. Wait, unless there's a different interpretation. "Exactly one black ball" and "Exactly two black balls"—if you have exactly two black balls, then you can't have exactly one, so they are mutually exclusive. So, C is exclusive.Similarly, D is exclusive because "At least one black" and "Both are red" can't happen together.So, the only possible non-exclusive pairs are A and B. Therefore, if the problem allows multiple answers, it's A and B, but if it's a single answer, there might be an error. However, considering that the original problem might have been in Chinese and perhaps there was a translation error, leading to confusion.Alternatively, let's check for a different approach. Maybe the problem is considering the probability of the intersection. For example, in some contexts, events are considered non-exclusive if the probability of their intersection is not zero, which we already established for A and B. So, again, both A and B are non-exclusive.Given that the user might expect a single answer, but according to the analysis, both A and B are correct. However, in the options given, perhaps the answer is A, B, or both.Wait, looking at the problem again, the user specifies "the non-exclusive events are", and gives four options. In the Chinese context, sometimes the term used is "compatible events" or "non-conflicting events", which is the same as non-exclusive. So, the answer should be both A and B. But the user instruction says to put the final answer in a single boxed {}.Alternatively, maybe there is a misunderstanding in the problem. If I re-express the events:A: "At least one black" and "Both are black"—these can co-occur.B: "At least one black" and "At least one red"—can co-occur.C: "Exactly one black" and "Exactly two black"—cannot co-occur.D: "At least one black" and "Both are red"—cannot co-occur.Therefore, the pairs that are non-exclusive are A and B. If the options are given as A, B, C, D, then the correct answers are A and B. However, in the context of a multiple-choice question where you select all that apply, but here, since the user hasn't specified multiple answers and the standard practice is to box one, maybe there's a trick.Alternatively, the problem might have a typo, and one of the options is different. For example, maybe option B was intended to be "At least one black" and "At least two reds", which would be exclusive, but as given, it's "At least one black" and "At least one red", which are non-exclusive.Alternatively, maybe in Chinese, the term "non-exclusive" is used differently. But according to standard probability terms, the answer should be both A and B. However, since the user's final answer requires a single box, perhaps there was a mistake in the problem, and the intended answer is A.Alternatively, maybe I need to check once more.Wait, let's check the counts:Total number of ways: 10.For event A: "At least one black" has 9 outcomes, "Both are black" has 3. Their intersection is 3 outcomes.For event B: "At least one black" has 9, "At least one red" has 7 (outcomes 1-7). Their intersection is 6 outcomes (2-7).So, both have significant overlaps.If the problem is from a source where multiple answers are possible, then both A and B are correct. However, if only one answer is expected, perhaps the problem had a typo. For example, if option B was "At least two black balls" and "At least one red ball", then they would be exclusive. But as given, it's "At least one black" and "At least one red", which are non-exclusive.Given that the problem is presented as is, and without further information, the correct pairs are A and B. But since the user is likely expecting one answer, perhaps the intended answer is A. However, given the analysis, I must conclude that both A and B are correct.But since the system here accepts only one boxed answer, maybe the problem expects only A. Alternatively, maybe there's a different approach.Wait, another angle: sometimes, events are considered non-exclusive if neither is a subset of the other. But in option A, "Both are black" is a subset of "At least one black", so technically, they are not independent, but they are still non-exclusive because they can co-occur. So, even though one is a subset of the other, they are still non-exclusive. So, A is correct.Option B: "At least one black" and "At least one red" are not subsets of each other. For example, outcome 1 is in "At least one red" but not in "At least one black". Outcome 8 is in "At least one black" but not in "At least one red". So, they are independent events? No, not necessarily. They just overlap in the middle. So, they are non-exclusive.Therefore, both A and B are non-exclusive, so if multiple answers are allowed, both are correct. However, if the problem expects a single answer, perhaps the answer is A, as it's more obvious, but according to strict definitions, both are correct.But since this is an exercise for the assistant, and the assistant is to provide a single answer, but in the analysis both A and B are correct. This is a dilemma.Wait, maybe the original problem in Chinese had a different structure. Alternatively, the answer options could be differently labeled.Alternatively, perhaps the answer is C? No, C is exclusive. D is exclusive.Wait, but according to the analysis, A and B are non-exclusive. But since the problem gives four options and the user expects one answer boxed, there might be a mistake. Alternatively, the problem might have intended to ask for "mutually exclusive" events, in which case C and D are correct.But the problem explicitly says "non-exclusive events are".Alternatively, maybe in Chinese, "non-exclusive" refers to events that do not completely overlap, i.e., they are neither mutually exclusive nor one is a subset of the other. But that's non-standard. In that case, B would be non-exclusive, because they partially overlap, while A is a subset. But that's not the standard definition.In standard terms, two events are non-exclusive if they can occur together, regardless of subset relations. So, A and B are both non-exclusive.Given the ambiguity, but given the problem's options, if only one answer is expected, maybe the intended answer is A. Alternatively, the problem might have a typo. But based on strict analysis, both A and B are correct.Since I need to provide an answer, and given that maybe the problem is from a source where the answer is A, but according to the analysis, both A and B are correct. However, given that B is also correct, but maybe in the problem's context, considering the possible answer options, it's better to check the probabilities.Wait, another way to check is to calculate the probabilities:For events to be non-exclusive, P(A ∩ B) ≠ 0.For option A: P(A1 ∩ A2) = P(A2) = 3/10 ≠ 0.For option B: P(B1 ∩ B2) = 6/10 ≠ 0.Therefore, both are non-zero, hence both A and B are non-exclusive.But since the problem gives four options and the user expects a single answer, but according to the math, two are correct. This is conflicting.Alternatively, maybe the problem is from a source where the answer is C, but that's incorrect.Alternatively, the problem might have a different definition. For instance, in some contexts, non-exclusive might mean that the occurrence of one affects the probability of the other, i.e., they are dependent events. But that's a different concept (independence vs exclusivity). If that's the case, then we need to check for independence.But the term used is "non-exclusive", which typically refers to mutual exclusivity, not independence.Assuming the problem uses standard terminology, both A and B are non-exclusive. However, since the user might expect a single answer, and if I have to choose based on the problem's structure, maybe the answer is A. But strictly speaking, both A and B are correct.Given that in the original problem, if this is a translation, perhaps the answer intended is A. Alternatively, maybe the user made a mistake in the options.But given the analysis, I should state that both A and B are non-exclusive. However, since the answer format is a single boxed letter, and assuming that the problem expects one answer, maybe there's an error. But according to proper analysis, the answer should include both A and B. However, since I can only provide one boxed answer, but according to the strict terms, both are correct. This is a problem.Alternatively, the problem might have a different answer. Let me check once more:Wait, perhaps the problem is asking for which pair of events are not mutually exclusive, i.e., non-exclusive. So, the answer is both A and B. However, in the options given, since both A and B are correct, but the system here allows only one boxed answer, perhaps the answer is A and B. But I can't box two letters. Therefore, maybe there's a mistake in the problem's options.Alternatively, if I have to choose the most appropriate answer, perhaps A is more obviously non-exclusive, but B is also correct.Alternatively, maybe the answer is C, but no, C is mutually exclusive.Given this confusion, and to adhere to the instructions, I think the problem might have intended the answer to be A, but according to the analysis, B is also correct. Given that the user might have mistyped or there's a translation error, but based on pure analysis, both A and B are correct. However, since only one answer can be boxed, and assuming the problem's intended answer is A, I will go with A.But wait, another thought: in option B, "At least one black" and "At least one red", when you draw two balls, having at least one of each requires that you have exactly one of each, which is possible. So, their intersection is not empty. Hence, they are non-exclusive. Therefore, both A and B are correct. However, since the options are separate and the user expects one answer, maybe the problem is from a source where multiple answers are possible. But given the way the question is structured here, with options A-D and a single boxed answer, there's inconsistency.Alternatively, perhaps the original problem had only one correct answer, and due to a translation error, it's presented differently here. Without more information, I think the mathematically correct answers are A and B. However, since I have to choose one, and given the options, I'll state both.But the system doesn't allow multiple boxed answers. So, this is a problem.Wait, maybe the answer is C. No, C is mutually exclusive.Alternatively, perhaps the answer expects the user to recognize that in option B, "At least one black" and "At least one red" can co-occur, so they are non-exclusive, which is also correct. So, if the problem allows multiple answers, the answer is A and B. If only one is allowed, then perhaps there's an error.Given that the user has not specified multiple answers and the standard format is a single boxed letter, but the correct answer is both A and B, I might need to mention both but follow the format.But since the system requires a single boxed answer, and given that both A and B are correct, but the options are given as separate, perhaps the answer is A and B. However, the standard way to represent multiple answers is boxed{A} and boxed{B}, but the user might not expect that.Alternatively, the problem might have a mistake, and the correct answer is A, while B is actually exclusive. Wait, no, in B, if you have one black and one red, both events occur. Hence, B is non-exclusive.In conclusion, strictly speaking, both A and B are correct. However, due to the answer format constraint, I might have to select both. But given the standard multiple-choice setup where only one answer is correct, perhaps the intended answer is A. But based on analysis, both A and B are correct.Given the time I've spent and to adhere to the instructions, I will state that both A and B are correct. However, since I can only provide one boxed answer, I will box both A and B. Although unconventional, it's the accurate conclusion.Final Answerboxed{A} and boxed{B}Note: In standard probability terms, both options A and B are non-exclusive. However, if only one answer is expected, there might be an error in the problem's options.</think>To determine which pair of events are non-exclusive when two balls are randomly selected from a pocket containing two red balls and three black balls, we need to check if the events can occur simultaneously (i.e., their intersection is not empty).1. Option A: "At least one black ball" and "Both are black balls" - "At least one black ball" includes outcomes with one or two black balls. - "Both are black balls" is a subset of "At least one black ball". - These events can occur together (e.g., selecting two black balls), so they are non-exclusive.2. Option B: "At least one black ball" and "At least one red ball" - "At least one black ball" includes outcomes with one or two black balls. - "At least one red ball" includes outcomes with one or two red balls. - These events can occur together (e.g., selecting one black and one red ball), so they are non-exclusive.3. Option C: "Exactly one black ball" and "Exactly two black balls" - These events cannot occur simultaneously (mutually exclusive).4. Option D: "At least one black ball" and "Both are red balls" - These events cannot occur simultaneously (mutually exclusive).Both options A and B are non-exclusive because they can occur together. However, since the problem might expect a single answer, the correct pairs are both A and B.boxed{A} and boxed{B},<think>Okay, let's see. The problem is about non-exclusive events when selecting two balls from a pocket containing two red and three black balls. The options are A, B, C, D. I need to figure out which pair of events are non-exclusive. Hmm. First, let me recall what non-exclusive events mean. Non-exclusive events are events that can occur at the same time, meaning their intersection is not empty. So, if two events have some overlap, they are non-exclusive. If they can't happen together, they are mutually exclusive.So, the task is to check each pair of events (A, B, C, D) and see if they can happen simultaneously. Let's go through each option one by one.First, let me figure out the sample space. There are 2 red balls and 3 black balls, total of 5 balls. When selecting two balls, the number of possible combinations is C(5,2) = 10. Let me list all possible outcomes:1. Red1 and Red22. Red1 and Black13. Red1 and Black24. Red1 and Black35. Red2 and Black16. Red2 and Black27. Red2 and Black38. Black1 and Black29. Black1 and Black310. Black2 and Black3So, these are all the possible pairs. Now, let's analyze each option.Option A: Events are "At least one black ball" and "Both are black balls". Let's denote them as Event A1 and Event A2.Event A1: At least one black ball. This would include all outcomes where there is one or two black balls. Looking at the list, that's outcomes 2-10. Because outcome 1 is both red, and the rest have at least one black.Event A2: Both are black balls. That's outcomes 8,9,10.So, the intersection of A1 and A2 is A2 itself, since all outcomes where both are black are already included in A1. So, they can occur together. Therefore, these events are non-exclusive because if both are black, then both events happen. So, A is a pair of non-exclusive events. But let me check the other options just in case.Option B: Events are "At least one black ball" (same as A1) and "At least one red ball" (let's call this Event B2). Let's see.Event B2: At least one red ball. This would be outcomes where there is one or two red balls. So, outcomes 1-7. Because outcomes 1 is both red, and 2-7 have one red and one black.Event A1 is outcomes 2-10. So, the intersection between A1 and B2 is outcomes 2-7. That is, the events where there is one red and one black. Since this intersection is not empty, these two events can occur at the same time. For example, if you pick one red and one black, both events are satisfied. So, B is also non-exclusive.Wait, but the question says "the non-exclusive events are" and gives four options. But maybe there are multiple correct answers? Wait, the problem is likely designed to have only one correct answer, so perhaps I need to check again.Wait, maybe I made a mistake. Let me verify.Option C: "Exactly one black ball" (Event C1) and "Exactly two black balls" (Event C2). Let's see.Event C1: Exactly one black ball. That's outcomes 2-7 (each has one black and one red).Event C2: Exactly two black balls. That's outcomes 8-10.These two events are mutually exclusive because you can't have exactly one and exactly two black balls at the same time. So, their intersection is empty. Therefore, they are exclusive. So, C is not non-exclusive. So, C is out.Option D: "At least one black ball" (Event D1, same as A1) and "Both are red balls" (Event D2).Event D2: Both are red balls. That's outcome 1.Event D1 is outcomes 2-10. So, the intersection of D1 and D2 is empty because D2 is outcome 1, which is not in D1. Therefore, these are mutually exclusive. So, D is exclusive, not non-exclusive.Wait, so going back. So, A is non-exclusive (since A1 and A2 can both happen if both are black). B is non-exclusive (since A1 and B2 can both happen if one red and one black). So both A and B are non-exclusive. But the options are given as A, B, C, D. So, the problem might be translated wrong? Wait, let me check again.Wait, the problem says "the non-exclusive events are", and gives four options. So, maybe in the original question, they are saying which pair is non-exclusive. So, possibly multiple answers? But in the problem statement, the way it's phrased is "the non-exclusive events are", which might imply only one correct answer. Maybe in Chinese, the original problem is clearer.Wait, perhaps I need to check the translation. But assuming the translation is correct, the problem is asking for which pair of events are non-exclusive. Since both A and B are non-exclusive, but in the options, maybe only one is correct? Wait, let me check again.Wait, Option A: "At least one black ball" and "Both are black balls". The intersection is "Both are black balls", which is non-empty. Therefore, they are non-exclusive.Option B: "At least one black ball" and "At least one red ball". Their intersection is "exactly one black and one red", which is non-empty. Therefore, they are non-exclusive.So, both A and B are non-exclusive. But let's look back at the problem statement. It says "the non-exclusive events are", and the options are A, B, C, D. So, if the question is asking which pairs are non-exclusive, then both A and B are correct, but if it's a multiple-choice question where only one is correct, maybe there's a mistake here.Wait, let me check again the problem statement:"When two balls are randomly selected from a pocket containing two red balls and three black balls, the non-exclusive events are:A: ... B: ... C: ... D: ..."So, the problem is presented as a multiple-choice question where the answer is one of A, B, C, D. But according to my analysis, both A and B are non-exclusive. However, maybe in the original problem, only one is correct. Let me double-check.Wait, maybe I miscalculated for B. Let's check B again.Event B1: At least one black ball (outcomes 2-10)Event B2: At least one red ball (outcomes 1-7)The intersection is outcomes that are in both, which is outcomes 2-7. So, that's selecting one red and one black. So, these events can occur together. So, B is non-exclusive.Similarly, in A, the intersection is outcomes where both are black, which is part of "at least one black". So, both A and B are non-exclusive. But the answer options are single-choice. Hmm. Maybe the problem is translated incorrectly or there's a misunderstanding.Wait, perhaps in the original problem, the term is "mutually non-exclusive", which is the same as non-exclusive. But maybe in Chinese, it's clearer. Alternatively, maybe in the options, there's another aspect.Wait, let's check C and D again. C is "Exactly one black" and "Exactly two black", which are mutually exclusive, so they are exclusive. D is "At least one black" and "Both red", which are also mutually exclusive. So, C and D are exclusive. So, only A and B are non-exclusive. But the problem is given as options A, B, C, D. So, perhaps in the original question, there are multiple correct answers, but here in the translation, the user expects to choose all that apply. But since the instructions here are to put the final answer within boxed{}, which is typically for a single answer, maybe there's a mistake in the problem.Wait, perhaps in the problem statement, it's "the non-exclusive events are" implying multiple answers, but the way it's presented here, as options A-D, perhaps in the original source, it's a multiple-select question. But since the user hasn't specified, and the initial problem is presented as a single-answer multiple-choice, perhaps there's a mistake in the options.Alternatively, maybe there's a misinterpretation in the definition. Wait, let me check the definition again. Non-exclusive events are events that can happen at the same time, i.e., their intersection is not empty. So, for each pair, check if there's an overlap.Option A: "At least one black" includes "both black". So, if you have both black, then both events A1 and A2 happen. So, they are non-exclusive.Option B: "At least one black" and "At least one red". If you have one black and one red, both events occur. So, they are non-exclusive.Option C: "Exactly one black" and "Exactly two black" – can't happen together, exclusive.Option D: "At least one black" and "Both red" – can't happen together, exclusive.So, both A and B are correct. But since the options are listed as A, B, C, D, perhaps the intended answer is both A and B. But the way the question is phrased "the non-exclusive events are", and the options are labeled A to D, each being a pair. In some multiple-choice questions, especially in Chinese examinations, sometimes multiple options can be correct, but here, if it's designed for a single answer, maybe there's an error. Alternatively, maybe the translation is off.Wait, let me check if "non-exclusive" is being used differently. Sometimes, in probability, people use "independent" vs "dependent" events. But non-exclusive is about mutual exclusivity. The term "non-exclusive" would mean that they can co-occur, i.e., not mutually exclusive. So, yes, A and B are non-exclusive.But the problem might be expecting only one correct answer. Let me check the pairs again.In option A: "At least one black" and "Both are black". The intersection is "both black", which is a subset of "at least one black", so they can co-occur. So, A is non-exclusive.In option B: "At least one black" and "At least one red". The intersection is "one black and one red". So, they can co-occur. So, B is also non-exclusive.But the problem is presented as a multiple-choice question with options A to D. If both A and B are correct, but the system here requires a single answer, maybe the problem has a typo or the original Chinese problem is different.Alternatively, maybe the question is which pair is NOT mutually exclusive, and the answer is both A and B, but since the user is expecting one boxed answer, perhaps there's a mistake.Alternatively, maybe in the original problem, the answer is C or D. Wait, no. C and D are exclusive.Wait, perhaps the term "non-exclusive" here refers to something else. Let me check.Wait, non-exclusive in Chinese might have a different translation. Maybe it's not about mutual exclusivity but something else. But given the context, mutual exclusivity is the standard term. Alternatively, maybe it's asking for independent events. Let me check.Wait, independent events are events where the probability of both occurring is the product of their probabilities. Let's check for that.For example, in option A:Probability of A1 (at least one black) is 1 - probability of both red. Probability of both red is C(2,2)/C(5,2) = 1/10. So, P(A1) = 9/10.Probability of A2 (both black) is C(3,2)/C(5,2) = 3/10.The intersection of A1 and A2 is A2, so P(A1 ∩ A2) = 3/10.If they were independent, then P(A1 ∩ A2) should equal P(A1)*P(A2) = (9/10)*(3/10) = 27/100. But 3/10 is 30/100 ≠ 27/100. So, they are not independent. So, non-exclusive here doesn't refer to independence.Alternatively, maybe the problem is using "non-exclusive" to mean overlapping, which is the same as not mutually exclusive. So, A and B are correct. But since the user might expect one answer, maybe in the original problem, there's a different approach.Wait, perhaps there's a misunderstanding in the problem statement. Maybe "non-exclusive" is meant to be "not mutually exclusive", which would mean the same as non-exclusive. So, if that's the case, both A and B are correct. But given that the options are A, B, C, D, and in Chinese tests often only one option is correct, maybe there's a mistake here.Alternatively, perhaps the problem is from the Chinese Gaokao or similar, and in the original, the answer is C. But according to the analysis, C is mutually exclusive.Wait, let me check once again.Event C: "Exactly one black" and "Exactly two black". These cannot happen at the same time. So, mutually exclusive. So, C is incorrect.Event D: "At least one black" and "Both red". These also can't happen together. So, D is also incorrect.Therefore, only A and B are non-exclusive. But the problem gives four options, each being a pair. If the answer expects multiple selections, but here it's presented as options A to D, each being a single pair, then perhaps the problem is designed to have multiple correct answers, but in the original context, maybe the answer is A and B. But since the user instruction says to put the final answer within boxed{}, which is typically one box, maybe the intended answer is A and B. However, given the options are A to D, each being a pair, perhaps the correct answers are both A and B, but in the original problem, maybe only one is considered correct.Alternatively, maybe I made a mistake in interpreting the problem. Let me think again.Wait, maybe "non-exclusive" is a mistranslation, and actually, it's asking for "dependent" events? But that's a different concept. However, dependent events are events where the occurrence of one affects the probability of the other. But the question specifically mentions "non-exclusive", which is more likely mutual exclusivity.Alternatively, the problem might be from probability where "non-exclusive" refers to events that are not mutually exclusive. In that case, both A and B are correct. However, given the answer format here, maybe the answer is both A and B. But since the user has to provide one boxed answer, perhaps there is a mistake in the problem's options, or maybe I need to look again.Wait, in the initial problem statement, it says "the non-exclusive events are", followed by four options. In Chinese exams, sometimes they use the term "non-exclusive" (非互斥事件) which directly translates to "not mutually exclusive". So, the question is asking which pair of events are not mutually exclusive. Therefore, the answer should be the pairs that can occur together.Since both A and B can occur together, they are non-exclusive (not mutually exclusive). However, in the options given, each is a separate pair. If the original question is multiple-select, but here presented as options A-D, perhaps the answer is A and B. But given that we can only provide one boxed answer, maybe the problem expects both A and B, but since that's not possible, there's a conflict.Alternatively, perhaps the problem is from a source where the correct answer is both A and B, but in the given options, since you can't select two, maybe there's an error. Alternatively, maybe I'm overcomplicating.Wait, let me check the pairs again:A: "At least one black" and "Both are black" – these can co-occur.B: "At least one black" and "At least one red" – these can co-occur (when you have one of each).C: "Exactly one black" and "Exactly two black" – cannot co-occur.D: "At least one black" and "Both red" – cannot co-occur.Therefore, the correct answers are A and B. However, given the way the question is presented, if it's a multiple-choice question with four options where only one is correct, then there must be a mistake. But in the original problem, maybe the options are labeled differently, or there is a different number of correct answers.Given that in the problem statement here, the user just copied it as options A-D, and asks to provide the answer boxed, maybe the intended answer is A and B. But in standard multiple-choice format, you can't choose two. So, perhaps the problem has a typo, and the correct answer is A and B, but in the given options, maybe the answer is C or D. But according to the analysis, C and D are exclusive.Alternatively, maybe the problem is from a source where the answer is C, but that doesn't make sense. Let me check once again.Wait, maybe the problem is in Chinese, and "non-exclusive" has a different meaning. For example, maybe it refers to events that are not subsets of each other or something. But in standard probability terms, non-exclusive is simply not mutually exclusive.Given that, the answer should be A and B, but since the options are single, perhaps the problem is incorrect. Alternatively, maybe the original problem had only A as the correct answer. Let me check the probability again.Wait, in option B, "At least one black" and "At least one red". The intersection is having one black and one red. So, they can co-occur. So, B is non-exclusive. A is also non-exclusive. So, if the problem allows multiple answers, A and B. If not, maybe the problem is wrong.Alternatively, maybe I need to look at the problem statement again. It says "the non-exclusive events are", so perhaps all pairs that are non-exclusive. But according to the analysis, A and B are non-exclusive, C and D are exclusive.Since the user is asking for the answer to be boxed, and typically one box, perhaps the intended answer is A and B. But given that, how to represent that? Maybe the problem expects multiple selections, but in the current format, it's unclear.Alternatively, perhaps there's a mistake in the problem, and the answer is A. But according to the analysis, B is also correct.Wait, maybe the problem is from a source where the answer is C, but that contradicts our analysis. Alternatively, maybe the answer is different.Wait, let me think in another way. Let's compute the probabilities.For option A:Event A1: At least one black. Probability = 1 - P(both red) = 1 - 1/10 = 9/10.Event A2: Both black. Probability = C(3,2)/C(5,2) = 3/10.Intersection: Both black, which is A2. So, P(A1 ∩ A2) = 3/10. Since this is not zero, they are non-exclusive.For option B:Event B1: At least one black. Probability = 9/10.Event B2: At least one red. Probability = 1 - P(both black) = 1 - 3/10 = 7/10.Intersection: One black and one red. Probability = C(3,1)*C(2,1)/C(5,2) = 6/10 = 3/5. So, 3/5 ≠ 0. Hence, non-exclusive.Therefore, both A and B are non-exclusive. So, if multiple answers are allowed, both are correct. If only one is allowed, perhaps the problem is faulty. Given the options, and in Chinese exams sometimes expecting multiple answers but written together, but here it's presented as separate options.But the user instruction says to put the final answer within boxed{}, which is typically for a single answer. Given that, maybe the problem expects both A and B to be correct, but since the options are given as A, B, C, D, and in the absence of clarification, the most precise answers are A and B. However, since we have to choose one, maybe there's a mistake.Alternatively, the problem might have a different intended answer. Let me check once again.Wait, perhaps in the problem statement, the term is "exclusive" instead of "non-exclusive". But the user wrote "non-exclusive". If it were exclusive, then C and D would be the answer. But since the user specified non-exclusive, it's A and B.But given the answer format, I need to provide one boxed answer. Maybe the answer is both A and B. But in standard multiple-choice notation, you can't. Alternatively, in some cases, options might be like (A,B), but here the options are separate.Given that, and given the problem's structure, perhaps the intended answer is A and B. But since I have to choose one, perhaps the problem is from a source where the answer is A and B, and the user expects both. But how to represent that.Alternatively, maybe the original problem had only A as the correct answer. Let's see. For instance, in option A, the intersection is non-empty, and in option B, the intersection is non-empty. So, both are correct. Therefore, the answer should be both A and B. However, given that the options are labeled A to D, and in the Chinese context sometimes answers are written together, maybe the answer is AB. But in the absence of such an option, maybe the problem is incorrect.But since in the problem given here, the options are A to D, each being a pair, and the user asks to put the final answer within boxed{}, which usually takes a single letter, perhaps there's a mistake in the problem. However, given the analysis, both A and B are correct. Therefore, the answer is A and B.But considering that in the problem's original context, maybe only A is considered correct. Alternatively, the answer is B.Wait, but the intersection of A's events is when both are black, which is allowed, so A is non-exclusive.The intersection of B's events is when there's one black and one red, which is allowed, so B is non-exclusive.So, both are correct.But the problem might have a typo. Alternatively, since the user has asked to select one, maybe the answer is A. But according to the reasoning, both are correct.Given that, I think the problem might have a mistake, but based on strict analysis, both A and B are non-exclusive. However, since the options are given as separate and the user expects one answer, I need to check again.Wait, maybe the answer is C? But no, C is mutually exclusive.Wait, let's check the definition again. Non-exclusive events can occur together. So, if two events have overlapping outcomes, they are non-exclusive. So, both A and B can occur together. Therefore, the answer should be both A and B.But in the given format, how is this represented? If the answer options are A, B, C, D, and multiple can be correct, but the system here allows only one boxed answer, there is a conflict.But the user might have intended the answer to be A and B, so maybe we need to write both. However, standard practice is to box a single answer. Given that, and since the problem is likely translated, maybe the intended answer is A.Alternatively, maybe in the original problem, "non-exclusive" refers to something else. For example, in some contexts, people might refer to non-exclusive as not being subsets. For example, in Option A, "Both are black" is a subset of "At least one black", so they are not non-exclusive in the sense of independence. But no, non-exclusive is about mutual exclusivity.Given all this, I think the correct answers are A and B. But since we can only provide one answer, maybe the answer is both A and B. However, in standard multiple-choice format, this is not possible. Therefore, there might be an error in the problem's options.But considering the strictest analysis and if forced to choose based on the problem's options, since both A and B are correct, but in the absence of a combined option, I think the problem is flawed. However, since I need to provide an answer, perhaps the intended answer is A and B, written as AB. But in the absence of such an option, I'm unsure.Alternatively, if the original problem intended only one correct answer, maybe A is the answer. Because in A, one event is a subset of the other, so they are definitely non-exclusive. In B, the events are overlapping but neither is a subset. Maybe in some contexts, overlapping but not subsets are considered non-exclusive. But both A and B are non-exclusive.Given that, and to comply with the user's request, I will state that the correct answers are A and B. But since I have to put one boxed answer, perhaps the problem expects both, but the standard format is boxed{A} and boxed{B}. However, instructions say to put the final answer within boxed{}.Given the ambiguity, but based on the analysis, both A and B are correct. Since the problem may have a mistake, but assuming the answer is one of them, perhaps A is the intended answer.But wait, let's see. In Chinese textbooks, sometimes when they ask for non-exclusive events, they might look for events where one is not a subset of the other. But no, non-exclusive just means not mutually exclusive.Given that, both A and B are correct. Therefore, the answer is both A and B. However, the standard way to present multiple answers in boxed format is to box each, but the user instruction says "put your final answer within boxed{}". Since the options are labeled A to D, and multiple answers are expected, maybe the answer is A and B. But how to represent that.Alternatively, the problem might have a typo, and the correct answer is A and B. But given the user's interface here, perhaps the answer is A and B.But since I can only write one boxed answer, and the problem may have intended a single answer, maybe the answer is B. Let me check once again.Wait, the difference between A and B:In A, "At least one black" includes "both black". So, if both are black, then both events A1 and A2 happen. So, they are non-exclusive.In B, "At least one black" and "At least one red" can happen together when there's one of each. So, they are also non-exclusive.Both are correct. Therefore, the answer is both A and B. Since the user has to input one answer, maybe they expect boxed{A} and boxed{B}, but the system might not allow. Alternatively, the problem is from a source where multiple answers are possible, and the correct answer is A and B.However, given the instructions here, I need to pick one. Given that, and the possibility that the original problem might have intended A as the answer, but in reality, both are correct. However, to be precise, according to the analysis, both A and B are non-exclusive. But in the absence of further information, I will proceed with both A and B.But since the user wants the answer in boxed{}, and typically in Chinese exams, multiple answers are boxed together. So, maybe boxed{A} and boxed{B}.But instructions say to put your final answer within boxed{}, which might imply a single box. Therefore, maybe the problem expects only A, but my analysis says both A and B.Given the problem's ambiguity, but based on strict probability theory, both A and B are correct. Therefore, the answer should be both A and B. If I must choose one, perhaps A is the answer, but strictly speaking, both are correct.Alternatively, the problem might have a different intended answer due to translation error.After careful consideration, I conclude that both A and B are non-exclusive events. Therefore, the correct answers are A and B. However, given the format required here, and if forced to choose one, perhaps the problem expects A, but I believe both are correct.But since in multiple-choice questions with single answers, sometimes the first correct option is the answer, so perhaps A is the answer. But I think both are correct.Given the time I've spent and to adhere to the instructions, I will state both A and B as the correct answers, boxed as boxed{A} and boxed{B}. However, since the platform might not accept two boxes, but the user specified "put your final answer within boxed{}", I will go with the standard answer format for multiple correct answers in Chinese exams, which is to write both letters inside one box: boxed{AB}.Final Answerboxed{AB}</think>To determine which pairs of events are non-exclusive when two balls are randomly selected from a pocket containing two red balls and three black balls, we need to check if the intersection of each pair of events is non-empty.1. Option A: "At least one black ball" and "Both are black balls" - Event A1 (At least one black ball): Includes outcomes 2-10 (one or two black balls). - Event A2 (Both are black balls): Includes outcomes 8-10. - Intersection: Outcomes 8-10 (both black balls), which is non-empty. Therefore, these events are non-exclusive.2. Option B: "At least one black ball" and "At least one red ball" - Event B1 (At least one black ball): Includes outcomes 2-10. - Event B2 (At least one red ball): Includes outcomes 1-7 (one or two red balls). - Intersection: Outcomes 2-7 (one black and one red ball), which is non-empty. Therefore, these events are non-exclusive.3. Option C: "Exactly one black ball" and "Exactly two black balls" - Event C1 (Exactly one black ball): Includes outcomes 2-7. - Event C2 (Exactly two black balls): Includes outcomes 8-10. - Intersection: Empty set. Therefore, these events are mutually exclusive.4. Option D: "At least one black ball" and "Both are red balls" - Event D1 (At least one black ball): Includes outcomes 2-10. - Event D2 (Both are red balls): Includes outcome 1. - Intersection: Empty set. Therefore, these events are mutually exclusive.Both options A and B are non-exclusive events because their intersections are non-empty. boxed{AB}]

🔑:To analyze the non-exclusive events among the given options, let's break down each pair of events to see if they can occur simultaneously or include one another.Option A: "At least one black ball" and "Both are black balls"- The event "At least one black ball" includes the possibilities of getting one black ball or two black balls.- The event "Both are black balls" is a subset of the event "At least one black ball" because if both balls selected are black, it certainly satisfies the condition of having at least one black ball.- Therefore, these two events are not exclusive, and they can occur in a manner where one event includes the other.Option B: "At least one black ball" and "At least one red ball"- The event "At least one black ball" can occur with the combinations of one black ball and one red ball, or two black balls.- The event "At least one red ball" can occur with the combinations of one red ball and one black ball, or two red balls.- Since there is a combination where one black ball and one red ball are selected, these two events can occur simultaneously.- Thus, these events are not exclusive.Option C: "Exactly one black ball" and "Exactly two black balls"- The event "Exactly one black ball" means one black ball and one red ball are selected.- The event "Exactly two black balls" means both selected balls are black.- These two events cannot occur at the same time because selecting exactly one black ball excludes the possibility of selecting exactly two black balls, and vice versa.- Therefore, these events are exclusive.Option D: "At least one black ball" and "Both are red balls"- The event "At least one black ball" requires at least one of the selected balls to be black.- The event "Both are red balls" means no black balls are selected.- These two events cannot occur simultaneously because if both balls selected are red, it's not possible to have at least one black ball.- Thus, these events are exclusive.Considering the analysis above, the correct options where the events are non-exclusive are A and B. Therefore, the correct answer is encapsulated as follows:boxed{text{AB}}