🧠:[<think>Okay, so I need to find the coordinates of the focus of the parabola given by the equation y squared equals negative six x. Hmm, let me recall what I know about parabolas. I remember that there are standard forms for the equations of parabolas, depending on whether they open up, down, left, or right. First, let me write down the equation again to visualize it: y² = -6x. Comparing this to the standard form of a parabola that opens to the left or right. I think the standard form is either y² = 4ax or y² = -4ax. Let me check. If the coefficient is positive, does it open to the right, and if it's negative, does it open to the left? Yes, that sounds right. So in this case, the equation is y² = -6x, which means it's similar to y² = -4ax. Therefore, 4a should be equal to 6 here. Wait, but in the standard form, it's y² = 4ax for opening to the right, so if it's negative, it's y² = -4ax, opening to the left. So comparing y² = -6x with y² = -4ax, we can set -4a equal to -6. Therefore, solving for a: -4a = -6 → a = (-6)/(-4) = 6/4 = 3/2. So a is 3/2. Now, the focus of a parabola in this form is at (a, 0) if it opens to the right, and (-a, 0) if it opens to the left. Wait, but in the standard form y² = 4ax, the focus is at (a, 0), and if it's y² = -4ax, then the focus is at (-a, 0). Since our equation is y² = -6x, which is similar to y² = -4ax, then 4a = 6, so a = 6/4 = 3/2. Therefore, the focus should be at (-a, 0) = (-3/2, 0). Let me verify that. So the standard parabola that opens to the left has the equation y² = -4ax with vertex at the origin. The focus is at (-a, 0), and the directrix is the line x = a. So in this problem, the coefficient on the x term is -6, which equals -4a. So solving for a, we get a = 6/4 = 3/2. Therefore, the focus is at (-3/2, 0). That seems right. Wait, let me check once more to make sure I didn't mix up the direction. If the equation is y² = -6x, then the parabola opens to the left because the coefficient is negative. So the focus should be to the left of the vertex. Since the vertex is at (0,0), the focus is at (-3/2, 0). Alternatively, maybe I can derive this from scratch. Let's recall that a parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. For a parabola that opens to the left, the directrix is a vertical line to the right of the vertex, and the focus is a point to the left of the vertex. Let me suppose the focus is at (h, k). Since the parabola opens to the left, h must be negative, and k should be 0 because the parabola is symmetric about the x-axis. The directrix would then be the line x = -h. For any point (x, y) on the parabola, the distance to the focus (h, k) should equal the distance to the directrix x = -h. So, setting up the equation: sqrt[(x - h)^2 + (y - k)^2] = |x - (-h)|. Since k is 0, this simplifies to sqrt[(x - h)^2 + y²] = |x + h|. Squaring both sides: (x - h)^2 + y² = (x + h)^2. Expanding both sides: x² - 2hx + h² + y² = x² + 2hx + h². Subtracting x² and h² from both sides: -2hx + y² = 2hx. Then, moving the terms: y² = 4hx. Wait, but our original equation is y² = -6x, which is y² = -6x. Comparing to y² = 4hx, it must be that 4h = -6, so h = -6/4 = -3/2. Therefore, the focus is at (h, k) = (-3/2, 0), which matches what I found earlier. Therefore, the coordinates of the focus are (-3/2, 0). To express this as a box, I need to put it in boxed{}. So the final answer is boxed{(-frac{3}{2}, 0)}. But just to make sure, let me recall another method. If I remember correctly, the general form of a parabola that opens horizontally is (y - k)^2 = 4p(x - h), where the vertex is at (h, k), and p is the distance from the vertex to the focus (and also to the directrix). If p is positive, it opens to the right; if negative, to the left. In our case, the equation is y² = -6x, so comparing to (y - 0)^2 = 4p(x - 0), so 4p = -6, so p = -6/4 = -3/2. Therefore, the focus is at (h + p, k) = (0 + (-3/2), 0) = (-3/2, 0). Yep, same result. I think that's solid. So, through multiple approaches: comparing to standard form, using the definition of a parabola, and using the formula with vertex at (h,k), all lead to the focus being at (-3/2, 0). So I'm confident that's the correct answer.Final AnswerThe coordinates of the focus are boxed{left( -frac{3}{2}, 0 right)}.</think>To find the coordinates of the focus of the parabola given by the equation ( y^2 = -6x ):1. Identify the standard form: The given equation ( y^2 = -6x ) is a parabola that opens to the left. The standard form of a parabola that opens left or right is ( y^2 = 4ax ) (opens right if positive, left if negative).2. Compare with the standard form: The given equation ( y^2 = -6x ) can be compared to ( y^2 = -4ax ). By equating the coefficients, we get: [ -4a = -6 ] Solving for ( a ): [ a = frac{-6}{-4} = frac{3}{2} ]3. Determine the coordinates of the focus: For the standard form ( y^2 = -4ax ), the focus is at ( (-a, 0) ). Substituting ( a = frac{3}{2} ): [ text{Focus} = left( -frac{3}{2}, 0 right) ]4. Verification using the definition of a parabola: By considering the definition of a parabola as the set of points equidistant from the focus and directrix, we confirm that the focus is indeed at ( left( -frac{3}{2}, 0 right) ).Thus, the coordinates of the focus are (boxed{left( -frac{3}{2}, 0 right)}).,<think>Okay, so I need to find the coordinates of the focus of the parabola given by the equation y² = -6x. Hmm, let's see. I remember that the standard form of a parabola can help with this. Parabolas can open up, down, left, or right, and the direction they open affects the equation and the focus location. First, let me recall the standard forms. If a parabola opens to the right or left, its equation is usually in the form y² = 4ax or y² = -4ax. Wait, right? So if it's y² = 4ax, it opens to the right, and the focus is at (a, 0). If it's y² = -4ax, then it opens to the left, and the focus is at (-a, 0). Similarly, if the parabola opens up or down, the equation is x² = 4ay or x² = -4ay, with the focus at (0, a) or (0, -a) respectively.In this problem, the equation is y² = -6x. Comparing this to the standard form y² = -4ax, which opens to the left. So here, the coefficient on the x term is -6. That must mean that -4a = -6. Let me solve for a.Starting with -4a = -6. If I divide both sides by -4, I get a = (-6)/(-4) = 6/4 = 3/2. So a is 3/2. But since the parabola opens to the left, the focus should be at (-a, 0). Wait, no. Wait, let's check. The standard form for a parabola opening to the left is y² = -4ax, and the focus is at (-a, 0). Wait, actually, wait, maybe I need to be careful here. Let me verify.Suppose the standard form is y² = 4ax. Then if 4a is positive, it opens to the right, and the focus is at (a, 0). If the equation is y² = -4ax, then it opens to the left, and the focus is at (-a, 0). So in the equation given, y² = -6x, which is similar to y² = -4ax. Therefore, equate -4a to -6. So -4a = -6. Solving for a, a = (-6)/(-4) = 6/4 = 3/2. Then, since the parabola opens to the left, the focus is at (-a, 0) = (-3/2, 0). Wait, but hold on. Let me confirm this with an example. Suppose we have y² = 4ax. If a is positive, it opens to the right, focus at (a,0). If I set 4a = 6, then a = 6/4 = 3/2. Then focus would be at (3/2, 0). Similarly, if the equation is y² = -6x, that's like y² = -4ax, so 4a = 6 here? Wait, maybe I made a miscalculation.Wait, no. Wait, the standard form is y² = 4ax. So if our equation is y² = -6x, then 4a = -6? No, that can't be. Because in the standard form, the coefficient is 4a, but here it's -6. So to make the standard form match, we have to set 4a equal to the coefficient of x. But in this case, the coefficient of x is -6, so 4a = -6. Therefore, a = -6/4 = -3/2. Wait, but that complicates things. Because in the standard form, a is the distance from the vertex to the focus. But if a is negative, then that would imply the focus is to the left if the parabola opens left. Wait, maybe my initial understanding was wrong.Let me step back. Let me think of the standard parabola that opens to the left. Its equation is y² = 4ax, but when a is negative, then it opens to the left. Wait, perhaps I need to clarify.No, actually, the standard form is: if you have y² = 4px, then if p > 0, it opens to the right, focus at (p, 0); if p < 0, it opens to the left, focus at (p, 0). Similarly, x² = 4py opens upward if p > 0, downward if p < 0. So in that case, p is the directed distance from the vertex to the focus. So maybe in this case, the given equation y² = -6x can be written as y² = 4px, where 4p = -6. Therefore, p = -6/4 = -3/2. Therefore, the focus is at (p, 0) = (-3/2, 0). So that's the same answer as before. So, to recap: compare the given equation y² = -6x to the standard form y² = 4px. Then 4p = -6. Solving for p, we get p = -6/4 = -3/2. Therefore, the focus is at (p, 0) = (-3/2, 0). Therefore, the coordinates of the focus are (-3/2, 0).Wait, let me check this again. Suppose the equation is y² = -6x. The vertex is at (0,0) because there are no shifts. The parabola opens to the left because of the negative coefficient. The distance from the vertex to the focus is |p|, where p is the coefficient in the standard form. Since p = -3/2, the focus is located at (-3/2, 0), which is 3/2 units to the left of the vertex. That makes sense because the parabola opens left, so the focus is inside the parabola on the left side.Alternatively, another way to remember is that for y² = ax, the focus is at (a/4, 0). Wait, no, that's not exactly. Wait, for example, if the equation is y² = 4px, then 4p is the coefficient, so focus at (p,0). So if the equation is y² = Ax, then A = 4p, so p = A/4. Therefore, the focus is at (A/4, 0). But in our case, the equation is y² = -6x, so A = -6, so p = -6/4 = -3/2. Therefore, focus at (-3/2, 0). That's the same answer. So that seems consistent.Therefore, I think that's the correct answer. To be thorough, let me also recall that the formula for the focus of a parabola in the form y² = 4ax is (a,0). So here, since the coefficient is -6, we have 4a = -6 => a = -6/4 = -3/2. Hence, the focus is at (a, 0) = (-3/2, 0). Alternatively, maybe there's a mnemonic or formula that I can refer to. Let me check. For a parabola in the form y² = 4px, focus is at (p,0). So in this case, y² = -6x is equivalent to y² = 4p x, so 4p = -6 => p = -6/4 = -3/2. Therefore, focus is at (p,0) = (-3/2, 0). So that's the same answer again.Alternatively, if I graph this parabola, y² = -6x, it's a parabola that opens to the left. The vertex is at (0,0). The focus is inside the parabola, so to the left of the vertex. The distance from the vertex to the focus is |p|. Here, p is -3/2, so the distance is 3/2. So moving 3/2 units to the left from the vertex (0,0), we reach (-3/2, 0). That seems correct.Let me also recall that the formula for the focus when the parabola is in the form (y - k)^2 = 4p(x - h). Then, the vertex is at (h,k), and the focus is at (h + p, k). In this case, our equation is y² = -6x, which can be written as (y - 0)^2 = 4p(x - 0). Therefore, h = 0, k = 0. Then, 4p = -6 => p = -6/4 = -3/2. Therefore, the focus is at (h + p, k) = (0 + (-3/2), 0) = (-3/2, 0). Yep, that's consistent.So all methods lead to the conclusion that the focus is at (-3/2, 0). Therefore, I believe that is the correct answer.But just to make sure, let me recall another way. For a parabola, the focus is located at a distance of |1/(4a)| from the vertex along the axis of symmetry. Wait, that might be confusing. Let me think. If the parabola is expressed as y = ax² + bx + c, then the focus is at a different formula. Wait, but that's for vertical parabolas. For horizontal parabolas, like this one, which open left or right, the standard form is x = a(y - k)^2 + h, but in this case, the equation is y² = -6x, which is already in a standard form.Alternatively, using the definition of a parabola: the set of all points equidistant from the focus and the directrix. For this parabola, since it opens left, the directrix is vertical, to the right of the vertex. The distance from the vertex to the focus is p, and the directrix is the line x = -p. Wait, let me verify.If the parabola opens to the left, then the directrix is a vertical line to the right of the vertex. The focus is at (-p, 0), and the directrix is the line x = p. Wait, maybe that's conflicting with earlier. Wait, let's use the standard formula.Given the equation y² = 4px. If p is positive, opens to the right, focus at (p, 0), directrix x = -p. If p is negative, opens to the left, focus at (p, 0) = (negative, 0), directrix x = -p (which would be positive, since p is negative). For example, if p = -3/2, then directrix is x = -(-3/2) = 3/2. So for the equation y² = -6x, which is y² = 4*(-3/2)x, so directrix is x = 3/2, focus at (-3/2, 0). So, if I pick a point on the parabola, say, let's take x = -1. Then y² = -6*(-1) = 6, so y = ±√6. So the point (-1, √6) is on the parabola. Let's check the distance from this point to the focus and to the directrix. Distance from (-1, √6) to focus (-3/2, 0):Using distance formula: sqrt[(-1 - (-3/2))² + (√6 - 0)²] = sqrt[(1/2)² + (√6)^2] = sqrt[1/4 + 6] = sqrt[25/4] = 5/2.Distance from (-1, √6) to directrix x = 3/2 is the horizontal distance: |x - 3/2| = |-1 - 3/2| = |-5/2| = 5/2.So the distances are equal, which confirms that the focus and directrix are correct. Therefore, focus at (-3/2, 0) is indeed correct.Therefore, after verifying through multiple approaches and even checking with a sample point, I can be confident that the coordinates of the focus are (-3/2, 0).Final AnswerThe coordinates of the focus are boxed{left( -dfrac{3}{2}, 0 right)}.</think>To find the coordinates of the focus of the parabola given by the equation ( y^2 = -6x ):1. Identify the standard form: The given equation ( y^2 = -6x ) is a horizontal parabola that opens to the left. The standard form of such a parabola is ( y^2 = 4px ), where ( p ) is the directed distance from the vertex to the focus.2. Compare with the standard form: By comparing ( y^2 = -6x ) with ( y^2 = 4px ), we see that ( 4p = -6 ).3. Solve for ( p ): [ 4p = -6 implies p = frac{-6}{4} = -frac{3}{2} ]4. Determine the coordinates of the focus: For a horizontal parabola that opens to the left, the focus is located at ( (p, 0) ). Substituting ( p = -frac{3}{2} ), the coordinates of the focus are: [ left( -frac{3}{2}, 0 right) ]Verification: Checking with a sample point on the parabola confirms that the distances from the point to the focus and to the directrix are equal, ensuring the correctness of the focus coordinates. Final AnswerThe coordinates of the focus are boxed{left( -dfrac{3}{2}, 0 right)}.]

🔑:AnalysisThis problem primarily tests the basic properties of parabolas. When finding the focus of a parabola, pay attention to the position of the focus and the direction in which the parabola opens. First, determine the coordinate axis on which the focus lies and the value of p based on the standard form of the parabolic equation, then find the coordinates of the focus.SolutionGiven the parabolic equation y^{2} = -6x,The focus is on the positive half of the x-axis, and p = 3,Therefore, the coordinates of the focus are boxed{(-frac{3}{2}, 0)}.

🧠:[<think>Okay, so I have this problem here where I need to find the maximum and minimum terms among the first 20 terms of the sequence ( a_n = frac{n - sqrt{98}}{n - sqrt{99}} ) where ( n ) is a positive integer. The answer choices are given, so I need to figure out which pair is correct. Let me start by understanding the sequence and then analyze its behavior.First, let me note that ( sqrt{98} ) and ( sqrt{99} ) are irrational numbers. Since ( sqrt{81} = 9 ), ( sqrt{100} = 10 ), so ( sqrt{98} ) is a bit less than 10, maybe around 9.899 (since 9.9 squared is 98.01), and ( sqrt{99} ) is approximately 9.949. So, both ( sqrt{98} ) and ( sqrt{99} ) are between 9.8 and 10. Let me confirm that:( 9.8^2 = 96.04 )( 9.9^2 = 98.01 )( 9.95^2 = 99.0025 )So, ( sqrt{98} ) is approximately 9.899 (since 9.9 squared is 98.01, so subtract a little), and ( sqrt{99} ) is approximately 9.949. Therefore, ( sqrt{98} approx 9.899 ) and ( sqrt{99} approx 9.9499 ).Therefore, the terms of the sequence are ( a_n = frac{n - 9.899}{n - 9.9499} ). So, this is a fraction where both numerator and denominator are linear functions of ( n ). Let me see if I can simplify or analyze this function.First, note that when ( n ) approaches the value ( sqrt{99} approx 9.9499 ), the denominator approaches zero. However, since ( n ) is a positive integer, the closest integer to 9.9499 is 10. Therefore, when ( n = 10 ), the denominator ( 10 - sqrt{99} ) is approximately ( 10 - 9.9499 = 0.0501 ), which is a small positive number. Similarly, the numerator when ( n = 10 ) is ( 10 - sqrt{98} approx 10 - 9.899 = 0.101 ). Therefore, ( a_{10} approx 0.101 / 0.0501 approx 2.016 ). So, ( a_{10} ) is a relatively large term. On the other hand, when ( n ) is 1, the first term, let's compute ( a_1 ):( a_1 = frac{1 - sqrt{98}}{1 - sqrt{99}} ). Let's compute numerator and denominator:Numerator: ( 1 - 9.899 approx -8.899 )Denominator: ( 1 - 9.9499 approx -8.9499 )Therefore, ( a_1 approx (-8.899)/(-8.9499) approx 0.994 ). So, ( a_1 ) is approximately 0.994, which is close to 1 but slightly less.Similarly, let's check ( a_{20} ):Numerator: ( 20 - sqrt{98} approx 20 - 9.899 = 10.101 )Denominator: ( 20 - sqrt{99} approx 20 - 9.9499 = 10.0501 )So, ( a_{20} approx 10.101 / 10.0501 approx 1.005 ). So, ( a_{20} ) is approximately 1.005, which is slightly more than 1.So, from these examples, ( a_1 approx 0.994 ), ( a_{10} approx 2.016 ), ( a_{20} approx 1.005 ). So, ( a_{10} ) seems to be the maximum so far. But let's check whether there are other terms that could be larger or smaller.But first, maybe I should analyze the function ( a(n) = frac{n - sqrt{98}}{n - sqrt{99}} ) as a function of real variable ( n ), and then see where it's increasing or decreasing, which could help find maxima or minima.So, let me consider ( a(n) = frac{n - sqrt{98}}{n - sqrt{99}} ). Let me denote ( sqrt{98} = s ), ( sqrt{99} = t ), so ( s approx 9.899 ), ( t approx 9.9499 ). Then, ( a(n) = frac{n - s}{n - t} ). Let's compute the derivative of this function with respect to ( n ):( a(n) = frac{n - s}{n - t} )The derivative ( a'(n) ) is:Using the quotient rule: ( frac{(1)(n - t) - (n - s)(1)}{(n - t)^2} = frac{(n - t - n + s)}{(n - t)^2} = frac{s - t}{(n - t)^2} )Since ( s = sqrt{98} approx 9.899 ), ( t = sqrt{99} approx 9.9499 ), so ( s - t approx -0.0501 ), which is negative. Therefore, the derivative ( a'(n) = frac{-0.0501}{(n - t)^2} ), which is negative for all ( n neq t ). Therefore, the function ( a(n) ) is strictly decreasing wherever it's defined (i.e., for ( n neq t )).But wait, since ( n ) is a positive integer, the function is only defined at integer points. However, the real function ( a(n) ) is decreasing everywhere except at the vertical asymptote ( n = t approx 9.9499 ). Therefore, for ( n < t approx 9.9499 ), the denominator ( n - t ) is negative, and the numerator ( n - s ) is also negative since ( n < t approx 9.95 < s approx 9.899? Wait, no, ( s = sqrt{98} approx 9.899 ), ( t = sqrt{99} approx 9.9499 ), so ( s < t ). Therefore, for ( n < s ), both numerator and denominator are negative (since ( n < s < t )), so the fraction is positive. For ( s < n < t ), numerator is positive, denominator is negative, so the fraction is negative. For ( n > t ), both numerator and denominator are positive, so the fraction is positive.But the derivative is negative everywhere, meaning that as ( n ) increases, the function ( a(n) ) decreases, except at the discontinuity point. But since ( n ) is an integer, the behavior around ( n = t ) is important.But let's think step by step.Given that ( a(n) ) is a strictly decreasing function on each interval where it's continuous. Since there's a vertical asymptote at ( n = t approx 9.9499 ), which is between 9 and 10. So, for ( n < t ), as ( n ) approaches ( t ) from the left, ( a(n) ) tends to negative infinity. For ( n > t ), as ( n ) approaches ( t ) from the right, ( a(n) ) tends to positive infinity. However, since ( n ) is an integer, the closest integer below ( t ) is 9, and the closest integer above is 10. Therefore, the term ( a_9 ) and ( a_{10} ) are the ones adjacent to the asymptote.Wait, let's compute ( a_9 ):Numerator: ( 9 - sqrt{98} approx 9 - 9.899 = -0.899 )Denominator: ( 9 - sqrt{99} approx 9 - 9.9499 = -0.9499 )Therefore, ( a_9 approx (-0.899)/(-0.9499) approx 0.947 ). Wait, that's positive because both numerator and denominator are negative.Wait, but if ( a(n) ) is decreasing for ( n < t ), then as ( n ) increases towards ( t ), ( a(n) ) would decrease towards negative infinity? But that's not possible since ( a(n) ) is positive for ( n < s ) and negative for ( s < n < t ). Wait, maybe my previous reasoning was off.Wait, let's clarify:If ( n < s approx 9.899 ), then both numerator and denominator are negative (since ( n < s < t )), so the ratio is positive.If ( s < n < t approx 9.9499 ), the numerator ( n - s ) is positive, and the denominator ( n - t ) is negative, so the ratio is negative.If ( n > t approx 9.9499 ), both numerator and denominator are positive, so the ratio is positive.But since ( s approx 9.899 ) and ( t approx 9.9499 ), so between 9.899 and 9.9499, the ratio is negative. But integers in this interval? The integers between 9.899 and 9.9499 are just 9. So, n=9 is 9, which is less than s (9.899), so actually n=9 is less than s. Wait, 9 is less than 9.899? Yes. So, n=9 is less than s. Therefore, when n=9, both numerator and denominator are negative. So, a_9 is positive.Wait, so the interval where the ratio is negative is when n is between s and t. But since s ≈9.899 and t≈9.9499, there's no integer n in that interval. Because the next integer after 9.899 is 10, but 10 is greater than t≈9.9499. Wait, no, 10 is greater than t≈9.9499. So, there's no integer n between s and t. Because s≈9.899 and t≈9.9499, the only integers in the vicinity are 9 and 10. So, 9 is less than s, 10 is greater than t.Therefore, for all integer n, the ratio is positive except when n is between s and t, but since there are no integers in that interval, all terms a_n are positive. Wait, but when n is between s and t, the ratio is negative, but since there are no integer n there, so all terms are positive. Let me check with n=10:n=10: numerator 10 - 9.899 ≈ 0.101, denominator 10 - 9.9499 ≈0.0501, so positive.n=9: numerator 9 -9.899≈-0.899, denominator 9 -9.9499≈-0.9499, so positive.n=10: positive.So, all terms are positive. Therefore, the sequence is positive for all n.But then, when I calculated a_1≈0.994, a_9≈0.947, a_{10}≈2.016, a_{20}≈1.005. So, a_9 is about 0.947, which is less than a_1. Hmm. So, the sequence might first decrease, then increase? But according to the derivative, the real function is decreasing everywhere except at the asymptote. But since n is integer, maybe the behavior is different.Wait, maybe let's consider the function as two separate parts: left of the asymptote and right of the asymptote.For n < t (≈9.9499), so n=1,2,...,9: Here, both numerator and denominator are negative (since n < t ≈9.9499, and s ≈9.899 < t). So, n < s ≈9.899 implies numerator and denominator both negative. For n between s and t (≈9.899 to ≈9.9499), numerator is positive and denominator is negative, but as we saw, there's no integer n there. So, for n=1 to 9, numerator and denominator are both negative, so the ratio is positive, and the function is decreasing (since the derivative is negative). Wait, but how is the function decreasing here? Let's check with actual values.Compute a_1: ≈0.994a_2: Let's compute:Numerator: 2 -9.899≈-7.899Denominator: 2 -9.9499≈-7.9499So, a_2≈ (-7.899)/(-7.9499)≈0.993. So, a_2≈0.993, which is less than a_1≈0.994. So, decreasing.a_3: numerator≈3 -9.899≈-6.899, denominator≈3 -9.9499≈-6.9499, so a_3≈ (-6.899)/(-6.9499)≈0.992. So, decreasing.Similarly, a_4≈0.991, a_5≈0.990, etc., each term decreasing by about 0.001 each time. Wait, but when we reach a_9:a_9≈0.947 as computed before. So, from n=1 to n=9, the terms are decreasing.Then, when n=10, the term jumps up to 2.016, which is much higher. Then, n=11:Numerator: 11 -9.899≈1.101Denominator:11 -9.9499≈1.0501So, a_{11}=1.101/1.0501≈1.048. Then, n=12:Numerator:12 -9.899≈2.101Denominator:12 -9.9499≈2.0501a_{12}=2.101/2.0501≈1.025.Similarly, n=13:3.101/3.0501≈1.016n=14:4.101/4.0501≈1.012n=15:5.101/5.0501≈1.010n=16:6.101/6.0501≈1.008n=17:7.101/7.0501≈1.007n=18:8.101/8.0501≈1.006n=19:9.101/9.0501≈1.005n=20:10.101/10.0501≈1.005So, from n=10 onwards, the terms start at around 2.016 (a_{10}), then decrease towards 1 as n increases. Therefore, the terms from n=10 to n=20 are decreasing, approaching 1 from above. Therefore, putting it all together:From n=1 to n=9, the terms are decreasing from approximately 0.994 to 0.947.At n=10, the term jumps up to 2.016.From n=10 to n=20, the terms decrease from 2.016 down to approximately 1.005.Therefore, the maximum term is a_{10}≈2.016, and the minimum term is a_9≈0.947. Then, looking at the options:A. a₁, a₂₀B. a₁, a₉C. a₁₀, a₉D. a₁₀, a₂₀So, the maximum is a_{10}, the minimum is a_9, which is option C. However, let me check again.Wait, in the options, C is a_{10}, a_9. So, maximum a_{10} and minimum a_9. That seems correct according to my calculations.But let me double-check the terms around n=9 and n=10 to make sure.a_9≈0.947, which is less than a₁≈0.994, so a_9 is indeed the minimum.a_{10}≈2.016, which is higher than a_{11}≈1.048, so yes, a_{10} is the maximum.Therefore, the answer should be C. But let me check the answer choices again.Options:A. a₁, a₂₀B. a₁, a₉C. a₁₀, a₉D. a₁₀, a₂₀Yes, C is a_{10}, a_9, which is the maximum and minimum respectively.But just to be thorough, let me check a_9 and a_{10} again.a_9: numerator=9 - sqrt(98)=9 - ~9.899≈-0.899Denominator=9 - sqrt(99)=9 - ~9.9499≈-0.9499So, ratio≈ (-0.899)/(-0.9499)=0.899/0.9499≈0.947.a_{10}: numerator=10 - sqrt(98)=10 -9.899≈0.101Denominator=10 - sqrt(99)=10 -9.9499≈0.0501Ratio≈0.101/0.0501≈2.016.Yes, that's correct. Then, from n=10 onwards, the terms decrease:a_{11}= (11 - sqrt(98))/(11 - sqrt(99))≈(11 -9.899)/(11 -9.9499)=1.101/1.0501≈1.048Similarly, a_{12}≈2.101/2.0501≈1.025, and so on.So, indeed, a_{10} is the maximum term in the first 20 terms.Similarly, the minimum term is a_9≈0.947, which is less than all other terms. Let me check a_1 is 0.994, which is higher than a_9. So, the minimum is a_9. So, the answer is C.But let me check why option D is a_{10}, a_{20}. If someone thinks the terms are decreasing from n=10 to 20, but the minimum would be a_{20}. But in reality, from n=10 onward, the terms are decreasing, but they approach 1 from above. So, the minimum term in the entire first 20 terms is a_9, which is much lower than a_{20}≈1.005. Similarly, the maximum is a_{10}.Therefore, the correct answer is C: a_{10}, a_9.Final Answerboxed{C}</think>Given the sequence ( a_n = frac{n - sqrt{98}}{n - sqrt{99}} ) where ( n ) is a positive integer, we need to determine the maximum and minimum terms among the first 20 terms.1. Approximate Values: - ( sqrt{98} approx 9.899 ) - ( sqrt{99} approx 9.9499 )2. Behavior of the Function: - The function ( a(n) = frac{n - sqrt{98}}{n - sqrt{99}} ) has a vertical asymptote at ( n = sqrt{99} approx 9.9499 ). - For ( n < sqrt{98} approx 9.899 ), both the numerator and denominator are negative, making the terms positive. - For ( sqrt{98} < n < sqrt{99} ), the numerator is positive and the denominator is negative, but there are no integers in this interval. - For ( n > sqrt{99} ), both the numerator and denominator are positive, making the terms positive.3. Derivative Analysis: - The derivative ( a'(n) = frac{sqrt{98} - sqrt{99}}{(n - sqrt{99})^2} ) is negative, indicating the function is decreasing on either side of the asymptote.4. Key Terms Calculation: - ( a_1 approx frac{1 - 9.899}{1 - 9.9499} approx 0.994 ) - ( a_9 approx frac{9 - 9.899}{9 - 9.9499} approx 0.947 ) - ( a_{10} approx frac{10 - 9.899}{10 - 9.9499} approx 2.016 ) - ( a_{20} approx frac{20 - 9.899}{20 - 9.9499} approx 1.005 )5. Behavior of Terms: - From ( n = 1 ) to ( n = 9 ), the terms decrease from approximately 0.994 to 0.947. - At ( n = 10 ), the term jumps to approximately 2.016. - From ( n = 10 ) to ( n = 20 ), the terms decrease towards 1.6. Conclusion: - The maximum term is ( a_{10} ) (approximately 2.016). - The minimum term is ( a_9 ) (approximately 0.947).Thus, the correct answer is (boxed{C}).,<think>Okay, let's see. I have this sequence a_n defined as (n - sqrt(98))/(n - sqrt(99)) where n is a positive integer. The question is asking among the first 20 terms, which are the maximum and minimum terms. The options are given as A, B, C, D with different pairs of terms. Hmm, I need to figure out which pair is correct. Let me start by understanding the sequence better.First, let me note the values of sqrt(98) and sqrt(99). Since sqrt(81) is 9, sqrt(100) is 10, so sqrt(98) must be a little less than 10, right? Let me calculate approximate values. sqrt(98) is approximately 9.899 (since 9.9^2 = 98.01), so sqrt(98) ≈ 9.899 and sqrt(99) ≈ 9.9499. So both denominators and numerators in the sequence terms are of the form n minus something close to 10. So the general term is (n - 9.899)/(n - 9.9499). Hmm. Let me write it as (n - s)/(n - t), where s = sqrt(98) ≈9.899 and t = sqrt(99)≈9.9499. So s is slightly less than t. Therefore, t is a bit closer to 10 than s is.Wait, so s ≈9.899 and t≈9.95. So s is about 9.899 and t is about 9.95, so s < t. So both numerator and denominator are n minus numbers slightly less than 10. Depending on n, when n is 10, the denominator is 10 - 9.95≈0.05, which is a small positive number, so a_10 would be (10 -9.899)/0.05≈0.101/0.05≈2.02. So that's a relatively large term. When n is 9, denominator is 9 -9.95≈-0.95, numerator is 9 -9.899≈-0.899. So a_9 is (-0.899)/(-0.95)≈0.946. Hmm, so a_9 is approximately 0.946, which is less than 1. When n is 1, a_1 is (1 -9.899)/(1 -9.95)= (-8.899)/(-8.95)≈0.994. So a_1 is roughly 0.994, which is slightly less than 1. When n is 20, a_20 is (20 -9.899)/(20 -9.95)=10.101/10.05≈1.005. So a_20 is approximately 1.005. So from these values, a_10 is about 2.02, which seems to be the maximum so far. The other terms: a_1 is ~0.994, a_9 is ~0.946, a_20 is ~1.005. So among these, the maximum is a_10 and the minimum is a_9. But wait, the options given are C. a10, a9; D. a10, a20. So maybe the answer is C. But let's check more carefully.Alternatively, maybe there's a systematic way to analyze this function. Let's consider the general term a_n = (n - s)/(n - t), where s = sqrt(98) and t = sqrt(99). Since s < t, the denominator is n - t, which is less than the numerator n - s for the same n. Wait, but only if n > t. If n < t, then both numerator and denominator are negative. Let's see.Let me see the function f(n) = (n - s)/(n - t). Let's consider this as a function of n, treating n as a real variable. Then, to find its extrema, we can take the derivative. Wait, but since n is a positive integer, maybe the maximum and minimum occur at certain integer points, but perhaps near where the derivative would be zero? Let me compute the derivative.Let’s differentiate f(n) with respect to n:f(n) = (n - s)/(n - t)f’(n) = [(1)(n - t) - (n - s)(1)]/(n - t)^2 = [ (n - t - n + s) ]/(n - t)^2 = (s - t)/(n - t)^2.Since s < t, the numerator s - t is negative, and the denominator is squared, so always positive. Therefore, f’(n) is negative for all n ≠ t. So the function f(n) is decreasing wherever it's defined (i.e., n ≠ t). However, when n approaches t from the left, the denominator approaches zero from the negative side, so f(n) approaches positive infinity (since numerator is n - s, and when n approaches t from below, numerator approaches t - s, which is positive, and denominator approaches 0 from negative, so overall approaches negative infinity). Wait, hold on. If n approaches t from below, denominator approaches 0 from the negative side (since n - t approaches 0-), numerator approaches t - s (positive because t > s). So (positive)/(approaching 0-) approaches negative infinity. Wait, but maybe I confused. Let's take n approaching t from above: denominator approaches 0+, numerator approaches t - s which is positive, so f(n) approaches positive infinity. If n approaches t from below, denominator approaches 0-, numerator is still t - s (positive), so f(n) approaches negative infinity. Wait, but in the case of our problem, n is a positive integer. So since t ≈9.95, so n can be 9 (which is less than t) and 10 (greater than t). So between n=9 and n=10, the function has a vertical asymptote at n = t ≈9.95. So when n approaches t from below (n=9), the value goes to negative infinity? Wait, but when n=9, the denominator is 9 - t ≈-0.95, and numerator is 9 - s ≈-0.899. So a_9 is positive because both numerator and denominator are negative. Wait, no: 9 - t is negative, 9 - s is also negative, so their ratio is positive. Similarly, for n=10, denominator is 10 - t ≈0.05, numerator is 10 - s ≈0.101, so positive. So in this case, the function f(n) crosses from positive to positive through the asymptote? Wait, no. Wait, for n=9, the value is positive, as both numerator and denominator are negative. For n=10, numerator and denominator are both positive. So the function actually jumps from a value of approximately 0.946 (n=9) to 2.02 (n=10). So there's a discontinuity at n=9.95, but since n is integer, we don't have that term. So between n=9 and n=10, the function jumps from ~0.946 to ~2.02. So that's a big increase. However, according to the derivative when treating n as a real variable, the function is decreasing on either side of the asymptote. Let me check: for n > t, so denominator is positive, and f(n) = (n - s)/(n - t). Since s < t, numerator is n - s, which is larger than denominator n - t. So (n - s)/(n - t) = 1 + (t - s)/(n - t). Since t - s is positive, and n - t is positive (for n > t), so (t - s)/(n - t) is positive, and as n increases, this term decreases. Therefore, for n > t, f(n) is decreasing from positive infinity towards 1 as n approaches infinity.For n < t, denominator is negative, and numerator is n - s. Since s ≈9.899, t≈9.95, so when n < t, n can be 1 to 9 (since n is positive integer). For these n, numerator is n - s, which is negative (since n ≤9 and s≈9.899), so numerator is negative, denominator is also negative (n - t <0), so their ratio is positive. Then, f(n) = (n - s)/(n - t) = (s - n)/(t - n) [multiplying numerator and denominator by -1]. So (s - n)/(t - n). Since t > s, so s - n < t - n. Therefore, (s - n)/(t - n) <1. So for n < t, f(n) is less than 1. Also, as n increases towards t from below, t - n approaches 0, so denominator approaches 0 from positive side (since t - n is positive when n < t), and numerator s - n approaches s - t ≈-0.051. Therefore, f(n) approaches (negative)/(approaching 0+) which tends to negative infinity? Wait, but earlier when n approaches t from below, with n as real variable, the denominator approaches 0 from the negative side, so f(n) approaches positive infinity? Wait, maybe I confused. Wait, if n approaches t from below, then n approaches 9.95 from below. Then, numerator is n - s ≈9.95 -9.899≈0.051, denominator is n - t≈-0.05. So the ratio is approximately 0.051 / (-0.05) ≈-1.02. Wait, no, if n approaches t from below, n approaches t=9.95 from the left, so n - s approaches t - s ≈0.051, and n - t approaches 0 from the negative side. Therefore, the ratio is (0.051)/(negative approaching 0), so it approaches negative infinity. But in our integer case, when n=9, which is less than t≈9.95, the value is positive, as both numerator and denominator are negative. Wait, maybe there's a confusion here. Let me re-examine.When n is an integer less than t (which is approximately 9.95), so n=1,...,9. Then, numerator is n - s ≈n -9.899, which is negative (since n≤9), and denominator is n - t ≈n -9.95, also negative. So negative divided by negative is positive. Therefore, for n=1 to 9, a_n is positive. When n=10, which is greater than t≈9.95, numerator is positive (10 -9.899≈0.101), denominator is positive (10 -9.95≈0.05), so a_10 is positive as well. For n>10, both numerator and denominator are positive, so a_n is positive. But according to the real variable analysis, the function f(n) = (n - s)/(n - t) has a vertical asymptote at n = t≈9.95. For n approaching t from below, f(n) approaches negative infinity, but since n is integer, we don't see that. Instead, the actual terms for n=9 and n=10 are both positive. So perhaps the sequence is increasing from n=1 to n=9, reaches a maximum at n=10, then decreases? Wait, but according to the derivative, when n > t, the function is decreasing. So for n=10,11,..., the terms are decreasing. But for n < t, how does the function behave? Let's consider n < t, so n=1,...,9. Let's write f(n) = (n - s)/(n - t). Let me rewrite this as [n - t + (t - s)]/(n - t) = 1 + (t - s)/(n - t). Since n - t is negative here (n < t), and t - s is positive (≈0.051), so (t - s)/(n - t) is negative. Therefore, f(n) = 1 + [negative number]. So for n < t, as n increases, n - t becomes less negative, so denominator (n - t) increases (towards zero), making the term (t - s)/(n - t) a negative number that approaches zero. Therefore, f(n) =1 + [negative approaching zero], which means f(n) increases towards 1 as n approaches t from below. Therefore, for n=1 to 9, the sequence is increasing, approaching 1. Then, at n=10, it jumps to a higher value (≈2.02), and then decreases towards 1 as n increases beyond 10. Therefore, the maximum term is a_10, and the minimum term is a_1? Wait, but earlier calculations showed a_1≈0.994, a_9≈0.946. So if from n=1 to n=9, the sequence is increasing towards 1, but since a_9 is 0.946 and a_1 is 0.994, that contradicts. Wait, maybe my analysis is wrong.Wait, perhaps I made a mistake in the analysis. Let's check for n < t. Let's take n=1: a_1=(1 - s)/(1 - t). Since s≈9.899, t≈9.95, both denominator and numerator are negative. So (1 - s)/(1 - t) = (s -1)/(t -1) ≈(9.899 -1)/(9.95 -1)=8.899/8.95≈0.994. Then, for n=2: (2 - s)/(2 - t)≈(2 -9.899)/(2 -9.95)=(-7.899)/(-7.95)≈0.993. Wait, that's slightly less than a_1. Hmm, so actually, it's decreasing. Wait, that contradicts the previous thought. Wait, what's going on here.Wait, maybe when n increases from 1 to 9, the value of a_n decreases, reaches a minimum at n=9, then jumps up at n=10. Let me compute a few terms numerically to check.Let me compute a_1: (1 - sqrt(98))/(1 - sqrt(99)) ≈(1 -9.899)/(1 -9.95)= (-8.899)/(-8.95)≈0.994.a_2: (2 -9.899)/(2 -9.95)= (-7.899)/(-7.95)≈0.993.a_3: (3 -9.899)/(3 -9.95)= (-6.899)/(-6.95)≈0.992.Continuing similarly:a_4≈(4 -9.899)/(4 -9.95)= (-5.899)/(-5.95)≈0.991a_5≈(5 -9.899)/(5 -9.95)= (-4.899)/(-4.95)≈0.989a_6≈(6 -9.899)/(6 -9.95)= (-3.899)/(-3.95)≈0.987a_7≈(7 -9.899)/(7 -9.95)= (-2.899)/(-2.95)≈0.982a_8≈(8 -9.899)/(8 -9.95)= (-1.899)/(-1.95)≈0.973a_9≈(9 -9.899)/(9 -9.95)= (-0.899)/(-0.95)≈0.946a_10≈(10 -9.899)/(10 -9.95)=0.101/0.05≈2.02a_11≈(11 -9.899)/(11 -9.95)=1.101/1.05≈1.048a_12≈(12 -9.899)/(12 -9.95)=2.101/2.05≈1.025a_13≈3.101/3.05≈1.016a_14≈4.101/4.05≈1.012a_15≈5.101/5.05≈1.010a_16≈6.101/6.05≈1.008a_17≈7.101/7.05≈1.007a_18≈8.101/8.05≈1.006a_19≈9.101/9.05≈1.005a_20≈10.101/10.05≈1.005Wait, so from n=1 to n=9, the terms are decreasing: 0.994, 0.993, 0.992, 0.991, 0.989, 0.987, 0.982, 0.973, 0.946. Then, at n=10, it jumps up to 2.02, then decreases again: 1.048, 1.025, 1.016, etc., approaching 1 as n increases. So the maximum term is a_10≈2.02, and the minimum term is a_9≈0.946. Then, looking at the options:A. a1, a20 (≈0.994 and ≈1.005) → No.B. a1, a9 (≈0.994 and ≈0.946) → Minimum is a9, maximum is a1? But a10 is larger than a1. So no.C. a10, a9 → Maximum a10, minimum a9. Yes.D. a10, a20 → Maximum a10, minimum a20 (≈1.005). But a9 is smaller than a20, so no.Therefore, the correct answer should be C. a10 and a9.But let me confirm once more. The sequence decreases from n=1 to n=9, reaching the minimum at n=9, then jumps up to a maximum at n=10, then decreases again towards 1. So among the first 20 terms, the maximum is at n=10 and the minimum at n=9. So the answer is C.Final Answerboxed{C}</think>Given the sequence (a_n = frac{n - sqrt{98}}{n - sqrt{99}}) where (n in mathbf{N}_+), we need to determine the maximum and minimum terms among the first 20 terms.1. Approximate Values: - (sqrt{98} approx 9.899) - (sqrt{99} approx 9.9499)2. General Term Analysis: - For (n = 1) to (9), both the numerator and denominator are negative, resulting in positive values. - For (n = 10), both the numerator and denominator are positive, resulting in a positive value. - For (n > 10), both the numerator and denominator are positive, and the term decreases as (n) increases.3. Key Terms Calculation: - (a_1 approx frac{1 - 9.899}{1 - 9.9499} approx 0.994) - (a_9 approx frac{9 - 9.899}{9 - 9.9499} approx 0.946) - (a_{10} approx frac{10 - 9.899}{10 - 9.9499} approx 2.02) - (a_{20} approx frac{20 - 9.899}{20 - 9.9499} approx 1.005)4. Behavior of the Sequence: - From (n = 1) to (n = 9), the sequence decreases. - At (n = 10), the sequence jumps to a higher value. - From (n = 10) to (n = 20), the sequence decreases towards 1.5. Conclusion: - The maximum term is (a_{10}). - The minimum term is (a_9).Thus, the correct answer is (boxed{C}).]

🔑:1. C.Since a_{n}=1+frac{sqrt{99}-sqrt{98}}{n-sqrt{99}}, when n=10 then n-sqrt{99} is the smallest positive number, so a_{10} is the largest; when n=9 then n-sqrt{99} is the largest negative number, so a_{9} is the smallest.

🧠:[<think>Okay, let's see. I need to find the value of this algebraic expression: (1 - (2ab - b²)/a²) divided by ( (a - b)/(a²b) ), given that 3ab - 3b² - 2 = 0. Hmm, that looks a bit complicated, but maybe I can simplify it step by step.First, let's write down the original equation: 3ab - 3b² = 2. Maybe I can factor that or solve for one variable in terms of the other. Let's try. The equation is 3ab - 3b² = 2. If I factor out 3b from the first two terms, I get 3b(a - b) = 2. So, 3b(a - b) = 2. Maybe that's useful later.Now, the expression to evaluate is (1 - (2ab - b²)/a²) ÷ ( (a - b)/(a²b) ). Let me parse this. The numerator of the main fraction is (1 - (2ab - b²)/a²), and the denominator is (a - b)/(a²b). Dividing by a fraction is the same as multiplying by its reciprocal, so maybe I can rewrite the entire expression as [1 - (2ab - b²)/a²] * [a²b/(a - b)]. That might be easier to handle.Let me work on the numerator first: 1 - (2ab - b²)/a². To combine these terms, I need a common denominator. Let's express 1 as a²/a². So that becomes (a² - 2ab + b²)/a². Wait, because 1 is a²/a², subtracting (2ab - b²)/a² gives (a² - 2ab + b²)/a². Hmm, the numerator here is a quadratic in terms of a and b. Let me check: a² - 2ab + b². That's a perfect square, right? Yes, (a - b)². So, the numerator simplifies to (a - b)² / a².So the first part simplifies to (a - b)² / a². Then, the entire expression becomes [(a - b)² / a²] * [a²b / (a - b)]. Let's see: multiplying these two fractions. The a² in the denominator cancels with the a² in the numerator of the second fraction. Then, we have (a - b)² times b divided by (a - b). The (a - b)² divided by (a - b) is just (a - b). So overall, this simplifies to (a - b) * b.Therefore, the entire expression simplifies to b(a - b). So, the value of the expression is b(a - b). But wait, the problem gives us the equation 3ab - 3b² - 2 = 0, which we can write as 3b(a - b) = 2. Therefore, 3b(a - b) = 2 implies that b(a - b) = 2/3. So, if the expression simplifies to b(a - b), then substituting from the given equation, it should be equal to 2/3.Let me double-check that. Starting from the given equation: 3ab - 3b² = 2. Factor out 3b: 3b(a - b) = 2. Therefore, dividing both sides by 3: b(a - b) = 2/3. Yes, that's exactly the same as the expression we have. So, the value of the algebraic expression is 2/3.Wait, but let me make sure that I didn't make a mistake in simplifying the original expression. Let me go through the steps again.Original expression: [1 - (2ab - b²)/a²] ÷ [(a - b)/(a²b)]First, the numerator: 1 - (2ab - b²)/a². Let's compute this:1 is a²/a², so subtract (2ab - b²)/a²:(a² - 2ab + b²)/a² = (a - b)² / a². Correct.Then, the denominator of the entire expression is (a - b)/(a²b). So, dividing by that is the same as multiplying by a²b/(a - b). So the entire expression becomes:[(a - b)² / a²] * [a²b / (a - b)] = [(a - b)² * a²b] / [a² * (a - b)] = (a - b) * b. Correct. So that's b(a - b).From the given equation: 3ab - 3b² = 2 => 3b(a - b) = 2 => b(a - b) = 2/3. Therefore, the value is 2/3. That seems right.But just to ensure I didn't skip any steps or make any miscalculations, let me verify with a different approach. Maybe plug in numbers that satisfy the equation and compute the expression.Suppose we can find values of a and b that satisfy 3ab - 3b² = 2, then compute the expression to see if it equals 2/3.Let me choose a value for b and solve for a. Let's take b = 1. Then, plugging into 3ab - 3b² = 2: 3a(1) - 3(1)^2 = 2 => 3a - 3 = 2 => 3a = 5 => a = 5/3.So, if b = 1, then a = 5/3. Let's compute the expression:First part: 1 - (2ab - b²)/a².Compute 2ab - b²: 2*(5/3)*1 - (1)^2 = 10/3 - 1 = 10/3 - 3/3 = 7/3.Divide by a²: (7/3) / ( (5/3)^2 ) = (7/3) / (25/9) = (7/3)*(9/25) = 21/25.So 1 - 21/25 = 4/25.Then the denominator part: (a - b)/(a²b) = (5/3 - 1)/( (25/9)*1 ) = (2/3)/(25/9) = (2/3)*(9/25) = 6/25.So the entire expression is (4/25) ÷ (6/25) = (4/25)*(25/6) = 4/6 = 2/3. Perfect, that's 2/3. So that checks out.Another example. Let's take b = 2. Then, 3ab - 3b² = 2 => 3a*2 - 3*4 = 2 => 6a - 12 = 2 => 6a = 14 => a = 14/6 = 7/3.So a = 7/3, b = 2. Compute the expression:First part: 1 - (2ab - b²)/a².Calculate 2ab - b²: 2*(7/3)*2 - (2)^2 = 28/3 - 4 = 28/3 - 12/3 = 16/3.Divide by a²: (16/3) / ( (7/3)^2 ) = (16/3) / (49/9) = (16/3)*(9/49) = 48/49.Then 1 - 48/49 = 1/49.Denominator part: (a - b)/(a²b) = (7/3 - 2)/( (49/9)*2 ) = (7/3 - 6/3)/(98/9) = (1/3)/(98/9) = (1/3)*(9/98) = 3/98.So the entire expression is (1/49) ÷ (3/98) = (1/49)*(98/3) = (98)/(49*3) = 2/3. Again, 2/3. So that works too.Therefore, it's safe to conclude that regardless of the values of a and b (as long as they satisfy the original equation), the expression simplifies to 2/3. So the answer should be 2/3.I think that's thorough. I checked algebraically and with numerical examples. Both methods give 2/3, so I feel confident that's the correct answer.Final Answerboxed{dfrac{2}{3}}</think>Given the equation (3ab - 3b^2 - 2 = 0), we need to find the value of the algebraic expression (left(1 - frac{2ab - b^2}{a^2}right) div frac{a - b}{a^2 b}).First, we simplify the given equation:[3ab - 3b^2 = 2 implies 3b(a - b) = 2 implies b(a - b) = frac{2}{3}]Next, we simplify the algebraic expression. Starting with the numerator:[1 - frac{2ab - b^2}{a^2}]we rewrite 1 as (frac{a^2}{a^2}):[frac{a^2}{a^2} - frac{2ab - b^2}{a^2} = frac{a^2 - 2ab + b^2}{a^2} = frac{(a - b)^2}{a^2}]The entire expression then becomes:[frac{(a - b)^2}{a^2} div frac{a - b}{a^2 b}]Dividing by a fraction is equivalent to multiplying by its reciprocal:[frac{(a - b)^2}{a^2} times frac{a^2 b}{a - b} = frac{(a - b)^2 cdot a^2 b}{a^2 cdot (a - b)} = (a - b) cdot b]From the given equation, we know (b(a - b) = frac{2}{3}). Therefore, the value of the algebraic expression is:[boxed{frac{2}{3}}],<think>Okay, so I have this algebra problem here. Let me read it again. The problem says: If 3ab - 3b² - 2 = 0, then find the value of the algebraic expression (1 - (2ab - b²)/a²) ÷ (a - b)/(a²b). Hmm, okay. I need to find the value of that expression given that equation. Let me start by understanding what's given and what's asked.First, the given equation is 3ab - 3b² - 2 = 0. Maybe I can solve for one variable in terms of the other here? Let's see. Let me try to rearrange the equation. Let's factor out 3b from the first two terms: 3b(a - b) - 2 = 0. So, 3b(a - b) = 2. Then, dividing both sides by 3b, assuming b is not zero, we get a - b = 2/(3b). Therefore, a = b + 2/(3b). Hmm, so maybe I can express a in terms of b here? That might be helpful. Let me write that down: a = b + 2/(3b). Alternatively, maybe I can write a - b = 2/(3b). Not sure yet which substitution will help, but maybe.Now, the expression we need to evaluate is (1 - (2ab - b²)/a²) ÷ (a - b)/(a²b). Let me first simplify this expression step by step. Let's handle the numerator of the big fraction first: 1 - (2ab - b²)/a². Let's compute that.First, compute (2ab - b²)/a². Let's split that into two terms: 2ab/a² - b²/a² = 2b/a - b²/a². Therefore, 1 - (2ab - b²)/a² becomes 1 - 2b/a + b²/a². Hmm, that looks like a quadratic in terms of b/a. Let me note that 1 - 2b/a + (b/a)² is the same as (1 - b/a)². Let me check: (1 - b/a)² = 1 - 2(b/a) + (b/a)². Yes, exactly. So the numerator simplifies to (1 - b/a)². That's a nice simplification.Now, the denominator of the big fraction is (a - b)/(a²b). So the entire expression is [(1 - b/a)²] ÷ [(a - b)/(a²b)]. Dividing by a fraction is the same as multiplying by its reciprocal, so this becomes [(1 - b/a)²] * [a²b/(a - b)]. Let me write that out: (1 - b/a)² * (a²b)/(a - b).Now, let's simplify (1 - b/a)². Let me write 1 - b/a as (a - b)/a. So, (a - b)/a squared is (a - b)²/a². Therefore, substituting back in, the expression becomes [(a - b)²/a²] * [a²b/(a - b)]. Let's see how that simplifies.First, the (a - b)² in the numerator and the (a - b) in the denominator will cancel one (a - b) term, leaving (a - b) in the numerator. The a² in the numerator from a²b and the a² in the denominator from the first term will cancel out. So simplifying:[(a - b)²/a²] * [a²b/(a - b)] = (a - b) * b.So the entire expression simplifies to (a - b) * b. So the value of the given algebraic expression is b(a - b). Wait, really? Let me check my steps again to make sure I didn't make a mistake.Starting with the original expression:(1 - (2ab - b²)/a²) ÷ (a - b)/(a²b)First, simplifying the numerator:(2ab - b²)/a² = 2ab/a² - b²/a² = 2b/a - (b/a)²Therefore, 1 - [2b/a - (b/a)²] = 1 - 2b/a + (b/a)² = (1 - b/a)^2.Then, the denominator is (a - b)/a²b.So dividing the two gives: (1 - b/a)^2 * (a²b)/(a - b)Expressing (1 - b/a) as (a - b)/a, then squaring gives (a - b)^2/a^2.So substituting back: [(a - b)^2/a^2] * [a²b/(a - b)] = [(a - b)^2 * a²b] / [a² * (a - b)] = [(a - b) * b] after canceling terms. Yes, that seems correct. So the expression simplifies to b(a - b).Therefore, the value of the expression is b(a - b). But we need to find its numerical value given that 3ab - 3b² - 2 = 0. Wait, so from the given equation, maybe we can find a relation between a and b, substitute into b(a - b), and find the value.Earlier, I had rearranged the equation 3ab - 3b² - 2 = 0 to get a - b = 2/(3b). Therefore, a - b is 2/(3b). Therefore, substituting into b(a - b) gives b * (2/(3b)) = 2/3. Because the b in the numerator and the b in the denominator cancel out. So the value is 2/3.Wait, so is the answer 2/3? Let me check that again.Given that 3ab - 3b² - 2 = 0, which simplifies to 3b(a - b) = 2. Therefore, a - b = 2/(3b). Then, substituting into the expression b(a - b) gives b*(2/(3b)) = 2/3. Yes, that cancels out the b. So the answer is 2/3. Hmm. That seems straightforward. Let me verify this with another approach to make sure.Alternatively, let's suppose I didn't simplify the original expression. Let's take the original expression:(1 - (2ab - b²)/a²) ÷ (a - b)/(a²b)We can compute this expression by plugging in a in terms of b from the given equation. Wait, but we might need to express a in terms of b first. From 3ab - 3b² = 2, we can write 3ab = 3b² + 2, so a = (3b² + 2)/(3b). Therefore, a = b + 2/(3b), as before.So let me compute a - b, which is 2/(3b). Then, let's compute the expression step by step.First, compute the numerator of the big fraction: 1 - (2ab - b²)/a².Let me compute 2ab - b². Since a = (3b² + 2)/(3b), substituting in:2ab = 2*( (3b² + 2)/(3b) )*b = 2*( (3b² + 2)/3 ) = (6b² + 4)/3.Then, 2ab - b² = (6b² + 4)/3 - b² = (6b² + 4 - 3b²)/3 = (3b² + 4)/3.So (2ab - b²)/a² = (3b² + 4)/3 divided by a².But a² is [ (3b² + 2)/(3b) ]² = (9b^4 + 12b² + 4)/(9b²).Therefore, (2ab - b²)/a² = [ (3b² + 4)/3 ] / [ (9b^4 + 12b² + 4)/(9b²) ] = [ (3b² + 4)/3 ] * [ 9b²/(9b^4 + 12b² + 4) ]Simplify numerator and denominator:The denominator 9b^4 +12b² +4 is equal to (3b² + 2)^2. Let me check: (3b² + 2)^2 = 9b^4 + 12b² +4. Yes, correct.Therefore, denominator is (3b² + 2)^2.So, (3b² + 4)/3 * 9b²/(3b² + 2)^2 = (3b² + 4) * 3b² / (3b² + 2)^2.Therefore, (2ab - b²)/a² = (3b² + 4) * 3b² / (3b² + 2)^2.Now, 1 - (2ab - b²)/a² = 1 - [ (3b² + 4)*3b² / (3b² + 2)^2 ].Let me compute this:1 = (3b² + 2)^2 / (3b² + 2)^2.So subtracting the two terms:[ (3b² + 2)^2 - 3b²(3b² + 4) ] / (3b² + 2)^2.Compute numerator:(3b² + 2)^2 - 3b²(3b² + 4) = (9b^4 + 12b² +4) - (9b^4 +12b²) = 9b^4 +12b² +4 -9b^4 -12b² = 4.Therefore, 1 - (2ab - b²)/a² = 4 / (3b² + 2)^2.Now, moving to the denominator of the original expression: (a - b)/(a²b). From earlier, a - b = 2/(3b). And a²b is [ (3b² + 2)/(3b) ]² * b = (9b^4 +12b² +4)/(9b²) * b = (9b^4 +12b² +4)/(9b).Therefore, (a - b)/(a²b) = [2/(3b)] / [ (9b^4 +12b² +4)/(9b) ] = [2/(3b)] * [9b/(9b^4 +12b² +4)] = [2/(3b)] * [9b/( (3b² + 2)^2 ) ].Simplify:The 3b in the denominator and 9b in the numerator: 9b/(3b) = 3. So 2 * 3 / (3b² + 2)^2 = 6 / (3b² + 2)^2.Therefore, the entire original expression is:[4 / (3b² + 2)^2 ] ÷ [6 / (3b² + 2)^2 ] = [4 / (3b² + 2)^2 ] * [ (3b² + 2)^2 /6 ] = 4/6 = 2/3.So, same result, 2/3. Therefore, regardless of the approach, the answer is 2/3. Therefore, the value of the expression is 2/3. So I think that's correct.Another way to check: if we suppose specific values of a and b that satisfy the original equation, and compute the expression. Let's pick a value for b and compute a accordingly. Let me choose b = 1. Then, from 3ab - 3b² -2 =0: 3a*1 -3*1 -2=0 => 3a -5=0 => a=5/3. Then, compute the expression:(1 - (2ab - b²)/a² ) ÷ (a - b)/(a²b)With a=5/3 and b=1:First compute 2ab -b² = 2*(5/3)*1 -1²= 10/3 -1=7/3.Then, divide by a²: (7/3)/( (25/9) )= (7/3)*(9/25)=21/25.So, 1 - 21/25=4/25.Then, compute (a - b)/(a²b)= (5/3 -1)/ ( (25/9)*1 )= (2/3)/(25/9)= (2/3)*(9/25)=6/25.Therefore, the entire expression is (4/25) ÷ (6/25)=4/25 *25/6=4/6=2/3.Yes, same result. So with specific numbers, it works out. So that's another verification. Therefore, the answer is 2/3.Final Answerboxed{dfrac{2}{3}}</think>Given the equation (3ab - 3b^2 - 2 = 0), we need to find the value of the algebraic expression (left(1 - frac{2ab - b^2}{a^2}right) div frac{a - b}{a^2b}).First, we simplify the given equation:[3ab - 3b^2 - 2 = 0 implies 3b(a - b) = 2 implies a - b = frac{2}{3b}]Next, we simplify the given expression. The numerator of the expression is:[1 - frac{2ab - b^2}{a^2}]We rewrite (2ab - b^2) as (2ab - b^2) and divide by (a^2):[frac{2ab - b^2}{a^2} = frac{2b}{a} - frac{b^2}{a^2}]Thus, the numerator becomes:[1 - left(frac{2b}{a} - frac{b^2}{a^2}right) = 1 - frac{2b}{a} + frac{b^2}{a^2} = left(1 - frac{b}{a}right)^2]The denominator of the expression is:[frac{a - b}{a^2b}]Combining these, the entire expression becomes:[left(1 - frac{b}{a}right)^2 div frac{a - b}{a^2b}]Rewriting (1 - frac{b}{a}) as (frac{a - b}{a}), we get:[left(frac{a - b}{a}right)^2 div frac{a - b}{a^2b} = left(frac{(a - b)^2}{a^2}right) times left(frac{a^2b}{a - b}right)]Simplifying this, we cancel out terms:[frac{(a - b)^2}{a^2} times frac{a^2b}{a - b} = (a - b) times b]Substituting (a - b = frac{2}{3b}) into the expression:[b times frac{2}{3b} = frac{2}{3}]Thus, the value of the algebraic expression is:[boxed{dfrac{2}{3}}]]

🔑:To solve the given problem, we start by simplifying the given algebraic expression step by step:[begin{align*}(1-frac{2ab-{b}^{2}}{{a}^{2}})div frac{a-b}{{a}^{2}b} &= frac{{a}^{2}-(2ab-{b}^{2})}{{a}^{2}}cdot frac{{a}^{2}b}{a-b} &= frac{{a}^{2}-2ab+{b}^{2}}{{a}^{2}}cdot frac{{a}^{2}b}{a-b} &= frac{(a-b)^{2}}{{a}^{2}}cdot frac{{a}^{2}b}{a-b} &= frac{(a-b)(a-b)}{a^2} cdot frac{a^2b}{a-b} &= (a-b) cdot b &= ab-b^{2}.end{align*}]Given that 3ab-3b^{2}-2=0, we can manipulate this equation to find a relationship between ab and b^2:[begin{align*}3ab-3b^{2}-2 &= 0 3ab-3b^{2} &= 2 ab-b^{2} &= frac{2}{3}.end{align*}]Therefore, substituting the value of ab-b^{2} into our original expression, we find that the value of the given algebraic expression is boxed{frac{2}{3}}.

🧠:[<think>Okay, let me try to solve these two problems. Hmm, starting with the first one. It looks like simplifying an expression with exponents. Let me recall the rules for exponents. When you multiply terms with the same base, you add their exponents, and when you divide, you subtract the exponents. Also, coefficients (the numbers in front) multiply and divide normally. Let me write that down step by step.Problem (1): (2a^{2/3}b^{1/2})(-6a^{1/2}b^{1/3}) ÷ (-3a^{1/6}b^{5/6})First, let me handle the multiplication part first, then the division. So, starting with the first two terms: (2a^{2/3}b^{1/2}) multiplied by (-6a^{1/2}b^{1/3}).Multiplying the coefficients: 2 * (-6) = -12.Now, multiplying the a terms: a^{2/3} * a^{1/2}. Add the exponents: 2/3 + 1/2. To add these, find a common denominator. 2/3 is 4/6 and 1/2 is 3/6, so 4/6 + 3/6 = 7/6. So that's a^{7/6}.Then the b terms: b^{1/2} * b^{1/3}. Adding those exponents: 1/2 + 1/3. Common denominator is 6, so 3/6 + 2/6 = 5/6. So that's b^{5/6}.So after multiplying the first two terms, we have -12a^{7/6}b^{5/6}.Now, this result is divided by (-3a^{1/6}b^{5/6}). So let's write that division:(-12a^{7/6}b^{5/6}) ÷ (-3a^{1/6}b^{5/6})Dividing the coefficients: -12 ÷ (-3) = 4.Now, dividing the a terms: a^{7/6} ÷ a^{1/6} = a^{7/6 - 1/6} = a^{6/6} = a^1 = a.Dividing the b terms: b^{5/6} ÷ b^{5/6} = b^{5/6 - 5/6} = b^0 = 1.So putting it all together, we have 4a * 1 = 4a.Wait, that seems straightforward. Let me check if I did the exponent arithmetic correctly.For the a exponents in multiplication: 2/3 + 1/2. Let me verify that again. 2/3 is approximately 0.666..., 1/2 is 0.5. Adding them gives 1.166..., which is 7/6. Yes, that's correct.Then, when dividing the a exponents: 7/6 - 1/6 = 6/6 = 1. Correct. And the b exponents: 5/6 - 5/6 = 0. So b^0 is 1. Also correct.Coefficients: 2 * (-6) = -12, then -12 ÷ (-3) = 4. Yup. So the final result for part (1) is 4a. That seems right.Moving on to problem (2): (lg2)^2 + lg2·lg5 + √{(lg2)^2 - 2lg2 +1}Hmm, "lg" is logarithm base 10, right? So lg2 is log₁₀2, and lg5 is log₁₀5. Let me note that.So the expression is:(log₁₀2)^2 + (log₁₀2)(log₁₀5) + √[(log₁₀2)^2 - 2log₁₀2 + 1]First, let's look at the square root part. The expression under the root is (log2)^2 - 2log2 + 1. That looks like a quadratic. Let me set x = log2 for simplicity. Then the expression under the root becomes x² - 2x + 1, which factors as (x - 1)^2. So the square root of (x - 1)^2 is |x - 1|. But since we're dealing with real numbers and square roots return non-negative results, it's the absolute value of (x - 1).So substituting back, the square root part is |(log2 - 1)|.But wait, log2 is approximately 0.3010, so log2 - 1 is negative. So |log2 - 1| = 1 - log2. Because the absolute value makes it positive. So that term simplifies to 1 - log2.Therefore, the entire expression becomes:(log2)^2 + (log2)(log5) + (1 - log2)Let me write that:(log2)^2 + log2·log5 + 1 - log2Hmm, can we simplify this further? Let's see. Maybe we can factor terms involving log2.Let me group the first two terms: (log2)^2 + log2·log5. Factor out log2:log2(log2 + log5)Then the remaining terms are +1 - log2.So:log2(log2 + log5) + 1 - log2Recall that log2 + log5 = log(2*5) = log10. And since we're using base 10, log10 = 1.Therefore, log2*(1) + 1 - log2 = log2 + 1 - log2 = 1.Wait, that's nice! So all the log terms cancel out, leaving 1. Let me check each step again to make sure.Original expression:(log2)^2 + log2·log5 + sqrt[(log2)^2 - 2log2 + 1]Simplify the square root:sqrt[(log2 - 1)^2] = |log2 - 1| = 1 - log2 (since log2 < 1)So expression becomes:(log2)^2 + log2·log5 + 1 - log2Factor log2 from the first two terms:log2(log2 + log5) + 1 - log2log2 + log5 = log(2*5) = log10 = 1So:log2*(1) + 1 - log2 = log2 + 1 - log2 = 1Yes, that's correct. So the entire expression simplifies to 1. That's a neat result. Let me verify with approximate values.Compute each term numerically:log2 ≈ 0.3010log5 ≈ 0.69897Compute (log2)^2 ≈ (0.3010)^2 ≈ 0.0906log2·log5 ≈ 0.3010 * 0.69897 ≈ 0.2105sqrt[(log2)^2 - 2log2 +1] = sqrt[0.0906 - 0.6020 + 1] = sqrt[0.4886] ≈ 0.699But 1 - log2 ≈ 1 - 0.3010 ≈ 0.699, which matches the square root term.Adding all terms: 0.0906 + 0.2105 + 0.699 ≈ 0.0906 + 0.2105 = 0.3011 + 0.699 ≈ 1.0001, which is approximately 1. So the numerical check works. Therefore, the answer is indeed 1.So problem (2) simplifies to 1.Final Answer(1) boxed{4a}(2) boxed{1}</think>(1) Simplify and evaluate: [(2a^{frac{2}{3}}b^{frac{1}{2}})(-6a^{frac{1}{2}}b^{frac{1}{3}}) div (-3a^{frac{1}{6}}b^{frac{5}{6}})]First, multiply the coefficients and add the exponents for the variables:[2 cdot (-6) = -12][a^{frac{2}{3} + frac{1}{2}} = a^{frac{7}{6}}, quad b^{frac{1}{2} + frac{1}{3}} = b^{frac{5}{6}}]Resulting in:[-12a^{frac{7}{6}}b^{frac{5}{6}}]Next, divide by (-3a^{frac{1}{6}}b^{frac{5}{6}}):[frac{-12}{-3} = 4][a^{frac{7}{6} - frac{1}{6}} = a^1 = a, quad b^{frac{5}{6} - frac{5}{6}} = b^0 = 1]Resulting in:[4a]Final answer for (1): (boxed{4a})(2) Simplify and evaluate: [(lg2)^2 + lg2 cdot lg5 + sqrt{(lg2)^2 - 2lg2 + 1}]First, simplify the square root term:[sqrt{(lg2 - 1)^2} = |lg2 - 1| = 1 - lg2 quad (text{since } lg2 < 1)]Substitute back into the expression:[(lg2)^2 + lg2 cdot lg5 + (1 - lg2)]Factor out (lg2) from the first two terms:[lg2(lg2 + lg5) + 1 - lg2]Since (lg2 + lg5 = lg10 = 1):[lg2 cdot 1 + 1 - lg2 = lg2 + 1 - lg2 = 1]Final answer for (2): (boxed{1}),<think>Okay, so I need to simplify and evaluate these two expressions. Let's start with the first one: (2a^(2/3)b^(1/2)) multiplied by (-6a^(1/2)b^(1/3)) divided by (-3a^(1/6)b^(5/6)). Hmm, that looks a bit complicated with all the exponents, but maybe if I break it down step by step, it will make more sense.First, let me rewrite the expression to make sure I understand it correctly. It's (2a^{2/3} b^{1/2}) * (-6a^{1/2} b^{1/3}) divided by (-3a^{1/6} b^{5/6}). So that's multiplication of the first two terms and then division by the third term.I remember that when you multiply terms with exponents, you can add the exponents of like bases, and when you divide, you subtract the exponents. Also, the coefficients (the numbers in front) can be multiplied and divided normally. Let me handle the coefficients first.The coefficients are 2, -6, and -3. So multiplying the first two coefficients: 2 * (-6) = -12. Then dividing by -3: -12 / (-3) = 4. Okay, so the coefficient simplifies to 4. That seems straightforward.Now, let's handle the variables with exponents. Starting with the a terms. In the numerator, we have a^{2/3} * a^{1/2}, and then in the denominator, we have a^{1/6}. So combining the exponents for a: 2/3 + 1/2 - 1/6. Let me compute that.First, add 2/3 and 1/2. To add these fractions, find a common denominator. The common denominator for 3 and 2 is 6. So 2/3 is 4/6 and 1/2 is 3/6. Adding those gives 4/6 + 3/6 = 7/6. Then subtract the denominator's exponent: 7/6 - 1/6 = 6/6 = 1. So the exponent for a is 1, which is just a.Now for the b terms. In the numerator, we have b^{1/2} * b^{1/3}, and in the denominator, b^{5/6}. So combining the exponents: 1/2 + 1/3 - 5/6. Let me calculate that.Again, common denominators. For 1/2 and 1/3, the common denominator is 6. So 1/2 = 3/6 and 1/3 = 2/6. Adding those gives 3/6 + 2/6 = 5/6. Then subtract the denominator's exponent: 5/6 - 5/6 = 0. So the exponent for b is 0. But anything to the power of 0 is 1, right? So the b term cancels out.Putting it all together, the coefficient is 4, the a term is a^1, and the b term is 1. So the simplified expression is 4a. Let me double-check that to make sure I didn't make a mistake.Coefficients: 2 * -6 = -12; -12 / -3 = 4. That's correct.For a exponents: 2/3 + 1/2 = (4/6 + 3/6) = 7/6; 7/6 - 1/6 = 6/6 = 1. Correct.For b exponents: 1/2 + 1/3 = 5/6; 5/6 - 5/6 = 0. Correct. So b term is gone. Yep, 4a. So problem (1) simplifies to 4a.Now, problem (2): (lg2)^2 + lg2 * lg5 + sqrt{(lg2)^2 - 2lg2 + 1}. Hmm. Let's parse this. "lg" here is logarithm base 10, right? So lg2 is log base 10 of 2, and similarly lg5 is log base 10 of 5.The expression is (log2)^2 + log2 * log5 + sqrt{(log2)^2 - 2 log2 + 1}. Let's see if we can simplify this step by step.First, let's look at the square root part: sqrt{(log2)^2 - 2 log2 + 1}. That looks like a quadratic in log2. Let me denote x = log2 for simplicity. Then the expression under the square root becomes x^2 - 2x + 1. But x^2 - 2x + 1 is (x - 1)^2. So the square root of (x - 1)^2 is |x - 1|. So sqrt{(log2)^2 - 2 log2 + 1} = |log2 - 1|.But log2 is approximately 0.3010, so 0.3010 - 1 = -0.699, so absolute value would be 0.699. But since we need to do this symbolically, maybe we can express 1 as log10, since log10 base 10 is 1. So 1 = log10. Therefore, log2 - log10 = log(2/10) = log(1/5) = -log5. But the absolute value of that is log5. Wait, let me check that.Wait, log10 is 1, so log2 - 1 = log2 - log10 = log(2/10) = log(1/5) = -log5. Therefore, |log2 - 1| = | - log5 | = log5, since log5 is positive. So sqrt{(log2)^2 - 2 log2 + 1} = log5.Wait, let me verify that step again. Let me suppose x = log2, then sqrt{(x - 1)^2} = |x - 1|. Since x = log2 ≈ 0.3010, which is less than 1, so x - 1 is negative, so absolute value is 1 - x. But 1 - log2 = log10 - log2 = log(10/2) = log5. Oh, right! Because log(a) - log(b) = log(a/b). So 1 - log2 = log5. Therefore, |log2 - 1| = 1 - log2 = log5. Wait, that's interesting. Let me confirm:log5 = log(10/2) = log10 - log2 = 1 - log2. Therefore, 1 - log2 = log5. Therefore, |log2 - 1| = log5. So the square root simplifies to log5.Okay, so the original expression becomes (log2)^2 + log2 * log5 + log5.Now, let's write that out: (log2)^2 + log2 * log5 + log5. Hmm. Maybe we can factor something here. Let me see.Notice that the first two terms have a common factor of log2. Let's factor that out: log2 (log2 + log5) + log5. Then, log2 + log5 is log(2*5) = log10 = 1. Therefore, this simplifies to log2 * 1 + log5 = log2 + log5. Which is again log(2*5) = log10 = 1. So the entire expression simplifies to 1?Wait, let me verify that step-by-step.Original expression: (log2)^2 + log2 * log5 + sqrt{(log2)^2 - 2 log2 +1}We simplified the sqrt part to log5. So the expression is (log2)^2 + log2 * log5 + log5.Then, factoring log2 from the first two terms: log2(log2 + log5) + log5.But log2 + log5 = log(2*5) = log10 = 1. So that part is log2*1 + log5 = log2 + log5 = log10 = 1. So yes, the entire expression simplifies to 1.Wait, that seems surprising but makes sense. Let me check with approximate values to verify.Let's compute each part numerically:First, log2 ≈ 0.3010, log5 ≈ 0.69897.Compute (log2)^2 ≈ (0.3010)^2 ≈ 0.0906.log2 * log5 ≈ 0.3010 * 0.69897 ≈ 0.2104.sqrt{(log2)^2 - 2 log2 +1} = sqrt{0.0906 - 2*0.3010 +1} = sqrt{0.0906 - 0.6020 +1} = sqrt{0.4886} ≈ 0.69897, which is log5.So adding all three parts: 0.0906 + 0.2104 + 0.69897 ≈ 0.0906 + 0.2104 = 0.3010, then +0.69897 ≈ 0.99997, which is approximately 1. So that checks out.Therefore, the second expression simplifies to 1.So, for problem (1), the simplified expression is 4a, and problem (2) evaluates to 1.Final Answer(1) boxed{4a}(2) boxed{1}</think>(1) To simplify the expression ((2a^{frac{2}{3}}b^{frac{1}{2}})(-6a^{frac{1}{2}}b^{frac{1}{3}}) div (-3a^{frac{1}{6}}b^{frac{5}{6}})):- Combine the coefficients: (2 times (-6) div (-3) = -12 div (-3) = 4).- Combine the exponents for (a): (frac{2}{3} + frac{1}{2} - frac{1}{6} = frac{4}{6} + frac{3}{6} - frac{1}{6} = frac{6}{6} = 1).- Combine the exponents for (b): (frac{1}{2} + frac{1}{3} - frac{5}{6} = frac{3}{6} + frac{2}{6} - frac{5}{6} = 0).Thus, the simplified expression is (4a).(2) To evaluate ((lg2)^2 + lg2 cdot lg5 + sqrt{(lg2)^2 - 2lg2 + 1}):- Simplify the square root term: (sqrt{(lg2 - 1)^2} = |lg2 - 1|). Since (lg2 approx 0.3010), we have (|lg2 - 1| = 1 - lg2 = lg5).- Substitute back into the expression: ((lg2)^2 + lg2 cdot lg5 + lg5).- Factor out (lg2) from the first two terms: (lg2(lg2 + lg5) + lg5). Since (lg2 + lg5 = lg10 = 1), this simplifies to (lg2 cdot 1 + lg5 = lg2 + lg5 = lg10 = 1).Thus, the evaluated result is (1).[(1) boxed{4a}][(2) boxed{1}]]

🔑:Solution: (1) (2a^{ frac {2}{3}}b^{ frac {1}{2}})(-6a^{ frac {1}{2}}b^{ frac {1}{3}})÷(-3a^{ frac {1}{6}}b^{ frac {5}{6}}) =4a^{ frac {2}{3}+ frac {1}{2}- frac {1}{6}}cdot b^{ frac {1}{2}+ frac {1}{3}- frac {5}{6}}=4a. So, the simplified result is boxed{4a}. (2) (lg2)^2+lg2cdotlg5+ sqrt {(lg2)^{2}-2lg2+1} =(lg2)^2+lg2cdotlg5+1-lg2 =lg2(lg2+lg5)+1-lg2 =lg2+1-lg2=1. Thus, the evaluated result is boxed{1}.

🧠:[<think>Okay, so I need to solve this problem where there are two different prime numbers p and q. They satisfy these two quadratic equations: p² - 2001p + m = 0 and q² - 2001q + m = 0. The question is asking for the value of p² + q². Hmm, let me think about how to approach this.First, since both p and q satisfy the same quadratic equation with the same m, that means they are both roots of the equation x² - 2001x + m = 0. Right? So if a quadratic equation has two roots, then those roots must satisfy certain relationships based on the coefficients. Specifically, Vieta's formulas. Vieta's formulas say that for a quadratic equation ax² + bx + c = 0, the sum of the roots is -b/a and the product is c/a. In this case, the equation is x² - 2001x + m = 0, so a = 1, b = -2001, c = m. Therefore, the sum of the roots should be 2001, and the product should be m.Since p and q are the two roots, that means p + q = 2001 and pq = m. The problem says that p and q are different prime numbers, so we need two primes that add up to 2001. Then, once we find those primes, we can compute p² + q². But how do we find p and q?First, since primes (except 2) are odd, and the sum of two odd numbers is even. But 2001 is an odd number. So if we have two primes adding up to an odd number, one of them must be even, which can only be 2 (since 2 is the only even prime). Therefore, one of the primes is 2 and the other is 2001 - 2 = 1999. So we need to check if 1999 is a prime number.Wait, is 1999 prime? Let me try to verify that. To check if 1999 is prime, we need to see if it has any divisors other than 1 and itself. The square root of 1999 is approximately 44.7, so we need to check primes less than or equal to 43.Let me divide 1999 by some primes:2: 1999 is odd, so not divisible by 2.3: Sum of digits is 1+9+9+9 = 28. 28 divided by 3 leaves a remainder of 1, so not divisible by 3.5: Ends with 9, so not divisible by 5.7: Let's do 1999 ÷ 7. 7*285=1995, so 1999-1995=4. Not divisible by 7.11: Alternating sum: (1 + 9) - (9 + 9) = 10 - 18 = -8. Not divisible by 11.13: 13*153 = 1989. 1999 - 1989 = 10. Not divisible by 13.17: 17*117 = 1989. 1999 - 1989 = 10. Not divisible by 17.19: 19*105 = 1995. 1999 - 1995 = 4. Not divisible by 19.23: 23*86= 1978. 1999 - 1978 =21. 21 is not divisible by 23.29: 29*68=1972. 1999 - 1972=27. 27 isn't divisible by 29.31: 31*64=1984. 1999-1984=15. Not divisible by 31.37: 37*54=1998. 1999 - 1998=1. Not divisible by 37.41: 41*48=1968. 1999 - 1968=31. 31 isn't divisible by 41.43: 43*46=1978. 1999 - 1978=21. 21 isn't divisible by 43.So none of the primes up to 43 divide 1999. Therefore, 1999 is a prime number. Therefore, the two primes are 2 and 1999.Therefore, p and q are 2 and 1999. Since they are different primes, that satisfies the problem's condition.Now, we need to compute p² + q². Let's calculate 2² + 1999². 2² is 4. What about 1999²? Let's compute that. 2000² is 4,000,000. Then, (2000 - 1)² = 2000² - 2*2000*1 + 1 = 4,000,000 - 4,000 + 1 = 3,996,001. Therefore, 1999² is 3,996,001. Adding 4 gives 3,996,001 + 4 = 3,996,005. Therefore, p² + q² = 3,996,005.But wait, let me confirm that this is correct. Let me check 1999 squared again. Let's compute 1999 times 1999:1999 * 1999= (2000 - 1)(2000 - 1)= 2000*2000 - 2000*1 - 1*2000 + 1*1= 4,000,000 - 2000 - 2000 + 1= 4,000,000 - 4000 + 1= 3,996,001. Yes, that's correct.So adding 4 gives 3,996,005. So the answer should be 3,996,005. But let me check if there's another way to compute p² + q² without directly calculating the squares. Since we know from Vieta's formula that p + q = 2001 and pq = m. Then, p² + q² can be expressed as (p + q)² - 2pq. So (2001)² - 2m. But we need to find m first. Wait, but m is pq. So m = pq. Therefore, p² + q² = (2001)² - 2m = (2001)² - 2pq. But since we already know p and q, we can compute this as well.Alternatively, if we use that formula, let's see:(2001)^2 = 2001*2001. Let me compute that.2000^2 = 4,000,0002*2000*1 = 4,0001^2 =1So (2000 +1)^2 = 4,000,000 + 4,000 +1 = 4,004,001.Therefore, p² + q² = 4,004,001 - 2pq.But pq is m, which is 2*1999 = 3998. Therefore, 4,004,001 - 2*3998 = 4,004,001 - 7996 = 4,004,001 - 7,996.Compute 4,004,001 - 7,000 = 3,997,001Then subtract 996: 3,997,001 - 996 = 3,997,001 - 1,000 + 4 = 3,996,001 + 4 = 3,996,005.Same result as before. So this cross-checks. Therefore, the answer is 3,996,005.But just to make sure, let me check if there's any other pair of primes that add up to 2001. Suppose there is another pair. For example, if p is another prime, then q = 2001 - p. But since 2001 is odd, one has to be even (2), and the other odd. So if there is another prime q, which is 2001 - 2 = 1999, which we already checked is prime. So there can't be another pair unless there's another even prime, but 2 is the only even prime. Therefore, the only possible pair is 2 and 1999. Therefore, p and q must be 2 and 1999.Hence, the answer is 2² + 1999² = 4 + 3,996,001 = 3,996,005.Final Answerboxed{3996005}</think>Given two different prime numbers ( p ) and ( q ) satisfying the quadratic equations ( p^2 - 2001p + m = 0 ) and ( q^2 - 2001q + m = 0 ), we need to find the value of ( p^2 + q^2 ).Since ( p ) and ( q ) are roots of the quadratic equation ( x^2 - 2001x + m = 0 ), by Vieta's formulas, we know:- The sum of the roots ( p + q = 2001 )- The product of the roots ( pq = m )Given that ( p ) and ( q ) are different primes and their sum is 2001 (an odd number), one of them must be 2 (the only even prime). Therefore, the other prime is ( 2001 - 2 = 1999 ).We verify that 1999 is a prime number by checking divisibility by all primes less than its square root (approximately 44.7), confirming it is indeed a prime.Thus, the primes are ( p = 2 ) and ( q = 1999 ). To find ( p^2 + q^2 ):- Calculate ( p^2 = 2^2 = 4 )- Calculate ( q^2 = 1999^2 ). Using the identity ( (2000 - 1)^2 = 2000^2 - 2 cdot 2000 cdot 1 + 1 ), we get: [ 1999^2 = 2000^2 - 2 cdot 2000 + 1 = 4,000,000 - 4,000 + 1 = 3,996,001 ]Adding these results:[p^2 + q^2 = 4 + 3,996,001 = 3,996,005]We can also verify using the identity ( p^2 + q^2 = (p + q)^2 - 2pq ):- ( (2001)^2 = 4,004,001 )- ( pq = 2 cdot 1999 = 3998 )- ( p^2 + q^2 = 4,004,001 - 2 cdot 3998 = 4,004,001 - 7996 = 3,996,005 )Thus, the value of ( p^2 + q^2 ) is (boxed{3996005}).,<think>Okay, let's try to tackle this problem step by step. So, we have two different prime numbers, p and q. They both satisfy the quadratic equation: p² - 2001p + m = 0 and q² - 2001q + m = 0. And we need to find p² + q². Hmm, okay.First, since both p and q satisfy the same quadratic equation, that means they are both roots of the equation x² - 2001x + m = 0. So, by Vieta's formulas, the sum of the roots should be equal to 2001, and the product should be equal to m. Vieta's formulas, right? For a quadratic equation ax² + bx + c = 0, the sum of the roots is -b/a and the product is c/a. Here, a is 1, so sum of roots is 2001 and product is m. So, p + q = 2001 and pq = m.But wait, p and q are primes, and primes except 2 are odd. Also, 2001 is an odd number. So, if p and q are primes adding up to 2001, which is odd, then one of them must be even because an odd plus an even is odd. But the only even prime number is 2. Therefore, one of the primes must be 2, and the other one must be 2001 - 2 = 1999. So, p and q are 2 and 1999.Let me check if 1999 is a prime number. Hmm, to confirm that, I need to see if it's divisible by any primes up to its square root. The square root of 1999 is approximately 44.7, so we need to check primes up to 43. Let's see:1999 divided by 2? No, it's odd. Divided by 3: 1+9+9+9 = 28, which is not divisible by 3. Divided by 5: ends with 9, so no. 7? 1999 divided by 7: 7*285 = 1995, remainder 4. Not divisible. 11: 1 - 9 + 9 - 9 = -8, not divisible by 11. 13: Let's see, 13*153 = 1989, remainder 10. Not divisible. 17: 17*117 = 1989, remainder 10. 19: 19*105 = 1995, remainder 4. 23: 23*86 = 1978, 1999 - 1978 = 21, which is not divisible by 23. 29: 29*68 = 1972, 1999 - 1972 = 27, not divisible. 31: 31*64 = 1984, 1999 - 1984 = 15, not divisible. 37: 37*54 = 1998, remainder 1. 41: 41*48 = 1968, 1999 - 1968 = 31, which is a prime, so no. 43: 43*46 = 1978, 1999 - 1978 = 21, which is not divisible by 43. So, 1999 is a prime number. Okay, that checks out.So, if p and q are 2 and 1999, then p² + q² would be 2² + 1999². Let's compute that. 2 squared is 4. 1999 squared is... Hmm, that's a big number. Let me compute that. (2000 - 1)² = 2000² - 2*2000*1 + 1² = 4,000,000 - 4,000 + 1 = 3,996,001. So, adding 4 gives 3,996,001 + 4 = 3,996,005. Therefore, the value of p² + q² is 3,996,005.But let me double-check. Since both primes satisfy the equation, let's verify with p = 2. Plugging into the first equation: 2² - 2001*2 + m = 0. So, 4 - 4002 + m = 0 => m = 4002 - 4 = 3998. Similarly, for q = 1999: 1999² - 2001*1999 + m = 0. Let's compute 1999² - 2001*1999. Factor out 1999: 1999*(1999 - 2001) = 1999*(-2) = -3998. So, -3998 + m = 0 => m = 3998. So, m is indeed 3998 in both cases, which matches. Therefore, the value of p² + q² is 4 + 3992001 (wait, 1999 squared was 3,996,001, right? So 4 + 3,996,001 = 3,996,005. Correct.Alternatively, since we know that p + q = 2001 and pq = m, which we found as 2*1999 = 3998. Then p² + q² can also be computed as (p + q)² - 2pq. So, that's 2001² - 2*3998. Let's compute 2001². 2000² is 4,000,000, and 2001² = (2000 + 1)^2 = 4,000,000 + 4,000 + 1 = 4,004,001. Then subtract 2*3998 = 7996. So, 4,004,001 - 7,996 = 4,004,001 - 7,996. Let's subtract step by step: 4,004,001 - 7,000 = 3,997,001; then subtract 996: 3,997,001 - 996 = 3,996,005. So, same result. So that's a good check.Therefore, the answer is 3,996,005. But let me just make sure there's no other pair of primes that add up to 2001. Since 2001 is odd, one has to be even (2) and the other odd (1999). Since 1999 is prime, as we checked earlier, that's the only possible pair. So, yes, this is the only solution.Hence, the value of p² + q² is 3,996,005.Final AnswerThe value of ( p^2 + q^2 ) is boxed{3996005}.</think>Given two different prime numbers ( p ) and ( q ) satisfying the quadratic equations ( p^2 - 2001p + m = 0 ) and ( q^2 - 2001q + m = 0 ), we need to find the value of ( p^2 + q^2 ).Since ( p ) and ( q ) are roots of the quadratic equation ( x^2 - 2001x + m = 0 ), by Vieta's formulas, we have:- The sum of the roots: ( p + q = 2001 )- The product of the roots: ( pq = m )Given that ( p ) and ( q ) are primes and their sum is 2001 (an odd number), one of them must be the even prime number 2. The other prime number is ( 2001 - 2 = 1999 ). We verify that 1999 is a prime number by checking divisibility by all primes up to its square root (approximately 44.7), confirming it is indeed prime.Thus, ( p = 2 ) and ( q = 1999 ). We calculate ( p^2 + q^2 ):- ( p^2 = 2^2 = 4 )- ( q^2 = 1999^2 = (2000 - 1)^2 = 2000^2 - 2 cdot 2000 cdot 1 + 1 = 4000000 - 4000 + 1 = 3996001 )Adding these results:[ p^2 + q^2 = 4 + 3996001 = 3996005 ]We can also verify using the identity ( p^2 + q^2 = (p + q)^2 - 2pq ):- ( (p + q)^2 = 2001^2 = 4004001 )- ( 2pq = 2 cdot 3998 = 7996 )- ( p^2 + q^2 = 4004001 - 7996 = 3996005 )Thus, the value of ( p^2 + q^2 ) is (boxed{3996005}).]

🔑:Subtracting the two equations, we get (p - q)(p + q - 2001) = 0,Since p neq q,Therefore, p + q = 2001. Since p and q are prime numbers,Therefore, one of p and q is 2, and the other is 1999.Therefore, p^2 + q^2 = 2^2 + 1999^2 = 3996005.Hence, the answer is boxed{3996005}.

🧠:[<think>Okay, let's see. I need to find the value of t such that the vector (2p + t r) is perpendicular to q. Hmm, vectors in the plane, so all are 2-dimensional. Let me recall that two vectors are perpendicular if their dot product is zero. Right, so if I can compute the dot product of (2p + t r) and q, set it equal to zero, and solve for t, that should do it.First, let me make sure I understand the vectors given. Vector p is (3,3), vector q is (-1,2), and vector r is (4,1). So, all these vectors are in the plane, which is straightforward.Let me write down the expression 2p + t r. Since p is (3,3), multiplying by 2 gives (6,6). Then, vector r is (4,1), so multiplying by t gives (4t, t). Adding these together component-wise: (6 + 4t, 6 + t). So the vector (2p + t r) is (6 + 4t, 6 + t).Now, this vector needs to be perpendicular to q, which is (-1,2). To check for perpendicularity, compute the dot product of (6 + 4t, 6 + t) and (-1, 2) and set it equal to zero.The dot product is calculated as (6 + 4t)*(-1) + (6 + t)*(2). Let me compute each term separately.First term: (6 + 4t)*(-1) = -6 -4tSecond term: (6 + t)*(2) = 12 + 2tAdding these two terms together: (-6 -4t) + (12 + 2t) = (-6 + 12) + (-4t + 2t) = 6 - 2tSo the equation is 6 - 2t = 0. Solving for t gives:6 - 2t = 0 -2t = -6 t = (-6)/(-2) t = 3Let me double-check my steps to make sure I didn't make a mistake. Calculated 2p correctly: 2*(3,3)=(6,6). Then t*r: t*(4,1)=(4t, t). Adding gives (6 +4t, 6 +t). Then dot product with q: (-1, 2). So (-1)(6 +4t) + 2*(6 +t). That is indeed -6 -4t +12 +2t, which simplifies to 6 -2t. Setting equal to zero gives t=3. Seems correct. Maybe I can plug t=3 back into the original vectors to verify.Compute 2p +3r: 2p is (6,6), 3r is (12,3). Adding gives (18,9). Then check if (18,9) is perpendicular to q=(-1,2). Dot product: 18*(-1) +9*2 = -18 +18 =0. Perfect, so yes, the dot product is zero. Therefore, t=3 is correct.Wait, just to make sure that my arithmetic wasn't wrong. Let me recheck:For 2p: (3*2, 3*2)=(6,6). Correct.t*r when t=3: (4*3, 1*3)=(12,3). Correct.Sum: (6+12, 6+3)=(18,9). Correct.Dot product with q: (-1)*18 +2*9= -18 +18=0. Perfect. So yes, t=3 is the correct answer.Therefore, the value of the real number t is 3.Final AnswerThe value of ( t ) is boxed{3}.</think>Given vectors (overrightarrow{p} = (3, 3)), (overrightarrow{q} = (-1, 2)), and (overrightarrow{r} = (4, 1)), we need to find the value of (t) such that ((2overrightarrow{p} + toverrightarrow{r}) perp overrightarrow{q}).First, compute the vector (2overrightarrow{p} + toverrightarrow{r}):- (2overrightarrow{p} = 2(3, 3) = (6, 6))- (toverrightarrow{r} = t(4, 1) = (4t, t))Adding these components together:[2overrightarrow{p} + toverrightarrow{r} = (6 + 4t, 6 + t)]Next, we need this vector to be perpendicular to (overrightarrow{q}). Two vectors are perpendicular if their dot product is zero. Compute the dot product of ((6 + 4t, 6 + t)) and (overrightarrow{q} = (-1, 2)):[(6 + 4t)(-1) + (6 + t)(2)]Calculate each term separately:- First term: ((6 + 4t)(-1) = -6 - 4t)- Second term: ((6 + t)(2) = 12 + 2t)Combine the terms:[-6 - 4t + 12 + 2t = 6 - 2t]Set the dot product equal to zero and solve for (t):[6 - 2t = 0 -2t = -6 t = 3]Verification by substituting (t = 3) back into the vector:- (2overrightarrow{p} + 3overrightarrow{r} = (6 + 12, 6 + 3) = (18, 9))- Dot product with (overrightarrow{q}): (18(-1) + 9(2) = -18 + 18 = 0)Thus, the value of (t) is (boxed{3}).,<think>Okay, let me try to figure out this problem step by step. So, we have three vectors: p, q, and r. Their coordinates are given as p = (3,3), q = (-1,2), and r = (4,1). The problem states that the vector (2p + t*r) is perpendicular to q, and we need to find the value of t. First, I remember that two vectors are perpendicular if their dot product is zero. So, if (2p + t*r) is perpendicular to q, then their dot product should equal zero. That gives me an equation to work with. Let me write that down:(2p + t*r) · q = 0Alright, next step is to compute 2p and t*r, add them together, and then take the dot product with q. Let's start by calculating 2p. Since p is (3,3), multiplying by 2 gives:2p = (2*3, 2*3) = (6,6)Then, t*r would be t*(4,1) = (4t, t). Adding these two vectors together:2p + t*r = (6 + 4t, 6 + t)Now, we need to take the dot product of this resulting vector with q, which is (-1,2). The dot product formula is (a,b)·(c,d) = a*c + b*d. Applying that here:(6 + 4t)*(-1) + (6 + t)*(2) = 0Let me compute each part separately. First, the first term:(6 + 4t)*(-1) = -6 -4tThen, the second term:(6 + t)*2 = 12 + 2tAdding those two results together:(-6 -4t) + (12 + 2t) = (-6 + 12) + (-4t + 2t) = 6 - 2tSo, the equation becomes:6 - 2t = 0Solving for t:-2t = -6t = (-6)/(-2) = 3Wait, that seems straightforward. Let me check if I did everything correctly. First, the calculation of 2p and t*r. Yes, 2*(3,3) is (6,6). t*(4,1) is (4t, t). Adding them gives (6+4t, 6+t). Then, dot product with (-1,2) is (6+4t)(-1) + (6+t)(2). That's right. Expanding that: -6 -4t +12 +2t. Combining like terms: (-6 +12) is 6, (-4t +2t) is -2t. So 6 -2t =0. Solving gives t=3. Hmm, seems correct. Let me verify once more. Let's plug t=3 back into the original vectors and check if they are indeed perpendicular. First, compute 2p +3r. 2p is (6,6). 3r is (12,3). Adding them gives (6+12, 6+3) = (18,9). Then, take the dot product with q=(-1,2). So (18)(-1) + (9)(2) = -18 +18 =0. Perfect, that works. So t=3 is indeed the correct answer. Wait, just to ensure I didn't make any arithmetic mistakes. Let me redo the dot product step:Original vectors after substitution:2p + tr = (6 +4t, 6 +t)q = (-1,2)Dot product:(6 +4t)(-1) + (6 +t)(2) = (-6 -4t) + (12 +2t) = (-6 +12) + (-4t +2t) = 6 -2tSet equal to zero: 6 -2t=0 => t=3. Yep, same result. Alright, so I think the answer is 3. But just to recap all steps once more:1. Multiply vector p by 2: 2*(3,3) = (6,6).2. Multiply vector r by t: t*(4,1) = (4t, t).3. Add the two vectors: (6 +4t,6 +t).4. Compute the dot product with q: (-1)*(6 +4t) + 2*(6 +t).5. Expand the dot product: -6 -4t +12 +2t = 6 -2t.6. Set equal to zero: 6 -2t =0.7. Solve for t: t=3.Every step checks out. The final answer is 3.Final AnswerThe value of the real number ( t ) is boxed{3}.</think>Given vectors (overrightarrow{p} = (3, 3)), (overrightarrow{q} = (-1, 2)), and (overrightarrow{r} = (4, 1)), we need to find the value of (t) such that (2overrightarrow{p} + toverrightarrow{r}) is perpendicular to (overrightarrow{q}).First, we compute (2overrightarrow{p}) and (toverrightarrow{r}):[2overrightarrow{p} = 2 cdot (3, 3) = (6, 6)][toverrightarrow{r} = t cdot (4, 1) = (4t, t)]Adding these vectors together:[2overrightarrow{p} + toverrightarrow{r} = (6 + 4t, 6 + t)]Next, we take the dot product of this resulting vector with (overrightarrow{q}) and set it to zero (since perpendicular vectors have a dot product of zero):[(6 + 4t, 6 + t) cdot (-1, 2) = 0]Calculating the dot product:[(6 + 4t)(-1) + (6 + t)(2) = 0]Expanding and simplifying:[-6 - 4t + 12 + 2t = 0][6 - 2t = 0]Solving for (t):[-2t = -6][t = 3]Thus, the value of the real number (t) is (boxed{3}).]

🔑:From the given condition, we can get 2 overrightarrow {p}+t overrightarrow {r} = (6+4t, 6+t). Since (2 overrightarrow {p}+t overrightarrow {r}) perp overrightarrow {q}, we have (2 overrightarrow {p}+t overrightarrow {r}) cdot overrightarrow {q} = 0, Substituting the given values, we get (6+4t)(-1) + 2(6+t) = 0, Solving this equation, we find t=3. Therefore, the answer is boxed{3}. This problem involves calculating the coordinates of vectors and using the dot product to determine the perpendicular relationship between vectors. It is a basic question that tests the understanding of vector operations and the concept of dot product.