answer：Okay, so I need to find the slope of the tangent line to a circle at the point (7,4). The center of the circle is given as (2,3). Hmm, I remember that the tangent line at any point on a circle is perpendicular to the radius drawn to that point. So, if I can find the slope of the radius, then the slope of the tangent line should be the negative reciprocal of that.First, let me find the slope of the radius. The radius connects the center (2,3) to the point of tangency (7,4). The formula for slope is (y2 - y1)/(x2 - x1). Plugging in the coordinates, that would be (4 - 3)/(7 - 2). So, 1 divided by 5, which is 1/5. Wait, so the slope of the radius is 1/5. Since the tangent line is perpendicular, its slope should be the negative reciprocal. The reciprocal of 1/5 is 5, and the negative of that is -5. So, the slope of the tangent line should be -5.Let me double-check that. If the radius has a slope of 1/5, then a line perpendicular to it would have a slope that, when multiplied together, equals -1. So, 1/5 multiplied by -5 is indeed -1, which confirms that they are perpendicular. That makes sense.I think I've got it right. The slope of the tangent line at (7,4) is -5.

answer：Alright, let's tackle this problem step by step. So, we have Xiao Ming, his father, and his mother playing a red envelope grabbing game during the Spring Festival. The father sent out 12 red envelopes with amounts ranging from 1 yuan to 12 yuan, one at a time. All three grabbed simultaneously, and in the end, each person grabbed 4 red envelopes. The father mentioned that he grabbed 1 yuan and 3 yuan. The mother said she grabbed 8 yuan and 9 yuan. Xiao Ming added that the sum of the amounts each of them grabbed is equal. Our goal is to determine the two amounts Xiao Ming definitely grabbed.First, let's understand the total sum of all the red envelopes. Since the amounts are from 1 to 12 yuan, we can calculate the total sum using the formula for the sum of the first n natural numbers:[text{Total Sum} = frac{n(n + 1)}{2} = frac{12 times 13}{2} = 78 text{ yuan}]Since there are three people and each grabbed 4 envelopes, the total sum each person grabbed should be:[text{Sum per person} = frac{78}{3} = 26 text{ yuan}]So, each person, including Xiao Ming, must have a total of 26 yuan from the four envelopes they grabbed.Now, let's analyze the information given by the father and the mother.**Father's Envelopes:**- The father grabbed 1 yuan and 3 yuan. Let's denote the other two envelopes he grabbed as ( a ) and ( b ).- Therefore, the total sum for the father is:[1 + 3 + a + b = 26][a + b = 26 - 4 = 22]So, the father's other two envelopes add up to 22 yuan. We need to find two numbers from the remaining amounts (excluding 1, 3, 8, 9, since the mother grabbed 8 and 9) that add up to 22.**Mother's Envelopes:**- The mother grabbed 8 yuan and 9 yuan. Let's denote her other two envelopes as ( c ) and ( d ).- Therefore, the total sum for the mother is:[8 + 9 + c + d = 26][c + d = 26 - 17 = 9]So, the mother's other two envelopes add up to 9 yuan. We need to find two numbers from the remaining amounts that add up to 9.**Remaining Envelopes:**The total envelopes are 1 to 12. Excluding the ones already grabbed by the father and mother, the remaining envelopes are:[{2, 4, 5, 6, 7, 10, 11, 12}]**Analyzing the Father's Remaining Envelopes:**We need two numbers from the remaining set that add up to 22. Let's list the possible pairs:- 10 + 12 = 22- 11 + 11 = 22 (But we don't have two 11s)- 9 + 13 = 22 (But 9 is already grabbed by the mother, and 13 isn't in the set)So, the only possible pair is 10 and 12.Therefore, the father grabbed 1, 3, 10, and 12 yuan.**Updating Remaining Envelopes:**After assigning 10 and 12 to the father, the remaining envelopes are:[{2, 4, 5, 6, 7, 11}]**Analyzing the Mother's Remaining Envelopes:**We need two numbers from the remaining set that add up to 9. Let's list the possible pairs:- 2 + 7 = 9- 4 + 5 = 9So, there are two possible pairs: (2, 7) and (4, 5).**Determining Xiao Ming's Envelopes:**Xiao Ming must have grabbed the remaining envelopes after the father and mother have taken theirs. Let's consider both possibilities for the mother's remaining envelopes.**Case 1: Mother grabbed 2 and 7 yuan.**- Mother's envelopes: 8, 9, 2, 7- Remaining envelopes for Xiao Ming: 4, 5, 6, 11Xiao Ming needs to have a total of 26 yuan. Let's check the sum:[4 + 5 + 6 + 11 = 26]Perfect, this adds up. So, Xiao Ming grabbed 4, 5, 6, and 11 yuan.**Case 2: Mother grabbed 4 and 5 yuan.**- Mother's envelopes: 8, 9, 4, 5- Remaining envelopes for Xiao Ming: 2, 6, 7, 11Xiao Ming needs to have a total of 26 yuan. Let's check the sum:[2 + 6 + 7 + 11 = 26]This also adds up. So, Xiao Ming grabbed 2, 6, 7, and 11 yuan.**Identifying the Common Envelopes:**In both cases, Xiao Ming grabbed 6 and 11 yuan. The other two envelopes differ based on the mother's choice, but 6 and 11 are common in both scenarios.Therefore, regardless of which pair the mother grabbed, Xiao Ming definitely grabbed 6 and 11 yuan.

answer：First, I need to determine the total sum of all seven numbers by multiplying the average by the number of numbers: 7 × 60 = 420.Next, I'll find the sum of the first three numbers by multiplying their average by the number of numbers: 3 × 45 = 135.Then, I'll calculate the sum of the next two numbers in the same way: 2 × 70 = 140.To find the sum of the last two numbers, I'll subtract the sum of the first five numbers from the total sum: 420 - (135 + 140) = 420 - 275 = 145.Finally, I'll determine the average of the last two numbers by dividing their sum by the number of numbers: 145 ÷ 2 = 72.5.

answer：Okay, so I have this problem where the diameter of a circle is divided into ( n ) equal parts, and on each part, a quarter circle is constructed. I need to figure out what the sum of the lengths of these quarter circles approaches as ( n ) becomes very large. The options are about comparing this sum to the semi-circumference, a fourth of the semi-circumference, the diameter, something in between, or infinite.First, let me visualize this. Imagine a circle with diameter ( D ). If I divide this diameter into ( n ) equal segments, each segment will have a length of ( frac{D}{n} ). On each of these small segments, we're constructing a quarter circle. So each quarter circle has a radius equal to half of ( frac{D}{n} ), which is ( frac{D}{2n} ).Now, the circumference of a full circle is ( 2pi r ), so the circumference of a quarter circle would be ( frac{1}{4} times 2pi r = frac{pi r}{2} ). Substituting ( r = frac{D}{2n} ), the length of each quarter circle is ( frac{pi (frac{D}{2n})}{2} = frac{pi D}{4n} ).Since there are ( n ) such quarter circles, the total length of all the arcs would be ( n times frac{pi D}{4n} = frac{pi D}{4} ).Wait, the semi-circumference of the original circle is ( frac{pi D}{2} ). So, ( frac{pi D}{4} ) is exactly one fourth of the semi-circumference. Hmm, that seems to match option (B). But let me double-check.Alternatively, maybe I made a mistake in calculating the circumference of the quarter circle. Let me go through it again. The radius of each small quarter circle is ( frac{D}{2n} ). The circumference of a full circle with this radius is ( 2pi times frac{D}{2n} = frac{pi D}{n} ). Therefore, a quarter of that circumference is ( frac{pi D}{4n} ). Multiplying by ( n ) gives ( frac{pi D}{4} ), which is indeed one fourth of the semi-circumference ( frac{pi D}{2} ).So, as ( n ) becomes very large, the sum of the lengths of the arcs approaches ( frac{pi D}{4} ), which is one fourth of the semi-circumference. Therefore, the answer should be (B).But wait, let me think about the limit as ( n ) approaches infinity. If ( n ) is very large, the small quarter circles approximate something. Maybe it's forming another shape? But each quarter circle is constructed on a segment of the diameter, so as ( n ) increases, these quarter circles become smaller and more numerous. Their total length approaches a finite value, not infinity, so (E) is out. It's definitely less than the semi-circumference because ( frac{pi D}{4} < frac{pi D}{2} ). So, (A) is not correct. It's not equal to the diameter either because ( frac{pi D}{4} ) is not equal to ( D ) unless ( pi = 4 ), which it's not. So, (C) is out. And it's definitely not greater than the semi-circumference, so (D) is out. Therefore, (B) must be the correct answer.Yeah, I think that makes sense. Each quarter circle contributes a small arc, and when you sum them all up, it's like integrating over the diameter, but in this case, it's a finite sum that converges to one fourth of the semi-circumference.**Final Answer**boxed{B}

answer：Okay, so I have this problem with three equations, and I need to figure out which of the options A to E applies to all of them. Let me take it step by step.First, let's look at the equations:1. (2x^2 - 5 = 20)2. ((3x - 2)^2 = (2x - 3)^2)3. (sqrt{x^2 - 16} = sqrt{2x - 2})And the options are:A. All roots are integers.B. No root greater than 4.C. At least one root less than 0.D. Exactly two distinct roots.E. One negative root and one positive root.I need to check each equation and see which of these options is true for all three.Starting with the first equation: (2x^2 - 5 = 20).Let me solve this. I'll add 5 to both sides:(2x^2 = 25)Then divide both sides by 2:(x^2 = frac{25}{2})Taking square roots:(x = pm sqrt{frac{25}{2}} = pm frac{5sqrt{2}}{2})So the roots are (frac{5sqrt{2}}{2}) and (-frac{5sqrt{2}}{2}). These are irrational numbers, approximately (pm 3.535). So, they are not integers. So option A is false for this equation.Looking at the roots, they are approximately (pm 3.535). So, both roots are less than 4 in absolute value. So, no root is greater than 4. So option B is true for this equation.Also, one root is negative and one is positive. So option E is true here. Also, since one root is negative, option C is true.Now, moving on to the second equation: ((3x - 2)^2 = (2x - 3)^2).Let me expand both sides:Left side: ((3x - 2)^2 = 9x^2 - 12x + 4)Right side: ((2x - 3)^2 = 4x^2 - 12x + 9)Set them equal:(9x^2 - 12x + 4 = 4x^2 - 12x + 9)Subtract (4x^2 - 12x + 9) from both sides:(5x^2 - 5 = 0)Factor out 5:(5(x^2 - 1) = 0)So, (x^2 - 1 = 0), which gives (x = pm 1).So the roots are 1 and -1. These are integers, so option A is true for this equation.Both roots are less than 4, so option B is true.One root is negative, so option C is true.There are exactly two distinct roots, so option D is true.Also, one negative and one positive root, so option E is true.Now, the third equation: (sqrt{x^2 - 16} = sqrt{2x - 2}).First, I need to square both sides to eliminate the square roots:((sqrt{x^2 - 16})^2 = (sqrt{2x - 2})^2)Which simplifies to:(x^2 - 16 = 2x - 2)Bring all terms to one side:(x^2 - 2x - 14 = 0)This is a quadratic equation. Let me try to factor it, but I don't think it factors nicely. So I'll use the quadratic formula:(x = frac{2 pm sqrt{(2)^2 - 4(1)(-14)}}{2(1)} = frac{2 pm sqrt{4 + 56}}{2} = frac{2 pm sqrt{60}}{2})Simplify (sqrt{60}):(sqrt{60} = sqrt{4 times 15} = 2sqrt{15})So,(x = frac{2 pm 2sqrt{15}}{2} = 1 pm sqrt{15})So the roots are (1 + sqrt{15}) and (1 - sqrt{15}).Calculating approximate values:(sqrt{15} approx 3.872)So,(1 + sqrt{15} approx 4.872)(1 - sqrt{15} approx -2.872)So, the roots are approximately 4.872 and -2.872.Now, checking the options:A. All roots are integers? No, they are irrational. So A is false.B. No root greater than 4? Wait, 4.872 is greater than 4, so B is false.C. At least one root less than 0? Yes, -2.872 is less than 0. So C is true.D. Exactly two distinct roots? Yes, two distinct roots. So D is true.E. One negative root and one positive root? Yes, one is positive (4.872) and one is negative (-2.872). So E is true.Now, let's summarize:Equation 1: Roots are (pm frac{5sqrt{2}}{2}) ≈ ±3.535- A: False- B: True- C: True- D: True (two distinct roots)- E: TrueEquation 2: Roots are 1 and -1- A: True- B: True- C: True- D: True- E: TrueEquation 3: Roots are ≈4.872 and ≈-2.872- A: False- B: False- C: True- D: True- E: TrueNow, the question is asking which of the options is true for all three equations.Looking at each option:A: All roots are integers. False for equations 1 and 3.B: No root greater than 4. False for equation 3.C: At least one root less than 0. True for all three equations.D: Exactly two distinct roots. True for all three equations.E: One negative root and one positive root. True for all three equations.Wait, so both C, D, and E are true for all three equations. But looking back at the options, only one can be chosen. Hmm, let me check.Wait, in the problem statement, it says "Each of the equations... has: (A)... (B)... etc." So it's asking which property is shared by all three equations.But in the options, both C, D, and E are true for all three equations. But in the original problem, the options are given as (A) to (E), and the user has to choose one.Wait, perhaps I made a mistake. Let me check each equation again.Equation 1: Roots are ±3.535. So, one negative, one positive. So E is true.Equation 2: Roots are 1 and -1. So, one negative, one positive. E is true.Equation 3: Roots are ≈4.872 and ≈-2.872. So, one negative, one positive. E is true.So E is true for all three.C: At least one root less than 0. All three have at least one negative root, so C is true.D: Exactly two distinct roots. All three equations have exactly two distinct roots, so D is true.But the options are A to E, and the user has to choose one. However, in the original problem, the options are given as (A) to (E), and the user has to choose one. But in the problem statement, it's asking which property is shared by all three equations.Wait, perhaps I misread the problem. Let me check again.The problem says: "Each of the equations... has: (A)... (B)... etc." So it's asking which property is true for each equation, i.e., which property is shared by all three equations.So, if multiple options are true, but the user has to choose one, perhaps the intended answer is E, as it's more specific.But in the original problem, the user has to choose one answer. So, perhaps E is the correct choice because all three equations have one negative and one positive root.Wait, but C is also true for all three. So, perhaps both C and E are correct, but since the options are single choice, maybe E is the better answer.Alternatively, perhaps the problem expects only one correct answer, and E is the intended one.Wait, let me check the original problem again.The problem is:Each of the equations (2x^2 - 5 = 20), ((3x - 2)^2 = (2x - 3)^2), and (sqrt{x^2 - 16} = sqrt{2x - 2}) has:(A) all roots are integers(B) no root greater than 4(C) at least one root less than 0(D) exactly two distinct roots(E) one negative root and one positive rootSo, the correct answer is E because all three equations have one negative and one positive root. While C is also true, E is more specific and directly addresses the nature of the roots in terms of sign, which is a stronger statement.Therefore, the answer is E.

answer：Okay, so I have this problem here about propositions and negations. Let me try to understand it step by step. The proposition p is given as: "There exists an x₀ in real numbers such that x₀ squared plus one is less than zero." Hmm, that sounds a bit strange because I know that any real number squared is always non-negative, right? So x₀ squared is at least zero, and adding one would make it at least one. Therefore, x₀ squared plus one should always be greater than or equal to one, which is definitely greater than zero. So, does that mean that the original proposition p is actually false?But the question isn't asking about the truth value of p; it's asking about the negation of p, which is written as ¬p. The options are A, B, C, and D, each giving a different form of ¬p. Let me recall how negations work in logic, especially for quantifiers. I remember that the negation of an existential quantifier ("there exists") is a universal quantifier ("for all"), and vice versa. So, if p is "there exists x such that P(x)," then ¬p should be "for all x, not P(x)."Applying that to the given proposition p: p is "there exists x₀ in ℝ such that x₀² + 1 < 0." So, the negation ¬p should be "for all x in ℝ, x² + 1 is not less than zero," which translates to "for all x in ℝ, x² + 1 ≥ 0." Let me check the options:A: ¬p: For all x in ℝ, x² + 1 > 0B: ¬p: There exists x in ℝ, x² + 1 > 0C: ¬p: For all x in ℝ, x² + 1 ≥ 0D: ¬p: There exists x in ℝ, x² + 1 ≥ 0So, according to my understanding, ¬p should be "for all x in ℝ, x² + 1 ≥ 0," which is option C. But wait, let me think again. The original statement p is about x₀² + 1 being less than zero. The negation would mean that for every x, x² + 1 is not less than zero, which is the same as saying x² + 1 is greater than or equal to zero. So, yes, that's exactly what option C says.But hold on, option A says "for all x in ℝ, x² + 1 > 0." Is that also true? Well, x² is always non-negative, so x² + 1 is always at least one, which is greater than zero. So, x² + 1 is always greater than zero for all real x. That means option A is also a true statement. But is it the correct negation of p?Wait, the negation of "there exists x such that P(x)" is "for all x, not P(x)." In this case, P(x) is "x² + 1 < 0." So, not P(x) is "x² + 1 ≥ 0." Therefore, the negation should be "for all x, x² + 1 ≥ 0," which is option C. Option A is a stronger statement; it says x² + 1 is strictly greater than zero, which is also true, but it's not the direct negation of p. The direct negation only requires that x² + 1 is not less than zero, which allows for the possibility of it being equal to zero. However, in reality, x² + 1 can never be zero because x² is always non-negative and adding one makes it at least one. So, in this specific case, both options A and C are true, but only option C is the correct negation of p.Let me double-check. If p is "there exists x such that x² + 1 < 0," and we negate it, we get "for all x, x² + 1 is not less than 0," which is "for all x, x² + 1 ≥ 0." That's option C. Option A is a true statement, but it's not the direct negation; it's a stronger statement. So, in terms of logical equivalence, the negation should match the structure of the original statement, changing the quantifier and negating the inner statement. Therefore, option C is the correct negation.Just to be thorough, let's look at options B and D. Option B says "there exists x in ℝ, x² + 1 > 0." Well, this is also true because, as we saw, x² + 1 is always greater than zero. But this is not the negation of p; it's a different statement. The negation of an existential statement is a universal statement, so option B is not the correct negation. Similarly, option D says "there exists x in ℝ, x² + 1 ≥ 0," which is also true, but again, it's not the correct negation because it's still an existential statement, whereas the negation should be universal.So, after considering all options, I'm confident that option C is the correct negation of the original proposition p.