💡First, I need to find three consecutive natural numbers whose least common multiple (LCM) is 660.To begin, I'll factorize 660 into its prime factors:660 = 2 × 2 × 3 × 5 × 11Next, I'll consider sets of three consecutive numbers and calculate their LCM to see if it matches 660.Starting with the numbers 10, 11, and 12:- 10 factors into 2 × 5- 11 is a prime number- 12 factors into 2 × 2 × 3Calculating the LCM of these numbers:LCM = 2 × 2 × 3 × 5 × 11 = 660Since the LCM of 10, 11, and 12 is indeed 660, these are the three consecutive natural numbers I'm looking for.

💡First, I need to expand the given dot product equation ((mathbf{a} + 2mathbf{b}) cdot (mathbf{a} - 3mathbf{b}) = -72). Using the distributive property of the dot product, this becomes:[mathbf{a} cdot mathbf{a} - 3mathbf{a} cdot mathbf{b} + 2mathbf{b} cdot mathbf{a} - 6mathbf{b} cdot mathbf{b} = -72]Simplifying further:[|mathbf{a}|^2 - mathbf{a} cdot mathbf{b} - 6|mathbf{b}|^2 = -72]Next, I substitute the known values. The magnitude of (mathbf{b}) is 4, so (|mathbf{b}|^2 = 16). The dot product (mathbf{a} cdot mathbf{b}) can be expressed using the cosine of the angle between them:[mathbf{a} cdot mathbf{b} = |mathbf{a}| |mathbf{b}| cos 60^circ = |mathbf{a}| times 4 times frac{1}{2} = 2|mathbf{a}|]Substituting these into the equation:[|mathbf{a}|^2 - 2|mathbf{a}| - 6 times 16 = -72][|mathbf{a}|^2 - 2|mathbf{a}| - 96 = -72][|mathbf{a}|^2 - 2|mathbf{a}| - 24 = 0]This is a quadratic equation in terms of (|mathbf{a}|). Solving for (|mathbf{a}|):[|mathbf{a}| = frac{2 pm sqrt{4 + 96}}{2} = frac{2 pm 10}{2}]This gives two solutions: (|mathbf{a}| = 6) or (|mathbf{a}| = -4). Since magnitude cannot be negative, the valid solution is:[|mathbf{a}| = 6]

💡Okay, so I have this problem here where I'm given two positive real numbers, x and y. There's this operation defined as x*y equals (x times y) divided by (x plus y). Then, I'm told that z is equal to x plus 2y, and I need to find the value of the operation x*z. The options are given as A, B, C, D, each with different expressions.Alright, let me break this down step by step. First, I need to understand what the operation * does. It takes two numbers, multiplies them together, and then divides by their sum. So, for any two numbers a and b, a*b is (a*b)/(a+b). That's like a kind of harmonic mean or something, maybe? Not sure, but it's a specific operation defined here.Now, z is given as x + 2y. So, z is just another expression in terms of x and y. I need to compute x*z using the * operation. So, substituting z into the operation, x*z would be (x*z)/(x + z). But z is x + 2y, so I can substitute that in.Let me write that out:x*z = (x * z) / (x + z)But z = x + 2y, so substituting:x*z = (x * (x + 2y)) / (x + (x + 2y))Okay, so let's compute the numerator and the denominator separately.First, the numerator: x multiplied by (x + 2y). That should be x*(x + 2y) = x^2 + 2xy.Now, the denominator: x plus (x + 2y). That's x + x + 2y, which simplifies to 2x + 2y.So, putting it together, x*z is (x^2 + 2xy) divided by (2x + 2y). Hmm, can I factor anything out here?Looking at the denominator, 2x + 2y, I can factor out a 2, so that becomes 2(x + y). Similarly, in the numerator, x^2 + 2xy, I don't think I can factor anything out except maybe an x, but that might not help. Let me see:Numerator: x^2 + 2xy = x(x + 2y). Hmm, but the denominator is 2(x + y). So, unless there's a common factor, I might just have to leave it as is.Wait, let me check if I can factor something else. The numerator is x^2 + 2xy, which is x(x + 2y). The denominator is 2(x + y). So, unless x + 2y and x + y have a common factor, which they don't, I think that's as simplified as it gets.So, x*z = (x^2 + 2xy) / [2(x + y)]. Let me look at the options given:A. (x² + 2xy)/(x + y)B. (x² + xy)/(2(x + y))C. (x² + 2xy)/(2(x + y))D. (x² + 3xy)/(3(x + y))So, my result matches option C exactly. Let me double-check my steps to make sure I didn't make a mistake.1. Defined the operation: x*y = (x*y)/(x + y). Correct.2. z = x + 2y. Correct.3. x*z = (x*z)/(x + z). Correct substitution.4. Substituted z into numerator and denominator: x*(x + 2y) and x + (x + 2y). Correct.5. Calculated numerator: x^2 + 2xy. Correct.6. Calculated denominator: 2x + 2y, factored to 2(x + y). Correct.7. So, x*z = (x^2 + 2xy)/(2(x + y)). Correct.Yep, that seems right. So, the answer should be option C.Just to be thorough, let me see if any of the other options could be equivalent or if I made a mistake in simplifying.Option A is (x² + 2xy)/(x + y). That's the same as my numerator but without the 2 in the denominator. So, unless I missed a factor of 2 somewhere, that's not correct. I don't think I did, because the denominator was 2x + 2y, which is 2(x + y). So, I think A is incorrect.Option B is (x² + xy)/(2(x + y)). The numerator here is x² + xy, which is less than my numerator x² + 2xy. So, unless I made a mistake in multiplying x*(x + 2y), which I don't think I did, B is incorrect.Option D is (x² + 3xy)/(3(x + y)). The numerator here is x² + 3xy, which is more than my numerator. So, that's not correct either.Therefore, I'm confident that option C is the correct answer.**Final Answer**boxed{C}

💡Alright, so I've got this problem about suitcases at the airport. There are 200 suitcases loaded onto a plane, and after the flight, passengers are waiting for their suitcases on the conveyor belt. The loader starts placing the suitcases on the conveyor belt one by one every two seconds in a random order. Among the passengers, there's a group of businesspeople waiting for their 10 suitcases.Part (a) asks for the probability that the businesspeople will have to wait exactly two minutes for their last suitcase. Okay, so two minutes is 120 seconds, and since the loader places a suitcase every two seconds, that means 120 divided by 2 is 60 suitcases. So, in two minutes, 60 suitcases will pass by on the conveyor belt.Now, the businesspeople are waiting for their 10 suitcases. For them to have to wait exactly two minutes for their last suitcase, that means their last suitcase must be the 60th one on the conveyor belt. So, the 10th suitcase of the businesspeople needs to be exactly at position 60.But how do we calculate the probability of that happening? Well, the suitcases are being loaded in a random order, so each suitcase has an equal chance of being in any position. So, we need to figure out how many ways the 10 businesspeople's suitcases can be arranged such that the last one is at position 60.To do that, we can think of it as choosing 9 positions out of the first 59 for the other 9 suitcases of the businesspeople, and then the 10th one is fixed at position 60. The number of ways to choose 9 positions out of 59 is given by the combination formula, which is C(59,9).On the other hand, the total number of ways to choose any 10 positions out of 200 for the businesspeople's suitcases is C(200,10). So, the probability that the last suitcase is exactly at position 60 is the ratio of these two combinations: C(59,9)/C(200,10).Wait, but let me make sure I'm not missing anything here. Is there any other factor I need to consider? Hmm, since the suitcases are loaded in a random order, each arrangement is equally likely, so I think this approach is correct.Moving on to part (b), which asks for the expected value of the time the businesspeople will have to wait for their last suitcase. Expected value means we need to find the average waiting time over all possible arrangements of the suitcases.This seems a bit more complex. I remember that for problems involving the expected maximum, there's a concept in probability called the "order statistics." Specifically, the expected value of the maximum of a set of uniformly distributed random variables.In this case, the positions of the 10 suitcases can be thought of as 10 randomly chosen positions out of 200. The expected value of the maximum position would give us the expected position of the last suitcase. Once we have that, we can convert it into time by multiplying by 2 seconds per suitcase.But how do we calculate the expected maximum position? I recall that for discrete uniform distributions, the expected maximum of k samples from n items is given by (k*(n+1))/(k+1). Let me verify that.Wait, actually, I think it's (k*(n+1))/(k+1). So, plugging in the numbers, k is 10 and n is 200. So, the expected maximum position would be (10*(200+1))/(10+1) = (10*201)/11 = 2010/11 ≈ 182.727 positions.But since each position corresponds to 2 seconds, the expected waiting time would be 182.727 * 2 ≈ 365.454 seconds.Hmm, that seems reasonable. But let me think if there's another way to approach this. Maybe using linearity of expectation or something else.Alternatively, we can think of the problem as inserting an additional "dummy" suitcase, making it 201 suitcases in total. Then, the expected position of the last business suitcase would be the same as the expected position of the 10th order statistic in 201 items, which is (10*(201+1))/(10+1) = (10*202)/11 = 2020/11 ≈ 183.636 positions.Wait, that's slightly different from before. Did I make a mistake? Let me check.Actually, the formula I used earlier was (k*(n+1))/(k+1), which for k=10 and n=200 gives (10*201)/11 ≈ 182.727. But when adding a dummy suitcase, making it 201, the formula becomes (k*(n+1))/(k+1) = (10*202)/11 ≈ 183.636.Hmm, so which one is correct? I think the first approach without adding a dummy suitcase is more accurate because we're dealing with exactly 200 suitcases. Adding a dummy suitcase might complicate things unnecessarily.Therefore, I'll stick with the first calculation: expected position is approximately 182.727, so the expected waiting time is approximately 365.454 seconds.But to express this exactly, it's 2010/11 seconds, which is approximately 365.4545 seconds.Wait, actually, 2010 divided by 11 is 182.727, and multiplying by 2 gives 365.4545. So, the exact value is 4020/11 seconds, which simplifies to approximately 365.4545 seconds.Okay, that makes sense. So, the expected waiting time is 4020/11 seconds.Let me just recap to make sure I didn't make any mistakes. For part (a), the probability is C(59,9)/C(200,10). For part (b), the expected time is 4020/11 seconds.I think that's it. I don't see any errors in my reasoning, so I'll go with these answers.

💡Okay, so I have this problem here about a hyperbola and a parabola. Let me try to figure it out step by step. First, the problem says that one focus of a hyperbola coincides with the focus of the parabola given by the equation ( y^2 = 4sqrt{7}x ). I remember that for a parabola in the form ( y^2 = 4px ), the focus is at ( (p, 0) ). So, in this case, comparing it to the standard form, ( 4p = 4sqrt{7} ), which means ( p = sqrt{7} ). Therefore, the focus of the parabola is at ( (sqrt{7}, 0) ). Since one focus of the hyperbola coincides with this, the hyperbola must also have a focus at ( (sqrt{7}, 0) ). I know that for a hyperbola, the distance from the center to each focus is ( c ), and the relationship between ( a ), ( b ), and ( c ) is ( c^2 = a^2 + b^2 ). So, if the hyperbola is centered at the origin (which I think it is because the focus is on the x-axis), then its equation would be ( frac{x^2}{a^2} - frac{y^2}{b^2} = 1 ).Next, the hyperbola intersects the line ( y = x - 1 ) at points M and N. The x-coordinate of the midpoint of MN is given as ( -frac{2}{3} ). I need to use this information to find ( a^2 ) and ( b^2 ).So, let me substitute ( y = x - 1 ) into the hyperbola equation:( frac{x^2}{a^2} - frac{(x - 1)^2}{b^2} = 1 ).Expanding this, I get:( frac{x^2}{a^2} - frac{x^2 - 2x + 1}{b^2} = 1 ).Let me combine the terms:( left( frac{1}{a^2} - frac{1}{b^2} right)x^2 + frac{2x}{b^2} - frac{1}{b^2} - 1 = 0 ).Multiplying through by ( a^2b^2 ) to eliminate denominators:( (b^2 - a^2)x^2 + 2a^2x - a^2 - a^2b^2 = 0 ).So, this is a quadratic equation in terms of ( x ). Let me denote this as:( (b^2 - a^2)x^2 + 2a^2x - (a^2 + a^2b^2) = 0 ).Now, for a quadratic equation ( Ax^2 + Bx + C = 0 ), the sum of the roots is ( -B/A ). In this case, the roots correspond to the x-coordinates of points M and N. The midpoint's x-coordinate is the average of the roots, which is ( (x_1 + x_2)/2 ).Given that the x-coordinate of the midpoint is ( -2/3 ), we have:( frac{x_1 + x_2}{2} = -frac{2}{3} ) ( x_1 + x_2 = -frac{4}{3} ).From the quadratic equation, the sum of the roots ( x_1 + x_2 = -B/A ). Here, ( A = b^2 - a^2 ) and ( B = 2a^2 ). So,( x_1 + x_2 = -frac{2a^2}{b^2 - a^2} ).Setting this equal to ( -4/3 ):( -frac{2a^2}{b^2 - a^2} = -frac{4}{3} ).Simplify:( frac{2a^2}{b^2 - a^2} = frac{4}{3} ).Multiply both sides by ( b^2 - a^2 ):( 2a^2 = frac{4}{3}(b^2 - a^2) ).Multiply both sides by 3 to eliminate the fraction:( 6a^2 = 4(b^2 - a^2) ).Expand the right side:( 6a^2 = 4b^2 - 4a^2 ).Bring all terms to one side:( 6a^2 + 4a^2 - 4b^2 = 0 ) ( 10a^2 - 4b^2 = 0 ).Simplify by dividing both sides by 2:( 5a^2 - 2b^2 = 0 ).So, equation (1): ( 5a^2 = 2b^2 ).Now, we also know that the hyperbola has a focus at ( (sqrt{7}, 0) ), so ( c = sqrt{7} ). For hyperbolas, ( c^2 = a^2 + b^2 ). Thus,( (sqrt{7})^2 = a^2 + b^2 ) ( 7 = a^2 + b^2 ).So, equation (2): ( a^2 + b^2 = 7 ).Now, we have two equations:1. ( 5a^2 = 2b^2 )2. ( a^2 + b^2 = 7 )Let me solve equation (1) for ( b^2 ):( b^2 = frac{5}{2}a^2 ).Substitute this into equation (2):( a^2 + frac{5}{2}a^2 = 7 ).Combine like terms:( left(1 + frac{5}{2}right)a^2 = 7 ) ( frac{7}{2}a^2 = 7 ).Multiply both sides by ( 2/7 ):( a^2 = 2 ).Now, substitute ( a^2 = 2 ) back into equation (1):( 5(2) = 2b^2 ) ( 10 = 2b^2 ) ( b^2 = 5 ).So, ( a^2 = 2 ) and ( b^2 = 5 ). Therefore, the equation of the hyperbola is:( frac{x^2}{2} - frac{y^2}{5} = 1 ).Looking at the options, that's option C.Wait, let me double-check my steps to make sure I didn't make a mistake. 1. Found the focus of the parabola correctly: ( (sqrt{7}, 0) ).2. Set up the hyperbola equation correctly with that focus.3. Substituted ( y = x - 1 ) into the hyperbola equation and expanded correctly.4. Formulated the quadratic equation in ( x ) correctly.5. Applied Vieta's theorem correctly to find the sum of roots.6. Set up the equation ( 5a^2 = 2b^2 ) correctly.7. Solved the system of equations with ( a^2 + b^2 = 7 ) correctly, leading to ( a^2 = 2 ) and ( b^2 = 5 ).8. Plugged back into the hyperbola equation correctly.Everything seems to check out. So, I think the answer is indeed option C.**Final Answer**boxed{C}

💡To determine the number of different convex quadrilaterals that can be formed from twelve distinct points on the circumference of a circle, we can use the combination formula. Since any four points chosen from the twelve will form a convex quadrilateral, the number of such quadrilaterals is equal to the number of ways to choose 4 points out of 12.The combination formula is given by:[{n choose k} = frac{n!}{k!(n - k)!}]Applying this formula with ( n = 12 ) and ( k = 4 ):[{12 choose 4} = frac{12!}{4!(12 - 4)!} = frac{12 times 11 times 10 times 9}{4 times 3 times 2 times 1} = frac{11880}{24} = 495]Therefore, the number of different convex quadrilaterals that can be formed is 495.