💡First, I need to determine the total number of persons who went to the hotel. Let's denote the number of persons as ( n ).Out of these ( n ) persons, 8 spent Rs 12 each on their meals. Therefore, the total expenditure by these 8 persons is ( 8 times 12 = Rs 96 ).The ninth person spent Rs 8 more than the average expenditure of all the persons. Let's denote the average expenditure per person as ( A ). Therefore, the expenditure by the ninth person is ( A + 8 ).The total money spent by all the persons is given as Rs 117. This gives us the equation:[ 96 + (A + 8) = 117 ]Solving for ( A ):[ A + 8 = 117 - 96 ][ A + 8 = 21 ][ A = 21 - 8 ][ A = 13 ]So, the average expenditure per person is Rs 13.The ninth person spent Rs 8 more than the average, which means he spent:[ 13 + 8 = Rs 21 ]Now, we can calculate the total number of persons by adding the 8 persons who spent Rs 12 each and the ninth person:[ n = 8 + 1 ][ n = 9 ]Therefore, 9 persons went to the hotel.

💡First, I will evaluate each term in the expression (247 + 5 cdot 39 + 7 cdot 143 + 4 cdot 15) modulo 13.1. **First Term: 247** - Since 247 is divisible by 13, its residue modulo 13 is 0. - (247 equiv 0 pmod{13})2. **Second Term: (5 cdot 39)** - 39 is divisible by 13, so (5 cdot 39) also has a residue of 0 modulo 13. - (5 cdot 39 equiv 0 pmod{13})3. **Third Term: (7 cdot 143)** - 143 is divisible by 13, so (7 cdot 143) will also have a residue of 0 modulo 13. - (7 cdot 143 equiv 0 pmod{13})4. **Fourth Term: (4 cdot 15)** - Calculate the residue of (4 cdot 15) modulo 13. - (4 cdot 15 = 60) - (60 equiv 8 pmod{13})Combining all these, the total residue of the expression modulo 13 is:[247 + 5 cdot 39 + 7 cdot 143 + 4 cdot 15 equiv 0 + 0 + 0 + 8 equiv 8 pmod{13}]Therefore, the modulo 13 residue of the given expression is (boxed{8}).

💡Okay, so I have this problem here: Given that ( f(x + y) = f(x) + f(y) ) and ( f(1) = 2 ), I need to figure out which of the options A, B, C, or D the expression ( f(1) + f(2) + ldots + f(n) ) cannot be equal to. First, I remember that the equation ( f(x + y) = f(x) + f(y) ) is known as Cauchy's functional equation. From what I recall, solutions to this equation are linear functions of the form ( f(x) = kx ) when we assume some regularity conditions like continuity, which I think we can safely assume here since we're dealing with integers or real numbers.Given that ( f(1) = 2 ), if the function is linear, then ( f(x) = 2x ). Let me verify that. If ( f(x) = 2x ), then ( f(x + y) = 2(x + y) = 2x + 2y = f(x) + f(y) ), which satisfies the given condition. So, I can confidently say that ( f(n) = 2n ) for integer values of ( n ).Now, the expression we're looking at is the sum ( f(1) + f(2) + ldots + f(n) ). Since each ( f(k) = 2k ), this sum becomes ( 2(1 + 2 + 3 + ldots + n) ). The sum of the first ( n ) natural numbers is ( frac{n(n + 1)}{2} ), so multiplying by 2 gives ( n(n + 1) ). Therefore, the sum ( f(1) + f(2) + ldots + f(n) = n(n + 1) ).Let me check each option one by one to see which one cannot be equal to this sum.**Option A:** ( f(1) + 2f(1) + 3f(1) + ldots + nf(1) )Breaking this down, each term is a multiple of ( f(1) ). Since ( f(1) = 2 ), this becomes ( 2(1 + 2 + 3 + ldots + n) ). As I calculated earlier, this is equal to ( 2 times frac{n(n + 1)}{2} = n(n + 1) ). So, this is equal to our sum. Therefore, option A is possible.**Option B:** ( fleft( frac{n(n + 1)}{2} right) )Since ( f(x) = 2x ), substituting ( x = frac{n(n + 1)}{2} ) gives ( 2 times frac{n(n + 1)}{2} = n(n + 1) ). So, this is also equal to our sum. Therefore, option B is possible.**Option C:** ( n(n + 1) )This is exactly the sum we calculated earlier, so it's equal to our expression. Therefore, option C is possible.**Option D:** ( n(n + 1)f(1) )Since ( f(1) = 2 ), this becomes ( 2n(n + 1) ). Comparing this to our sum ( n(n + 1) ), it's clear that ( 2n(n + 1) ) is twice as large. Therefore, this cannot be equal to our sum. So, option D is not possible.Wait, let me make sure I didn't make a mistake. I know that ( f(1) = 2 ), so ( f(1) ) is 2. So, ( n(n + 1)f(1) = 2n(n + 1) ). But our sum is ( n(n + 1) ). So, unless ( n(n + 1) = 0 ), which isn't the case for positive integers ( n ), these two expressions are not equal. Therefore, option D is indeed not equal to our sum.So, after checking all the options, the expression ( f(1) + f(2) + ldots + f(n) ) cannot be equal to option D.**Final Answer**boxed{D}

💡Okay, so I have this problem here where I need to find the sum of all real numbers ( x ) such that the median of the numbers ( 3, 5, 7, 18, ) and ( x ) is equal to their mean. The options given are A) -8, B) 0, C) 2, D) 5, E) 7. Hmm, let me try to figure this out step by step.First, I remember that the median is the middle number when the numbers are arranged in order, and the mean is the average. So, I need to find ( x ) such that when I arrange these five numbers, the middle one (the median) is equal to the average (the mean) of all five numbers.Let me start by calculating the mean. The mean is given by the sum of all numbers divided by the number of numbers. So, the sum of the numbers without ( x ) is ( 3 + 5 + 7 + 18 = 33 ). Adding ( x ), the total sum becomes ( 33 + x ). Since there are five numbers, the mean is ( frac{33 + x}{5} ).Now, the median. Since there are five numbers, the median will be the third number when they are arranged in order. So, I need to consider where ( x ) can be placed in the order of ( 3, 5, 7, 18 ).Let me think about different cases for ( x ):1. **Case 1: ( x ) is less than or equal to 3.** If ( x ) is the smallest number, the order becomes ( x, 3, 5, 7, 18 ). The median here is 5.2. **Case 2: ( x ) is between 3 and 5.** If ( x ) is between 3 and 5, the order becomes ( 3, x, 5, 7, 18 ). The median is still 5.3. **Case 3: ( x ) is between 5 and 7.** If ( x ) is between 5 and 7, the order becomes ( 3, 5, x, 7, 18 ). Now, the median is ( x ).4. **Case 4: ( x ) is between 7 and 18.** If ( x ) is between 7 and 18, the order becomes ( 3, 5, 7, x, 18 ). The median is 7.5. **Case 5: ( x ) is greater than or equal to 18.** If ( x ) is the largest number, the order becomes ( 3, 5, 7, 18, x ). The median is 7.So, depending on where ( x ) falls, the median can be 5, ( x ), or 7. Now, I need to set the median equal to the mean and solve for ( x ) in each case.**Case 1 and 2: Median = 5**If the median is 5, then the mean should also be 5. So, set the mean equal to 5:[frac{33 + x}{5} = 5]Multiply both sides by 5:[33 + x = 25]Subtract 33 from both sides:[x = 25 - 33 = -8]So, ( x = -8 ) is a possible solution. Let me check if this makes sense. If ( x = -8 ), the numbers in order are ( -8, 3, 5, 7, 18 ). The median is indeed 5, and the mean is ( frac{-8 + 3 + 5 + 7 + 18}{5} = frac{25}{5} = 5 ). So, this works.**Case 3: Median = ( x )**If the median is ( x ), then ( x ) must be between 5 and 7. So, set the mean equal to ( x ):[frac{33 + x}{5} = x]Multiply both sides by 5:[33 + x = 5x]Subtract ( x ) from both sides:[33 = 4x]Divide both sides by 4:[x = frac{33}{4} = 8.25]Wait, ( x = 8.25 ) is not between 5 and 7. It's actually between 7 and 18, which falls into Case 4. So, this solution doesn't work because it doesn't satisfy the condition for Case 3. Therefore, there's no solution in this case.**Case 4 and 5: Median = 7**If the median is 7, then the mean should also be 7. So, set the mean equal to 7:[frac{33 + x}{5} = 7]Multiply both sides by 5:[33 + x = 35]Subtract 33 from both sides:[x = 35 - 33 = 2]So, ( x = 2 ) is another possible solution. Let me check if this makes sense. If ( x = 2 ), the numbers in order are ( 2, 3, 5, 7, 18 ). The median is 5, not 7. Wait, that's a problem. So, even though I set the median to 7, when I plug ( x = 2 ) back in, the median becomes 5. That means this solution doesn't work because it doesn't satisfy the condition for Case 4 or 5.Hmm, so ( x = 2 ) doesn't actually make the median 7. Therefore, this is not a valid solution. So, only ( x = -8 ) is valid.Wait, but the problem asks for the sum of all real numbers ( x ) that satisfy the condition. So, if only ( x = -8 ) works, then the sum is just -8.But let me double-check to make sure I didn't miss anything. Did I consider all possible cases?- When ( x ) is less than or equal to 3: median 5, got ( x = -8 ).- When ( x ) is between 3 and 5: median 5, same as above.- When ( x ) is between 5 and 7: median ( x ), which led to ( x = 8.25 ), which doesn't fit.- When ( x ) is between 7 and 18: median 7, which led to ( x = 2 ), which doesn't fit.- When ( x ) is greater than or equal to 18: median 7, same as above.So, indeed, only ( x = -8 ) works. Therefore, the sum of all such ( x ) is just -8.But wait, the options include -8 as option A. So, is the answer A) -8?Alternatively, maybe I missed another case where ( x ) could be equal to 5 or 7, but let me check.If ( x = 5 ), the numbers are ( 3, 5, 5, 7, 18 ). The median is 5, and the mean is ( frac{3 + 5 + 5 + 7 + 18}{5} = frac{38}{5} = 7.6 ). So, median is 5, mean is 7.6, not equal.If ( x = 7 ), the numbers are ( 3, 5, 7, 7, 18 ). The median is 7, and the mean is ( frac{3 + 5 + 7 + 7 + 18}{5} = frac{30}{5} = 6 ). So, median is 7, mean is 6, not equal.So, no, ( x = 5 ) or ( x = 7 ) don't work either.Therefore, the only valid solution is ( x = -8 ), so the sum is -8.But wait, the question says "the sum of all real numbers ( x )". So, if there are multiple solutions, we need to add them up. But in this case, only one solution exists, which is -8. So, the sum is -8.Alternatively, maybe I made a mistake in considering the cases. Let me try another approach.Let me consider that the median is equal to the mean. So, the median is either 5, 7, or ( x ). Let me set each of these equal to the mean and solve for ( x ), then check if the solution is valid.1. **Median = 5:** Mean = 5 So, ( frac{33 + x}{5} = 5 ) ( 33 + x = 25 ) ( x = -8 ) Check: Numbers are ( -8, 3, 5, 7, 18 ). Median is 5, mean is 5. Valid.2. **Median = 7:** Mean = 7 So, ( frac{33 + x}{5} = 7 ) ( 33 + x = 35 ) ( x = 2 ) Check: Numbers are ( 2, 3, 5, 7, 18 ). Median is 5, not 7. Invalid.3. **Median = ( x ):** So, ( x ) is the median, meaning it's the third number when sorted. So, ( x ) must be between 5 and 7. Mean = ( x ) So, ( frac{33 + x}{5} = x ) ( 33 + x = 5x ) ( 33 = 4x ) ( x = 8.25 ) But 8.25 is not between 5 and 7, so invalid.So, again, only ( x = -8 ) is valid. Therefore, the sum is -8.Wait, but the options include 0, 2, 5, 7 as well. Maybe I need to consider if there are multiple solutions where the median is equal to the mean, but in different positions.Wait, another thought: when ( x ) is exactly 5 or 7, does that affect the median? Let me check.If ( x = 5 ), the numbers are ( 3, 5, 5, 7, 18 ). Median is 5, mean is ( frac{3 + 5 + 5 + 7 + 18}{5} = frac{38}{5} = 7.6 ). Not equal.If ( x = 7 ), the numbers are ( 3, 5, 7, 7, 18 ). Median is 7, mean is ( frac{3 + 5 + 7 + 7 + 18}{5} = frac{30}{5} = 6 ). Not equal.So, no, those don't work.Wait, maybe I need to consider that when ( x ) is equal to the median, but in a different position. For example, if ( x ) is equal to 5, but that doesn't change the median.Alternatively, maybe I need to consider that when ( x ) is equal to the mean, but that's a different approach.Wait, no, the problem states that the median is equal to the mean. So, the median must be equal to the mean, which is a value, not necessarily one of the numbers.Wait, but in our earlier cases, we considered the median being 5, 7, or ( x ). But maybe ( x ) could be such that it's the median, but also equal to the mean. So, in that case, ( x ) would be both the median and the mean.But in our earlier calculation, when we set ( x ) as the median, we got ( x = 8.25 ), which doesn't fit because it's not between 5 and 7. So, that solution is invalid.Therefore, only ( x = -8 ) is valid.Wait, but let me think again. Maybe I missed a case where ( x ) is equal to the mean, but not necessarily the median. But no, the problem states that the median is equal to the mean, so the median must be equal to the mean, which is a specific value.So, in summary, the only valid solution is ( x = -8 ), so the sum is -8.But wait, the options include -8 as option A, so that should be the answer.Wait, but the problem says "the sum of all real numbers ( x )". So, if there are multiple solutions, we need to add them. But in this case, only one solution exists, which is -8. So, the sum is -8.Alternatively, maybe I made a mistake in considering the cases. Let me try another approach.Let me consider all possible positions of ( x ) and see if the median equals the mean.1. **x ≤ 3:** Order: ( x, 3, 5, 7, 18 ) Median: 5 Mean: ( frac{33 + x}{5} ) Set equal: ( frac{33 + x}{5} = 5 ) ( 33 + x = 25 ) ( x = -8 ) Valid, as checked before.2. **3 < x ≤ 5:** Order: ( 3, x, 5, 7, 18 ) Median: 5 Mean: ( frac{33 + x}{5} ) Set equal: ( frac{33 + x}{5} = 5 ) ( x = -8 ) But ( x ) is supposed to be between 3 and 5 here, but ( x = -8 ) doesn't fit. So, no solution in this interval.3. **5 < x ≤ 7:** Order: ( 3, 5, x, 7, 18 ) Median: ( x ) Mean: ( frac{33 + x}{5} ) Set equal: ( frac{33 + x}{5} = x ) ( 33 + x = 5x ) ( 33 = 4x ) ( x = 8.25 ) But ( x ) is supposed to be between 5 and 7 here, but 8.25 doesn't fit. So, no solution in this interval.4. **7 < x ≤ 18:** Order: ( 3, 5, 7, x, 18 ) Median: 7 Mean: ( frac{33 + x}{5} ) Set equal: ( frac{33 + x}{5} = 7 ) ( 33 + x = 35 ) ( x = 2 ) But ( x ) is supposed to be between 7 and 18 here, but 2 doesn't fit. So, no solution in this interval.5. **x > 18:** Order: ( 3, 5, 7, 18, x ) Median: 7 Mean: ( frac{33 + x}{5} ) Set equal: ( frac{33 + x}{5} = 7 ) ( x = 2 ) But ( x ) is supposed to be greater than 18 here, but 2 doesn't fit. So, no solution in this interval.So, after checking all intervals, only ( x = -8 ) is a valid solution. Therefore, the sum of all such ( x ) is -8.Wait, but the problem says "the sum of all real numbers ( x )". So, if there are multiple solutions, we need to add them. But in this case, only one solution exists, which is -8. So, the sum is -8.Therefore, the answer is A) -8.

💡Alright, so I've got this problem here about coordinate systems and parametric equations. It's divided into two parts, (Ⅰ) and (Ⅱ). Let me try to work through each part step by step. I'm a bit nervous because coordinate geometry can sometimes trip me up, but I'll take it slow.Starting with part (Ⅰ): It says that in the Cartesian coordinate system xOy, there's a line l with parametric equations:x = 1 - (√2 / 2) * t y = 4 - (√2 / 2) * twhere t is the parameter. Then, there's a polar coordinate system set up with the origin as the pole and the positive x-axis as the polar axis. The unit of length is the same as in the Cartesian system. The equation of circle C in polar coordinates is given as ρ = -4 cos θ.I need to find the Cartesian coordinate equation of circle C. Hmm, okay. I remember that converting polar equations to Cartesian involves using the relationships between polar and Cartesian coordinates. Specifically, ρ is the radius, θ is the angle, and the conversions are:x = ρ cos θ y = ρ sin θ ρ² = x² + y²So, given ρ = -4 cos θ, I can try to manipulate this equation to get it into Cartesian form. Let me write down the equation:ρ = -4 cos θI can multiply both sides by ρ to make it easier to convert:ρ² = -4 ρ cos θNow, substituting ρ² with x² + y² and ρ cos θ with x:x² + y² = -4xHmm, that looks like the equation of a circle. Let me rearrange it to standard form. I'll move the -4x to the left side:x² + 4x + y² = 0To complete the square for the x terms, I'll take the coefficient of x, which is 4, divide by 2 to get 2, and square it to get 4. So, I'll add and subtract 4:(x² + 4x + 4) + y² = 4 Which simplifies to: (x + 2)² + y² = 4Okay, so that's the Cartesian equation of circle C. It's a circle centered at (-2, 0) with a radius of 2. That makes sense because the original polar equation was ρ = -4 cos θ, which is a circle with diameter along the x-axis, shifted to the left.So, part (Ⅰ) seems manageable. Now, moving on to part (Ⅱ). It says that circle C intersects line l at points A and B. The coordinates of point M are (-2, 1). I need to find the value of |MA| * |MB|.First, let me visualize this. We have circle C centered at (-2, 0) with radius 2, and line l given by the parametric equations. Point M is at (-2, 1), which is just above the center of the circle. So, point M is inside the circle because the distance from M to the center is 1 unit, which is less than the radius 2.Wait, actually, the distance from M to the center is sqrt[(-2 - (-2))² + (1 - 0)²] = sqrt[0 + 1] = 1. Yes, so M is inside the circle. So, when line l intersects the circle at points A and B, point M is inside the circle, and we need to find the product of the distances from M to A and M to B.I remember something called the power of a point with respect to a circle. The power of a point M with respect to circle C is equal to |MA| * |MB|, where A and B are the intersection points of any line through M with the circle. The formula for the power of a point is:Power = |MO|² - r²where MO is the distance from M to the center O of the circle, and r is the radius of the circle.Wait, hold on. Let me recall correctly. The power of a point M inside the circle is equal to |MA| * |MB|, and it's equal to r² - |MO|². If M is outside, it's |MO|² - r². Since M is inside, it's r² - |MO|².Let me confirm that. Yes, for a point inside the circle, the power is negative, but since we're taking the absolute value, it's r² - |MO|².So, in this case, the center O is at (-2, 0), and point M is at (-2, 1). So, the distance between M and O is:|MO| = sqrt[(-2 - (-2))² + (1 - 0)²] = sqrt[0 + 1] = 1So, |MO| = 1, and the radius r is 2. Therefore, the power of point M is:Power = r² - |MO|² = 2² - 1² = 4 - 1 = 3Therefore, |MA| * |MB| = 3.Wait, that seems straightforward. But let me double-check by actually computing it using the parametric equations.Given the parametric equations of line l:x = 1 - (√2 / 2) * t y = 4 - (√2 / 2) * tI can write this in terms of t. Let me see if I can express t from one equation and substitute into the other.From the x equation:x = 1 - (√2 / 2) * t => t = (1 - x) * (2 / √2) = (1 - x) * √2Similarly, from the y equation:y = 4 - (√2 / 2) * t => t = (4 - y) * (2 / √2) = (4 - y) * √2Since both expressions equal t, I can set them equal:(1 - x) * √2 = (4 - y) * √2 Divide both sides by √2:1 - x = 4 - y => y = x + 3So, the Cartesian equation of line l is y = x + 3. Let me confirm that. If I plug in t = 0, x = 1, y = 4, which is a point on the line. If t = 2, x = 1 - √2, y = 4 - √2, which should lie on y = x + 3. Let's see: x + 3 = (1 - √2) + 3 = 4 - √2, which matches y. So, yes, y = x + 3 is correct.Now, circle C has equation (x + 2)² + y² = 4. Let me substitute y = x + 3 into this equation to find the points of intersection A and B.Substituting:(x + 2)² + (x + 3)² = 4 Expand both squares:(x² + 4x + 4) + (x² + 6x + 9) = 4 Combine like terms:2x² + 10x + 13 = 4 Subtract 4:2x² + 10x + 9 = 0 Divide by 2:x² + 5x + 4.5 = 0Hmm, that seems a bit messy. Maybe I made a mistake in substitution. Let me check.Wait, (x + 2)² is x² + 4x + 4, and (x + 3)² is x² + 6x + 9. Adding them gives 2x² + 10x + 13. Setting equal to 4: 2x² + 10x + 13 = 4, so 2x² + 10x + 9 = 0. That's correct.Let me try to solve this quadratic equation:2x² + 10x + 9 = 0 Using the quadratic formula:x = [-10 ± sqrt(100 - 72)] / 4 = [-10 ± sqrt(28)] / 4 = [-10 ± 2√7] / 4 = [-5 ± √7] / 2So, the x-coordinates of A and B are (-5 + √7)/2 and (-5 - √7)/2. Then, the corresponding y-coordinates are:y = x + 3 So, for x = (-5 + √7)/2, y = (-5 + √7)/2 + 3 = (1 + √7)/2 Similarly, for x = (-5 - √7)/2, y = (-5 - √7)/2 + 3 = (1 - √7)/2Therefore, points A and B are:A: [(-5 + √7)/2, (1 + √7)/2] B: [(-5 - √7)/2, (1 - √7)/2]Now, point M is at (-2, 1). Let me compute the distances |MA| and |MB|.First, compute |MA|:Coordinates of M: (-2, 1) Coordinates of A: [(-5 + √7)/2, (1 + √7)/2]Difference in x: [(-5 + √7)/2 - (-2)] = [(-5 + √7)/2 + 2] = [(-5 + √7) + 4]/2 = (-1 + √7)/2 Difference in y: [(1 + √7)/2 - 1] = [(1 + √7) - 2]/2 = (-1 + √7)/2So, |MA| = sqrt[ ( (-1 + √7)/2 )² + ( (-1 + √7)/2 )² ] = sqrt[ 2 * ( (1 - 2√7 + 7)/4 ) ] = sqrt[ 2 * (8 - 2√7)/4 ] = sqrt[ (8 - 2√7)/2 ] = sqrt[4 - √7]Similarly, compute |MB|:Coordinates of B: [(-5 - √7)/2, (1 - √7)/2]Difference in x: [(-5 - √7)/2 - (-2)] = [(-5 - √7)/2 + 2] = [(-5 - √7) + 4]/2 = (-1 - √7)/2 Difference in y: [(1 - √7)/2 - 1] = [(1 - √7) - 2]/2 = (-1 - √7)/2So, |MB| = sqrt[ ( (-1 - √7)/2 )² + ( (-1 - √7)/2 )² ] = sqrt[ 2 * ( (1 + 2√7 + 7)/4 ) ] = sqrt[ 2 * (8 + 2√7)/4 ] = sqrt[ (8 + 2√7)/2 ] = sqrt[4 + √7]Therefore, |MA| * |MB| = sqrt[4 - √7] * sqrt[4 + √7]Hmm, that's the product of two square roots. Let me compute that:sqrt[(4 - √7)(4 + √7)] = sqrt[16 - (√7)²] = sqrt[16 - 7] = sqrt[9] = 3So, |MA| * |MB| = 3.Wait, that's the same result I got earlier using the power of a point. So, both methods give me 3. That's reassuring.But let me think again. The parametric equations of line l were given, and point M is (-2, 1). I could also approach this problem by parametrizing the line l with point M as the starting point.Let me try that approach to confirm.Given the parametric equations of line l:x = 1 - (√2 / 2) * t y = 4 - (√2 / 2) * tI can see that when t = 0, the point is (1, 4). But point M is (-2, 1). Let me see if M lies on line l.Plugging x = -2 into the x equation:-2 = 1 - (√2 / 2) * t => -3 = - (√2 / 2) * t => t = (-3) / (-√2 / 2) = (3 * 2) / √2 = 6 / √2 = 3√2Similarly, plugging y = 1 into the y equation:1 = 4 - (√2 / 2) * t => -3 = - (√2 / 2) * t => t = 3√2So, point M corresponds to t = 3√2 on line l. Therefore, I can write the parametric equations of line l starting from M by letting s = t - 3√2. But maybe it's simpler to just use the parameter t with M as a point on the line.Alternatively, I can write the parametric equations of line l passing through M with the same direction vector.The direction vector of line l can be found from the parametric equations. The coefficients of t in x and y are both -√2 / 2. So, the direction vector is (-√2 / 2, -√2 / 2). Alternatively, we can write it as (-1, -1) scaled by √2 / 2.So, parametrizing line l with point M as the starting point:x = -2 + (-√2 / 2) * s y = 1 + (-√2 / 2) * swhere s is a parameter.Now, substitute these into the equation of circle C: (x + 2)² + y² = 4Plugging in:[(-2 + (-√2 / 2)s + 2)]² + [1 + (-√2 / 2)s]² = 4 Simplify:[(-√2 / 2)s]² + [1 - (√2 / 2)s]² = 4 Compute each term:First term: ( (-√2 / 2)s )² = (2 / 4)s² = (1/2)s² Second term: [1 - (√2 / 2)s]² = 1 - √2 s + (2 / 4)s² = 1 - √2 s + (1/2)s²Adding them together:(1/2)s² + 1 - √2 s + (1/2)s² = 4 Combine like terms:(1/2 + 1/2)s² - √2 s + 1 = 4 => s² - √2 s + 1 = 4 Subtract 4:s² - √2 s - 3 = 0So, the quadratic equation is s² - √2 s - 3 = 0. Let me solve for s:s = [√2 ± sqrt( (√2)^2 + 12 ) ] / 2 = [√2 ± sqrt(2 + 12)] / 2 = [√2 ± sqrt(14)] / 2So, the two values of s are [√2 + sqrt(14)] / 2 and [√2 - sqrt(14)] / 2.These correspond to the parameters s1 and s2 for points A and B. The distances |MA| and |MB| are the absolute values of these parameters multiplied by the scaling factor of the direction vector.Wait, actually, in parametric equations, the parameter s represents a scaled distance along the line. The direction vector has a magnitude. Let me compute the magnitude of the direction vector.The direction vector is (-√2 / 2, -√2 / 2). Its magnitude is sqrt[ ( (-√2 / 2)^2 + (-√2 / 2)^2 ) ] = sqrt[ (2/4 + 2/4) ] = sqrt[1] = 1.Oh, interesting. So, the parameter s actually represents the actual distance from point M along the line. Therefore, |MA| = |s1| and |MB| = |s2|.But wait, s1 and s2 are [√2 + sqrt(14)] / 2 and [√2 - sqrt(14)] / 2. Let me compute their absolute values.First, compute s1: [√2 + sqrt(14)] / 2. Since both √2 and sqrt(14) are positive, s1 is positive. So, |s1| = s1.s2: [√2 - sqrt(14)] / 2. Since sqrt(14) > √2, this is negative. So, |s2| = | [√2 - sqrt(14)] / 2 | = [sqrt(14) - √2] / 2.Therefore, |MA| * |MB| = s1 * |s2| = [ (√2 + sqrt(14)) / 2 ] * [ (sqrt(14) - √2) / 2 ]Multiply numerator and denominator:Numerator: (√2 + sqrt(14))(sqrt(14) - √2) = (√2)(sqrt(14)) - (√2)^2 + (sqrt(14))^2 - sqrt(14)(√2) Simplify term by term:First term: √2 * sqrt(14) = sqrt(28) = 2 sqrt(7) Second term: - (√2)^2 = -2 Third term: (sqrt(14))^2 = 14 Fourth term: - sqrt(14) * √2 = - sqrt(28) = -2 sqrt(7)So, adding them up:2 sqrt(7) - 2 + 14 - 2 sqrt(7) = (2 sqrt(7) - 2 sqrt(7)) + (-2 + 14) = 0 + 12 = 12Denominator: 2 * 2 = 4So, |MA| * |MB| = 12 / 4 = 3Again, I get 3. So, all three methods—using the power of a point, solving the quadratic directly, and parametrizing from point M—give me the same result. That gives me confidence that the answer is indeed 3.Wait, but in the second method, when I solved for x and y, I had to compute the distances |MA| and |MB|, which involved square roots and then multiplying them. It was a bit more involved, but it still led to the same answer. So, I think 3 is correct.Just to recap:1. Converted the polar equation of the circle to Cartesian and found it's (x + 2)^2 + y^2 = 4.2. Recognized that point M is inside the circle and used the power of a point theorem to find |MA| * |MB| = 3.3. Verified by solving the system of equations and computing the distances, which also gave 3.4. Another verification by parametrizing the line from point M and solving the quadratic, again resulting in 3.So, I think I'm solid here. The key takeaway is remembering the power of a point theorem, which simplifies the problem significantly. It's always good to recall such theorems because they can save a lot of computation time.I also need to remember that when a point is inside a circle, the power of the point is equal to the product of the lengths from the point to the intersection points, and it's calculated as r² - |MO|². If the point were outside, it would be |MO|² - r², but since M is inside, it's the other way around.Another thing I noticed is that the parametric equations of line l can be converted to a Cartesian equation by eliminating the parameter t. That was useful in substituting into the circle's equation. It's a good skill to be able to switch between parametric and Cartesian forms depending on what's needed.Also, when dealing with parametric equations, sometimes it's helpful to adjust the parameter to start from a specific point, like M in this case. By doing that, the parameter s directly relates to the distance from M, which made calculating |MA| and |MB| more straightforward.I think I've covered all the steps and checked them thoroughly. I feel confident that both parts (Ⅰ) and (Ⅱ) are correctly solved.**Final Answer**Ⅰ. The Cartesian coordinate equation of circle ( C ) is (boxed{(x + 2)^2 + y^2 = 4}).Ⅱ. The value of ( |MA| cdot |MB| ) is (boxed{3}).

💡To find the distance between the two mountains on the map, we can use the scale determined by Ram's location.First, calculate the scale by dividing the actual distance Ram is from the base of the mountain by the map distance:Scale = Actual distance / Map distanceScale = 12.205128205128204 km / 28 inchesNext, use this scale to find the map distance between the two mountains by dividing the actual distance between the mountains by the scale:Map distance between mountains = Actual distance between mountains / ScaleMap distance between mountains = 136 km / (12.205128205128204 km / 28 inches)Finally, calculate the map distance:Map distance between mountains = 312 inches