🧠:<think>Alright, so I've got this problem about a high-security lock system that involves both number theory and geometry. It's broken down into two parts: Prime Encoding and Geometric Configuration. Let me tackle each part step by step.Starting with the first part, Prime Encoding. The lock uses a code that's the product of the first ( n ) prime numbers, denoted as ( P_n = p_1 times p_2 times p_3 times cdots times p_n ). I need to find the smallest ( n ) such that ( P_n > 10^{10} ). Hmm, okay, so I need to calculate the product of primes until it exceeds 10 billion.First, let me recall the sequence of prime numbers. The primes start at 2, then 3, 5, 7, 11, 13, 17, 19, 23, 29, and so on. I think I can list them out and keep multiplying until the product surpasses ( 10^{10} ).Let me write down the primes and compute the product step by step:1. ( p_1 = 2 ) - ( P_1 = 2 )2. ( p_2 = 3 ) - ( P_2 = 2 times 3 = 6 )3. ( p_3 = 5 ) - ( P_3 = 6 times 5 = 30 )4. ( p_4 = 7 ) - ( P_4 = 30 times 7 = 210 )5. ( p_5 = 11 ) - ( P_5 = 210 times 11 = 2310 )6. ( p_6 = 13 ) - ( P_6 = 2310 times 13 = 30030 )7. ( p_7 = 17 ) - ( P_7 = 30030 times 17 = 510510 )8. ( p_8 = 19 ) - ( P_8 = 510510 times 19 = 9699690 )9. ( p_9 = 23 ) - ( P_9 = 9699690 times 23 ) - Let me compute that: 9699690 * 20 = 193,993,800 and 9699690 * 3 = 29,099,070. Adding them together: 193,993,800 + 29,099,070 = 223,092,870 - So, ( P_9 = 223,092,870 )10. ( p_{10} = 29 ) - ( P_{10} = 223,092,870 times 29 ) - Let me compute this: 223,092,870 * 30 = 6,692,786,100. Subtract 223,092,870 to get 6,692,786,100 - 223,092,870 = 6,469,693,230 - So, ( P_{10} = 6,469,693,230 )11. ( p_{11} = 31 ) - ( P_{11} = 6,469,693,230 times 31 ) - Let me compute this: 6,469,693,230 * 30 = 194,090,796,900 and 6,469,693,230 * 1 = 6,469,693,230. Adding them: 194,090,796,900 + 6,469,693,230 = 200,560,490,130 - So, ( P_{11} = 200,560,490,130 )Wait, let me check if I did that correctly. 6,469,693,230 multiplied by 31 is indeed 200,560,490,130. That's way above ( 10^{10} ), which is 10,000,000,000. So, ( P_{11} ) is 200 billion, which is much larger than 10 billion.But wait, let me double-check the previous step. ( P_{10} ) was 6,469,693,230, which is about 6.47 billion, still less than 10 billion. So, ( P_{10} ) is 6.47 billion, which is less than 10 billion, and ( P_{11} ) is 200.56 billion, which is way more.Therefore, the smallest ( n ) such that ( P_n > 10^{10} ) is 11. So, n=11.Wait, hold on. Let me verify the calculations because sometimes when multiplying large numbers, it's easy to make a mistake.Let me recalculate ( P_9 ):( P_8 = 9,699,690 )Multiply by 23: 9,699,690 * 20 = 193,993,800; 9,699,690 * 3 = 29,099,070. Adding them: 193,993,800 + 29,099,070 = 223,092,870. That seems correct.( P_9 = 223,092,870 )Multiply by 29: 223,092,870 * 29.Let me compute 223,092,870 * 30 = 6,692,786,100. Subtract 223,092,870: 6,692,786,100 - 223,092,870 = 6,469,693,230. Correct.So, ( P_{10} = 6,469,693,230 ), which is approximately 6.47 billion.Multiply by 31: 6,469,693,230 * 31.Compute 6,469,693,230 * 30 = 194,090,796,900Add 6,469,693,230: 194,090,796,900 + 6,469,693,230 = 200,560,490,130.Yes, that's correct. So, ( P_{11} ) is indeed 200.56 billion, which is more than 10 billion. So, n=11 is the smallest n where ( P_n > 10^{10} ).Wait, but just to be thorough, let me check if ( P_{10} ) is indeed less than 10^10.10^10 is 10,000,000,000.( P_{10} = 6,469,693,230 ) is approximately 6.47 billion, which is less than 10 billion. So, yes, n=11 is the smallest n where the product exceeds 10^10.Okay, so part 1 is solved. The smallest n is 11.Moving on to part 2, Geometric Configuration. The lock has n gears arranged in a circular pattern, and the angle θ between each consecutive pair of gears is given by θ = 360° / n. Given that n is 11 from part 1, I need to calculate θ in degrees and radians.First, in degrees: θ = 360° / 11.Let me compute that. 360 divided by 11.11 goes into 36 three times (33), remainder 3. Bring down the 0: 30. 11 goes into 30 two times (22), remainder 8. Bring down the next 0: 80. 11 goes into 80 seven times (77), remainder 3. Bring down the next 0: 30 again. So, it's repeating.So, 360 / 11 = 32.727272..., which is 32.727272... degrees.To express this as a fraction, it's 32 and 8/11 degrees because 0.727272... is 8/11.Now, converting degrees to radians. The formula is radians = degrees * (π / 180).So, θ in radians is (360 / 11) * (π / 180).Simplify that: 360 / 11 divided by 180 is (360 / 180) / 11 = 2 / 11.So, θ = (2π) / 11 radians.Alternatively, as a decimal, 2π is approximately 6.283185307, so 6.283185307 / 11 ≈ 0.5712 radians.But since the problem asks to express it in both degrees and radians, I think it's better to leave the radians in terms of π unless a decimal approximation is specifically requested.So, summarizing:θ in degrees is 360/11°, which is approximately 32.727°, and in radians, it's 2π/11, approximately 0.5712 radians.Wait, let me double-check the conversion from degrees to radians.Yes, θ = 360° / 11 = (360/11)°. To convert to radians, multiply by π/180:(360/11) * (π/180) = (360π)/(11*180) = (2π)/11. Correct.So, θ = 2π/11 radians.Therefore, the angle between each consecutive pair of gears is 360/11 degrees or 2π/11 radians.Just to make sure I didn't make any mistakes in the calculations:- For part 1, I listed primes up to the 11th prime and calculated their product step by step, confirming each multiplication. It seems correct.- For part 2, the angle calculation is straightforward division and conversion, which I double-checked.I think I'm confident with these answers.Final Answer1. The smallest ( n ) is boxed{11}.2. The measure of ( theta ) is ( frac{360}{11}^circ ) or ( frac{2pi}{11} ) radians. So, in degrees, it's boxed{frac{360}{11}^circ} and in radians, it's boxed{frac{2pi}{11}}.</think>

🧠:<think>Okay, so I have this problem about a fighter pilot with lung cancer undergoing treatment, and I need to solve two parts. Let me start with the first part about the tumor volume.The tumor is modeled as an irregular ellipsoid with semi-major axes a, b, and c. The volume formula is given as V = (4/3)πabc. During treatment, each semi-major axis decreases exponentially over time t (in months). The equations are:a(t) = a0 * e^(-λ1 * t)b(t) = b0 * e^(-λ2 * t)c(t) = c0 * e^(-λ3 * t)I need to find the time t when the volume is reduced to 10% of its original volume.First, let me recall that the original volume V0 is (4/3)πa0b0c0. The volume at time t is V(t) = (4/3)πa(t)b(t)c(t). So, I can write V(t) as:V(t) = (4/3)π * a0 e^(-λ1 t) * b0 e^(-λ2 t) * c0 e^(-λ3 t)Simplify that:V(t) = (4/3)πa0b0c0 * e^(- (λ1 + λ2 + λ3) t)Because when you multiply exponentials with the same base, you add the exponents. So, e^(-λ1 t) * e^(-λ2 t) * e^(-λ3 t) = e^(- (λ1 + λ2 + λ3) t).So, V(t) = V0 * e^(- (λ1 + λ2 + λ3) t)We need V(t) = 0.1 V0. So:0.1 V0 = V0 * e^(- (λ1 + λ2 + λ3) t)Divide both sides by V0:0.1 = e^(- (λ1 + λ2 + λ3) t)Take natural logarithm on both sides:ln(0.1) = - (λ1 + λ2 + λ3) tSolve for t:t = - ln(0.1) / (λ1 + λ2 + λ3)Compute ln(0.1). Since ln(1/10) = -ln(10), so ln(0.1) = -ln(10). Therefore:t = ln(10) / (λ1 + λ2 + λ3)So, the time t when the volume is reduced to 10% is ln(10) divided by the sum of the decay constants λ1, λ2, λ3.Wait, let me double-check. So, V(t) = V0 * e^(-kt), where k = λ1 + λ2 + λ3. So, setting V(t) = 0.1 V0:0.1 = e^(-kt)Take ln: ln(0.1) = -ktSo, t = -ln(0.1)/k = ln(10)/k, since ln(0.1) = -ln(10). Yes, that seems correct.So, the answer is t = ln(10)/(λ1 + λ2 + λ3). I think that's it for part 1.Moving on to part 2. The pilot is running with a velocity that's a sinusoidal function. The distance covered is the integral of velocity over time. The velocity is given as v(τ) = v0 sin(ωτ + φ). Given v0 = 10 km/h, ω = π/2 rad/h, and φ = 0. So, v(τ) = 10 sin(πτ/2 + 0) = 10 sin(πτ/2).We need to find the total distance he runs after 8 hours, which is d(8). Also, find the maximum speed he achieves during this period.First, let's compute d(t) = ∫₀ᵗ v(τ) dτ. So, d(t) = ∫₀ᵗ 10 sin(πτ/2) dτ.Let me compute this integral.The integral of sin(ax) dx is (-1/a) cos(ax) + C. So, applying that here:d(t) = 10 * ∫₀ᵗ sin(πτ/2) dτ = 10 * [ (-2/π) cos(πτ/2) ] from 0 to tCompute the bounds:At t: (-2/π) cos(πt/2)At 0: (-2/π) cos(0) = (-2/π)(1) = -2/πSo, subtracting:d(t) = 10 * [ (-2/π cos(πt/2)) - (-2/π) ] = 10 * [ (-2/π cos(πt/2) + 2/π) ] = 10 * (2/π)(1 - cos(πt/2)) = (20/π)(1 - cos(πt/2))So, d(t) = (20/π)(1 - cos(πt/2))Now, evaluate this at t = 8:d(8) = (20/π)(1 - cos(π*8/2)) = (20/π)(1 - cos(4π))Compute cos(4π). Since cos(2π) = 1, so cos(4π) = 1 as well. So:d(8) = (20/π)(1 - 1) = 0Wait, that can't be right. If he's running for 8 hours, the distance can't be zero. Maybe I made a mistake.Wait, let's think about the integral again. The velocity is 10 sin(πτ/2). The integral is the distance, but since sine is positive and negative, the integral might result in some cancellation. But wait, over 8 hours, let's see.Wait, the period of the sine function is T = 2π / ω = 2π / (π/2) = 4 hours. So, the period is 4 hours. So, over 8 hours, it's two full periods.But the integral over each period is zero because the area above the x-axis cancels the area below. So, integrating over two periods, the total distance would be zero? That doesn't make sense because distance should be the total area covered, regardless of direction.Wait, hold on. Wait, no. Wait, in physics, distance is the integral of speed, which is the absolute value of velocity. But in the problem, it's given as the integral of velocity, which is displacement, not distance. So, maybe the question is actually about displacement, not total distance.Wait, the problem says: "the distance d(t) he covers in kilometers in time t (in hours) is given by the integral: d(t) = ∫₀ᵗ v(τ) dτ"But in physics, displacement is the integral of velocity, while distance is the integral of speed (absolute value of velocity). So, perhaps the problem is using "distance" incorrectly, and it's actually displacement.But if that's the case, then over two full periods, the displacement is zero because it's symmetric. So, that would make sense why d(8) is zero.But the question also asks for the maximum speed he achieves during this period. So, maximum speed is the maximum of |v(τ)|, which is 10 km/h, since v(τ) = 10 sin(πτ/2). The maximum of sin is 1, so maximum speed is 10 km/h.Wait, but the problem says "maximum speed", which is the maximum of |v(τ)|, which is 10 km/h. So, that's straightforward.But let me double-check the integral. Maybe I did something wrong.Wait, the integral of velocity is displacement, not distance. So, if the question is about displacement, then yes, over 8 hours, which is two full periods, the displacement is zero. But if it's about total distance, then we need to integrate the absolute value of velocity.But the problem says "distance d(t) he covers... is given by the integral of v(τ) dτ". So, perhaps in this context, they are using "distance" to mean displacement. So, the answer is zero.But that seems odd because usually, distance covered would be total distance, not displacement. Maybe the problem is misworded. Alternatively, perhaps the velocity is always positive, but given that it's a sinusoidal function, it goes positive and negative.Wait, let's think again. The velocity function is v(τ) = 10 sin(πτ/2). Let's plot this function.At τ = 0: sin(0) = 0At τ = 1: sin(π/2) = 1At τ = 2: sin(π) = 0At τ = 3: sin(3π/2) = -1At τ = 4: sin(2π) = 0And so on.So, the velocity is positive from τ = 0 to τ = 2, then negative from τ = 2 to τ = 4, and so on.So, integrating from 0 to 8, which is two full periods, the positive and negative areas cancel, giving a displacement of zero.But if we were to compute the total distance, we would need to integrate the absolute value of velocity, which would be twice the integral over half a period.But since the problem defines d(t) as the integral of velocity, which is displacement, then d(8) is indeed zero.But the question says "the distance he covers", which is ambiguous. In common language, distance usually refers to total distance, but in physics, displacement is different. So, perhaps the problem is using "distance" as displacement.Alternatively, maybe the velocity is always positive? Let me check.Wait, v(τ) = 10 sin(πτ/2). So, sin(πτ/2) is positive when πτ/2 is between 0 and π, i.e., τ between 0 and 2, and negative between 2 and 4, etc. So, the velocity is positive for the first two hours, negative for the next two, etc.So, over 8 hours, the displacement is zero, but the total distance is the sum of the absolute areas.But since the problem defines d(t) as the integral of velocity, which is displacement, then d(8) is zero.So, the total displacement after 8 hours is zero, and the maximum speed is 10 km/h.Alternatively, if the problem had meant total distance, it would have been different. Let me compute that just in case.Total distance would be ∫₀⁸ |v(τ)| dτ = ∫₀⁸ |10 sin(πτ/2)| dτSince the function is symmetric, over each period, the integral of |sin| is 4/π times the amplitude. Wait, the integral of |sin(ax)| over one period is 2/a.Wait, let me recall. The integral of |sin(ax)| over 0 to 2π/a is 4/a.Wait, let me compute ∫₀⁴ |10 sin(πτ/2)| dτ. Since the period is 4, over 0 to 4, the integral is 20*(2/π) = 40/π? Wait, let's compute it.Compute ∫₀⁴ |10 sin(πτ/2)| dτFrom 0 to 2, sin is positive, so integral is 10 ∫₀² sin(πτ/2) dτ= 10 * [ (-2/π) cos(πτ/2) ] from 0 to 2= 10 * [ (-2/π)(cos(π) - cos(0)) ]= 10 * [ (-2/π)(-1 - 1) ]= 10 * [ (-2/π)(-2) ] = 10 * (4/π) = 40/πFrom 2 to 4, sin is negative, so |sin| is positive, so integral is same as above.So, total over 0 to 4 is 40/π + 40/π = 80/πSimilarly, over 4 to 8, it's another 80/πSo, total distance over 8 hours is 160/π ≈ 50.93 kmBut the problem says d(t) is the integral of v(τ) dτ, which is displacement, so d(8) is zero.But maybe the problem is using "distance" incorrectly. Hmm.Wait, the problem says: "the distance d(t) he covers in kilometers in time t (in hours) is given by the integral: d(t) = ∫₀ᵗ v(τ) dτ"So, according to this, distance is displacement, which is not standard. So, perhaps the answer is zero.But let me check the question again: "determine the total distance he runs after 8 hours."Hmm, that suggests total distance, not displacement. So, maybe the problem is misworded, and they actually mean displacement. But if they mean total distance, it's 160/π.But since the integral is given as displacement, perhaps we have to go with that.Alternatively, perhaps the velocity is always positive? Let me check.Wait, v(τ) = 10 sin(πτ/2 + 0). So, sin(πτ/2) is positive for τ in (0, 2), negative for τ in (2,4), etc. So, over 8 hours, it's positive for 0-2, negative for 2-4, positive for 4-6, negative for 6-8.So, integrating from 0 to 8, the positive and negative areas cancel, giving zero displacement.But total distance would be the sum of the absolute areas, which is 160/π.But since the problem defines d(t) as the integral of velocity, which is displacement, then d(8) is zero.But the question says "total distance he runs", which is ambiguous. If it's displacement, it's zero. If it's total distance, it's 160/π.But given that the integral is defined as displacement, perhaps the answer is zero.Alternatively, maybe I misread the velocity function. Let me check.v(τ) = 10 sin(πτ/2 + 0). So, yes, it's a sine wave with period 4 hours.Wait, another thought: maybe the pilot is running in a circular path, so displacement is zero, but distance is non-zero. But the problem doesn't specify direction, so perhaps it's just a scalar distance, which would be the integral of speed, i.e., |v(τ)|.But the problem says "distance he covers... is given by the integral of v(τ) dτ", which is displacement. So, perhaps the answer is zero.But to be thorough, let me compute both.Displacement: d(8) = 0Total distance: 160/π ≈ 50.93 kmBut the problem says "distance he covers... is given by the integral of v(τ) dτ", so it's displacement. So, the answer is zero.But the question also asks for the maximum speed he achieves during this period. Since speed is the magnitude of velocity, the maximum speed is 10 km/h.Wait, but velocity is 10 sin(πτ/2), so the maximum of |v(τ)| is 10 km/h. So, maximum speed is 10 km/h.So, to summarize:1. Time to reduce tumor volume to 10%: t = ln(10)/(λ1 + λ2 + λ3)2. Total distance after 8 hours: 0 km (if displacement), or 160/π km (if total distance). But since the problem defines distance as the integral of velocity, which is displacement, the answer is 0 km. However, the maximum speed is 10 km/h.But wait, the problem says "distance he covers", which is usually total distance, not displacement. So, maybe the problem is misworded, and they actually mean displacement. Alternatively, perhaps the velocity is always positive, but given the function, it's not.Wait, let me think again. If the velocity is given as 10 sin(πτ/2), then it's positive for τ in (0,2), negative for τ in (2,4), etc. So, over 8 hours, the displacement is zero, but the total distance is 160/π.But the problem says "distance he covers... is given by the integral of v(τ) dτ", which is displacement. So, perhaps the answer is zero.Alternatively, maybe the problem is using "distance" incorrectly, and they actually mean total distance, so the answer is 160/π.But since the integral is defined as displacement, I think the answer is zero.But to be safe, maybe I should compute both and see which one makes sense.Wait, let me compute the integral again.d(t) = ∫₀ᵗ 10 sin(πτ/2) dτ = (20/π)(1 - cos(πt/2))At t=8:d(8) = (20/π)(1 - cos(4π)) = (20/π)(1 - 1) = 0So, displacement is zero.Total distance is ∫₀⁸ |10 sin(πτ/2)| dτ = 160/πBut the problem says "distance he covers... is given by the integral of v(τ) dτ", so it's displacement. So, the answer is zero.But the question also asks for the maximum speed, which is 10 km/h.So, to conclude:1. Time t = ln(10)/(λ1 + λ2 + λ3)2. Total distance after 8 hours: 0 km, maximum speed: 10 km/hBut I'm a bit confused because "distance covered" usually refers to total distance, not displacement. Maybe the problem is using "distance" incorrectly. Alternatively, perhaps the velocity is always positive, but given the function, it's not.Wait, another thought: maybe the phase shift φ is such that the sine function is always positive. But in this case, φ=0, so it's just sin(πτ/2), which is positive and negative.Alternatively, maybe the velocity is the absolute value of the sine function, but the problem says v(τ) = 10 sin(πτ/2 + φ), so it's not.So, I think the answer is displacement is zero, and maximum speed is 10 km/h.But to be thorough, let me check the integral again.Yes, the integral of sin over a full period is zero, so over two periods, it's zero.So, I think that's the answer.Final Answer1. The time at which the tumor volume is reduced to 10% is boxed{dfrac{ln(10)}{lambda_1 + lambda_2 + lambda_3}} months.2. The total distance run after 8 hours is boxed{0} kilometers, and the maximum speed achieved is boxed{10} km/h.</think>

🧠:<think>Okay, so I have this problem where a parent is helping their teenager write personal narratives and wants to anonymize names using a mathematical model. The model involves a 3x3 matrix and some linear algebra. Hmm, interesting. Let me try to break this down step by step.First, part 1 asks me to find the resulting vector w when the matrix A is multiplied by the vector v. The matrix A is given, and the vector v is [65, 66, 67]^T, which are the ASCII codes for 'A', 'B', and 'C'. So, I need to perform the matrix multiplication A * v.Let me recall how matrix multiplication works. Each element of the resulting vector is the dot product of the corresponding row of matrix A with the vector v. So, for a 3x3 matrix and a 3x1 vector, the result will be a 3x1 vector.Let me write down matrix A and vector v:A = [2 -1 3][0 4 1][-2 5 7]v = [65][66][67]So, to compute w = A * v, I need to compute each component:First component of w:2*65 + (-1)*66 + 3*67Second component:0*65 + 4*66 + 1*67Third component:-2*65 + 5*66 + 7*67Let me compute each one step by step.First component:2*65 = 130-1*66 = -663*67 = 201Adding them together: 130 - 66 + 201 = (130 - 66) + 201 = 64 + 201 = 265Second component:0*65 = 04*66 = 2641*67 = 67Adding them: 0 + 264 + 67 = 331Third component:-2*65 = -1305*66 = 3307*67 = 469Adding them: -130 + 330 + 469 = ( -130 + 330 ) + 469 = 200 + 469 = 669So, putting it all together, the resulting vector w is [265, 331, 669]^T.Wait, let me double-check my calculations to make sure I didn't make any arithmetic errors.First component:2*65 = 130-1*66 = -663*67 = 201130 - 66 = 64; 64 + 201 = 265. That seems correct.Second component:0*65 = 04*66 = 2641*67 = 670 + 264 = 264; 264 + 67 = 331. Correct.Third component:-2*65 = -1305*66 = 3307*67 = 469-130 + 330 = 200; 200 + 469 = 669. Correct.Alright, so part 1 seems done. Now, moving on to part 2.Part 2 asks whether matrix A is invertible. If it is, I need to find its inverse. To determine if a matrix is invertible, I need to check if its determinant is non-zero. If the determinant is zero, the matrix is singular and not invertible; otherwise, it is invertible.So, let me compute the determinant of A.The determinant of a 3x3 matrix can be calculated using the rule of Sarrus or the general method of expansion by minors. I think I'll use the expansion by minors for clarity.Given matrix A:[2 -1 3][0 4 1][-2 5 7]The determinant is calculated as:2 * det(minor of 2) - (-1) * det(minor of -1) + 3 * det(minor of 3)Where the minor of an element is the determinant of the 2x2 matrix that remains after deleting the row and column containing that element.So, let's compute each minor.First term: 2 * det(minor of 2)Minor of 2 is the submatrix:[4 1][5 7]Determinant: (4*7) - (1*5) = 28 - 5 = 23Second term: -(-1) * det(minor of -1) = 1 * det(minor of -1)Minor of -1 is:[0 1][-2 7]Determinant: (0*7) - (1*(-2)) = 0 + 2 = 2Third term: 3 * det(minor of 3)Minor of 3 is:[0 4][-2 5]Determinant: (0*5) - (4*(-2)) = 0 + 8 = 8Now, putting it all together:Determinant = 2*23 + 1*2 + 3*8 = 46 + 2 + 24 = 72Since the determinant is 72, which is not zero, the matrix A is invertible.Now, I need to find the inverse of matrix A. The inverse of a matrix A is given by (1/det(A)) * adjugate(A), where adjugate(A) is the transpose of the cofactor matrix of A.So, first, I need to compute the cofactor matrix of A.The cofactor matrix is computed by taking each element of A, computing its minor, then multiplying by (-1)^(i+j) where i and j are the row and column indices.Let me write down the matrix A again:Row 1: 2, -1, 3Row 2: 0, 4, 1Row 3: -2, 5, 7I'll compute each cofactor one by one.Cofactor of element a11 (2):Minor is the determinant of:[4 1][5 7] which is 23 as before.Multiply by (-1)^(1+1) = 1. So, cofactor is 23.Cofactor of a12 (-1):Minor is determinant of:[0 1][-2 7] which is 2 as before.Multiply by (-1)^(1+2) = -1. So, cofactor is -2.Cofactor of a13 (3):Minor is determinant of:[0 4][-2 5] which is 8 as before.Multiply by (-1)^(1+3) = 1. So, cofactor is 8.Cofactor of a21 (0):Minor is determinant of:[-1 3][5 7]Determinant: (-1*7) - (3*5) = -7 -15 = -22Multiply by (-1)^(2+1) = -1. So, cofactor is (-1)*(-22) = 22.Cofactor of a22 (4):Minor is determinant of:[2 3][-2 7]Determinant: (2*7) - (3*(-2)) = 14 + 6 = 20Multiply by (-1)^(2+2) = 1. So, cofactor is 20.Cofactor of a23 (1):Minor is determinant of:[2 -1][-2 5]Determinant: (2*5) - (-1*(-2)) = 10 - 2 = 8Multiply by (-1)^(2+3) = -1. So, cofactor is -8.Cofactor of a31 (-2):Minor is determinant of:[-1 3][4 1]Determinant: (-1*1) - (3*4) = -1 -12 = -13Multiply by (-1)^(3+1) = 1. So, cofactor is -13.Cofactor of a32 (5):Minor is determinant of:[2 3][0 1]Determinant: (2*1) - (3*0) = 2 - 0 = 2Multiply by (-1)^(3+2) = -1. So, cofactor is -2.Cofactor of a33 (7):Minor is determinant of:[2 -1][0 4]Determinant: (2*4) - (-1*0) = 8 - 0 = 8Multiply by (-1)^(3+3) = 1. So, cofactor is 8.So, the cofactor matrix is:[23 -2 8][22 20 -8][-13 -2 8]Now, the adjugate of A is the transpose of this cofactor matrix. So, let's transpose it.Original cofactor matrix:Row 1: 23, -2, 8Row 2: 22, 20, -8Row 3: -13, -2, 8Transposing it, we get:Column 1 becomes Row 1: 23, 22, -13Column 2 becomes Row 2: -2, 20, -2Column 3 becomes Row 3: 8, -8, 8So, the adjugate matrix is:[23 22 -13][-2 20 -2][8 -8 8]Now, the inverse of A is (1/det(A)) * adjugate(A). Since det(A) is 72, we have:A⁻¹ = (1/72) * adjugate(A)So, each element of the adjugate matrix is divided by 72.Let me write that out:A⁻¹ = [23/72 22/72 -13/72][-2/72 20/72 -2/72][8/72 -8/72 8/72]Simplify the fractions:23/72 can't be simplified.22/72 = 11/36-13/72 can't be simplified.-2/72 = -1/3620/72 = 5/18-2/72 = -1/368/72 = 1/9-8/72 = -1/98/72 = 1/9So, substituting these simplified fractions:A⁻¹ = [23/72 11/36 -13/72][-1/36 5/18 -1/36][1/9 -1/9 1/9]Let me double-check my calculations to make sure I didn't make any mistakes.First, the cofactor matrix:- For a11: minor determinant 23, cofactor 23. Correct.- For a12: minor determinant 2, cofactor -2. Correct.- For a13: minor determinant 8, cofactor 8. Correct.- For a21: minor determinant -22, cofactor 22. Correct.- For a22: minor determinant 20, cofactor 20. Correct.- For a23: minor determinant 8, cofactor -8. Correct.- For a31: minor determinant -13, cofactor -13. Correct.- For a32: minor determinant 2, cofactor -2. Correct.- For a33: minor determinant 8, cofactor 8. Correct.Transposing the cofactor matrix: yes, the rows become columns. So, the adjugate is correct.Then, dividing each element by 72:23/72, 22/72=11/36, -13/72.-2/72=-1/36, 20/72=5/18, -8/72=-1/9. Wait, hold on, in the third row, the last element was 8, so 8/72=1/9, not -1/9. Wait, in the adjugate matrix, the third row is [8, -8, 8], so when divided by 72, it becomes [1/9, -1/9, 1/9]. So, that's correct.Wait, in the second row, the third element was -8, so -8/72 = -1/9. Correct.So, the inverse matrix looks correct.Just to make sure, let me verify if A * A⁻¹ equals the identity matrix.But that might be time-consuming, but let me try at least one element.Compute the (1,1) element of A * A⁻¹:Row 1 of A: [2, -1, 3]Column 1 of A⁻¹: [23/72, -1/36, 1/9]Dot product: 2*(23/72) + (-1)*(-1/36) + 3*(1/9)Calculate each term:2*(23/72) = 46/72 = 23/36(-1)*(-1/36) = 1/363*(1/9) = 3/9 = 1/3Adding them together: 23/36 + 1/36 + 12/36 = (23 + 1 + 12)/36 = 36/36 = 1Good, that's correct for the (1,1) element.Let me check another element, say (2,3):Row 2 of A: [0, 4, 1]Column 3 of A⁻¹: [-13/72, -1/36, 1/9]Dot product: 0*(-13/72) + 4*(-1/36) + 1*(1/9)Compute each term:0 + (-4/36) + 1/9 = (-1/9) + 1/9 = 0Which is correct for the (2,3) element of the identity matrix.One more, (3,2):Row 3 of A: [-2, 5, 7]Column 2 of A⁻¹: [11/36, 5/18, -1/9]Dot product: (-2)*(11/36) + 5*(5/18) + 7*(-1/9)Compute each term:-22/36 = -11/1825/18-7/9 = -14/18Adding them: (-11/18) + 25/18 + (-14/18) = (-11 + 25 -14)/18 = 0/18 = 0Which is correct for the (3,2) element.So, it seems that the inverse is correct.Therefore, the matrix A is invertible, and its inverse is as calculated above.Final Answer1. The resulting vector ( mathbf{w} ) is boxed{begin{pmatrix} 265 331 669 end{pmatrix}}.2. The matrix ( A ) is invertible, and its inverse is boxed{begin{pmatrix} dfrac{23}{72} & dfrac{11}{36} & -dfrac{13}{72} -dfrac{1}{36} & dfrac{5}{18} & -dfrac{1}{36} dfrac{1}{9} & -dfrac{1}{9} & dfrac{1}{9} end{pmatrix}}.</think>

🧠:<think>Okay, so I have this problem where a cultural anthropologist is studying the impact of language education on refugee communities. She has two groups: Group A with intensive language education and Group B with standard education. Both groups have 100 students each, and their proficiency improvement scores are normally distributed. The first part asks for the probability that a randomly selected student from Group A has a higher improvement score than a randomly selected student from Group B. Hmm, okay, so I need to find P(A > B), where A and B are the improvement scores from each group.Let me recall that if A and B are independent normal variables, then the difference D = A - B is also normally distributed. The mean of D would be the difference of the means, and the variance would be the sum of the variances since they are independent.So, for Group A, the mean improvement is 15 points with a standard deviation of 5. For Group B, the mean is 10 points with a standard deviation of 7. First, let's compute the mean of D: μ_D = μ_A - μ_B = 15 - 10 = 5 points.Next, the variance of D: Var(D) = Var(A) + Var(B) = (5)^2 + (7)^2 = 25 + 49 = 74. So, the standard deviation σ_D is sqrt(74) ≈ 8.6023.Now, we need to find P(D > 0), which is the probability that the difference is positive, meaning A > B.Since D is normally distributed with μ = 5 and σ ≈ 8.6023, we can standardize this to a Z-score:Z = (0 - 5) / 8.6023 ≈ -0.5814.Looking at the standard normal distribution table, the probability that Z is less than -0.5814 is approximately 0.2810. Therefore, the probability that Z is greater than -0.5814 is 1 - 0.2810 = 0.7190.So, the probability that a randomly selected student from Group A has a higher improvement score than a student from Group B is approximately 71.9%.Wait, let me double-check my calculations. The mean difference is 5, which is positive, so the probability should be more than 50%, which aligns with 71.9%. The Z-score calculation seems correct: (0 - 5)/8.6023 ≈ -0.5814. The corresponding probability from the Z-table is indeed around 0.2810 for Z < -0.58, so 1 - 0.2810 is 0.7190. That seems right.Moving on to the second part. The anthropologist wants to model the relationship between time spent in language education (x) and improvement in proficiency scores (y) using linear regression for each group. The model is y = β0 + β1x + ε.We are given the average time spent for Group A is 120 hours with a variance of 400 hours², and for Group B, it's 80 hours with a variance of 250 hours². Also, the correlation between time spent and improvement score is 0.8 for Group A and 0.6 for Group B.We need to estimate the coefficients β0 and β1 for each group. I remember that in linear regression, the slope coefficient β1 can be calculated using the formula:β1 = r * (sy / sx),where r is the correlation coefficient, sy is the standard deviation of y, and sx is the standard deviation of x.But wait, do we have the standard deviations of y? The problem doesn't directly give us the standard deviations of the improvement scores for each group in terms of their relationship with time. Hmm, maybe I need to make some assumptions here.Wait, actually, in the first part, we know the standard deviations of the improvement scores for each group. For Group A, it's 5 points, and for Group B, it's 7 points. So, perhaps sy for Group A is 5 and for Group B is 7.But hold on, in the first part, the improvement scores are given as normal distributions with those standard deviations. So, if we are modeling the improvement score y as a function of time x, then the standard deviation of y (sy) is given as 5 for Group A and 7 for Group B.Similarly, the standard deviation of x (time spent) can be calculated from the variances given. For Group A, variance of x is 400, so sx = sqrt(400) = 20. For Group B, variance is 250, so sx = sqrt(250) ≈ 15.8114.So, let's compute β1 for each group.For Group A:r = 0.8, sy = 5, sx = 20β1 = 0.8 * (5 / 20) = 0.8 * 0.25 = 0.2For Group B:r = 0.6, sy = 7, sx ≈ 15.8114β1 = 0.6 * (7 / 15.8114) ≈ 0.6 * 0.4427 ≈ 0.2656Now, to find β0, the intercept, we use the formula:β0 = ȳ - β1 * x̄,where ȳ is the mean of y and x̄ is the mean of x.But wait, in the first part, the mean improvement for Group A is 15 and for Group B is 10. So, ȳ for Group A is 15, and for Group B is 10.For Group A:β0 = 15 - 0.2 * 120 = 15 - 24 = -9For Group B:β0 = 10 - 0.2656 * 80 ≈ 10 - 21.248 ≈ -11.248So, the regression equations are:For Group A: y = -9 + 0.2xFor Group B: y ≈ -11.248 + 0.2656xWait, let me verify these calculations again.For Group A:β1 = 0.8 * (5 / 20) = 0.8 * 0.25 = 0.2. That seems correct.β0 = 15 - 0.2 * 120 = 15 - 24 = -9. Correct.For Group B:sy = 7, sx = sqrt(250) ≈ 15.8114β1 = 0.6 * (7 / 15.8114) ≈ 0.6 * 0.4427 ≈ 0.2656. Correct.β0 = 10 - 0.2656 * 80 ≈ 10 - 21.248 ≈ -11.248. Correct.So, rounding off, for Group B, β0 ≈ -11.25 and β1 ≈ 0.2656.Alternatively, if we want to express β1 more precisely, it's approximately 0.2656, which is roughly 0.266.So, summarizing:Group A:β0 = -9β1 = 0.2Group B:β0 ≈ -11.25β1 ≈ 0.266I think that's it. Let me just make sure I didn't mix up any numbers. For Group A, the mean improvement is higher, and the correlation is stronger, so the slope should be positive and relatively steeper. Wait, no, actually, the slope is 0.2 for Group A and about 0.266 for Group B. So, Group B has a slightly steeper slope, which might seem counterintuitive because Group A had a higher mean improvement. But since Group A also had a higher mean time spent (120 vs 80), the slopes are different.But actually, the slope is the change in y per unit change in x. So, even though Group A has a higher mean y, the slope is smaller because the standard deviation of x is larger (20 vs ~15.81). So, the effect per hour is smaller for Group A. That makes sense.Yes, I think the calculations are correct.</think>