💡Okay, so I have this problem here: Given f(x) = x² + 3x f''(1), and I need to find f''(2). The options are A:1, B:2, C:4, D:8. Hmm, let me try to figure this out step by step.First, I know that f''(x) is the second derivative of the function f(x). So, I need to find the second derivative of f(x) and then evaluate it at x=2. But wait, the function f(x) itself is given in terms of f''(1). That seems a bit tricky because f''(1) is a constant, right? It's the value of the second derivative at x=1.So, maybe I can find f''(1) first and then use that to find f''(2). Let me write down the function again:f(x) = x² + 3x f''(1)Alright, let's find the first derivative f'(x). The derivative of x² is 2x, and the derivative of 3x f''(1) is 3 f''(1) because f''(1) is a constant. So,f'(x) = 2x + 3 f''(1)Now, let's find the second derivative f''(x). The derivative of 2x is 2, and the derivative of 3 f''(1) is 0 because it's a constant. So,f''(x) = 2Wait, that's interesting. The second derivative is just 2, which is a constant. That means no matter what x is, f''(x) is always 2. So, f''(1) is 2, and f''(2) is also 2.But let me double-check to make sure I didn't make a mistake. Starting from the beginning:f(x) = x² + 3x f''(1)First derivative:f'(x) = 2x + 3 f''(1)Second derivative:f''(x) = 2Yes, that seems correct. Since f''(x) is 2 for all x, then f''(1) = 2 and f''(2) = 2. So, the answer should be B:2.I think that makes sense because the second derivative of x² is 2, and the other term involving f''(1) disappears when taking the second derivative since it's multiplied by x, but its derivative becomes a constant, which then disappears in the second derivative.Alternatively, maybe I can plug in x=1 into the original equation to find f''(1) and then use that to find f''(2). Let's try that approach.Given f(x) = x² + 3x f''(1)If I plug in x=1:f(1) = (1)² + 3*(1)*f''(1) = 1 + 3 f''(1)But I don't know f(1), so that might not help directly. Maybe I need another equation. Let's take the first derivative:f'(x) = 2x + 3 f''(1)If I plug in x=1:f'(1) = 2*(1) + 3 f''(1) = 2 + 3 f''(1)Again, I don't know f'(1), so that might not help either. Maybe I need to use the second derivative.We already found that f''(x) = 2, so f''(1) = 2. Therefore, f''(2) = 2 as well.Yeah, that seems consistent. So, the answer is B:2.

💡Let me try to figure out this problem step by step. So, we have a band that can be arranged in a rectangular formation. When they arrange themselves in 10 rows, there are 4 unoccupied positions. When they arrange themselves in 12 rows, there are 6 unoccupied positions. We need to find the number of band members, and it's given that this number is between 200 and 300.First, let's try to understand what it means to have a rectangular formation. If the band members are arranged in a rectangle, that means the number of members can be represented as the product of the number of rows and the number of columns. However, in both cases mentioned, there are some unoccupied positions. So, the number of band members is less than the total number of positions in the rectangle.Let me denote the number of band members as ( n ).When arranged in 10 rows, there are 4 unoccupied positions. So, the total number of positions in this case is ( n + 4 ). Since it's a rectangular formation, the number of columns must be an integer. Let's denote the number of columns as ( c_1 ). Therefore, we can write the equation:[10 times c_1 = n + 4]Similarly, when arranged in 12 rows, there are 6 unoccupied positions. So, the total number of positions here is ( n + 6 ). Let the number of columns in this case be ( c_2 ). Thus, we have:[12 times c_2 = n + 6]Now, we have two equations:1. ( 10c_1 = n + 4 )2. ( 12c_2 = n + 6 )Our goal is to find ( n ) such that it satisfies both equations and lies between 200 and 300.Let me rearrange both equations to express ( n ) in terms of ( c_1 ) and ( c_2 ):1. ( n = 10c_1 - 4 )2. ( n = 12c_2 - 6 )Since both expressions equal ( n ), we can set them equal to each other:[10c_1 - 4 = 12c_2 - 6]Let me simplify this equation:[10c_1 - 12c_2 = -6 + 4][10c_1 - 12c_2 = -2]Hmm, this looks like a Diophantine equation, which is an equation that seeks integer solutions. Let me see if I can find integer values for ( c_1 ) and ( c_2 ) that satisfy this equation.First, let me simplify the equation by dividing both sides by 2:[5c_1 - 6c_2 = -1]Now, we have:[5c_1 - 6c_2 = -1]I need to find integers ( c_1 ) and ( c_2 ) that satisfy this equation. Let me try to express one variable in terms of the other. Let's solve for ( c_1 ):[5c_1 = 6c_2 - 1][c_1 = frac{6c_2 - 1}{5}]Since ( c_1 ) must be an integer, ( 6c_2 - 1 ) must be divisible by 5. Let me write this condition:[6c_2 - 1 equiv 0 mod 5][6c_2 equiv 1 mod 5]Since 6 is congruent to 1 modulo 5 (because 6 - 5 = 1), we can simplify:[c_2 equiv 1 mod 5]This means that ( c_2 ) can be written as:[c_2 = 5k + 1]where ( k ) is a non-negative integer.Now, let's substitute ( c_2 = 5k + 1 ) back into the equation for ( c_1 ):[c_1 = frac{6(5k + 1) - 1}{5}][c_1 = frac{30k + 6 - 1}{5}][c_1 = frac{30k + 5}{5}][c_1 = 6k + 1]So, ( c_1 = 6k + 1 ) and ( c_2 = 5k + 1 ).Now, let's substitute ( c_1 ) back into the expression for ( n ):[n = 10c_1 - 4][n = 10(6k + 1) - 4][n = 60k + 10 - 4][n = 60k + 6]So, ( n = 60k + 6 ). Now, we need to find the value of ( k ) such that ( n ) is between 200 and 300.Let's solve for ( k ):[200 < 60k + 6 < 300][200 - 6 < 60k < 300 - 6][194 < 60k < 294][frac{194}{60} < k < frac{294}{60}][3.233... < k < 4.9]Since ( k ) must be an integer, the possible values for ( k ) are 4.Let me check ( k = 4 ):[n = 60 times 4 + 6 = 240 + 6 = 246]Wait, but 246 is within the range of 200 to 300. Let me verify if this satisfies both original conditions.First condition: 10 rows with 4 unoccupied positions.Total positions: ( n + 4 = 246 + 4 = 250 ).Number of columns: ( c_1 = frac{250}{10} = 25 ). So, 10 rows and 25 columns, which is a valid rectangular formation.Second condition: 12 rows with 6 unoccupied positions.Total positions: ( n + 6 = 246 + 6 = 252 ).Number of columns: ( c_2 = frac{252}{12} = 21 ). So, 12 rows and 21 columns, which is also a valid rectangular formation.Therefore, ( n = 246 ) satisfies both conditions.Wait a minute, but earlier when I set ( c_2 = 5k + 1 ), and ( k = 4 ), let's check what ( c_2 ) would be:[c_2 = 5 times 4 + 1 = 21]Which matches our earlier calculation. So, everything seems consistent.But hold on, earlier when I solved for ( k ), I found that ( k ) must be between approximately 3.233 and 4.9, so ( k = 4 ) is the only integer solution. Therefore, ( n = 246 ) is the only solution in the given range.But wait, let me double-check my calculations because I initially thought the answer might be 254, but according to this, it's 246.Let me go back to the equation:[n = 60k + 6]For ( k = 4 ):[n = 60 times 4 + 6 = 240 + 6 = 246]Yes, that's correct.But let me check if 246 satisfies both conditions:1. 10 rows: ( 246 = 10 times 25 - 4 ). Wait, 10 rows with 25 columns would have 250 positions, so 250 - 4 = 246. Correct.2. 12 rows: ( 246 = 12 times 21 - 6 ). 12 rows with 21 columns would have 252 positions, so 252 - 6 = 246. Correct.Therefore, 246 is indeed the correct number of band members.But wait, earlier I thought the answer was 254. Let me see where I might have gone wrong.Looking back, I think I might have made a mistake in my initial assumption when setting up the equations. Let me re-examine the problem.The problem states that when arranged in 10 rows, there are 4 unoccupied positions, and when arranged in 12 rows, there are 6 unoccupied positions. So, the total number of positions in each case is ( n + 4 ) and ( n + 6 ), respectively.Therefore, the equations should be:1. ( 10c_1 = n + 4 )2. ( 12c_2 = n + 6 )Which leads to:1. ( n = 10c_1 - 4 )2. ( n = 12c_2 - 6 )Setting them equal:[10c_1 - 4 = 12c_2 - 6][10c_1 - 12c_2 = -2][5c_1 - 6c_2 = -1]Which is the same as before. Then, solving for ( c_1 ):[c_1 = frac{6c_2 - 1}{5}]So, ( 6c_2 - 1 ) must be divisible by 5, leading to ( c_2 equiv 1 mod 5 ), so ( c_2 = 5k + 1 ).Substituting back:[c_1 = 6k + 1][n = 10c_1 - 4 = 60k + 6]So, ( n = 60k + 6 ). For ( k = 4 ), ( n = 246 ). For ( k = 5 ), ( n = 306 ), which is above 300, so not acceptable.Wait, but earlier I thought the answer was 254. Let me see if 254 fits into this equation.If ( n = 254 ), then:[254 = 60k + 6][60k = 248][k = 248 / 60 ≈ 4.133...]Which is not an integer, so 254 is not a solution according to this equation.Therefore, my initial thought of 254 was incorrect, and the correct answer is 246.But let me double-check with the original problem statement to ensure I haven't misread anything."The members of a band are arranged in a rectangular formation. If arranged in 10 rows, there are 4 unoccupied positions. If arranged in 12 rows, there are 6 unoccupied positions. What is the number of band members, assuming it is between 200 and 300?"Yes, that's correct. So, 246 is the number of band members.Wait, but let me check if there are any other possible values of ( k ) that could give ( n ) within the range.We have ( n = 60k + 6 ). For ( k = 3 ):[n = 60 times 3 + 6 = 186]Which is below 200, so not acceptable.For ( k = 4 ):[n = 246]Which is within the range.For ( k = 5 ):[n = 306]Which is above 300, so not acceptable.Therefore, the only solution is ( n = 246 ).Wait, but earlier I thought the answer was 254. Let me see if 254 could be a solution.If ( n = 254 ), then:1. For 10 rows: ( 254 + 4 = 258 ). So, ( c_1 = 258 / 10 = 25.8 ), which is not an integer. Therefore, 254 does not satisfy the first condition.Similarly, for 12 rows: ( 254 + 6 = 260 ). ( c_2 = 260 / 12 ≈ 21.666... ), which is also not an integer. Therefore, 254 is not a valid solution.Therefore, the correct answer is 246.But wait, let me check if there are any other possible solutions by considering different values of ( k ).We have ( n = 60k + 6 ). Let's see for ( k = 4 ), ( n = 246 ). For ( k = 5 ), ( n = 306 ), which is too high. For ( k = 3 ), ( n = 186 ), which is too low. So, 246 is the only solution in the given range.Therefore, the number of band members is 246.

💡To find the magnitude of the complex number ( z ) that satisfies ( iz = 3 + i ), I start by solving for ( z ).First, I divide both sides of the equation by ( i ) to isolate ( z ):[z = frac{3 + i}{i}]To simplify this expression, I multiply the numerator and the denominator by ( i ) to eliminate the imaginary unit in the denominator:[z = frac{(3 + i) cdot i}{i cdot i} = frac{3i + i^2}{i^2}]Since ( i^2 = -1 ), the equation becomes:[z = frac{3i - 1}{-1} = -3i + 1 = 1 - 3i]Now that I have ( z = 1 - 3i ), I calculate its magnitude using the formula for the magnitude of a complex number ( a + bi ):[|z| = sqrt{a^2 + b^2}]Substituting ( a = 1 ) and ( b = -3 ):[|z| = sqrt{1^2 + (-3)^2} = sqrt{1 + 9} = sqrt{10}]Therefore, the magnitude of ( z ) is ( sqrt{10} ).

💡First, I will simplify each fraction individually. For the fraction (frac{15}{5}), I can divide both the numerator and the denominator by 5, which gives me 3.Next, for the fraction (frac{25}{-75}), I can divide both the numerator and the denominator by 25. This simplifies to (-frac{1}{3}).Now, I will multiply all the simplified terms together: (4 times 3 times left(-frac{1}{3}right)).Multiplying 4 and 3 gives me 12. Then, multiplying 12 by (-frac{1}{3}) results in (-4).Therefore, the simplified expression is (-4).

💡First, I need to determine the effective width of each swath Lily cuts. The swath is 30 inches wide, but she overlaps each cut by 6 inches. This means the effective width of each cut is 30 inches minus 6 inches, which is 24 inches. Converting this to feet, since there are 12 inches in a foot, the effective swath width is 24 divided by 12, which equals 2 feet.Next, I need to calculate how many strips Lily will need to mow the entire lawn. The lawn is 180 feet wide, and each effective swath is 2 feet wide. Dividing the total width of the lawn by the effective swath width gives the number of strips: 180 feet divided by 2 feet per strip equals 90 strips.Now, I need to find the total distance Lily will walk while mowing. Each strip is 120 feet long, and there are 90 strips. Multiplying the length of each strip by the number of strips gives the total distance: 120 feet multiplied by 90 strips equals 10,800 feet.Finally, to determine how long it will take Lily to mow the lawn, I divide the total distance by her walking rate. Lily walks at a rate of 6,000 feet per hour. Dividing 10,800 feet by 6,000 feet per hour gives approximately 1.8 hours.Therefore, it will take Lily approximately 1.8 hours to mow her lawn.

💡Хорошо, давайте разберемся с этой задачей шаг за шагом. Нам нужно найти четырехугольные числа, которые кратны 14. Сначала вспомним, что такое четырехугольные числа.Четырехугольные числа, также известные как тетрагональные числа, можно выразить с помощью формулы:[u_n = frac{n(n+1)(n+2)}{6}]где ( n ) — это индекс четырехугольного числа.Теперь нам нужно найти такие ( n ), при которых ( u_n ) будет кратно 14. То есть:[frac{n(n+1)(n+2)}{6} equiv 0 mod 14]Это равносильно тому, что:[n(n+1)(n+2) equiv 0 mod 84](поскольку ( 14 times 6 = 84 )).Теперь рассмотрим, как ( n(n+1)(n+2) ) может быть кратно 84. Для этого нужно, чтобы в произведении ( n(n+1)(n+2) ) были все простые делители 84, то есть 2, 3 и 7.1. **Кратность 2**: В любом трех последовательных числах есть хотя бы одно четное число, поэтому это условие автоматически выполняется.2. **Кратность 3**: В трех последовательных числах есть ровно одно число, которое делится на 3, поэтому это условие также выполняется.3. **Кратность 7**: Здесь нужно, чтобы хотя бы одно из чисел ( n ), ( n+1 ) или ( n+2 ) было кратно 7.Таким образом, основное условие — это кратность 7. Значит, ( n equiv 0 mod 7 ), ( n equiv 6 mod 7 ) или ( n equiv 5 mod 7 ) (поскольку ( n+2 equiv 0 mod 7 ) означает ( n equiv 5 mod 7 )).Однако, помимо этого, нужно убедиться, что ( n(n+1)(n+2) ) кратно 84, то есть, помимо 7, также кратно 12 (поскольку ( 84 = 12 times 7 )). Для этого:- Если ( n equiv 0 mod 7 ), то ( n ) должно быть кратно 12.- Если ( n equiv 6 mod 7 ), то ( n+1 ) должно быть кратно 12.- Если ( n equiv 5 mod 7 ), то ( n+2 ) должно быть кратно 12.Это приводит к следующим формулам для ( n ):- ( n = 28t ) (поскольку 28 — это НОК 4 и 7)- ( n = 28t + 6 )- ( n = 28t + 7 )- ( n = 28t + 12 )- ( n = 28t + 14 )- ( n = 28t - 9 )- ( n = 28t - 8 )- ( n = 28t - 2 )- ( n = 28t - 1 )Теперь, подставляя различные значения ( t ), мы можем найти конкретные значения ( n ), которые удовлетворяют всем условиям.