answer:To determine the correct answer, we need to analyze the given equation ( n^6 + 3a = k^3 ), where ( k ) is a positive integer. We want to find conditions under which ( a ) is a positive integer for any given natural number ( n ).We can rewrite the equation as:[ 3a = k^3 - n^6 ][ a = frac{k^3 - n^6}{3} ]Now, let's solve this step by step in Python using SymPy to determine if ( a ) can be a positive integer for infinitely many values of ( n ) and ( k ). Step-by-Step Solution:1. Define the variables and equation: Set up the variables ( n ), ( k ), and ( a ) and establish the equation ( n^6 + 3a = k^3 ).2. Solve for ( a ): Determine the conditions under which ( a ) is a positive integer.3. Check the nature of ( a ): Verify if there are infinitely many, finitely many, or no solutions where ( a ) is a positive integer.Let's write the Python code to solve this problem.```pythonimport sympy as sp# Define variablesn, k, a = sp.symbols('n k a', integer=True, positive=True)# Define the equationequation = sp.Eq(n6 + 3*a, k3)# Solve the equation for 'a'solution_a = sp.solve(equation, a)# Function to check if 'a' is a positive integer for any given n and kdef check_positive_integer_solutions(n_max, k_max): positive_integer_solutions = [] for n_val in range(1, n_max + 1): for k_val in range(1, k_max + 1): a_val = solution_a[0].subs({n: n_val, k: k_val}) if a_val.is_integer and a_val > 0: positive_integer_solutions.append((n_val, k_val, a_val)) return positive_integer_solutions# Check for a range of values (up to n_max and k_max)n_max = 10k_max = 100solutions = check_positive_integer_solutions(n_max, k_max)print(solutions)print("Number of solutions:", len(solutions))``````output[(1, 4, 21), (1, 7, 114), (1, 10, 333), (1, 13, 732), (1, 16, 1365), (1, 19, 2286), (1, 22, 3549), (1, 25, 5208), (1, 28, 7317), (1, 31, 9930), (1, 34, 13101), (1, 37, 16884), (1, 40, 21333), (1, 43, ... (output truncated)```The output shows multiple pairs of ((n, k)) and corresponding (a) values that are positive integers. Since there is a pattern where ( a ) repeatedly becomes a positive integer for increasing values of (k), this indicates that there are indeed infinitely many solutions where (a) is a positive integer.Therefore, the correct conclusion for the given problem is:[boxed{text{(A) There are infinitely many such } a.}]

answer:To solve this problem, we need to use the formula for the volume of a sphere using a definite integral. We'll integrate the cross-sectional area of the sphere along its diameter.The equation of a sphere of radius ( R ) centered at the origin is ( x^2 + y^2 + z^2 = R^2 ).The cross-sectional area at a slice ( z ) is a circle with radius ( sqrt{R^2 - z^2} ), and its area is ( pi left( sqrt{R^2 - z^2} right)^2 = pi (R^2 - z^2) ).To find the volume, we integrate this area from ( z = -R ) to ( z = R ):[ V = int_{-R}^{R} pi (R^2 - z^2) , dz ]Let's solve this using Python and SymPy.```pythonimport sympy as sp# Define the variable and constantsz = sp.symbols('z')R = sp.symbols('R', positive=True)# Define the integrand: π(R^2 - z^2)integrand = sp.pi * (R2 - z2)# Compute the definite integral from -R to Rvolume_integral = sp.integrate(integrand, (z, -R, R))# Simplify the resultvolume_simplified = sp.simplify(volume_integral)print(volume_simplified)``````output4*pi*R3/3```The volume ( V ) of a sphere with radius ( R ) calculated using the definite integral is:[ V = frac{4}{3} pi R^3 ]Thus, the final answer is (boxed{frac{4}{3} pi R^3}).

answer:To find the area of the region ( S' ) after applying the matrix transformation, we need to determine how the area scale factor is affected by the transformation matrix.Given the matrix[A = begin{pmatrix} 2 & 1 7 & -3 end{pmatrix}]The area of a transformed region is given by the absolute value of the determinant of the transformation matrix times the original area. Therefore, if the original area of the region ( S ) is 10, and we apply the matrix ( A ) to this region, the area of the transformed region ( S' ) will be:[text{Area of } S' = | det(A) | times text{Area of } S]First, let's calculate the determinant of matrix ( A ):[det(A) = begin{vmatrix} 2 & 1 7 & -3 end{vmatrix} = (2 cdot -3) - (1 cdot 7) = -6 - 7 = -13]So, the absolute value of the determinant is:[| det(A) | = 13]Thus, the area of ( S' ) is:[text{Area of } S' = 13 times 10 = 130]We can verify this computation using Python.```pythonimport numpy as np# Define the transformation matrixA = np.array([[2, 1], [7, -3]])# Compute the determinant of the matrixdet_A = np.linalg.det(A)# The original area of the regionarea_S = 10# Calculate the area of the transformed regionarea_S_prime = abs(det_A) * area_Sprint(area_S_prime)``````output130.0```The area of the transformed region ( S' ) is (boxed{130}).

answer:Let's break down the given expression step by step:[frac{0.128 div 3.2 + 0.86}{frac{5}{6} cdot 1.2 + 0.8} cdot frac{left(1 frac{32}{63} - frac{13}{21}right) cdot 3.6}{0.505 cdot frac{2}{5} - 0.002}]We can evaluate the expression in smaller parts.First, we evaluate:[0.128 div 3.2 + 0.86]Second, we evaluate:[frac{5}{6} cdot 1.2 + 0.8]Third, we evaluate:[1 frac{32}{63} - frac{13}{21}]Fourth, we evaluate:[0.505 cdot frac{2}{5} - 0.002]Then, we use these results to compute the two big fractions and their product:[frac{0.128 div 3.2 + 0.86}{ frac{5}{6} cdot 1.2 + 0.8 }]and[frac{left(1 frac{32}{63} - frac{13}{21}right) cdot 3.6}{ 0.505 cdot frac{2}{5} - 0.002 }]Finally, multiply the two fractions to get the final answer. Let's evaluate each part using Python.```python# Evaluate each part of the expression step-by-stepfrom fractions import Fraction# First part: 0.128 / 3.2 + 0.86part1_numerator = 0.128 / 3.2 + 0.86# Second part: (5/6) * 1.2 + 0.8part1_denominator = (5/6) * 1.2 + 0.8# Third part: 1 (32 / 63) - 13 / 21mixed_fraction_part = 1 + Fraction(32, 63)third_part_result = float(mixed_fraction_part - Fraction(13, 21))# Fourth part: 0.505 * (2/5) - 0.002denom_part_2 = 0.505 * (2/5) - 0.002# The main complex fractionsfraction1 = part1_numerator / part1_denominatorfraction2 = (third_part_result * 3.6) / denom_part_2# Final resultfinal_result = fraction1 * fraction2print(final_result)``````output7.999999999999999```The evaluated expression results in a value of approximately:[boxed{8}]This result is obtained by step-by-step computation of each part of the expression using Python.

answer:To solve for the real number ( m ) such that the complex number ( z = frac{m + i}{1 + mi} ) is purely imaginary, we need to follow these steps:1. Understand that for ( z ) to be purely imaginary, its real part must be zero.2. Express ( z ) in the form ( frac{a + bi}{c + di} ) and separate real and imaginary parts.3. Set the real part to zero and solve for ( m ).Let's proceed step-by-step using sympy in Python:1. Define the complex number ( z ).2. Separate the real and imaginary parts of ( z ).3. Set the real part to zero and solve for ( m ).Here is the implementation:```pythonimport sympy as sp# Define the symbol for mm = sp.symbols('m', real=True)i = sp.I# Define the complex number znumerator = m + idenominator = 1 + m*iz = numerator / denominator# Separate the real and imaginary parts of zreal_part = sp.re(z)imaginary_part = sp.im(z)# Set the real part to zero and solve for msolution = sp.solve(real_part, m)print(solution)``````output[0]```The value of ( m ) such that the complex number ( z = frac{m + i}{1 + mi} ) is purely imaginary is (boxed{0}).

answer:To solve the problem, we need to define a function ( f(x) ) that satisfies the given properties:1. ( f(x) = f(4 - x) )2. ( f(x - 2) + f(2 - x) = 0 )We'll use these properties to determine ( f(x) ) and then find ( f(508) ).Let's break it down into manageable steps:1. Symmetry Property: The equation ( f(x) = f(4 - x) ) indicates that the function is symmetric about ( x = 2 ).2. Functional Equation Property: The equation ( f(x - 2) + f(2 - x) = 0 ) tells us that the function is odd when shifted by 2 units.We'll use these properties to deduce the form of ( f(x) ).Let's start by considering the implications of these properties and see if we can derive a general form for ( f(x) ). We need to find if there is a specific pattern or function that matches these properties.Here's the plan:- Check the values of ( f(x) ) at a few points using the given properties.- Attempt to derive a general form that satisfies both properties.- Evaluate ( f(508) ) using the deduced form.Let's write the Python code using `sympy` to help us in this process.```pythonimport sympy as sp# Define the variable xx = sp.symbols('x')# Define the function f(x)f = sp.Function('f')# Define the propertiesprop1 = sp.Eq(f(x), f(4 - x))prop2 = sp.Eq(f(x - 2) + f(2 - x), 0)# Solve for specific values of f(x)# Let's start by exploring the properties# We'll test a few values and see if we can deduce a pattern# Test valuesx_values = [0, 1, 2, 3, 4]results = {}for x_val in x_values: result1 = prop1.subs(x, x_val) result2 = prop2.subs(x, x_val) results[x_val] = (result1, result2)results``````output{0: (Eq(f(0), f(4)), Eq(f(-2) + f(2), 0)), 1: (Eq(f(1), f(3)), Eq(f(-1) + f(1), 0)), 2: (True, Eq(2*f(0), 0)), 3: (Eq(f(3), f(1)), Eq(f(-1) + f(1), 0)), 4: (Eq(f(4), f(0)), Eq(f(-2) + f(2), 0))}```From the output, we can make some observations:1. ( f(0) = f(4) )2. ( f(1) = f(3) )3. ( f(2) ) must satisfy ( f(2 - 2) + f(2 - 2) = 0 Rightarrow f(0) + f(0) = 0 Rightarrow 2f(0) = 0 Rightarrow f(0) = 0 )4. Similarly, ( f(-2) + f(2) = 0 ), which implies ( f(2) = 0 )5. ( f(1) + f(-1) = 0 Rightarrow f(1) = -f(-1) )6. ( f(3) + f(-1) = 0 Rightarrow f(3) = -f(-1) )From these observations, we can deduce:- ( f(0) = f(4) = 0 )- ( f(2) = 0 )- ( f(1) = f(3) )- ( f(1) = -f(-1) )- ( f(3) = -f(-1) )Based on the symmetry ( f(x) = f(4 - x) ) and the antisymmetry around 2, a possible candidate for ( f(x) ) could be ( f(x) = k sinleft(frac{pi x}{2}right) ). However, since ( f(0) = 0 ) and ( f(2) = 0 ), we may just use a simplified function where ( f(x) ) is periodic with period 4.Considering ( f(x) = k sinleft(frac{pi x}{2}right) ) is a valid candidate, let's use this function to find ( f(508) ).Given ( f(508) = k sinleft(frac{pi cdot 508}{2}right) ):```pythonimport sympy as spimport math# Define the parameter k (we can assume k = 1 without loss of generality because we only care about the function form)k = 1# Define the function f(x)def f(x): return k * sp.sin(sp.pi * x / 2)# Evaluate f(508)result = f(508)print(result)``````output0```From the output, we find that ( f(508) = 0 ).Thus, the function ( f(x) = k sinleft(frac{pi x}{2}right) ) satisfies both properties, and specifically, ( f(508) = 0 ).Hence, the value of ( f(508) ) is ( boxed{0} ).