answer:To solve the problem, we need to find the possible values for (|z|) where (z) is a complex number. We can solve the quadratic equation (z^2 - 8z + 37 = 0), and then determine the magnitude of the roots.Step-by-step solution:1. Solve the quadratic equation (z^2 - 8z + 37 = 0).2. Compute the roots, which may be complex.3. Calculate the magnitude (|z|) of each root.4. Determine the distinct values of (|z|).Let's start by solving the quadratic equation. For a quadratic equation of the form (az^2 + bz + c = 0), the roots can be found using the quadratic formula:[z = frac{-b pm sqrt{b^2 - 4ac}}{2a}]Here:[a = 1][b = -8][c = 37]Let's find the roots using Python.```pythonimport cmath# Coefficients of the quadratic equationa = 1b = -8c = 37# Calculate the discriminantdiscriminant = b2 - 4*a*c# Calculate the two roots using the quadratic formularoot1 = (-b + cmath.sqrt(discriminant)) / (2*a)root2 = (-b - cmath.sqrt(discriminant)) / (2*a)print((root1, root2))``````output((4+4.58257569495584j), (4-4.58257569495584j))```The roots of the quadratic equation (z^2 - 8z + 37 = 0) are:[ z_1 = 4 + 4.58257569495584i ][ z_2 = 4 - 4.58257569495584i ]Next, we need to find the magnitude (|z|) of each root. The magnitude of a complex number (z = x + yi) is given by:[ |z| = sqrt{x^2 + y^2} ]Let's compute the magnitudes for both roots.```python# Calculate the magnitude of each rootmagnitude_root1 = abs(root1)magnitude_root2 = abs(root2)print((magnitude_root1, magnitude_root2))``````output(6.082762530298219, 6.082762530298219)```The magnitudes of both roots are:[ |z_1| = |4 + 4.58257569495584i| = 6.082762530298219 ][ |z_2| = |4 - 4.58257569495584i| = 6.082762530298219 ]Since both roots have the same magnitude, there is only one distinct possible value for (|z|).Thus, the number of possible values for (|z|) is (boxed{1}).

answer:To solve this problem, we need to use the binomial theorem and the properties of binomial coefficients. Let's break it down step-by-step.1. Understand the Binomial Expansion: The binomial expansion of ((3 + x)^n) is given by: [ (3 + x)^n = sum_{k=0}^{n} binom{n}{k} 3^{n-k} x^k ]2. Identify the Coefficients: - The constant term (when (k = 0)) is (binom{n}{0} 3^n). - The coefficient of the (x^2) term (when (k = 2)) is (binom{n}{2} 3^{n-2}).3. Set Up the Given Condition: We are given that the coefficient of the (x^2) term is 5 times the coefficient of the constant term. So, we have: [ binom{n}{2} 3^{n-2} = 5 cdot binom{n}{0} 3^n ]4. Simplify the Equation: Simplify the equation to solve for (n): [ frac{binom{n}{2} 3^{n-2}}{3^n} = 5 cdot binom{n}{0} ] [ binom{n}{2} 3^{-2} = 5 ] [ binom{n}{2} = 5 cdot 9 ] [ binom{n}{2} = 45 ]5. Solve for (n): We need to solve (binom{n}{2} = 45). Recall that: [ binom{n}{2} = frac{n(n-1)}{2} ] Therefore: [ frac{n(n-1)}{2} = 45 ] [ n(n-1) = 90 ] Solve the quadratic equation (n^2 - n - 90 = 0).Let's implement this in Python using SymPy:```pythonimport sympy as sp# Define the variablen = sp.symbols('n')# Define the equationequation = sp.Eq(n*(n-1)/2, 45)# Solve the equationsolution = sp.solve(equation, n)print(solution)``````output[-9, 10]```The solution to the equation (n^2 - n - 90 = 0) gives us two values: (-9) and (10). Since (n) must be a positive integer in the context of binomial expansions, we discard (-9).Hence, the value of (n) is ( boxed{10} ).

answer:To find the limit of the function [ lim _{x rightarrow 0} frac{(1+x)^{3}-(1+3 x)}{x+x^{5}} ]we can approach it by expanding ((1+x)^3) using the binomial expansion and then simplifying the expression.1. Expand ((1+x)^3) using the binomial theorem.2. Subtract (1 + 3x) from the expansion.3. Simplify the resulting expression.4. Divide by (x + x^5).5. Take the limit as (x) approaches 0.Let's perform these steps using Python to ensure accuracy.```pythonfrom sympy import symbols, limit, expand # Define the variable x = symbols('x') # Define the function numerator = (1 + x)3 - (1 + 3*x) denominator = x + x5 function = numerator / denominator # Expand the numerator using binomial expansion expanded_numerator = expand((1 + x)3) - (1 + 3*x) # Simplify the function simplified_function = expanded_numerator / denominator # Take the limit as x approaches 0 result = limit(simplified_function, x, 0) print(result)``````output0```The limit of the given function[ lim _{x rightarrow 0} frac{(1+x)^{3}-(1+3 x)}{x+x^{5}} ]is (boxed{0}).

answer:To solve the given equation ( p^2 - 6pq + q^2 + 3q - 1 = 0 ) in prime numbers, we can follow these steps:1. Understand the Problem: We need to find prime numbers ( p ) and ( q ) that satisfy the equation ( p^2 - 6pq + q^2 + 3q - 1 = 0 ).2. Formulate the Approach: - We can solve this problem by iterating over pairs of prime numbers and checking if they satisfy the equation. - For this purpose, we can generate a list of prime numbers up to a certain limit using the SymPy library.3. Check Each Pair: - For each pair ((p, q)) of prime numbers, substitute them into the equation. - Check if the equation equals zero for that pair.Let's implement this step-by-step in Python:```pythonimport sympy as sp# Define the equationp, q = sp.symbols('p q')equation = p2 - 6*p*q + q2 + 3*q - 1# Generate a list of prime numbers up to a certain limitlimit = 100 # You can increase this limit if necessaryprimes = list(sp.primerange(1, limit))# Find pairs (p, q) that satisfy the equationsolution_pairs = []for prime_p in primes: for prime_q in primes: if equation.subs({p: prime_p, q: prime_q}) == 0: solution_pairs.append((prime_p, prime_q))print(solution_pairs)``````output[(17, 3)]```The solution to the equation ( p^2 - 6pq + q^2 + 3q - 1 = 0 ) in prime numbers is the pair ((17, 3)).Thus, the pair of prime numbers ((p, q)) that satisfies the given equation is (boxed{(17, 3)}).

answer:To determine the value of (c) such that the quadratic equation (x^2 + 2x + 4c = 0) has two distinct real roots, we need to examine the discriminant of the quadratic equation.For a quadratic equation (ax^2 + bx + c = 0), the discriminant (Delta) is given by:[Delta = b^2 - 4ac]The roots of the quadratic equation are distinct and real if and only if (Delta > 0).For the given equation (x^2 + 2x + 4c = 0):- (a = 1)- (b = 2)- (c = 4c)The discriminant is then:[Delta = b^2 - 4ac = 2^2 - 4 cdot 1 cdot 4c = 4 - 16c]We need (Delta > 0) for the equation to have two distinct real roots:[4 - 16c > 0]Let's solve this inequality to find the range of (c).```pythonimport sympy as sp# Define the variablec = sp.symbols('c')# Define the discriminant inequalitydiscriminant_inequality = 4 - 16*c > 0# Solve the inequality for csolution = sp.solve(discriminant_inequality, c)print(solution)``````output(-oo < c) & (c < 1/4)```The solution to the inequality (4 - 16c > 0) is:[c < frac{1}{4}]Therefore, (c) must be less than (frac{1}{4}) for the quadratic equation (x^2 + 2x + 4c = 0) to have two distinct real roots.To provide a specific value that satisfies this condition, we can choose any value of (c) that is less than (frac{1}{4}). For instance, we can select (c = 0).Thus, (c = boxed{0}) would satisfy the given condition.

answer:To solve the problem, we need to compute the expected value of the random number ( X = 0.x_1 x_2 x_3 ldots x_{1000} ), where ( x_i ) are random digits. The expected value of ( X ) can be calculated by considering the contribution of each digit ( x_i ) to ( X ).1. Each digit ( x_i ) has a uniform distribution from 0 to 9. Hence, the expected value ( E(x_i) ) of each ( x_i ) is the average of all digits from 0 to 9, which is 4.5.2. The digit ( x_i ) appears at the ( i )-th place after the decimal point, and hence it contributes ( x_i cdot 10^{-i} ) to ( X ).3. The linearity of expectation allows us to consider the expected value of ( X ) as the sum of the expected values of each ( x_i cdot 10^{-i} ).Let's write the Python code using sympy to calculate this expected value.```pythonimport sympy as sp# Define the number of digitsn = 1000# Calculate the expected value of Xexpected_value = sum(4.5 * 10-i for i in range(1, n + 1))print(expected_value)``````output0.5```The expected value of the random number ( X = 0.x_1 x_2 x_3 ldots x_{1000} ) is (boxed{0.5}).Here's a detailed explanation for clarity:1. Each digit ( x_i ) can take any value from 0 to 9 with equal probability.2. The expected value of each digit ( x_i ) is the average of all possible values: [ E(x_i) = frac{0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9}{10} = 4.5 ]3. Each ( x_i ) contributes ( x_i cdot 10^{-i} ) to the number ( X ).By linearity of expectation, the expected value ( mathrm{E}X ) is:[mathrm{E}X = sum_{i=1}^{1000} E(x_i cdot 10^{-i}) = sum_{i=1}^{1000} left( E(x_i) cdot 10^{-i} right)]Substituting ( E(x_i) = 4.5 ):[mathrm{E}X = 4.5 cdot sum_{i=1}^{1000} 10^{-i}]The series ( sum_{i=1}^{1000} 10^{-i} ) is a geometric series with the first term ( a = 10^{-1} = 0.1 ) and the common ratio ( r = 0.1 ). The sum of the infinite geometric series is given by:[sum_{i=1}^{infty} 10^{-i} = frac{0.1}{1 - 0.1} = frac{0.1}{0.9} = frac{1}{9}]However, since we only sum up to 1000 terms, it is a finite geometric series:[sum_{i=1}^{1000} 10^{-i} = frac{0.1 cdot (1 - 0.1^{1000})}{1 - 0.1} approx frac{0.1}{0.9} = frac{1}{9}]Thus,[mathrm{E}X = 4.5 cdot frac{1}{9} = 4.5 cdot frac{1}{9} = 0.5]Therefore, the expected value ( mathrm{E}X ) is (boxed{0.5}).