XA ) -
Calculate the volume of the prism by first finding the total number of half-unit
cubes that will fill it. There are 8 half-unit cubes in every unit cube.
23
A. Number of half-unit cubes = 10
V = 2 cubic units
B. Number of half-unit cubes = 20
V=2 cubic units
O C. Number of half-unit cubes = 20
V = 10 cubic units
D. Number of half-unit cubes = 10
V = 5 cubic units

(A) The given matrices are A=def=2 and B=[3a 3b 3c 19 h il I-a +4g -b + 4h-C + 4i].

Solution: AB = 2 [3a 3b 3c 19 h il I-a +4g -b + 4h-C + 4i] = [6a 6b 6c 38 h 2il 2a+8g-2b+8h-2c+8i]AB-4 = [6a 6b 6c 38 h 2il 2a+8g-2b+8h-2c+8i] - 4 [1 0 0 0 0 0 0 1 0 0 0 0 0 1] = [6a-4 6b 6c 38 h 2il 2a+8g-2b+8h-2c+8i-4] |AB-4| = |-4 0 0 0 0 0 0 -3 0 0 0 0 0 -2|=24

(B) For this part, we are required to find A²B". Let's first compute A² and then multiply it with B".A² = AA = 2 [2] = [4] We are to multiply [4] with B". B" = [1 0 0 0 0 0 0 1 0 0 0 0 0 1] [3a 3b 3c 19 h il I-a +4g -b + 4h-C + 4i] [3a 3b 3c 19 h il I-a +4g -b + 4h-C + 4i] = [3a+I-a 3b-4b 3c+4c 19 h+4h il+4i I-a+4g-b+4h-C+4i] A²B" = [4] [3a+I-a 3b-4b 3c+4c 19 h+4h il+4i I-a+4g-b+4h-C+4i] = [12a+2-2a+8g-2b+8h-2c+8i 12b-16b 12c+16c 76h+16h 4il+16i 4a+16g-4b+16h-4c+16i] The value of A²B" is [12a+2-2a+8g-2b+8h-2c+8i 12b-16b 12c+16c 76h+16h 4il+16i 4a+16g-4b+16h-4c+16i].

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The given costs are : Cost of pen = $1.85Cost of marker = $0.47 less than a pen

Cost of marker = $1.85 - $0.47 = $1.38Cost of 2 pens and a marker= 2($1.85) + $1.38 = $3.72 + $1.38 = $5.10

If Jon gives a ten-dollar bill, then he gets a change of $10 - $5.10 = $4.90

Thus, Jon will get $4.90 change from a ten-dollar bill.

Pens are more harmful to the environment than pencils. Pens are always ready to write, whereas pencils need to be sharpened. A pencil becomes harder to use and shorter the more you sharpen it. Pencils cannot be used to write on skin. Ballpoint pens are one of the most common and well-known kinds of pens.

Ballpoint pens use an oil-based ink that dries more quickly than other types of ink. When you write, you'll notice less smudging as a result. Since the ink is thick, ballpoint pens utilize less ink as you compose, enduring longer than other pen types.

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a) Jackson's bill can be represented by the polynomial f(m) = 5 + m.

b) Cherie's bill can be represented by the polynomial g(m) = 3 + 2m.

c) The combined fare for Jackson and Cherie can be represented by the polynomial h(m) = 8 + 3m.

d) If they both drove 22 miles, their combined fares would be $74.

a) Jackson's bill consists of a flat fee of $5 per fare plus an additional $1 per mile.

This can be represented by the polynomial f(m) = 5 + m, where m represents the number of miles between the airport and the hotel.

b) Cherie's bill consists of a flat fee of $3 per fare plus an additional $2 per mile.

This can be represented by the polynomial g(m) = 3 + 2m, where m represents the number of miles between the airport and the hotel.

c) To calculate the combined fare for Jackson and Cherie, we add their individual polynomial representations.

Therefore, the combined fare polynomial is h(m) = f(m) + g(m) = (5 + m) + (3 + 2m) = 8 + 3m.

d) If both Jackson and Cherie drove 22 miles, we can calculate their combined fares by substituting m = 22 into the combined fare polynomial, h(m) = 8 + 3m.

Thus, h(22) = 8 + 3(22) = 8 + 66 = 74.

Therefore, their combined fares would be $74.

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Answer : 5xy + 4 = 20ab + 5a + 5b + 9 is odd, as odd + odd = even and even + odd = odd.This proves the contrapositive of the given statement. Hence, the given statement is true.

Explanation :

We are given that for every pair of integers x and y, if 5xy + 4 is even, then at least one of x or y must be even.

We need to prove that this statement is true.Let's start by proving the contrapositive of this statement.

Contrapositive of this statement is "If both x and y are odd, then 5xy + 4 is odd".

Let's consider two odd integers x and y. Hence we can write them as x = 2a + 1 and y = 2b + 1 where a and b are integers.

Now substituting these values of x and y in the given expression we get,                                                                                                      5xy + 4 = 5(2a + 1)(2b + 1) + 4= 20ab + 5a + 5b + 9                                                                                                                                                                                                          Here,20ab + 5a + 5b is clearly an odd number, as it can be written as 5(4ab + a + b).

Therefore,5xy + 4 = 20ab + 5a + 5b + 9 is odd, as odd + odd = even and even + odd = odd.This proves the contrapositive of the given statement. Hence, the given statement is true.

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we have proven that either G is abelian or Z(G) = 1.

What is the equivalent expression?

Equivalent expressions are expressions that perform the same function despite their appearance. If two algebraic expressions are equivalent, they have the same value when we use the same variable value.

To prove the statement, we will use the previous problem, which states that if |G| = pq for primes p and q, then either G is cyclic or G has an element of order p and an element of order q.

Let's consider the two cases separately:

Case 1: G is cyclic.

If G is cyclic, then G is abelian since all cyclic groups are abelian.

Case 2: G is not cyclic.

If G is not cyclic, we know from the previous problem that G has an element of order p and an element of order q. Let's denote these elements as a and b, respectively.

Consider the subgroup H generated by a:

H = {e, a, a², ..., [tex]a^{(p-1)}[/tex]}

Since the order of a is p, H has p elements. Similarly, consider the subgroup K generated by b:

K = {e, b, b², ..., [tex]b^{(q-1)}[/tex]}

Since the order of b is q, K has q elements.

Now, let's consider the intersection of H and K, denoted as HK:

HK = {h * k | h ∈ H, k ∈ K}

Since G is not cyclic, the intersection HK cannot be equal to G. Therefore, |HK| < |G|.

Now, let's consider the order of the product h * k for arbitrary elements h ∈ H and k ∈ K. By the property of cyclic groups, we know that:

[tex](h * k)^p = h^p * k^p = e * k^p = k^p[/tex]

Since p is a prime, the order of [tex]k^p[/tex] can only be 1 or q. If the order is 1, then [tex]k^p = e[/tex], and thus h * k = h for any h ∈ H. This implies that HK is a subset of H.

Similarly, if the order of [tex]k^p[/tex] is q, then [tex]k^p = e[/tex], and thus h * k = k for any h ∈ H. This implies that HK is a subset of K.

In both cases, HK is a proper subset of either H or K. Therefore, |HK| < p or |HK| < q.

Since |HK| is a divisor of |G| = pq, and |HK| < p or |HK| < q, the only possibility is that |HK| = 1.

This implies that the intersection HK contains only the identity element e. Therefore, all elements of H commute with all elements of K.

Now, let's consider an arbitrary element g ∈ G. Since g ∈ H and g ∈ K, g commutes with all elements in H and all elements in K. This means that g commutes with all elements in the subgroup generated by H and all elements in the subgroup generated by K.

Therefore, every element of G commutes with every element in G, and thus G is abelian.

In summary, we have shown that if |G| = pq for primes p and q, then either G is abelian (Case 1) or G is not cyclic, and in this case, the intersection of the subgroups generated by the elements of order p and q is trivial (HK = {e}), implying that Z(G) = 1 (Case 2).

Therefore, we have proven that either G is abelian or Z(G) = 1.

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It will prompt you to enter a number between 0 and 100. The program will then generate random guesses and provide feedback until it matches the goal number. Finally, it will display the guessed number, indicate a match, and show the total number of guesses made.

Here's a notebook cell containing the program for the number guessing game:

import random

def guess_number():

   goal_number = int(input("Enter a number between 0 and 100: "))

   guesses = 0

   guessed_number = random.randint(0, 100)

   while guessed_number != goal_number:

       print("Guessed number:", guessed_number)

        if guessed_number < goal_number:

           print("Too low!")

       else:

           print("Too high!")

        guesses += 1

       guessed_number = random.randint(0, 100)

   print("Guessed number:", guessed_number)

   print("Match!")

   print("Total guesses:", guesses)

guess_number()

To use the program, simply run the cell. It will prompt you to enter a number between 0 and 100. The program will then generate random guesses and provide feedback until it matches the goal number. Finally, it will display the guessed number, indicate a match, and show the total number of guesses made.

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The probability that the system will work is approximately 0.7267, or 72.67%.

To calculate the probability that the system will work, we need to consider the probabilities of each component working and combine them using the principles of probability theory.

Let's break down the problem step by step:

Probability of the key working: The given probability of the key working is 0.826. This means there is an 82.6% chance that the key will function properly.

Probability of the battery working: The given probability of the battery working is 0.971. This means there is a 97.1% chance that the battery will function properly.

Probability of the wire working: The given probability of the wire working is 0.890. This means there is an 89% chance that the wire will function properly.

Probability of the plug working: The given probability of the plug working is 0.954. This means there is a 95.4% chance that the plug will function properly.

To calculate the probability that all components work together, we multiply these individual probabilities:

Probability of the system working = Probability of key working× Probability of battery working× Probability of wire working× Probability of plug working

Probability of the system working = 0.826× 0.971× 0.890 ×0.954

Calculating this expression, we find:

Probability of the system working ≈ 0.726656356

Therefore, the probability that the system will work is approximately 0.7267, or 72.67%.

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The equation of motion for the described system is: x(t) = (-1/2) * sin(2t) + (-1/4) * cos(2t).

To find the resulting equation of motion for the system described, we need to solve the differential equation that governs the behavior of the mass-spring system.

The equation of motion for a mass-spring system can be written as:

m * d^2x/dt^2 + k * x = f(t),

where m is the mass, d^2x/dt^2 is the acceleration, k is the spring constant, x is the displacement from the equilibrium position, and f(t) is the external force applied to the system.

m = 2 kg (mass of the object),

k = 18 N/m (spring constant), and

f(t) = 2sin(2t) (external force).

Substituting these values into the equation of motion, we have:

2 * d^2x/dt^2 + 18 * x = 2sin(2t).

To solve this differential equation, we'll assume a solution of the form:

x(t) = A * sin(2t) + B * cos(2t),

where A and B are constants to be determined.

Taking the derivatives of x(t) with respect to time:

dx/dt = 2A * cos(2t) - 2B * sin(2t),

d^2x/dt^2 = -4A * sin(2t) - 4B * cos(2t).

Substituting these derivatives into the equation of motion:

2 * (-4A * sin(2t) - 4B * cos(2t)) + 18 * (A * sin(2t) + B * cos(2t)) = 2sin(2t).

Simplifying and collecting like terms:

(-8A + 18B) * sin(2t) + (-8B - 18A) * cos(2t) = 2sin(2t).

To satisfy this equation for all t, the coefficients of sin(2t) and cos(2t) on both sides of the equation must be equal. This leads to the following equations:

-8A + 18B = 2,

-8B - 18A = 0.

Solving this system of equations, we find:

A = -1/2,

B = -1/4.

Substituting these values back into the assumed solution x(t):

x(t) = (-1/2) * sin(2t) + (-1/4) * cos(2t).

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The given statement is correct.

Hence it is true.

We have a statement regarding the quadratic equations.

We have to verify whether it is true or not.

Since we know that,

A quadratic equation is an equation with a single variable of degree 2. Its general form is ax² + bx + c = 0, where x is variable and a, b, and c are constants, and a ≠ 0.

According to the question, we are provided with the standard form of the quadratic equation as -  ax² + bx + c = 0.

If we compare the statement given in the question with the definition discussed above, then it can be concluded that the given statement is true. Equation ax² + bx + c = 0 is the standard form of a quadratic equation with a, b, and c as constant real numbers.

The constant 'a' cannot be 0, as this would reduce the degree of the equation to 1.

Hence, the given statement is correct.

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Considering the random variables X and Y, the covariance coefficient Cov(U,V) = E[(U - E[U])(V - E[V])] is given by E(X²) - E(Y²).

Given that the random variables X and Y are identically distributed random variables (not necessarily independent).

We are to compute the covariance coefficient between U and V where U = X + Y and V = X-Y.

Covariance between U and V is given by;

            Cov (U,V) = E [(U- E(U)) (V- E(V))]

The expected values of U and V can be obtained as follows;

             E (U) = E(X+Y)E(U) = E(X) + E(Y) [Since X and Y are identically distributed]

             E(U) = 2E(X).....................(1)

Similarly,

               E(V) = E(X-Y)E(V) = E(X) - E(Y) [Since X and Y are identically distributed]

               E(V) = 0.........................(2)

Covariance can also be expressed as follows;

              Cov (U,V) = E (UX) - E(U)E(X) - E(UY) + E(U)E(Y) - E(VX) + E(V)E(X) + E(VY) - E(V)E(Y)

Since X and Y are identically distributed random variables, we have;

      E(UX) = E(X²) + E(X)E(Y)E(UY) = E(Y²) + E(X)E(Y)E(VX) = E(X²) - E(X)E(Y)E(VY) = E(Y²) - E(X)E(Y)

On substituting the respective values, we have;

      Cov (U,V) = E(X²) - [2E(X)]²

On simplifying further, we obtain;

  Cov (U,V) = E(X²) - 4E(X²)

    Cov (U,V) = -3E(X²)

Therefore, the covariance coefficient

    Cov(U,V) = E[(U - E[U])(V - E[V])] is given by;

    Cov(U,V) = E(UV) - E(U)E(V)

                     = [E{(X+Y)(X-Y)}] - 2E(X) × 0

      Cov(U,V) = [E(X²) - E(Y²)]

       Cov(U,V) = E(X²) - E(Y²)

Hence, the covariance coefficient Cov(U,V) = E[(U - E[U])(V - E[V])] is given by E(X²) - E(Y²).

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The scale factor from the pizza in Tallahassee to the pizza in Jaco, Costa Rica, is approximately 0.684.

Part A: To calculate the area of the pizza from Tallahassee, we need to use the formula for the area of a circle:

Area = π * (radius)^2

The given information is the diameter, so we first need to find the radius. The diameter is 16 inches, so the radius is half of that:

Radius = 16 inches / 2 = 8 inches

Now we can calculate the area:

Area = π * (8 inches)^2

Using the approximation of π as 3.14, we can substitute the values and calculate:

Area ≈ 3.14 * (8 inches)^2

    ≈ 3.14 * 64 square inches

    ≈ 200.96 square inches

Therefore, the pizza from Tallahassee has an area of approximately 200.96 square inches.

Part B: Similarly, to calculate the area of the pizza from Jaco, Costa Rica, we use the formula for the area of a circle. The given information is the diameter of 27.8 centimeters, so we find the radius:

Radius = 27.8 centimeters / 2 = 13.9 centimeters

Now we can calculate the area:

Area = π * (13.9 centimeters)^2

Using the approximation of π as 3.14:

Area ≈ 3.14 * (13.9 centimeters)^2

    ≈ 3.14 * 192.21 square centimeters

    ≈ 603.7954 square centimeters

Therefore, the pizza from Jaco, Costa Rica, has an area of approximately 603.7954 square centimeters.

Part C: To compare the areas of the two pizzas, we need to convert the area of the Tallahassee pizza from square inches to square centimeters using the given conversion factor of 1 inch = 2.54 centimeters:

Area in square centimeters = Area in square inches * (2.54 centimeters/inch)^2

Substituting the value of the area of the Tallahassee pizza:

Area in square centimeters = 200.96 square inches * (2.54 centimeters/inch)^2

                          ≈ 200.96 * 6.4516 square centimeters

                          ≈ 1296.159616 square centimeters

Since the area of the pizza from Jaco, Costa Rica, is approximately 603.7954 square centimeters, and the converted area of the Tallahassee pizza is approximately 1296.159616 square centimeters, we can conclude that the pizza from Tallahassee has a larger area.

Part D: The scale factor from the pizza in Tallahassee to the pizza in Jaco, Costa Rica, can be calculated by dividing the diameter of the Jaco pizza by the diameter of the Tallahassee pizza:

Scale factor = Diameter of Jaco pizza / Diameter of Tallahassee pizza

Using the given diameters of 27.8 centimeters and 16 inches:

Scale factor = 27.8 centimeters / 16 inches

To compare the two measurements, we need to convert inches to centimeters using the conversion factor of 1 inch = 2.54 centimeters:

Scale factor = 27.8 centimeters / (16 inches * 2.54 centimeters/inch)

           = 27.8 centimeters / 40.64 centimeters

           ≈ 0.684

Therefore, the scale factor from the pizza in Tallahassee to the pizza in Jaco, Costa Rica, is approximately 0.684.

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Matrix Equation Of The Form AX=B is [6 -4 -6; 1 3 7; -1 5 0] [x; y; z] = [-7; 8; 7]

To write the given system of equations as a matrix equation of the form AX = B, given the system of equations:

6x - 4y - 6z = -7x + 3y + 7z = 8-x + 5y = 7

we will first need to form a matrix containing all the coefficients of x, y, and z.

Then, we will form the B matrix by placing the constants on the right side of the equal sign.

Then we will combine the two matrices to form the desired matrix equation.

This process can be better understood by following these steps:

Step 1: Coefficient Matrix (A)

The first step is to find the coefficient matrix (A) by just taking the coefficients of x, y, and z and placing them in a 3x3 matrix.

A = [6 -4 -6

       1  3   7

      -1  5  0]

This will be the coefficient matrix.

Step 2: Right-Hand Side Matrix (B)

The next step is to form the right-hand side matrix (B).

To create the B matrix, we just take the constants on the right side of each equation and place them in a column vector. B = [-7; 8; 7]

This will be the right-hand side matrix.

Step 3: Matrix Equation

We can now combine the coefficient matrix and the right-hand side matrix to form the matrix equation.

AX = B [6 -4 -6; 1 3 7; -1 5 0] [x; y; z] = [-7; 8; 7]

Using the equation 6x - 4y - 6z=-7, we can now substitute and write the augmented matrix as[A|B] = [6 -4 -6|-7; 1 3 7|8; -1 5 0|7]

Therefore, the matrix equation of the form AX = B for the given system of equations is:

[6 -4 -6; 1 3 7; -1 5 0] [x; y; z] = [-7; 8; 7]

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Multiple linear regression analysis is a statistical technique used to analyze the relationship between a dependent variable and two or more independent variables.

It extends the concept of simple linear regression analysis by incorporating multiple predictors to explain and predict the variability in the dependent variable.

In simple linear regression, there is only one independent variable, whereas multiple linear regression allows for the inclusion of multiple independent variables.

In multiple linear regression analysis, the relationship between the dependent variable (Y) and independent variables (X₁, X₂, ..., Xₚ) is represented by the equation:

Y = β₀ + β₁X₁ + β₂X₂ + ... + βₚXₚ + ε

Here, Y represents the dependent variable, X₁, X₂, ..., Xₚ are the independent variables, β₀, β₁, β₂, ..., βₚ are the coefficients (also known as regression weights), and ε represents the error term.

The main difference between simple linear regression and multiple linear regression is the number of independent variables included in the analysis.

Simple linear regression has one independent variable, resulting in a linear relationship between the dependent and independent variables.

On the other hand, multiple linear regression incorporates multiple independent variables, allowing for the examination of their individual and combined effects on the dependent variable.

For example, in a simple linear regression analysis, we might examine the relationship between a person's years of experience (X) and their salary (Y).

However, in multiple linear regression, we can consider additional predictors such as education level, job title, or age, to better understand the factors influencing salary.

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The solution to the system of linear equations is:

[tex]\(x_1 = 14\)\\\(x_2 = -1\)\\\(x_3 = 11\)\\[/tex]

To solve the system of linear equations by modifying it to row echelon form (REF) and then to reduced row echelon form (RREF), we'll perform row operations on the augmented matrix.

Given the system of equations:

[tex]\(x_1 - 5x_2 + x_3 = 2\)\\\(-3x_1 + x_2 + 2x_3 = 9\)\\\(-x_1 - 7x_2 + 2x_3 = -1\)\\[/tex]

Let's construct the augmented matrix by writing down the coefficients and the constants:

[tex]\[\begin{bmatrix}1 & -5 & 1 & | & 2 \\-3 & 1 & 2 & | & 9 \\-1 & -7 & 2 & | & -1 \\\end{bmatrix}\][/tex]

To obtain row echelon form (REF), we'll use row operations to eliminate the coefficients below the main diagonal.

Row 2 = Row 2 + 3 * Row 1:

[tex]\[\begin{bmatrix}1 & -5 & 1 & | & 2 \\0 & -14 & 5 & | & 15 \\-1 & -7 & 2 & | & -1 \\\end{bmatrix}\][/tex]

Row 3 = Row 3 + Row 1:

[tex]\[\begin{bmatrix}1 & -5 & 1 & | & 2 \\0 & -14 & 5 & | & 15 \\0 & -12 & 3 & | & 1 \\\end{bmatrix}\][/tex]

Next, we'll perform row operations to eliminate the coefficient below the main diagonal in the second column.

Row 3 = Row 3 - (12/14) * Row 2:

[tex]\[\begin{bmatrix}1 & -5 & 1 & | & 2 \\0 & -14 & 5 & | & 15 \\0 & 0 & -1/7 & | & -11/7 \\\end{bmatrix}\][/tex]

Next, we'll perform row operations to obtain leading 1's in each row.

Row 1 = (1/14) * Row 1:

[tex]\[\begin{bmatrix}1/14 & -5/14 & 1/14 & | & 1/7 \\0 & -14 & 5 & | & 15 \\0 & 0 & -1/7 & | & -11/7 \\\end{bmatrix}\][/tex]

Row 2 = (-1/14) * Row 2:

[tex]\[\begin{bmatrix}1/14 & -5/14 & 1/14 & | & 1/7 \\0 & 1 & -5/14 & | & -15/14 \\0 & 0 & -1/7 & | & -11/7 \\\end{bmatrix}\][/tex]

Next, we'll perform row operations to eliminate the coefficients above and below the main diagonal in the third column.

Row 1 = Row 1 - (1/14) * Row 3:

[tex]\[\begin{bmatrix}1/14 & -5/14 & 0 & | & 8/7 \\0 & 1 & -5/14 & | & -15/14 \\0 & 0 & -1/7 & | & -11/7 \\\end{bmatrix}\][/tex]

Row 2 = Row 2 + (5/14) * Row 3:

[tex]\[\begin{bmatrix}1/14 & -5/14 & 0 & | & 8/7 \\0 & 1 & 0 & | & -10/7 \\0 & 0 & -1/7 & | & -11/7 \\\end{bmatrix}\][/tex]

Next, we'll perform row operations to obtain a leading 1 in the third row.

Row 3 = (-7) * Row 3:

[tex]\[\begin{bmatrix}1/14 & -5/14 & 0 & | & 8/7 \\0 & 1 & 0 & | & -10/7 \\0 & 0 & 1 & | & 11 \\\end{bmatrix}\][/tex]

Next, we'll perform row operations to eliminate the coefficients above the main diagonal in the second column.

Row 1 = Row 1 + (5/14) * Row 2:

[tex]\[\begin{bmatrix}1/14 & 0 & 0 & | & 1 \\0 & 1 & 0 & | & -10/7 \\0 & 0 & 1 & | & 11 \\\end{bmatrix}\][/tex]

Row 2 = (7/14) * Row 2:

[tex]\[\begin{bmatrix}1/14 & 0 & 0 & | & 1 \\0 & 1 & 0 & | & -5/7 \\0 & 0 & 1 & | & 11 \\\end{bmatrix}\][/tex]

The augmented matrix is now in row echelon form (REF).

To obtain the reduced row echelon form (RREF), we'll perform row operations to obtain leading 1's and zeros above each leading 1.

Row 1 = 14 * Row 1:

[tex]\[\begin{bmatrix}1 & 0 & 0 & | & 14 \\0 & 1 & 0 & | & -5/7 \\0 & 0 & 1 & | & 11 \\\end{bmatrix}\][/tex]

Row 2 = (7/5) * Row 2:

[tex]\[\begin{bmatrix}1 & 0 & 0 & | & 14 \\0 & 1 & 0 & | & -1 \\0 & 0 & 1 & | & 11 \\\end{bmatrix}\][/tex]

The augmented matrix is now in reduced row echelon form (RREF).

Therefore, the solution to the system of linear equations is:

[tex]\(x_1 = 14\)\\\(x_2 = -1\)\\\(x_3 = 11\)\\[/tex]

Note: Each row in the augmented matrix corresponds to an equation, and the values in the rightmost column are the solutions for the variables [tex]\(x_1\)[/tex],[tex]\(x_2\)[/tex], and [tex]\(x_3\)[/tex] respectively.

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The values of f'(x), f''(x) and f''(x) for the discontinuous function are 3x² + 4x, 6x + 4 and 6; 4x³ + 1, 12x² and 24x respectively. The value of the integral is e -1.

1. For the first function, f(x) has a different expression for x < 1 and x > 1. To calculate the distributional derivatives, we evaluate the derivatives separately for each interval. For x < 1, we differentiate f(x) = x³ + 2x² - 1 with respect to x,

- f'(x) = d/dx (x³ + 2x² - 1) = 3x² + 4x

Taking the derivative once again, we get,

- f"(x) = d/dx (3x² + 4x) = 6x + 4

Finally, the third derivative is,

- f"'(x) = d/dx (6x + 4) = 6

For x > 1, we differentiate f(x) = x⁴ + x + 1 with respect to x,

- f'(x) = d/dx (x⁴ + x + 1) = 4x³ + 1

Taking the derivative once again, we get,

- f"(x) = d/dx (4x³ + 1) = 12x²

Finally, the third derivative is,

- f"'(x) = d/dx (12x²) = 24x

These derivatives represent the distributional derivatives of the given functions, accounting for the discontinuity at x = 1.

2. Simplifying the integral and evaluating it,

I = ∫[∞ to 1] eˣ[δ(x) + δ(x – 1)] dx

Since the limits of integration are from ∞ to 1, the only non-zero contribution will be from the δ(x - 1) term when x = 1. Now we can evaluate the integral,

I = ∫[∞ to 1] e dx

i = [e] from ∞ to 1

i = e - e⁰

i = e - 1

So, the value of the integral is e - 1.

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Complete question - 1. Evaluate the distributional derivatives f'(x),f"(x),f"'(x) for the following discontinuous functions.

1) f(x) = x³ + 2x²- 1; x<1

f(x) = x⁴ + x + 1; x>1

2. Evaluate the following integrals by first simplifying the integrands. 2.) I = ∫[∞ to 1] eˣ[δ(x) + δ(x – 1]dx.

Taking the given data into consideration we reach the conclusion that
a) the probability that a randomly selected runner will take between 2.35 and 2.85 hours to complete the half-marathon is 0.1974, or about 19.74%.
This means that if we randomly select a runner, there is a 19.74% chance that their completion time will be between 2.35 and 2.85 hours.
b) randomly select 5 customers from the restaurant, there is a 63.68% chance that the proportion of customers who wait longer than 25 minutes for their order will be greater than 0.1276.

a) To solve this problem, we can use the binomial probability formula:
[tex]P(X \leq 2) = \sum n=0,1,2 (nC_x) * p^n * (1-p)^{(n-x)}[/tex]
where:
X =  count  of incorrect bills in the sample
n = sample size, which is 12 in this case
p = evaluated probability of an incorrect bill, which is 1/10 or 0.1
[tex]nC_x[/tex] =  count of combinations of n things taken x at a time
Applying this formula, we can calculate the probability of no more than two incorrect bills in the sample:
[tex]P(X \leq 2) = (0C_0 * 0.1^0 * 0.9^{12} ) + (1C_1 * 0.1^1 * 0.9^{11} ) + (2C_2 * 0.1^2 * 0.9^{10} )[/tex]
[tex]P(X \leq 2) = (1 * 1 * 0.2824) + (12 * 0.1 * 0.2824) + (66 * 0.01 * 0.3487)[/tex]
[tex]P(X \leq 2) = 0.2824 + 0.3389 + 0.0229[/tex]
[tex]P(X \leq 2) = 0.6442[/tex]
Hence , the probability that no more than two of the bills will be incorrect is 0.6442, or about 64.42%. This means that if we randomly select a sample of 12 bills, there is a 64.42% chance that no more than two of them will be incorrect.
The time taken to complete a half-marathon is normally distributed, with an average time (m) of 3.15 hours and a standard deviation ([tex]\sigma[/tex]) of 0.95 hours.
To solve this problem, we need to standardize the range of times using the z-score formula:
[tex]z = (x - m) / \sigma[/tex]
where:
x = time taken to complete the half-marathon
m =  average time, which is 3.15 hours
[tex]\sigma[/tex] =  standard deviation, which is 0.95 hours
For the lower bound of 2.80 hours:
[tex]z = (2.80 - 3.15) / 0.95[/tex]
[tex]z = -0.3684[/tex]
For the upper bound of 3.25 hours:
[tex]z = (3.25 - 3.15) / 0.95[/tex]
[tex]z = 0.1053[/tex]
Applying a standard normal distribution table , we can find the area under the curve between these two z-scores:
[tex]P(-0.3684 \leq z \leq 0.1053) = 0.3557[/tex]
Then, the probability that a randomly selected runner will take between 2.80 and 3.25 hours to complete the half-marathon is 0.3557, or about 35.57%. This means that if we randomly select a runner, there is a 35.57% chance that their completion time will be between 2.80 and 3.25 hours.
To evaluate this problem, we need to standardize the range of times using the z-score formula:
[tex]z_1 = (2.35 - 3.15) / 0.95[/tex]
[tex]z_1 = -0.8421[/tex]
[tex]z_2 = (2.85 - 3.15) / 0.95[/tex]
[tex]z_2 = -0.3158[/tex]
Applying a standard normal distribution table , we can find the area under the curve between these two z-scores:
[tex]P(-0.8421 \leq z \leq -0.3158) = 0.1974[/tex]

Now for the second question
b) In the given problem, we are told that 12% of customers at a particular restaurant wait longer than 25 minutes for their order, and we assume normal distribution. We randomly select 5 customers.
The standard error for a sample proportion is given by the formula:
[tex]SE = \sqrt(p * (1 - p) / n)[/tex]
where:
p is the sample proportion, which is 0.12 in this case
n is the sample size, which is 5 in this case
Substituting the values, we get:
[tex]SE = \sqrt(0.12 * (1 - 0.12) / 5)[/tex]
SE = 0.219
Therefore, the standard error for this sample is 0.219.
To evaluate this problem, we need to standardize the sample proportion using the z-score formula:
[tex]z = (p - P) / SE[/tex]
where:
p = sample proportion, which is 0.12 in this case
P = population proportion, which is 0.1276 in this case
SE = standard error, which we calculated to be 0.219
Staging the values, we get:
[tex]z = (0.12 - 0.1276) / 0.219[/tex]
[tex]z = -0.346[/tex]
Applying a standard normal distribution table or calculator, we can find the area under the curve to the right of this z-score:
[tex]P(z > -0.346) = 0.6368[/tex]
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we find that the Trivia genre provides the highest profitability per additional user ($0.8). Therefore, the correct answer is: Trivia

To determine which genre would provide the most profitability per additional user, we need to calculate the profit per additional user for each genre.

The profit per additional user can be calculated using the following formula:

Profit per additional user = (ARPU - CPI) * Conversion rate

Let's calculate the profit per additional user for each genre:

For the Action genre:

Profit per additional user = ($2 - $1) * 50% = $0.5

For the Casino genre:

Profit per additional user = ($8 - $5) * 20% = $0.6

For the Trivia genre:

Profit per additional user = ($4 - $2) * 70% = $0.8

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A relation a on N/(a,b) e Rif and only if EN the properties that R satisfy is a. Reflexive

Checking whether R is reflexive requires seeing if (n, n) exists for every natural integer n. R is defined as "a is related to b if and only if an is an element of N," which implies that every natural number is connected to itself. R is reflexive as a result. As per definition of R, "a is related to b if and only if an is an element of N." As a result, if a and b are connected, an is an element of N. However, this does not necessarily indicate that b is a component of N. R is not symmetric.

Since a is related to b if and only if it is an element of N, applying to R, this indicates that the presence of (a, b) in R implies that an is an element of N. Nevertheless, this says nothing about whether or not (b, a) is in R. R is not symmetric or antisymmetric as a result. Since the statement "a is related to b if and only if an is an element of N," applies to R, then the presence of (a, b) in R indicates that an is an element of N. R's transitivity cannot be ascertained because this does not reveal whether or not relation (b, c) is in R.

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Complete Question:

Define a relation a on N/(a,b) e Rif and only if EN. Which of the following properties does R satisfy?

a. Reflexive

b. Symmetric

c. Antisymmetric

d. Transitive

The 95% confidence interval for the true population mean yield per plant based on the given data is estimated to be between 38.4 and 41.6 ounces. This means that we can be 95% confident that the true mean yield per plant falls within this range.

To construct the confidence interval, we can use the formula: Confidence interval = sample mean ± (critical value * standard error)In this case, the sample mean is 40 ounces per plant. The critical value can be obtained from the standard normal distribution for a 95% confidence level, which is approximately 1.96. The standard error can be calculated as the population standard deviation divided by the square root of the sample size, which in this case is 4.2 / √61.

Plugging in these values, we find that the confidence interval is 40 ± (1.96 * (4.2 / √61)), which simplifies to 38.4 to 41.6 ounces. This means that we can be 95% confident that the true mean yield per plant in the population lies within this interval.

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f'(2) = -112. To find f'(2), the derivative of f(x) with respect to x, we can use the product rule since f(x) is the product of x^5 and g(x).Let's start by finding the derivative of g(x):

g'(x) is the derivative of g(x). Given that g(2) = -3 and g'(2) = 4, we have some information about g(x) at x = 2. Now, let's use the product rule to find f'(x): f(x) = x^5 * g(x). Using the product rule: f'(x) = (x^5 * g'(x)) + (5x^4 * g(x)). Now, let's evaluate f'(2) using the given information: f'(2) = (2^5 * g'(2)) + (5 * 2^4 * g(2))

Substituting the values we know: f'(2) = (32 * 4) + (5 * 16 * -3). Simplifying: f'(2) = 128 - 240, f'(2) = -112. Therefore, f'(2) = -112.

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Distributive property also known as FOIL i.e. First, Outer, Inner and Last is an algebraic expression used to multiply two or more terms together.

Using distributive property (FOIL) to determine each product:

A. (2 + 5y)²

= (2 + 5y)² = (2 + 5y)(2 + 5y)

= 2 * 2 + 2 * 5y + 5y * 2 + 5y * 5y

= 4 + 10y + 10y + 25y²

= 4 + 20y + 25y²

B. 2(2a + 3b)²

= 2(2a + 3b)² = 2(2a + 3b)(2a + 3b)

= 2 * 2a * 2a + 2 * 2a * 3b + 2 * 3b * 2a + 2 * 3b * 3b

= 4a² + 12ab + 12ab + 18b²

= 4a² + 24ab + 18b²

C. 2x(x²+ x - 1)

= 2x(x² + x - 1) = 2x * x² + 2x * x + 2x * (-1)

= 2x³ + 2x² + (-2x)

= 2x³ + 2x² - 2x

D. 3x(x - 2y)(x + y)

= 3x(x - 2y)(x + y) = 3x * x * x + 3x * x * y + 3x * (-2y) * x + 3x * (-2y) * y

= 3x³ + 3x²y - 6xy² - 6x²y

E. (2a - 3)(3a² + 5a - 2)

= (2a - 3)(3a² + 5a - 2) = 2a * 3a² + 2a * 5a + 2a * (-2) - 3 * 3a² - 3 * 5a - 3 * (-2)

= 6a³ + 10a² - 4a - 9a² - 15a + 6

= 6a³ + (10a² - 9a²) + (-4a - 15a) + 6

= 6a³ + a² - 19a + 6

F. (x² + 2x - 1)(x² - 2x + 1)

= (x² + 2x - 1)(x² - 2x + 1) = x² * x² + x² * (-2x) + x² * 1 + 2x * x² + 2x * (-2x) + 2x * 1 - 1 * x² - 1 * (-2x) - 1 * 1

= x⁴ - 2x³ + x² + 2x³ - 4x² + 2x - x² + 2x - 1

= x⁴ - 3x² + 4x - 1

G. (2x + 3) - 4x(x + 4)(3x - 1)

= 4x(x + 4)(3x - 1) = 4x * 3x² + 4x * (-1) + 4x * 12x + 4x * 4

= 12x³ - 4x + 48x² + 16x

= (2x + 3) - 4x(x + 4)(3x - 1) = 2x + 3 - (12x³ - 4x + 48x² + 16x)

= 2x + 3 - 12x³ + 4x - 48x² - 16x

= -12x³ - 44x² - 10x + 3

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The solution to the system of equations is x = 3 and y = 4.

To solve the system of equations, we can use the method of substitution or elimination. In this case, let's use the method of elimination to find the solution.

Given the system of equations:

x + 3y = 15

4x + 2y = 30

We can multiply the first equation by 2 and the second equation by -3 to eliminate the x term:

2(x + 3y) = 2(15) --> 2x + 6y = 30

-3(4x + 2y) = -3(30) --> -12x - 6y = -90

Adding these two equations together, we get:

(2x + 6y) + (-12x - 6y) = 30 + (-90)

-10x = -60

x = 6

Substituting this value of x into the first equation, we can solve for y:

6 + 3y = 15

3y = 9

y = 3

Therefore, the solution to the system of equations is x = 6 and y = 3.

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The total surface area of the cylinder with a height of 5 cm and radius of 2 cm is 28π cm^2.

The total surface area of a cylinder can be calculated using the formula: 2πrh + 2πr^2, where r is the radius and h is the height of the cylinder.

Given that the height (h) of the cylinder is 5 cm and the radius (r) is 2 cm, we can substitute these values into the formula:

Total Surface Area = 2πrh + 2πr^2

= 2π(2)(5) + 2π(2)^2

= 20π + 8π

= 28π cm^2

Therefore, the total surface area of the cylinder with a height of 5 cm and radius of 2 cm is 28π cm^2.

The correct option is:

a. 28π cm^2

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The correct answer is  9.591, 34.170. (option-d)

To find the critical [tex]x^2[/tex] values for a sample of size 21 and a 95% confidence level, we need to use a chi-square distribution table with 20 degrees of freedom (df = n-1, where n is the sample size).

From the table, we can find the values of chi-square that correspond to a 95% level of confidence. The critical values are the chi-square values that include 95% of the area under the curve, with 2.5% in each tail.

According to the chi-square distribution table, the critical [tex]x^2[/tex] values for a 95% confidence level with 20 degrees of freedom are 9.591 and 34.170.

Answer A, B, and C are incorrect, as they do not match the critical values for a 95% confidence level with 20 degrees of freedom. It's important to note that the critical values for chi-square increase as the degrees of freedom increase, so it's crucial to use the correct degrees of freedom when looking up these values in a distribution table.(option-d)

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Based on the given information that AD is congruent to BC and AC is congruent to BD, we can prove that ED is congruent to EC.

To prove that ED is congruent to EC, we will use the concept of triangle congruence. We know that AD is congruent to BC (given) and AC is congruent to BD (given).

Now, let's consider triangle ACD and triangle BDC. According to the given information, we have AD ≅ BC and AC ≅ BD.

By the Side-Side-Side (SSS) congruence criterion, if the corresponding sides of two triangles are congruent, then the triangles are congruent.

Therefore, triangle ACD is congruent to triangle BDC.

Now, let's focus on segment DE. Since triangle ACD is congruent to triangle BDC, the corresponding parts of congruent triangles are congruent. Therefore, segment ED is congruent to segment EC.

Hence, we have proved that ED is congruent to EC using the given information about the congruence of AD with BC and AC with BD.

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The amount of interest earned is $2,400.

To calculate the interest earned, we can use the simple interest formula:

Interest = Principal × Rate × Time

Given:

Principal (P) = $96,000

Rate (R) = 5.0% or 0.05 (decimal)

Time (T) = From December 5, 2019, to May 6, 2020

First, we need to calculate the time in terms of years. The time period is approximately 5 months or 5/12 years (from December to May).

Now, we can substitute the values into the formula:

Interest = $96,000 × 0.05 × (5/12)

Calculating this expression will give us the interest earned over the given time period.

Explanation:

The interest earned can be calculated using the simple interest formula, which considers the principal amount, the interest rate, and the time period. By substituting the given values into the formula and performing the necessary calculations, we can determine the interest earned.

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The difference between the observed value and the estimated value is referred to as a B. residual.

Residual refers to the difference between the predicted value and the observed value. In the field of statistics, it's typically used in the context of regression analysis.

A residual can be calculated by subtracting the predicted value from the actual value. Residuals are utilized to determine if a model is a good fit for a given dataset, as well as to identify any patterns or trends in the data that were not accounted for by the model.

Option A: Heteroscedasticity refers to a situation where the variance of the errors in a regression model is not constant. Therefore, A is not correct.

Option B: Residual refers to the difference between the predicted value and the observed value, and hence, B is correct.

Option C: Error, also known as the noise term, refers to the difference between the true model and the observed value. In statistics, we refer to the difference between the observed value and the predicted value as a residual, as described in the previous paragraph. Hence, C is not correct.

Option D: Regression is a statistical approach that is utilized to establish a relationship between a dependent variable and one or more independent variables.

Regression models estimate the relationship between the dependent and independent variables in the form of a linear equation. It's an umbrella term that includes a variety of regression models, including linear regression, logistic regression, and so on.

The term regression is not appropriate for the definition provided in the question; hence, D is incorrect.

Therefore, the correct option is option B: Residual is the difference between the observed value and the estimated value.

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Given F=22.

For n ≥ 1, consider the code C = {0^n, 1^n}, where 0=00...0. n2 veces. We are required to prove that the given code C performs singlet dimensioning and if n=2m+1 is odd, then it is also a perfect code.

A code is a set of symbols or characters that can be arranged or combined to represent or convey information in a specific format or pattern. Codification refers to the process of converting data, information, or knowledge into codes or symbols. The main purpose of codification is to simplify the presentation, interpretation, and analysis of data. Decodification is the process of converting coded information or symbols back into data or information that can be understood or analyzed. The main purpose of decodification is to retrieve and interpret the encoded information from a given code. Now, let us prove that the given code C performs singlet dimensioning. To prove that the code C performs singlet dimensioning, we have to show that there is a unique decoding for every code word in C.

Let us consider a code word C∈C.

If C=0^n,

then the corresponding message is M=0^n.

If C=1^n,

then the corresponding message is M=1^n.

In both cases, there is only one possible message for each code word. Therefore, the code C performs singlet dimensioning. Now, let us prove that if n=2m+1 is odd, then the code C is a perfect code. To prove that the code C is a perfect code, we have to show that it is a singlet dimensioning code and that it meets the sphere-packing bound. Let us first show that the code C is a singlet dimensioning code. We have already shown that in the previous proof. Let us now show that the code C meets the sphere-packing bound. Let d(C) be the minimum distance of the code C. Since C is a singlet dimensioning code, we have d(C)=2. Let V_r be the volume of a ball of radius r in the n-dimensional Hamming space. Since the Hamming distance is the number of positions in which two n-bit strings differ, the number of balls of radius r that can be placed in the n-dimensional Hamming space is given by V_r = (nC_r)2^r. (Here, nC_r denotes the number of ways to choose r positions out of n.) If d(C)=2, then we can place only one code word in each ball of radius 1. Therefore, the maximum number of code words that can be placed in the n-dimensional Hamming space is given by N = V_1 = (nC_1)2 = 2n. We can now calculate the packing density of the code C as the ratio of the number of code words to the number of balls that can be placed in the Hamming space. This is given by δ(C) = (number of code words)/(number of balls) = 2/N = 2/2n = 1/2^(n-1).Therefore, the code C meets the sphere-packing bound, and it is a perfect code.

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To derive an equation for the time ta it takes for rock a to hit the ground in terms of v0, tb, and physical constants, we can start by considering the equations of motion for both rocks.

For rock b, we can use the equation of motion for vertical free fall:

yb = 1/2 * g * tb^2

where yb is the vertical position of rock b at time tb, g is the acceleration due to gravity, and tb is the time it takes for rock b to hit the ground.

For rock a, we know that it is launched with an initial velocity v0 and undergoes vertical free fall as well. Using the same equation of motion, we have:

ya = v0 * ta - 1/2 * g * ta^2

where ya is the vertical position of rock a at time ta and ta is the time we want to find.

Since both rocks hit the ground, their vertical positions are zero when they land. Therefore, we can set both equations equal to zero:

yb = 1/2 * g * tb^2 = 0

ya = v0 * ta - 1/2 * g * ta^2 = 0

Now we can solve the second equation for ta:

v0 * ta - 1/2 * g * ta^2 = 0

ta * (v0 - 1/2 * g * ta) = 0

Solving for ta, we find two solutions: ta = 0 (which corresponds to the time when rock a is launched) and ta = (2 * v0) / g.

Therefore, the equation for the time ta it takes for rock a to hit the ground in terms of v0, tb, and physical constants is ta = (2 * v0) / g.

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The equation 1 - 0.99c - 1.1 = 0 can be rearranged as 0.99c + 1.1 = 1. This equation shows that the function g(x) = √(0.99x + 1.1) satisfies the conditions of the main statement on convergence of the Fixed-Point Iteration (FPI) method on the interval [1, 2]. To verify this, we can plot the graphs of y = √(0.99x + 1.1), y = 1, and y = 2 for x in the range [1, 2] on the Wolfram Alpha website.

Upon plotting the graphs, we can observe that the graph of y = √(0.99x + 1.1) intersects with y = 1 and y = 2 in the interval [1, 2]. This intersection indicates that the function g(x) has a fixed point within this interval. Therefore, the Fixed-Point Iteration method is expected to converge for this problem.

To find an approximation of the fixed point p of g(2) using the Fixed-Point Iteration method, we can start with an initial approximation p₀ = 1. We can iteratively calculate the values of pₙ for n = 1, 2, 3, ... until the relative error RE(pₙpₙ₋₁) is less than 10⁻⁷.

Using the formula pₙ = √(0.99pₙ₋₁ + 1.1), we can perform the calculations as shown in the following table:

After several iterations, we can see that the relative error becomes smaller than 10⁻⁷. Therefore, the approximation pₙ is a satisfactory solution for the fixed point of g(2), which corresponds to the root of the polynomial f(x) = 24 - 0.99x² - 1.1 in the interval [1, 2].

In conclusion, the Fixed-Point Iteration method converges for the given problem, and the approximation pₙ provides a suitable estimate for the root of the polynomial within the specified tolerance.

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