Find (if possible) a nonsingular matrix P such that P-¹AP is diagonal. Verify that P-¹AP is a diagonal matrix with the eigenvalues on the main diagonal. I [53-11 2 A = 0 0 lo 2 0 7. (20%) Find a matrix P such that PT AP orthogonally diagonalizes A. Verify that PT AP gives the correct diagonal form. [9 30 01 3900 A = 0 09 3 0 3 9

1. The parametric equation of the line passing through the points (x+6, y-3) and (2, -3) can be found by subtracting the coordinates of the two points and expressing the equations in terms of a parameter t.

2. The z-intercept of the plane given by the equation (x, y, z) = (4, 6, 5) + s(2, 3, 1) + t(3, 6, 12) can be found by substituting x = 0 and y = 0 into the equation and solving for z.

1. To find the parametric equation of the line, we subtract the coordinates of the two points: (x+6, y-3) - (2, -3). This gives us the direction vector of the line. We then express the equations in terms of a parameter t to obtain the parametric equations for x and y.

2. To find the z-intercept of the plane, we substitute x = 0 and y = 0 into the equation (x, y, z) = (4, 6, 5) + s(2, 3, 1) + t(3, 6, 12). This simplifies the equation to z = 5 + s + 2t. Since we are looking for the z-intercept, we set x = 0 and y = 0, and solve for z. The resulting value of z gives us the coordinates of the z-intercept.

3a. To determine the Cartesian equation of the plane with x, y, and z-intercepts at -2, 4, and -3 respectively, we can use the intercept form of the equation of a plane, which states that (x/a) + (y/b) + (z/c) = 1, where a, b, and c are the intercepts. Substituting the given values, we obtain the Cartesian equation of the plane.

3b. The Cartesian equation of the plane that passes through the origin and is parallel to the plane in part a can be obtained by shifting the intercepts by the corresponding amount. Since the plane passes through the origin, the intercepts remain the same, resulting in the Cartesian equation of the parallel plane.

3c. The vector equation of the line of intersection between the plane in part b and the xy-plane can be found by setting z = 0 in the Cartesian equation of the plane. This simplifies the equation to an equation involving x and y, representing the line of intersection in vector form.

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To find the slopes of the secant lines, we need to calculate the average rate of change of the function over different intervals.

Given: f(x) = 23 cos(x) and x = 2

Let's fill in the table with the intervals and corresponding secant line slopes:

Interval | Slope of Secant Line

(x, π/2) | (f(π/2) - f(x)) / (π/2 - x)

We will calculate the secant line slopes for each interval using the givenfunction.

Interval | Slope of Secant Line

(x, π/2) | (f(π/2) - f(x)) / (π/2 - x)

(1, π/2) | (f(π/2) - f(1)) / (π/2 - 1)

(1.5, π/2) | (f(π/2) - f(1.5)) / (π/2 - 1.5)

(1.8, π/2) | (f(π/2) - f(1.8)) / (π/2 - 1.8)

(1.9, π/2) | (f(π/2) - f(1.9)) / (π/2 - 1.9)

(1.99, π/2) | (f(π/2) - f(1.99)) / (π/2 - 1.99)

(1.999, π/2) | (f(π/2) - f(1.999)) / (π/2 - 1.999)

(2, π/2) | (f(π/2) - f(2)) / (π/2 - 2)

Let's evaluate these values:

Interval | Slope of Secant Line

(1, π/2) | (f(π/2) - f(1)) / (π/2 - 1)

(1.5, π/2) | (f(π/2) - f(1.5)) / (π/2 - 1.5)

(1.8, π/2) | (f(π/2) - f(1.8)) / (π/2 - 1.8)

(1.9, π/2) | (f(π/2) - f(1.9)) / (π/2 - 1.9)

(1.99, π/2) | (f(π/2) - f(1.99)) / (π/2 - 1.99)

(1.999, π/2) | (f(π/2) - f(1.999)) / (π/2 - 1.999)

(2, π/2) | (f(π/2) - f(2)) / (π/2 - 2)

Now, let's substitute the function values and calculate the slopes:

Interval | Slope of Secant Line

(1, π/2) | (23 cos(π/2) - 23 cos(1)) / (π/2 - 1)

(1.5, π/2) | (23 cos(π/2) - 23 cos(1.5)) / (π/2 - 1.5)

(1.8, π/2) | (23 cos(π/2) - 23 cos(1.8)) / (π/2 - 1.8)

(1.9, π/2) | (23 cos(π/2) - 23 cos(1.9)) / (π/2 - 1.9)

(1.99, π/2) | (23 cos(π/2) - 23 cos(1.99)) / (π/2 - 1.99)

(1.999, π/2) | (23 cos(π/2) - 23 cos(1.999)) / (π/2 - 1.999)

(2, π/2) | (23 cos(π/2) - 23 cos(2)) / (π/2 - 2)

Evaluating these expressions, we get:

Interval | Slope of Secant Line

(1, π/2) | 20.2621

(1.5, π/2) | 20.4202

(1.8, π/2) | 20.4471

(1.9, π/2) | 20.4522

(1.99, π/2) | 20.4528

(1.999, π/2) | 20.4529

(2, π/2) | 20.4529

By observing the values in the table, we can make a conjecture about the slope of the tangent line at x = 2. The slopes of the secant lines seem to be approaching the value 20.4529 as the interval gets closer to x = 2. Therefore, we can conjecture that the slope of the tangent line at x = 2 is approximately 20.4529.

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The equation of the tangent line to the curve f(x) = 3x² + 1 at the point (4, 49) is y = 24x - 47.

To find the equation of the tangent line to the curve f(x) = 3x² + 1 at the point (4, 49), we need to determine the slope of the tangent line and the coordinates of the point of tangency.

First, let's find the derivative of the function f(x) with respect to x, which will give us the slope of the tangent line at any point on the curve:

f'(x) = d/dx (3x² + 1)

     = 6x

Next, let's find the slope of the tangent line at the point (4, 49) by evaluating f'(x) at x = 4:

f'(4) = 6(4)

     = 24

So, the slope of the tangent line at the point (4, 49) is 24.

Now that we have the slope, we can use the point-slope form of a linear equation to find the equation of the tangent line. The point-slope form is given by:

y - y₁ = m(x - x₁),

where (x₁, y₁) is a point on the line and m is the slope of the line.

Using the point (4, 49) and the slope 24, the equation of the tangent line is:

y - 49 = 24(x - 4).

We can simplify this equation to obtain the final form:

y - 49 = 24x - 96.

Rearranging this equation, we have:

y = 24x - 96 + 49,

y = 24x - 47.

Therefore, the equation of the tangent line to the curve f(x) = 3x² + 1 at the point (4, 49) is y = 24x - 47.

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The converse of the statement "If a figure is a square, its diagonals divide it into isosceles triangles" would be:

"If a figure's diagonals divide it into isosceles triangles, then the figure is a square."

The converse statement is not necessarily true. While it is true that in a square, the diagonals divide it into isosceles triangles, the converse does not hold. There are other shapes, such as rectangles and rhombuses, whose diagonals also divide them into isosceles triangles, but they are not squares. Therefore, the converse of the statement is not always true.

Therefore, the converse of the given statement is not true. The existence of diagonals dividing a figure into isosceles triangles does not guarantee that the figure is a square. It is possible for other shapes to exhibit this property as well.

In conclusion, the converse statement does not hold for all figures. It is important to note that the converse of a true statement is not always true, and separate analysis is required to determine the validity of the converse in specific cases.

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The average monthly sales volume S(x), in thousands of dollars, for a firm is given by the equation S(x) = 40x + 100.

We are asked to find the maximum value of S(x) and the minimum value of S(x) in terms of thousands of dollars.

(a) To find the maximum value of S(x), we look for the highest possible value of x.

Since the coefficient of x in the equation S(x) = 40x + 100 is positive, the function increases as x increases. Therefore, there is no maximum value for S(x).

(b) To find the minimum value of S(x), we look for the lowest possible value of x. Again, since the coefficient of x is positive, the function increases as x increases. Thus, there is no minimum value for S(x).

In summary, the average monthly sales volume S(x) = 40x + 100 does not have a maximum or minimum value. The function increases indefinitely as x increases, and there is no lowest or highest point in the range of the function.

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Among the given options, the only variable that is a discrete random variable is a) the number of patients in a hospital.

a. the number of patients in a hospital

A discrete random variable is a variable that can only take on a finite or countably infinite set of distinct values. In this case, the number of patients in a hospital can only be whole numbers (e.g., 0, 1, 2, 3, etc.), which is a countable set of values. Therefore, it is a discrete random variable.

b. the average amount of electricity consumed

The average amount of electricity consumed is not a discrete random variable but a continuous random variable. It can take on any real number value within a certain range, and it is not restricted to specific distinct values.

c. the amount of paint used in repainting a building

The amount of paint used in repainting a building can be measured in continuous quantities (e.g., liters or gallons). It is not restricted to specific distinct values, and therefore, it is not a discrete random variable.

d. the average weight of female athletes

Similar to the average amount of electricity consumed, the average weight of female athletes is not a discrete random variable but a continuous random variable. It can take on any real number value within a certain range and is not restricted to specific distinct values.

Among the given options, the only variable that is a discrete random variable is a) the number of patients in a hospital.

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Approximately 7.3% is the annual rate of simple interest charged on the loan amount of $4,500 for 160 days.

Kelly was charged interest of $114 for a loan amount of $4,500 that he borrowed for 160 days.

The annual rate of simple interest was charged?

Given, Principal (P) = $4500

                           Time (t) = 160 days

                            Interest (I) = $114

We can use the simple interest formula: Simple Interest formula is given as:I = P × R × tWhere,I is the InterestP is the principalR is the rate of interestt is the time in years.

To find the rate of interest, let's rearrange the formula.R = I / P × t

Substituting the values in the above formula,R = 114 / (4500 × 160 / 365) = 114 / 1.25R = 91.2%

Therefore, the annual rate of simple interest charged on the loan amount of $4,500 for 160 days was 91.2%.

Kelly borrowed a sum of $4,500 for 160 days and he was charged an interest of $114.

We need to find the annual rate of simple interest charged on the loan.

Using the simple interest formula, we get:I = P × R × tWhere,I is the InterestP is the principalR is the rate of interestt is the time in years.

Substituting the given values,I = 114, P = 4500, t = 160 days

Simple interest formula can be modified as,R = I / P × tR

                                                                   = 114 / 4500 × (160 / 365)R

                                                                 = 0.032 × 100R

                                                                 = 3.2%

As the above value is for 160 days, we need to find the annual rate of interest.

There are 365 days in a year, therefore:Annual rate = 3.2 × 365 / 160Annual rate = 7.3%

Approximately 7.3% is the annual rate of simple interest charged on the loan amount of $4,500 for 160 days.

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The solution to the given partial differential equation is u(x, y) = (1 - y) sin(2x), where x ranges from 0 to 1 and y ranges from 0 to 1. u(0, y) = u(1, y) = 0, u(x, 0) = 0, and u(x, b) = sin(2x), where b is a constant.

To solve the partial differential equation u_xx + u_2 = 0, where u_xx denotes the second derivative of u with respect to x and u_2 denotes the second derivative of u with respect to y, we assume a separable solution of the form u(x, y) = X(x)Y(y). Substituting this into the equation, we get X''(x)Y(y) + X(x)Y''(y) = 0. Dividing both sides by X(x)Y(y), we obtain X''(x)/X(x) = -Y''(y)/Y(y). Since the left side depends only on x and the right side depends only on y, both sides must be equal to a constant -λ^2. This gives us two ordinary differential equations: X''(x) + λ^2X(x) = 0 and Y''(y) + λ^2Y(y) = 0.

Solving the first equation, we find that X(x) must be a linear combination of sine and cosine functions: X(x) = A sin(λx) + B cos(λx), where A and B are constants. Applying the boundary conditions u(0, y) = u(1, y) = 0, we obtain B = 0 and λ = nπ, where n is an integer. Therefore, X(x) = A_n sin(nπx).

Solving the second equation, we find that Y(y) must be a linear combination of exponential functions: Y(y) = C e^(-λy) + D e^(λy), where C and D are constants. Applying the boundary conditions u(x, 0) = 0 and u(x, b) = sin(2x), we obtain C = 0 and λ = 2. Therefore, Y(y) = D e^(-2y).

Combining the solutions for X(x) and Y(y), we get u(x, y) = Σ A_n sin(nπx) e^(-2y). To satisfy the condition u(x, b) = sin(2x), we set n = 2 and A_2 = 1. Thus, the final solution is u(x, y) = (1 - y) sin(2x), which satisfies all the given conditions.

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the velocity of the rock when it hits the ground is approximately 43.69 m/s.

(a) To find the average velocity of the rock for the first 2 seconds, we need to calculate the displacement of the rock during that time and divide it by the time. The displacement is given as 4.91² m, and the time is 2 seconds. Therefore, the average velocity is 4.91²/2 ≈ 9.62 m/s.

(b) To determine how long it takes for the rock to hit the ground, we can use the equation for the displacement of a falling object: d = 1/2 gt², where d is the distance (88.6 m) and g is the acceleration due to gravity (9.8 m/s²). Solving for t, we get t = √(2d/g) ≈ 4.46 seconds.

(c) The average velocity during the fall can be calculated by dividing the total displacement (88.6 m) by the total time (4.46 seconds). The average velocity during the fall is 88.6/4.46 ≈ 19.88 m/s.

(d) When the rock hits the ground, its velocity will be equal to the final velocity, which can be determined using the equation v = u + gt, where u is the initial velocity (0 m/s), g is the acceleration due to gravity (9.8 m/s²), and t is the time it takes to hit the ground (4.46 seconds). Substituting the values, we get v = 0 + (9.8)(4.46) ≈ 43.69 m/s.

Therefore, the velocity of the rock when it hits the ground is approximately 43.69 m/s.

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The given differential equation is, [tex]$$\frac{d^5y}{dx^5}-\frac{d^4y}{dx^4}-\frac{d^3y}{dx^3}+\frac{d^2y}{dx^2}=3$$[/tex based on given details.

The characteristic equation is given as:

A differential equation is a type of mathematical equation that connects the derivatives of an unknown function. The function itself, as well as the variables and their rates of change, may be involved. These equations are employed to model a variety of phenomena in the domains of engineering, physics, and other sciences.

Depending on whether the function and its derivatives are with regard to one variable or several variables, respectively, differential equations can be categorised as ordinary or partial.

Finding a function that solves the equation is the first step in solving a differential equation, which is sometimes done with initial or boundary conditions. There are numerous approaches for resolving these equations, including numerical methods, integrating factors, and variable separation.

[tex]$$m^5-m^4-m^3+m^2=0$$ $$\implies m^2(m^3-m^2-m+1)=0$$[/tex]

Solving the cubic factor, $m^3-m^2-m+1=0$ by synthetic division, we get,[tex]$$(m-1)(m^2-m-1)=0$$[/tex]

Therefore the characteristic equation is given as,[tex]$$m^2(m-1)(m^2-m-1)=0$$[/tex]

Hence the general solution is given by[tex]$$y=C_1+C_2x+C_3e^x+C_4e^{-\frac{1}{2}x}(cos\frac{\sqrt{3}}{2}x+sin\frac{\sqrt{3}}{2}x)+C_5e^{-\frac{1}{2}x}(cos\frac{\sqrt{3}}{2}x-sin\frac{\sqrt{3}}{2}x)$$[/tex]

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The dimensions of the rectangle with the greatest possible area are length (L) = 2√(4/3) and width (W) = 8/3. The exact numerical value of the maximum area can be calculated as A = 2(√(4/3)) * (8/3).

To find the dimensions of the rectangle with the greatest possible area, we need to maximize the area function.

Let's denote the dimensions of the rectangle as length (L) and width (W). Since the base of the rectangle is on the x-axis, the length of the rectangle will be equal to 2 times the x-coordinate of the upper corner. So, L = 2x.

The area of the rectangle is given by the product of its length and width: A = L * W.

Substituting L = 2x, we have A = 2x * W.

To maximize the area, we can differentiate A with respect to x and set the derivative equal to zero:

[tex]dA/dx = 2(4 - x^2) - 2x(2x)\\dA/dx = 8 - 2x^2 - 4x^2\\dA/dx = 8 - 6x^2\\[/tex]

Setting dA/dx = 0, we have:

[tex]8 - 6x^2 = 0\\6x^2 = 8\\x^2 = 8/6\\x^2 = 4/3\\[/tex]

x = ±√(4/3)

Since we're interested in the dimensions of the rectangle, we take the positive value of x. So, x = √(4/3).

Substituting this value of x back into the width equation [tex]W = 4 - x^2[/tex], we have:

W = 4 - 4/3

W = 8/3

Therefore, the dimensions of the rectangle with the greatest possible area are:

Length (L) = 2x

= 2√(4/3)

Width (W) = 8/3

Please note that the area can also be calculated by substituting the value of x into the area equation A = 2x * W:

A = 2(√(4/3)) * (8/3)

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The correct symbol used for a column alias containing spaces is C. " " (quotation marks).

In SQL, when we want to assign a column alias containing spaces, we enclose the alias within double quotation marks. This is done to differentiate the alias from other SQL keywords or to handle cases where the alias includes special characters, spaces, or is case-sensitive.

For example, consider the following SQL query:

SELECT column_name AS "Column Alias"

FROM table_name;

In this query, we are selecting a column named "column_name" from the table "table_name" and assigning it the alias "Column Alias" containing spaces. By enclosing the alias within double quotation marks, we indicate to the database that it should treat the entire string as a single identifier or alias.

Using other symbols such as '', ||, or // will not achieve the desired result of creating an alias with spaces. These symbols have different meanings in SQL.

'' (two single quotation marks) typically represents an empty string or a string literal in SQL.

|| (double vertical bars) is the concatenation operator in some SQL dialects, used to combine strings or values.

// (double forward slashes) is commonly used for comments in various programming languages and does not have any special meaning for column aliases in SQL.

Therefore, the correct symbol to use for a column alias containing spaces is C. " " (quotation marks).

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For confidence interval: (a) support of the association rule A → B is 0.5 and the confidence of the association rule A → B is 0.5. (b) The rule A B given above is strong if min_sup = 30% and min_conf = 50%.

(a) Support and confidence of the association rule A → BIn order to find the support and the confidence of the association rule A → B, where A = {1, }, B = {1}, we use the formulas given below:Support(A → B) = frequency of (A, B) / NConfidence(A → B) = frequency of (A, B) / frequency of Awhere N is the number of transactions in the database.Let us first find the frequency of (A, B) and the frequency of A.Frequency of (A, B) = 1Since there is only one transaction in the database where both A and B occur, the frequency of (A, B) is 1.Frequency of A = 2The itemset {1, } occurs in two transactions T₁ and T₂. Therefore, the frequency of A is 2.

Now, let us use the above formulas to find the support and the confidence of the association rule A → B.Support(A → B) = frequency of (A, B) / N = 1 / 2 = 0.5Confidence(A → B) = frequency of (A, B) / frequency of A = 1 / 2 = 0.5Therefore, the support of the association rule A → B is 0.5 and the confidence of the association rule A → B is 0.5.

(b) Is the rule A B given above strong if min_sup = 30% and min_conf = 50%?To check whether the rule A B given above is strong if min_sup = 30% and min_conf = 50%, we compare its support and confidence with the minimum support and confidence thresholds respectively.

Minimum support threshold = 30% = 0.3Since the support of the association rule A → B is 0.5, which is greater than the minimum support threshold of 0.3, the rule satisfies the minimum support requirement.Minimum confidence threshold = 50% = 0.5Since the confidence of the association rule A → B is 0.5, which is equal to the minimum confidence threshold of 0.5, the rule satisfies the minimum confidence requirement.

Therefore, the rule A B given above is strong if min_sup = 30% and min_conf = 50%.

Hence, the correct answers are:Support = 0.50;

Confidence = 0.50.The rule A B given above is strong if min_sup = 30% and min_conf = 50%.

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(a) By multiplying [tex]$\begin{pmatrix}3 & 5 & 1 \\\ -2 & 0 & 2\end{pmatrix}$[/tex] and [tex]$\begin{pmatrix}2 & 1 \\\ 1 & 3 \\\ 4 & 1\end{pmatrix}$[/tex], the resulting matrix is [tex]$\begin{pmatrix}15 & 20 \\\ 4 & 0\end{pmatrix}$[/tex].

(b) By multiplying [tex]$\begin{pmatrix}4 & -2 \ \\ 6 & -4 \\\ 8 & -6\end{pmatrix}$[/tex]and [tex]$\begin{pmatrix}1 & 2 & 3\end{pmatrix}$[/tex], the resulting matrix is [tex]$\begin{pmatrix}0 \\ \ -2 \\ \ -4\end{pmatrix}$[/tex].

The first problem involves multiplying two matrices.

Let's denote the first matrix as A and the second matrix as B.

(a) A = [tex]$\begin{pmatrix}3 & 5 & 1 \\\ -2 & 0 & 2\end{pmatrix}$[/tex]

B = [tex]$\begin{pmatrix}2 & 1 \\\ 1 & 3 \\\ 4 & 1\end{pmatrix}$[/tex]

To compute the product AB, we need to ensure that the number of columns in A is equal to the number of rows in B.

In this case, A has 3 columns and B has 3 rows, so the multiplication is possible.

The resulting matrix C will have dimensions (2 rows x 2 columns) since the number of rows from matrix A and the number of columns from matrix B determine the dimensions of the resulting matrix.

To calculate the product, we multiply the corresponding elements of each row in A by the corresponding elements of each column in B, and sum the results.

C = AB = [tex]$\begin{pmatrix}3 & 5 & 1 \\\ -2 & 0 & 2\end{pmatrix}\begin{pmatrix}2 & 1 \\\ 1 & 3 \\\ 4 & 1\end{pmatrix}$[/tex]

Evaluating the product, we get:

C = [tex]$\begin{pmatrix}(3 \cdot 2 + 5 \cdot 1 + 1 \cdot 4) & (3 \cdot 1 + 5 \cdot 3 + 1 \cdot 1) \ \\ (-2 \cdot 2 + 0 \cdot 1 + 2 \cdot 4) & (-2 \cdot 1 + 0 \cdot 3 + 2 \cdot 1)\end{pmatrix}$[/tex]

C = [tex]$\begin{pmatrix}15 & 20 \\\ 4 & 0\end{pmatrix}$[/tex]

The resulting matrix C is [tex]$\begin{pmatrix}15 & 20 \\\ 4 & 0\end{pmatrix}$[/tex].

(b) For the second problem, we need to multiply a 3x2 matrix by a 1x3 matrix.

Let's denote the first matrix as D and the second matrix as E.

D = [tex]$\begin{pmatrix}4 & -2 \ \\ 6 & -4 \\\ 8 & -6\end{pmatrix}$[/tex]

E = [tex]$\begin{pmatrix}1 & 2 & 3\end{pmatrix}$[/tex]

To compute the product DE, we need to ensure that the number of columns in D is equal to the number of rows in E.

In this case, D has 2 columns and E has 1 row, so the multiplication is possible.

The resulting matrix F will have dimensions (3 rows x 1 column) since the number of rows from matrix D and the number of columns from matrix E determine the dimensions of the resulting matrix.

To calculate the product, we multiply the corresponding elements of each row in D by the corresponding elements of each column in E, and sum the results.

F = DE = [tex]$\begin{pmatrix}4 & -2 \\\ 6 & -4 \\\ 8 & -6\end{pmatrix}\begin{pmatrix}1 & 2 & 3\end{pmatrix}$[/tex]

Evaluating the product, we get:

F = [tex]$\begin{pmatrix}(4 \cdot 1 + -2 \cdot 2) \\ \ (6 \cdot 1 + -4 \cdot 2) \\ \ (8 \cdot 1 + -6 \cdot 2)\end{pmatrix}$[/tex]

F = [tex]$\begin{pmatrix}0 \\\ -2 \\\ -4\end{pmatrix}$[/tex]

The resulting matrix F is [tex]$\begin{pmatrix}0 \\ \ -2 \\ \ -4\end{pmatrix}$[/tex].

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The complete question is:

Compute the following problems

(a) [tex]$\begin{pmatrix}3 & 5 & 1 \\\ -2 & 0 & 2\end{pmatrix}\begin{pmatrix}2 & 1 \\\ 1 & 3 \\\ 4 & 1\end{pmatrix}$[/tex]

(b) [tex]$\begin{pmatrix}4 & -2 \\\ 6 & -4 \\\ 8 & -6\end{pmatrix}\begin{pmatrix}1 & 2 & 3\end{pmatrix}$[/tex]

The integral of 5 times the inverse sine of the square root of x with respect to x can be evaluated using the table of integrals, which gives us the result of -5x sin^(-1)(√x) + 5/2 √(1 - x) + C.

To evaluate the integral ∫[5 sin^(-1)(√x)] dx, we can use the table of integrals. According to the table, the integral of sin^(-1)(u) with respect to u is u sin^(-1)(u) + √(1 - u^2) + C. In this case, we substitute u with √x, so we have sin^(-1)(√x) as our u.

Now we can substitute u back into the equation and multiply by the coefficient 5:

∫[5 sin^(-1)(√x)] dx = 5(√x sin^(-1)(√x) + √(1 - x) + C).

This simplifies to:

-5x sin^(-1)(√x) + 5/2 √(1 - x) + C.

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The solution of the four differential equations is as follows: 1. y(2) = 1.17227, 2. y(2) = 0.999999, 3. y(2) = 2860755979.73702 and 4. y(2) = 1.057037e+106.

The solution of a differential equation is a solution that can be found by directly applying the differential equation to the initial conditions. In this case, the initial conditions are given as y(2) = -1, y(1) = 0, y(0) = 3, and y(-1) = 1. The differential equations are then solved using Euler's method, which is a numerical method for solving differential equations. Euler's method uses a step size to approximate the solution at a particular value of x. In this case, the step size is 0.5.

The results of the solution show that the value of y at x = 2 varies depending on the differential equation. The value of y is smallest for the first differential equation, and largest for the fourth differential equation. This is because the differential equations have different coefficients, which affect the rate of change of y.

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Therefore, The resulting matrix [x₁ x₂ x₃] will contain the coefficients for the first column vector of A as a linear combination of C₁, C₂, and C₃.

Let's denote the column vectors of A as A₁, A₂, and A₃. We want to find the coefficients x₁, x₂, x₃, y₁, y₂, y₃, z₁, z₂, and z₃ such that:

A₁ = C₁ * x₁ + C₂ * y₁ + C₃ * z₁

A₂ = C₁ * x₂ + C₂ * y₂ + C₃ * z₂

A₃ = C₁ * x₃ + C₂ * y₃ + C₃ * z₃

For the given values:

A = [4 8 0

-3 6 2

6 4 4]

C₁ = [1 0 0]

C₂ = [0 1 0]

C₃ = [0 0 1]

We can solve the system of equations using matrix operations. Writing the system in matrix form, we have:

[A₁ A₂ A₃] = [C₁ C₂ C₃] * [x₁ x₂ x₃

y₁ y₂ y₃

z₁ z₂ z₃]

To find the coefficients, we can compute the inverse of the coefficient matrix [C₁ C₂ C₃] and multiply it with the matrix [A₁ A₂ A₃]. The resulting matrix will have the coefficients in its columns.

Using this method, we can find the coefficients for each column vector of A as follows:

First column:

[A₁ A₂ A₃] = [1 0 0

-3 6 4

6 4 4]

Inverse of [C₁ C₂ C₃] = [1 0 0

0 1 0

0 0 1]

Multiplying the inverse by [A₁ A₂ A₃]:

[x₁ x₂ x₃] = [1 0 0

0 1 0

0 0 1] * [4 8 0

-3 6 2

6 4 4]

The resulting matrix [x₁ x₂ x₃] will contain the coefficients for the first column vector of A as a linear combination of C₁, C₂, and C₃. Similarly, you can perform the same calculations for the second and third columns to express them as linear combinations of C₁, C₂, and C₃.

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S does not have a finite infimum. Therefore, we can say that S is not bounded below. This is a contradiction to our assumption that S is bounded below.

(a) Finding the infimum and supremum of the set S:

Given the set,

S= -{2-(-2":nEN}

First, we need to find the set S. It can be found that S is {-2, 4}. The infimum of S is the greatest lower bound, and the supremum is the least upper bound of the set S. It can be seen that the infimum of S is -2 and the supremum of S is 4.

Therefore, Inf(S) = -2 and Sup(S) = 4

(b) Proving or disproving:

If a set SCR has a finite infimum, then there is a point a € S such that for any given € > 0, then inf S+ea2 inf S. Let S be a set with a finite infimum. Let α be the infimum of the set S. Take any ε > 0. Since α is the infimum of S, we can say that α ≤ s for all s ∈ S. Now, we can add ε/2 to α and get α + ε/2. It can be seen that α + ε/2 > α, and hence there is at least one element in S that is greater than α. Let us call this element as a. Now; we can say that α ≤ a < α + ε/2.

We can square both sides of the inequality and get

α^2 ≤ a^2 < (α + ε/2)^2

Rewriting this inequality as

α^2 ≤ a^2 < α^2 + αε + ε^2/4

Since α is the infimum of S, we can say that α ≤ s for all s ∈ S. Thus,α^2 ≤ s^2 for all s ∈ S.

Adding ε^2/4 to both sides of the inequality, we get

α^2 + ε^2/4 ≤ s^2 + ε^2/4 for all s ∈ S.

Therefore, we have shown that

inf S + ε^2/4 ≤ inf{s^2 + ε^2/4: s ∈ S}.

Hence proved.

However, we assumed that S does not have a finite infimum. Therefore, we can say that S is not bounded below. This is a contradiction to our assumption that S is specified below. Consequently, we can conclude that if S is a non-empty set determined below, it has a finite infimum. Therefore, we have proved that the statement is true.

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To verify that the given function y = x³ + 5 is a solution to the differential equation y' = 3x², we need to substitute the function into the differential equation and check if both sides are equal.

The given differential equation is y' = 3x².

Substituting y = x³ + 5 into the differential equation, we have:

(y)' = (x³ + 5)' = 3x².

Both sides are equal, so y = x³ + 5 is indeed a solution to the differential equation y' = 3x².

Therefore, the correct choices are:

Step to verify: B. Substitute the given function into the differential equation.

How the result can be used to verify the solution: B. Both sides of the equation are equal, which means y = x³ + 5 is a solution to the differential equation.

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The velocity of the truck is v(t) = 15t^2 + 60t. We can write the rate of change of velocity as a differential equation proportional to the acceleration.

Using t = 0 to solve for a general solution of v(t), we get v(t) = at^2 + b, where a and b are constants. We know that v(0) = 0, so b = 0. We also know that the acceleration is proportional to -2 + 60, so a = (-2 + 60)/t^2. Plugging this into our equation for v(t), we get v(t) = 15t^2 + 60t.

Here are the steps involved in solving the differential equation:

1. We start by writing the rate of change of velocity as a differential equation. This is done by taking the derivative of velocity with respect to time. In this case, the rate of change of velocity is acceleration, so we can write the differential equation as v'(t) = a(t).

2. We know that the acceleration is proportional to -2 + 60, so we can write a(t) = k(-2 + 60), where k is a constant.

3. We need to find the value of k. We can do this by using the fact that v(0) = 0. This means that when t = 0, the velocity is 0. We can plug this into the differential equation to get 0 = k(-2 + 60). This tells us that k = 1/30.

4. Now that we know the value of k, we can plug it back into the differential equation to get v'(t) = (-2 + 60)/30t^2.

5. To find the velocity, we need to integrate the differential equation. This gives us v(t) = 15t^2 + 60t + C, where C is an arbitrary constant.

6. We know that v(0) = 0, so we can plug this into the equation to get C = 0.

7. This leaves us with v(t) = 15t^2 + 60t. This is the velocity of the truck at any time t.

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Using Euler's method, we have the following iterations:

For h = 0.25: 1st iteration: [tex]x_1[/tex]=0.25, [tex]y_1[/tex]=3.25 and 2nd iteration: [tex]x_2[/tex]=0.5, [tex]y_2[/tex]=3.75

For h = 0.1: 1st iteration: [tex]x_1[/tex]=0.1, [tex]y_1[/tex]=3.2 and 2nd iteration: [tex]x_2[/tex]=0.2, [tex]y_2[/tex]=3.36

On comparing the three-decimal-place values of the two approximations, it is observed that both approximations underestimate the value of 2y(1) of the actual solution.

To approximate the solution to the initial value problem using Euler's method, we will first compute the values at two different step sizes: h = 0.25 and h = 0.1.

The initial value is y(0) = 4, and the differential equation is y' = y - 2x - 3.

For h = 0.25:

Using Euler's method, we have the following iterations:

1st iteration: [tex]x_1[/tex] = 0 + 0.25 = 0.25

[tex]y_1[/tex] = 4 + (0.25)(4 - 2(0) - 3) = 3.25

2nd iteration: [tex]x_2[/tex] = 0.25 + 0.25 = 0.5

[tex]y_2[/tex] = 3.25 + (0.25)(3.25 - 2(0.25) - 3) = 3.75

For h = 0.1:

Using Euler's method, we have the following iterations:

1st iteration: [tex]x_1[/tex] = 0 + 0.1 = 0.1

[tex]y_1[/tex] = 4 + (0.1)(4 - 2(0) - 3) = 3.2

2nd iteration: [tex]x_2[/tex] = 0.1 + 0.1 = 0.2

[tex]y_2[/tex] = 3.2 + (0.1)(3.2 - 2(0.2) - 3) = 3.36

Now, we will compare the three-decimal-place values of the two approximations ([tex]y_2[/tex] for h = 0.25 and [tex]y_2[/tex] for h = 0.1) with the value of 2y(1) of the actual solution.

Actual solution: y(x) = 5 + 2x - [tex]e^x[/tex]

y(1) = 5 + 2(1) - [tex]e^1[/tex] ≈ 5 + 2 - 2.718 ≈ 4.282

Comparing the values:

Approximation for h = 0.25: [tex]y_2[/tex] ≈ 3.75

Approximation for h = 0.1: [tex]y_2[/tex] ≈ 3.36

Actual solution at x = 1: 2y(1) ≈ 2(4.282) ≈ 8.564

We observe that both approximations underestimate the value of 2y(1) of the actual solution.

The approximation with a smaller step size, h = 0.1, is closer to the actual solution compared to the approximation with a larger step size, h = 0.25.

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The function F(x)= (x + 1)² Below is the graph of the function .From the graph, it can be observed that the function is increasing on the interval (-1, ∞) and decreasing on the interval (-∞, -1).

Analytically, the first derivative of the function will give us the intervals on which the function is increasing or decreasing. F(x)= (x + 1)² Differentiating both sides with respect to x, we get; F'(x) = 2(x + 1)The derivative is equal to zero when 2(x + 1) = 0x + 1 = 0x = -1The critical value is x = -1.Therefore, the intervals are increasing on (-1, ∞) and decreasing on (-∞, -1).

The open intervals on which the function is increasing are (-1, ∞) and the open interval on which the function is decreasing is (-∞, -1).

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The graph the equation in order to determine the intervals over which it is increasing on (2,∞) and decreasing on (−∞,2).

The graph of y = −(x + 2)² has a parabolic shape, with a minimum point of (2,−4). This means that the function is decreasing on the open interval (−∞,2) and increasing on the open interval (2,∞).

Therefore, the open intervals on which the function is increasing or decreasing can be expressed analytically as follows:

Decreasing on (−∞,2)

Increasing on (2,∞)

Hence, the graph the equation in order to determine the intervals over which it is increasing on (2,∞) and decreasing on (−∞,2).

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The regulation that is NOT covered under RESPA (Real Estate Settlement Procedures Act) is "Made bait-and-switch advertising a federal offense."

RESPA primarily focuses on regulating certain practices and disclosures in real estate transactions, particularly those related to mortgage loans. It aims to protect consumers by promoting transparency and preventing unfair practices in the settlement process.

The correct statement regarding RESPA would be: RESPA does not include a specific provision that makes bait-and-switch advertising a federal offense.

Bait-and-switch advertising generally refers to a deceptive marketing tactic where a seller advertises a product or service at a low price to attract customers, but then attempts to switch them to a different, often more expensive option. While bait-and-switch practices may be regulated under other consumer protection laws, RESPA primarily addresses issues such as escrow requirements, title insurance, kickbacks, and settlement service disclosures in real estate transactions.

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The inverse of the given matrix A is calculated to be:

A-1 = [13, 13, 13; -13, -13, -13; 6, 6, 0]

To find the inverse of a matrix, we need to use the formula A-1 = (1/det(A)) * adj(A), where det(A) represents the determinant of matrix A and adj(A) represents the adjugate of matrix A.

In this case, the given matrix A is:

A = [i, !, i; i, i, !; i, i, i]

To calculate the determinant of A, we use the formula det(A) = (i * (i * i - ! * i)) - (! * (i * i - i * i)) + (i * (i * i - i * !)), which simplifies to det(A) = i * (i^2 - i) - ! * (i^2 - i) + i * (i^2 - !).

The determinant of A is non-zero, indicating that the matrix is invertible. Therefore, we can proceed to calculate the adjugate of A, which is obtained by taking the transpose of the cofactor matrix of A.

The adjugate of A is:

adj(A) = [tex][i^2 - i, -(! * i), i^2 - !; -(! * i), i^2 - i, -(! * i); i^2 - !, -(! * i), i^2 - i][/tex]

Finally, using the formula for the inverse, we obtain:  

A-1 = (1/det(A)) * adj(A)

Substituting the values, we get:  

A-1 = [13, 13, 13; -13, -13, -13; 6, 6, 0]

To check the answer, we can multiply the original matrix A with its inverse A-1. If the result is the identity matrix, then the inverse is correct.

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1. The second player has to choose the column where they have the highest payoff given the strategies chosen by the first and third players.

The pair of strategies ((1½, ½), (1½, ½)) chosen by the first and third players is the mixed strategy with a probability 1/2 of choosing the first row and a probability 1/2 of choosing the third row.

Thus, the payoff for the second player is:

Column 1: 2*1½ + 3*½ + 2*1½ + 0*½ = 5

Column 2: 4*½ + 0*½ + 3*½ + 1*½ = 4

Column 3: 1*½ + 1*½ + 2*½ + 3*½ = 3

The second player's best response is to choose column 1.

2. Player 3 has a strictly dominated strategy.

In the fourth row, the first and second columns both have a higher payoff for player 3 than the third column.

Thus, player 3 will never choose the third column.

We can remove that column and the last row:

( (2, 4, 1) (3, 0, 1) (2, 3, 2) ) ( (4, 3, 0) (3, 4, 0) (2, 1, 1) )

The game is now a two-player game.

Player 1 chooses the row and player 2 chooses the column.

3. To find all mixed Nash equilibria, we can use the following

steps: Write the payoff matrix as a sum of matrices: one matrix for player 1, one matrix for player 2, and one matrix for the constant payoffs.

In this case: (2, 4, 1, 3, 0, 1, 2, 3, 2, 4, 3, 0, 3, 4, 0, 2, 1, 1) = (2, 4, 1) + (3, 0, 1) + (2, 3, 2) + (4, 3, 0) + (3, 4, 0) + (2, 1, 1) + (0, 1, 3) + (0, 7, 2) + (0, 0, 0)

Write down the best response condition for each player:

the probability of playing each strategy should make the other player indifferent between their strategies.

For example, for player 1:

2p1 + 3(1-p1) + 2p2 + 4(1-p2) + 3(1-p2) + 2(1-p1) = 3p1 + 4(1-p1) + p2 + 3p2 + 4(1-p2) + p2

Simplify and rearrange the best response conditions:

-p1 + 5p2 = 9-4p1 + 6p2

= 8

For player 2:

4p1 + 3(1-p1) + 1p2 + 3(1-p2) + 0p2 + 2(1-p1) = 2p1 + 4(1-p1) + 3p2 + 4(1-p2) + 1p2 + 1p1

Simplify and rearrange the best response conditions:

3p1 - 2p2 = 5-p1 + p2

= 1

Solve the system of equations:

p1 = 5/12, p2 = 23/36.

Thus, the mixed Nash equilibrium is:

Player 1 chooses the first row with a probability of 5/12 and the third row with a probability of 7/12.

Player 2 chooses the first column with a probability of 23/36 and the second column with a probability of 13/36.

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the basis for the column space of A is {[1; 2; 5]}.

To find a basis for the null space and column space of matrix A, we need to perform row reduction to its reduced row echelon form (RREF) or find the pivot columns.

Matrix A:

A = [1 3; 2 2; 5 6]

To find the basis for the null space, we solve the system of equations represented by the matrix equation A * X = 0, where X is a column vector.

A * X = [1 3; 2 2; 5 6] * [x; y] = [0; 0; 0]

We can set up the augmented matrix [A | 0] and perform row reduction:

[1 3 | 0]

[2 2 | 0]

[5 6 | 0]

Performing row reduction:R2 = R2 - 2R1

R3 = R3 - 5R1

[1 3 | 0]

[0 -4 | 0]

[0 -9 | 0]

R3 = R3 - (9/4)R2

[1 3 | 0]

[0 -4 | 0]

[0 0 | 0]

The RREF of the matrix shows that there are two pivot columns (leading 1's). Let's denote the variables corresponding to the columns as x and y.

The system of equations can be represented as:

x + 3y = 0

-4y = 0

From the second equation, we get y = 0. Substituting this into the first equation, we get x + 3(0) = 0, which simplifies to x = 0.

So the null space of A is spanned by the vector [0; 0]. Therefore, the basis for the null space is {[0; 0]}.

To find the basis for the column space, we look for the pivot columns in the RREF of the matrix A. In this case, the first column is a pivot column.

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The rate at which the volume of the cone is changing when the radius is 102 in and the height is 158 in is `148.4 in³/s`.2)

1) Given thatThe rate at which the radius of a right circular cone increases, `dr/dt = 1.4 in/s`.The rate at which the height of a right circular cone decreases, `dh/dt = -2.7 in/s`.Radius `r = 102 in`, and Height `h = 158 in`.

We need to find the rate at which the volume of the cone is changing.

Volume of a right circular cone is given as `V = (1/3)πr²h`.

Differentiating both sides with respect to t, we get;`dV/dt = (1/3)π * 2rh * (dr/dt) + (1/3)πr² * (dh/dt)`

Substituting the values of `r`, `h`, `dr/dt` and `dh/dt`, we get;dV/dt = (1/3)π * 2 * (102) * (158) * (1.4) + (1/3)π * (102)² * (-2.7)dV/dt = 9428.8 - 9280.4dV/dt = 148.4 in³/s

Therefore, the rate at which the volume of the cone is changing when the radius is 102 in and the height is 158 in is `148.4 in³/s`.2)

Given thatOne side of a triangle is increasing at a rate of `9 cm/s` and a second side is decreasing at a rate of `2 cm/s`.

If the area of the triangle remains constant, we need to find the rate at which the angle between the sides change.

Angle `θ = T/3`, one side of the triangle `a = 26 cm`, and the second side `b = 39 cm`.

Area of a triangle is given as `A = (1/2)ab sin θ`.

We know that the area of the triangle remains constant, therefore `dA/dt = 0`.

Differentiating both sides with respect to t, we get;`dA/dt = (1/2)(b sin θ) da/dt + (1/2)(a sin θ) db/dt + (1/2)ab cos θ dθ/dt = 0`

Substituting the values of `a`, `b`, `da/dt` and `db/dt`, we get;`(1/2)(39 sin(T/3))(9) - (1/2)(26 sin(T/3))(2) + (1/2)(26)(39) cos(T/3) dθ/dt = 0`

Multiplying both sides by `2/(26)(39)`, we get;`(39/52)sin(T/3)(9) - (13/26)sin(T/3)(2) + cos(T/3) dθ/dt = 0`

Substituting the value of `θ = T/3`, we get`(39/52)sin(T/3)(9) - (13/26)sin(T/3)(2) + cos(T/3) d(T/3)/dt = 0`

Simplifying, we get;`d(T/3)/dt = [(13/26)(2) - (39/52)(9)]/[cos(T/3)]`

Evaluating the values, we get;`d(T/3)/dt = -2.386 rad/s`

Therefore, the rate at which the angle between the sides change when the first side is 26 cm long, the second side is 39 cm, and the angle is T/3 is `2.386 rad/s`.

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(a) The Laplace transform of f(t) = 2t sin(3t) – 3te^5t is F(s) = (12s^2 - 30s + 30) / ((s - 3)^2 (s + 5)^2).

(b) The Laplace transform of f(t) = 6 sin(t) cos(t) is F(s) = 3 / (s^2 - 1).

(a) To find the Laplace transform of f(t) = 2t sin(3t) – 3te^5t, we apply the linearity property of the Laplace transform. We know that the Laplace transform of t^n is n! / s^(n+1), and the Laplace transform of sin(at) is a / (s^2 + a^2). Using these properties, we can find the Laplace transform of each term separately and then combine them. Applying the Laplace transform, we get F(s) = 2(3!)/(s^2 - 3^2) - 3((1!)/(s^2 - (-5)^2)).

(b) For The function f(t) = 6 sin(t) cos(t), we can use the double-angle formula for sine, sin(2t) = 2sin(t)cos(t). Rearranging this equation, we have sin(t)cos(t) = (1/2)sin(2t). We know that the Laplace transform of sin(at) is a / (s^2 + a^2), so applying the Laplace transform to (1/2)sin(2t), we get F(s) = (1/2)(2) / (s^2 + 2^2) = 1 / (s^2 - 1).

Therefore, the Laplace transforms of the given functions are:

(a) F(s) = (12s^2 - 30s + 30) / ((s - 3)^2 (s + 5)^2)

(b) F(s) = 3 / (s^2 - 1)

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To evaluate the line integral ∫ F · dr, where F = <2ay, 2³¹ + 32², 3y²>, and C is the boundary of the triangle with vertices P = (2,0,0), Q = (0,3,0), and R = (0,0,5) oriented from P to Q to R and back to P, we can split the integral into three segments: PQ, QR, and RP.

Segment PQ:
For this segment, we parameterize the line as r(t) = (2 - 2t, 3t, 0), where 0 ≤ t ≤ 1.
dr = (-2, 3, 0)dt.

Substituting r(t) and dr into F, we have F(r(t)) = <2a(3t), 2³¹ + 32², 3(3t)²> = <6at, 2³¹ + 32², 9t²>.

The integral over PQ becomes:
∫PQ F · dr = ∫[0^1] <6at, 2³¹ + 32², 9t²> · (-2, 3, 0)dt.

Segment QR:
For this segment, we parameterize the line as r(t) = (0, 3 - 3t, 5t), where 0 ≤ t ≤ 1.
dr = (0, -3, 5)dt.

Substituting r(t) and dr into F, we have F(r(t)) = <0, 2³¹ + 32², 9(3 - 3t)²> = <0, 2³¹ + 32², 9(9 - 18t + 9t²)>.

The integral over QR becomes:
∫QR F · dr = ∫[0^1] <0, 2³¹ + 32², 9(9 - 18t + 9t²)> · (0, -3, 5)dt.

Segment RP:
For this segment, we parameterize the line as r(t) = (2t, 0, 5 - 5t), where 0 ≤ t ≤ 1.
dr = (2, 0, -5)dt.

Substituting r(t) and dr into F, we have F(r(t)) = <2a(0), 2³¹ + 32², 3(0)²> = <0, 2³¹ + 32², 0>.

The integral over RP becomes:
∫RP F · dr = ∫[0^1] <0, 2³¹ + 32², 0> · (2, 0, -5)dt.

Finally, we evaluate each integral segment separately, and then sum them up to obtain the overall line integral.

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Here {u₁, u₂} is not an orthogonal set since their dot product u₁ • u₂ = 8 is not equal to zero.

To verify that {u₁, u₂} is an orthogonal set, we need to find the dot product of u₁ and u₂, denoted as u₁ • u₂. Given the vectors:

u₁ = [-3, 5, 2]

u₂ = [4, 4, 0]

The dot product is calculated as follows: u₁ • u₂ = (-3)(4) + (5)(4) + (2)(0) = -12 + 20 + 0 = 8, Since the dot product is not zero, u₁ • u₂ ≠ 0, the vectors u₁ and u₂ are not orthogonal. Therefore, {u₁, u₂} is not an orthogonal set.

In order for a set of vectors to be considered orthogonal, the dot product of every pair of vectors in the set must be zero. In this case, we computed the dot product of u₁ and u₂ and obtained a non-zero value of 8. This means that the vectors u₁ and u₂ are not orthogonal.

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