Solve, write your answer in a+bi form. (3+4i)^20

both functions are linear and increasing

If you are putting a quadratic function in the form of [tex]ax^2 + bx + c[/tex] into the quadratic formula [tex]x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex] and the b value in the function is negative, then you still write it as negative in the quadratic formula.

The reason is that the b term in the quadratic formula is being added or subtracted, depending on whether it is positive or negative.The quadratic formula is used to solve quadratic equations that are difficult to solve using factoring or other methods. The formula gives the values of x that are the roots of the quadratic equation.

The quadratic formula [tex]x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex] can be used for any quadratic equation in the form of [tex]ax^2 + bx + c = 0[/tex].

In the formula, a, b, and c are coefficients of the quadratic equation. The value of a cannot be zero, otherwise, the equation would not be quadratic.

The discriminant [tex]b^2-4ac[/tex] determines the nature of the roots of the quadratic equation. If the discriminant is positive, then there are two real roots, if it is zero, then there is one real root, and if it is negative, then there are two complex roots.

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This is the general solution to the given differential equation, where C is the arbitrary constant.

general solution to the given differential equation, we can follow these steps:

Step 1: Put the problem in standard form:

Rearrange the equation to have the derivative term on the left side and the other terms on the right side:

dy/dx - x + x^2y = x^2 - x.

Step 2: Find the integrating factor:

The integrating factor, p(x), can be found by multiplying the coefficient of the y term by -1:

p(x) = -x^2.

Step 3: Rewrite the equation using the integrating factor:

Multiply both sides of the equation by the integrating factor, p(x):

-x^2(dy/dx) + x^3y = x^3 - x^2.

Step 4: Simplify the equation further:

Rearrange the equation to isolate the derivative term on one side:

x^2(dy/dx) + x^3y = x^3 - x^2.

Step 5: Apply the integrating factor:

The left side of the equation can be rewritten using the product rule:

d/dx (x^3y) = x^3 - x^2.

Step 6: Integrate both sides:

Integrating both sides of the equation with respect to x:

∫ d/dx (x^3y) dx = ∫ (x^3 - x^2) dx.

Integrating, we get:

x^3y = (1/4)x^4 - (1/3)x^3 + C,

where C is the unknown constant.

Step 7: Solve for y(x):

Divide both sides of the equation by x^3 to solve for y(x):

y = (1/4)x - (1/3) + C/x^3.

This is the general solution to the given differential equation, where C is the arbitrary constant.

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Answer:

37 more 6th graders than seventh graders signed up for PE

Step-by-step explanation:

number of 6th graders = n = 220

number of 7th graders = m = 190

Now, 60% of 6th graders registered for PE,

Now, 60% of 220 is,

(0.6)(220) = 132

132 6th graders signed up for PE,

Also, 50% of 7th graders signed up for PE,

Now, 50% of 190 is,

(50/100)(190) = (0.5)(190) = 95

so, 95 7th graders signed up for PE,

We have to find how many more 6th graders than seventh graders signed up for PE, the number is,

Number of 6th graders which signed up for PE - Number of 7th graders which signed up for PE

which gives,

132 - 95 = 37

Hence, 37 more 6th graders than seventh graders signed up for PE

In conclusion, the eigenvalues of A^(-1)BA and ABA^(-1) are the same as the eigenvalues of B.

To show that the eigenvalues of A^(-1)BA and ABA^(-1) are the same as the eigenvalues of B, we can use the fact that similar matrices have the same eigenvalues.

First, let's consider A^(-1)BA. We know that A and A^(-1) are invertible, which means they are similar matrices. Therefore, A^(-1)BA and B are similar matrices. Since similar matrices have the same eigenvalues, the eigenvalues of A^(-1)BA are the same as the eigenvalues of B.

Next, let's consider ABA^(-1). Again, A and A^(-1) are invertible, so they are similar matrices. This means ABA^(-1) and B are also similar matrices. Therefore, the eigenvalues of ABA^(-1) are the same as the eigenvalues of B.

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Permutation is the arrangement of objects in a definite order while Combination is the arrangement of objects where the order in which the objects are selected does not matter.

How to determine this

Using the permutation term

[tex]_nP_{r}[/tex] = n!/(n-r)!

Where n = 7

r = 5

[tex]_7P_{5}[/tex] = 7!/(7-5)!

[tex]_7P_{5}[/tex] = 7 * 6 * 5 * 4 * 3 * 2 * 1/ 2 * 1

[tex]_7P_{5}[/tex] = 5040/2

[tex]_7P_{5}[/tex] = 2520

Using the combination term

[tex]_{n} C_{k}[/tex] = n!/k!(n-k)!

Where n = 12

k = 4

[tex]_{12} C_{4}[/tex] = 12!/4!(12-4)!

[tex]_{12} C_{4}[/tex] = 12!/4!(8!)

[tex]_{12} C_{4}[/tex] = 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 *4 *3 * 2 * 1/4 * 3 *2 * 1 * 8 *7 * 6 * 5 * 4 * 3 *2 * 1

[tex]_{12} C_{4}[/tex] = 479001600/24 * 40320

[tex]_{12} C_{4}[/tex] = 479001600/967680

[tex]_{12} C_{4}[/tex] = 495

Therefore, [tex]_7P_{5}[/tex] and [tex]_{12} C_{4}[/tex] are 2520 and 495 respectively

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To solve the expression 2 + 3 × 4 + 5, we follow the order of operations, also known as the PEMDAS rule (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction):

First, we perform the multiplication: 3 × 4 = 12.

Then, we add the remaining numbers: 2 + 12 + 5.

Finally, we perform the addition: 2 + 12 + 5 = 19.

Therefore, the correct solution to the expression 2 + 3 × 4 + 5 is 19, not 45. It's important to note that the order of operations dictates that multiplication and division should be performed before addition and subtraction. So, in this case, the multiplication (3 × 4) is evaluated first, followed by the addition (2 + 12), and then the final addition (14 + 5).

If you obtained a result of 45, it's possible that there was an error in the calculation or a misunderstanding of the order of operations.

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Answer:

[tex]\alpha=54^o[/tex]

Step-by-step explanation:

[tex]\alpha+36^o=90^o\\\mathrm{or,\ }\alpha=90^o-36^o=54^o[/tex]

f(x) from the given equation, we get: xv' = -2v + C²² (C₂²-1) 1 – Cx Cx - + D X

To show that the substitution u = y' leads to a Bernoulli equation, we need to substitute y' with u in the given equation:

xy" = y' + (y')³ C²² (C₂²-1) 1 – Cx Cx - + D X

Substituting y' with u, we get:

xu' = u + u³ C²² (C₂²-1) 1 – Cx Cx - + D X

Now, we have an equation in terms of x and u.

To solve this equation, we can rearrange it by dividing both sides by x:

u' = (u + u³ C²² (C₂²-1) 1 – Cx Cx - + D X) / x

Next, we can multiply both sides by x to eliminate the denominator:

xu' = u + u³ C²² (C₂²-1) 1 – Cx Cx - + D X

This is the same equation we obtained earlier after the substitution.

Now, we have a Bernoulli equation in the form of xu' = u + u^n f(x), where n = 3 and f(x) = C²² (C₂²-1) 1 – Cx Cx - + D X.

To solve the Bernoulli equation, we can use the substitution v = u^(1-n), where n = 3. This leads to the equation:

xv' = (1-n)v + f(x)

Substituting the value of n and f(x) from the given equation, we get:

xv' = -2v + C²² (C₂²-1) 1 – Cx Cx - + D X

This is now a first-order linear differential equation. We can solve it using standard techniques, such as integrating factors or separating variables, depending on the specific form of f(x).

Please note that the specific solution of this equation would depend on the exact form of f(x) and any initial conditions given. It is advisable to use appropriate techniques and methods to solve the equation accurately and obtain the solution in a desired form.

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If y=-2x+6 were changed to y= 3x+2, how would the graph of the new line compare with the first one: A. The new graph would be steeper than the original-graph, and the y-intercept would shift down 4 units.

In Mathematics and Geometry, a steeper slope simply means that the slope of a line is bigger than the slope of another line. This ultimately implies that, a graph with a steeper slope has a greater (faster) rate of change in comparison with another graph.

In this context, we can reasonably infer and logically deduce that the graph of the new line would be steeper than the original graph because a slope of 3 is greater than a slope of -2.

Also, the y-intercept would shift down 4 units;

y-intercept = 6 - 2

y-intercept = 4 units.

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Answer:

Step-by-step explanation:

Its rectangular prism trust me I did the quiz

At the beginning of the school year, Oak Hill Middle School has a total of 480 students. Out of these students, there are 270 seventh graders and 210 eighth graders.

To determine the total number of students in the school, we add the number of seventh graders and eighth graders:

270 seventh graders + 210 eighth graders = 480 students

So, the number of students matches the total given at the beginning, which is 480.

Additionally, we can verify the accuracy of the information by adding the number of seventh graders and eighth graders separately:

270 seventh graders + 210 eighth graders = 480 students

This confirms that the total number of students at Oak Hill Middle School is indeed 480.

Therefore, at the beginning of the school year, Oak Hill Middle School has 270 seventh graders, 210 eighth graders, and a total of 480 students.

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if B is a symmetric matrix, then the matrix C = [tex]\rm A^TBA[/tex] is also symmetric. The correct answer is: C. Symmetric.

It means that [tex]\rm B^T[/tex]= B, where [tex]\rm B^T[/tex] denotes the transpose of matrix B.

Now let's consider the matrix C = [tex]\rm A^TBA[/tex].

To determine whether C is symmetric or not, we need to check if C^T = C.

Taking the transpose of C:

[tex]\rm C^T = (A^TBA)^T[/tex]

[tex]\rm = A^T (B^T)^T (A^T)^T[/tex]

[tex]\rm = A^TB^TA[/tex]

Since B is symmetric ([tex]\rm B^T = B[/tex]), we have:

[tex]\rm C^T = A^TB^TA[/tex]

[tex]\rm = A^TB(A^T)^T[/tex]

[tex]\rm = A^TBA[/tex]

Comparing [tex]\rm C^T[/tex] and C, we can see that [tex]\rm C^T[/tex] = C.

As a result, if matrix B is symmetric, then matrix [tex]\rm C = A^TBA[/tex] is also symmetric. The right response is C. Symmetric.

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We have found that the solutions of the given equation satisfying x > 0, y > 0, and z > 0 are (2, 1, 2√2) and (6, 1, 2√3).

The given equation is x² + 3y² = z², and the conditions are x > 0, y > 0, and z > 0. We need to find all the solutions of this equation that satisfy these conditions.

To solve the equation, let's consider odd values of x and y, where x > y.

Let's start with x = 1 and y = 1. Substituting these values into the equation, we get:

1² + 3(1)² = z²

1 + 3 = z²

4 = z²

z = 2√2

As x and y are odd, x² is also odd. This means the value of z² should be even. Therefore, the value of z must also be even.

Let's check for another set of odd values, x = 3 and y = 1:

3² + 3(1)² = z²

9 + 3 = z²

12 = z²

z = 2√3

So, the solutions for the given equation with x > 0, y > 0, and z > 0 are (2, 1, 2√2) and (6, 1, 2√3).

Therefore, the solutions to the given equation that fulfil x > 0, y > 0, and z > 0 are (2, 1, 22) and (6, 1, 23).

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Answer:

5

Step-by-step explanation:

The required number of integer solutions is 820. To determine the number of integer solutions (x, y, z, w) to the equation x + y + z + w = 40 that satisfy x ≥ 0, y ≥ 0, z ≥ 6, and w ≥ 4, we can use the concept of generating functions.

Let's define four generating functions as follows:

f(x) = (1 + x + x^2 + ... + x^40)     -> generating function for x

g(x) = (1 + x + x^2 + ... + x^40)     -> generating function for y

h(x) = (x^6 + x^7 + x^8 + ... + x^40) -> generating function for z, since z ≥ 6

k(x) = (x^4 + x^5 + x^6 + ... + x^40) -> generating function for w, since w ≥ 4

The coefficient of x^n in the product of these generating functions represents the number of solutions (x, y, z, w) to the equation x + y + z + w = 40 with the given constraints.

We need to find the coefficient of x^40 in the product f(x) * g(x) * h(x) * k(x).

By multiplying these generating functions, we can find the desired coefficient.

Coefficient of x^40 = [x^40] (f(x) * g(x) * h(x) * k(x))

Now, let's calculate this coefficient.

Since f(x) and g(x) are the same, their product is (f(x))^2.

(x^40) is obtained by choosing x^0 from f(x), x^0 from g(x), x^34 from h(x), and x^6 from k(x).

Therefore, the coefficient of x^40 is:

[x^40] (f(x))^2 * x^34 * x^6

[x^40] (f(x))^2 * x^40

[x^0] (f(x))^2

The coefficient of x^0 in (f(x))^2 represents the number of solutions to the equation x + y + z + w = 40 with the given constraints.

To find the coefficient of x^0 in (f(x))^2, we can use the binomial coefficient.

The coefficient of x^0 in (f(x))^2 is given by:

C(40 + 2 - 1, 2) = C(41, 2) = 820

Therefore, the number of integer solutions (x, y, z, w) to the equation x + y + z + w = 40 that satisfy x ≥ 0, y ≥ 0, z ≥ 6, and w ≥ 4 is 820.

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The construction on page 734 involves a step-by-step process to solve a specific problem or demonstrate a mathematical concept.

The construction on page 734 is a methodical procedure used in mathematics to solve a particular problem or illustrate a concept. It typically involves a series of steps that are carefully chosen and executed to achieve the desired outcome.

The purpose of the construction can vary depending on the specific context, but it generally aims to provide a visual representation, demonstrate a theorem, or solve a given problem.

In the explanation provided on page 734, the construction steps are detailed and justified. Each step is crucial to the overall process and contributes to the final result.

The author likely presents the reasoning behind each step to help the reader understand the underlying principles and logic behind the construction.

It is important to note that without specific details about the construction mentioned on page 734, it is challenging to provide a more specific explanation. However, it is essential to carefully follow the given steps and their justifications, as they are likely designed to ensure accuracy and validity in the mathematical context.

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Step-by-step explanation:

There are 6 possible rolls

  4 5 6   are greater than 3

   1  and 3   are odd rolls to include in the count

     so 5 rolls out of 6  =   5/6

Answer:

the answer is -109

Step-by-step explanation:

To factorize 4x2 + 9x - 13 completely, we will make use of splitting the middle term method. Let's start by multiplying the coefficient of the x2 term and the constant

term 4(-13) = -52. Our aim is to find two

numbers that multiply to give -52 and add up to 9. The numbers are +13 and

-4Therefore, 4x2 + 13x - 4x - 13 = ONow,

group the first two terms together and the last two terms together and factorize them out4x(x + 13/4) - 1(× + 13/4) = 0(x + 13/4)(4x - 1)

= OTherefore, the fully factorised form of 4x2 + 9x - 13 is (x + 13/4)(4x - 1).

The product of [tex]\(3A^2 - A + 5I\)[/tex] is [tex]\[\begin{bmatrix}308 & -78 & -126 \\-90 & 282 & -39 \\-50 & -42 & 99\end{bmatrix}\][/tex]

To find the product 3A² - A + 5I, where A is the given matrix:

[tex]\[A = \begin{bmatrix} -4 & 2 & 3 \\ 1 & -5 & 0 \\ 2 & 3 & -1 \end{bmatrix}\][/tex]

1. A² (A squared):

A² = A.A

[tex]\[A \cdot A = \begin{bmatrix} -4 & 2 & 3 \\ 1 & -5 & 0 \\ 2 & 3 & -1 \end{bmatrix} \cdot \begin{bmatrix} -4 & 2 & 3 \\ 1 & -5 & 0 \\ 2 & 3 & -1 \end{bmatrix}\][/tex]

Multiplying the matrices, we get,

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

Simplifying, we have,

[tex]\[A \cdot A = \begin{bmatrix} 31 & -8 & -13 \\ -9 & 29 & -4 \\ -5 & -4 & 11 \end{bmatrix}\][/tex]

2. 3A²,

Multiply the matrix A² by 3,

[tex]\[3A^2 = 3 \cdot \begin{bmatrix} 31 & -8 & -13 \\ -9 & 29 & -4 \\ -5 & -4 & 11 \end{bmatrix}\]3A^2 = \begin{bmatrix} 3(31) & 3(-8) & 3(-13) \\ 3(-9) & 3(29) & 3(-4) \\ 3(-5) & 3(-4) & 3(11) \end{bmatrix}\]3A^2 = \begin{bmatrix} 93 & -24 & -39 \\ -27 & 87 & -12 \\ -15 & -12 & 33 \end{bmatrix}\][/tex]

3. -A,

Multiply the matrix A by -1,

[tex]\[-A = -1 \cdot \begin{bmatrix} -4 & 2 & 3 \\ 1 & -5 & 0 \\ 2 & 3 & -1 \end{bmatrix}\]-A = \begin{bmatrix} 4 & -2 & -3 \\ -1 & -5 & 0 \\ -2 & -3 & 1 \end{bmatrix}\][/tex]

4. 5I,

[tex]5I = \left[\begin{array}{ccc}5&0&0\\0&5&0\\0&0&5\end{array}\right][/tex]

The product becomes,

The product 3A² - A + 5I is equal to,

[tex]= \[\begin{bmatrix} 93 & -24 & -39 \\ -27 & 87 & -12 \\ -15 & -12 & 33 \end{bmatrix} - \begin{bmatrix} -4 & 2 & 3 \\ 1 & -5 & 0 \\ 2 & 3 & -1 \end{bmatrix} + \begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix}\][/tex]

[tex]= \[\begin{bmatrix}308 & -78 & -126 \\-90 & 282 & -39 \\-50 & -42 & 99\end{bmatrix}\][/tex]

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Complete question -  If

A = [tex]\left[\begin{array}{ccc}-4&2&3\\1&-5&0\\2&3&-1\end{array}\right][/tex]

find the product 3A² − A + 5I

The solution set for each case are:

1) (-∞, ∞)

2) [-1, 1]

3)  (-∞, 0]

4)  {∅}

5)  {∅}

6) [0, ∞)

The first inequality is:

1) |x| > -1

Remember that the absolute value is always positive, so the solution set here is the set of all real numbers (-∞, ∞)

2) Here we have:

0 ≤ |x|≤ 1

The solution set will be the set of all values of x with an absolute value between 0 and 1, so the solution set is:

[-1, 1]

3) |x| = -x

Remember that |x| is equal to -x when the argument is 0 or negative, so the solution set is (-∞, 0]

4) |x| = -1

This equation has no solution, so we have an empty set {∅}

5) |x| ≤ 0

Again, no solutions here, so an empty set {∅}

6) Finally, |x| = x

This is true when x is zero or positive, so the solution set is:

[0, ∞)

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The uncoded row matrices for the message "SELL CONSOLIDATED" with a row matrix size of 1 × 3 and encoding matrix A = 1 0 −1 −2 1 2 are:

1 -1 1

-2 0 0

-1 1 2

To obtain the uncoded row matrices for the given message, we need to multiply the message matrix with the encoding matrix. The message "SELL CONSOLIDATED" has a row matrix size of 1 × 3, which means it has one row and three columns.

The encoding matrix A has a size of 3 × 3, which means it has three rows and three columns.

To perform the matrix multiplication, we multiply each element in the first row of the message matrix with the corresponding elements in the columns of the encoding matrix, and then sum the results.

This process is repeated for each row of the message matrix.

For the first row of the message matrix [1 -1 1], the multiplication with the encoding matrix A gives us:

(1 × 1) + (-1 × -2) + (1 × -1) = 1 + 2 - 1 = 2

(1 × 0) + (-1 × 1) + (1 × 1) = 0 - 1 + 1 = 0

(1 × -1) + (-1 × 2) + (1 × 2) = -1 - 2 + 2 = -1

Therefore, the first row of the uncoded row matrix is [2 0 -1].

Similarly, we can calculate the remaining rows of the uncoded row matrices using the same process. Matrix multiplication and encoding matrices to gain a deeper understanding of the calculations involved in obtaining uncoded row matrices.

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The forecast error in this situation is negative, indicating that the forecast was too high. To obtain the absolute value of the error, we ignore the minus sign. Therefore, the answer is 4.67 (rounded to two decimal places).

A moving average is a forecasting technique that uses a rolling time frame of data to estimate the next time frame's value. A three-period moving average can be calculated by adding the values of the three most recent time frames and dividing by three.

Let's calculate the three-period moving averages for the given periods:

To calculate the forecast error for period 5, we use the formula: Error = Actual - Forecast. In this case, the actual value is 18.

Let's calculate the forecast error for period 5:

In this case, the forecast error is negative, indicating that the forecast was overly optimistic. We disregard the minus sign to determine the absolute value of the error. As a result, the answer is 4.67 (rounded to the nearest two decimal points).

In summary, using a three-period moving average for forecasting, the forecast for period 4 is 23.33, the forecast for period 7 is 21.33, the forecast for period 13 is 22.33, and the forecast error for period 5 is 4.67.

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The value of m is for i) No solution: m = 0

ii) Many solutions: m ≠ 0

iii) Unique solution: m = 2/9

To determine the values of m for which the system of linear equations has no solution, many solutions, or a unique solution, we need to analyze the coefficients and the resulting augmented matrix of the system.

Let's rewrite the system of equations in matrix form:

⎡ 1   4   -1 ⎤ ⎡ x ⎤   ⎡ 0 ⎤

⎢ 0  -9    5  ⎥ ⎢ y ⎥ = ⎢-1⎥

⎣ 0  -2   -m ⎦ ⎣ z ⎦   ⎣ m ⎦

Now, let's analyze the possibilities:

i) No solution:

This occurs when the system is inconsistent, meaning that the equations are contradictory and cannot be satisfied simultaneously. In other words, the rows of the augmented matrix do not reduce to a row of zeros on the left side.

ii) Many solutions:

This occurs when the system is consistent but has at least one dependent equation or redundant information. In this case, the rows of the augmented matrix reduce to a row of zeros on the left side.

iii) Unique solution:

This occurs when the system is consistent and all the equations are linearly independent, meaning that each equation provides new information and there are no redundant equations. In this case, the augmented matrix reduces to the identity matrix on the left side.

Now, let's perform row operations on the augmented matrix to determine the conditions for each case.

R2 = (1/9)R2

R3 = (1/2)R3

⎡ 1   4   -1 ⎤ ⎡ x ⎤   ⎡ 0 ⎤

⎢ 0   1 -5/9 ⎥ ⎢ y ⎥ = ⎢-1/9⎥

⎣ 0   1  -m/2⎦ ⎣ z ⎦   ⎣ m/2⎦

R3 = R3 - R2

⎡ 1   4   -1 ⎤ ⎡ x ⎤   ⎡ 0 ⎤

⎢ 0   1 -5/9 ⎥ ⎢ y ⎥ = ⎢-1/9⎥

⎣ 0   0  -m/2⎦ ⎣ z ⎦   ⎣ m/2 - 1/9⎦

From the last row, we can see that the value of m will determine the outcome of the system.

i) No solution:

If m = 0, the last row becomes [0 0 0 | -1/9], which is inconsistent. Thus, there is no solution when m = 0.

ii) Many solutions:

If m ≠ 0, the last row will not reduce to a row of zeros. In this case, we have a dependent equation and the system will have infinitely many solutions.

iii) Unique solution:

If the system has a unique solution, m must be such that the last row reduces to [0 0 0 | 0]. This means that the right-hand side of the last row, m/2 - 1/9, must equal zero:

m/2 - 1/9 = 0

Simplifying this equation:

m/2 = 1/9

m = 2/9

Therefore, for m = 2/9, the system will have a unique solution.

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The first six terms of the sequence defined by the formula an = n² + 1 are 2, 5, 10, 17, 26, and 37.

The first six terms of the sequence defined by the formula an = n² + 1 are:

a1 = 1² + 1 = 2

a2 = 2² + 1 = 5

a3 = 3² + 1 = 10

a4 = 4² + 1 = 17

a5 = 5² + 1 = 26

a6 = 6² + 1 = 37

The sequence starts with 2, and each subsequent term is obtained by squaring the term number and adding 1. For example, a2 is obtained by squaring 2 (2² = 4) and adding 1, resulting in 5. Similarly, a3 is obtained by squaring 3 (3² = 9) and adding 1, resulting in 10.

This pattern continues for the first six terms, where the term number is squared and 1 is added. The resulting sequence is 2, 5, 10, 17, 26, 37.

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The probability of both days being dry is 0.48 (48%), the probability of both days being wet is 0.08 (8%), and the probability of exactly one dry day is 0.44 (44%).

In the given scenario, we have two events: Monday being dry or wet, and Tuesday being dry or wet. We can represent this situation using a tree diagram:

```

         Dry (0.6)

       /         \

  Dry (0.8)    Wet (0.2)

    /               \

Dry (0.8)       Wet (0.4)

```

The branches represent the probabilities of each event occurring. Now we can answer the questions:

1. The probability of both days being dry is the product of the probabilities along the path: 0.6 ˣ 0.8 = 0.48 (or 48%).

2. The probability of both days being wet is the product of the probabilities along the path: 0.4ˣ  0.2 = 0.08 (or 8%).

3. The probability of exactly one dry day is the sum of the probabilities of the two mutually exclusive paths: 0.6 ˣ  0.2 + 0.4 ˣ  0.8 = 0.12 + 0.32 = 0.44 (or 44%).

By using the tree diagram and calculating the appropriate probabilities, we can determine the likelihood of different outcomes based on the given conditional and independent probabilities.

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The speed of light in diamond is approximately 1.24 x 10⁸ meters per second.

The index of refraction (n) of a given media affects how fast light travels through it. The refractive is given as the speed of light divided by the speed of light in the medium.

n = c / v

Rearranging the equation, we can solve for the speed of light in the medium,

v = c / n

The refractive index of the diamond is given to e 2.42 so we can now replace the values,

v = c / 2.42

Thus, the speed of light in diamond is approximately 1.24 x 10⁸ meters per second.

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x-3 is not a factor of f(x).Hence, the remainder when f(x) is divided by x-3 is -14, and x-3 is not a factor of f(x).

Remainder theorem and factor theorem for f(x)The given polynomial is

$f(x) = 3x^4 - 7x^3 - 1$.

To find the remainder when f(x) is divided by x-3 and to determine whether x-3 is a factor of f(x), we will use the remainder theorem and factor theorem respectively. Remainder Theorem: It states that the remainder of the division of any polynomial f(x) by a linear polynomial of the form x-a is equal to f(a).Here, we have to find the remainder when f(x) is divided by x-3.

Therefore, using remainder theorem, the remainder will be:

f(3)=3(3)^4-7(3)^3-1

= 3*81-7*27-1

= 243-189-1

= -14.

The remainder when f(x) is divided by x-3 is -14.Factor Theorem: It states that if a polynomial f(x) is divisible by a linear polynomial x-a, then f(a) = 0. In other words, if a is a root of f(x), then x-a is a factor of f(x).Here, we have to determine whether x-3 is a factor of f(x).Therefore, using factor theorem, we need to find f(3) to check whether it is equal to zero or not. From above, we have already found that f(3)=-14.The remainder is not equal to zero,

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A. Yes, someone can create a design on Desmos on the topic "zero hunger" using at least one of each of the listed functions.

B. To create a design on Desmos related to "zero hunger" using the specified functions, you can follow these steps:

1. Start by creating a set of points that form the outline of a plate or a food-related shape using a polynomial function of an even degree (greater than 2).

For example, you can use a quadratic function like y = ax^2 + bx + c to shape the plate.

Certainly! Here's an example design on Desmos related to the topic "zero hunger" using the given functions:

Polynomial function of even degree (greater than 2):

[tex]\(f(x) = x^4 - 2x^2 + 3\)[/tex]

Polynomial function of odd degree (greater than 1):

[tex]\(f(x) = x^3 - 4x\)[/tex]

Exponential function:

[tex]\(h(x) = e^{0.5x}\)[/tex]

Logarithmic function:

[tex]\(j(x) = \ln(x + 1)\)[/tex]

Trigonometric function:

[tex]\(k(x) = \sin(2x) + 1\)[/tex]

Rational function:

[tex]\(m(x) = \frac{x^2 + 2}{x - 1}\)[/tex]

Sum/difference/product/quotient of two functions:

[tex]\(n(x) = f(x) + g(x)\)[/tex]

These equations represent various functions related to zero hunger. You can plug these equations into Desmos and adjust the parameters as needed to create a design that visually represents the topic.

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Jennifer and her family hiked approximately 5 miles each day during their week-long hiking trip.

To find out how many miles were hiked each day during the week-long trip, we can divide the total distance of 34 miles by the number of days in a week, which is 7.

Distance hiked per day = Total distance / Number of days

Distance hiked per day = 34 miles / 7 days

Calculating this division gives us:

Distance hiked per day ≈ 4.8571 miles

Since it is not possible to hike a fraction of a mile, we can round this value to the nearest whole number.

Rounded distance hiked per day = 5 miles

1. The problem states that Jennifer went on a 34-mile hiking trip with her family.

2. Since they decided to hike an equal amount each day, we need to determine the distance hiked per day.

3. To find the distance hiked per day, we divide the total distance of 34 miles by the number of days in a week, which is 7.

  Distance hiked per day = Total distance / Number of days

  Distance hiked per day = 34 miles / 7 days

4. Performing the division, we get approximately 4.8571 miles per day.

5. Since we cannot hike a fraction of a mile, we need to round this value to the nearest whole number.

6. Rounding 4.8571 to the nearest whole number gives us 5.

7. Therefore, Jennifer and her family hiked approximately 5 miles each day during their week-long hiking trip.

By dividing the total distance by the number of days in a week, we can determine the equal distance hiked per day during the week-long trip. Rounding to the nearest whole number ensures that we have a practical and realistic estimate.

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