a):Proofs by contradiction.
For all integers x and y, x2−4y≠2.
You can use the following fact in your proof: If n2 is an even integer, then n is also an even integer.
1(b): Computing exponents mod m.
Compute each quantity below using the methods outlined in this section. Show your steps, and remember that you should not use a calculator.
(a) 4610 mod 7
(b) 345 mod 9

Answer:

Step-by-step explanation:

You want the translation that maps f(x) = x² to g(x) = x² -6x +6.

A graph of the two functions shows g(x) is right 3 units and down 3 units from f(x).

We know the vertex of f(x) = x² is the origin (0, 0). The vertex of g(x) will tell us the translation. Putting that function in vertex form, we have ...

  g(x) = x² -6x +6

  g(x) = (x² -6x) +6

  g(x) = (x² -6x +9) +6 -9 . . . . . add and subtract 9 to complete the square

  g(x) = (x -3)² -3

Compare this to ...

  y = (x -h)² +k . . . . . . has vertex (h, k)

We see that (h, k) = (3, -3).

g(x) is translated right 3 units and down 3 units.

The value of the angle is 34 degrees

To determine the value of the variable, we need to the following;

From the information given, we have that the angles are;

73 degrees

73 degrees

x degrees

Equate the angles

73 + 73 +x = 180

collect the like terms

x = 180 - 146

subtract the values

x = 34 degrees

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Answer:245/36 π

Step-by-step explanation: you do 50/360 times π(7)^2

To convert the integral to cylindrical coordinates, we use the following conversions:

x = r cos(theta)

y = r sin(theta)

z = z

And we also replace dV with r dz dr d(theta).

The limits of integration are:

0 ≤ r ≤ 2 (since the bounds on x and y are from 0 to 2)

0 ≤ theta ≤ 2pi (since we integrate over the entire circle)

0 ≤ z ≤ 16 - r^2 (since the bounds on z are from 0 to 16 - x^2 - y^2, which in cylindrical coordinates is 16 - r^2)

Thus, the integral becomes:

∫^(2pi)_0 ∫^2_0 ∫^(16-r^2)_0 r dz dr d(theta)

Integrating with respect to z, we get:

∫^(2pi)_0 ∫^2_0 (16 - r^2)r dr d(theta)

Integrating with respect to r, we get:

∫^(2pi)_0 [8r^2 - (1/3)r^4]∣_0^2 d(theta)

= ∫^(2pi)_0 (32/3) d(theta)

= (32/3) ∫^(2pi)_0 d(theta)

= (32/3)(2pi)

= (64/3)pi

Therefore, the value of the iterated integral in cylindrical coordinates is (64/3)pi.

Answer:

£75 and £125

Step-by-step explanation:

To divide £200 in the ratio 3:5, you need to first find the total parts of the ratio, which is 3+5=8.

Then you can divide £200by 8 to find the value of each part of the ratio:

£200/8 =£25

So, each part of the ratio 3:5 is worth £25.

To find the share of each part of the ratio, you can multiply the value of each part by the corresponding ratio number:

The share of the first part (3) is £25*3=£75

The share pf the second part (5) is £25*5=£125

Therefore, the £200 is divided into the ratio of 3:5 as £75 for the first part and £125 for the second part.

Answer:

78.65

Step-by-step explanation:

The GLB of the set {x : |x - 4| < 1}  is option (d) 3

The set {x : |x - 4| < 1} can be written as the open interval (3, 5), which contains all values of x that satisfy the inequality |x - 4| < 1.

The greatest lower bound (GLB), also known as the infimum, is a concept in mathematics that applies to sets of numbers or other mathematical objects that are partially ordered.

To find the GLB (greatest lower bound) of this interval, we need to look for the greatest value that is less than or equal to every element in the interval.

Since the interval contains all real numbers greater than 3 and less than 5, its GLB is 3. Therefore, the answer is option (d) 3.

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The values for x(1) and x(-1) are both 0.

To find the values of x(1) and x(-1) given that x(t) = 2·tri(t/4)*δ(t – 2), we will evaluate the function at these points.

a. x(1):
To find the value of x(1), we need to substitute t = 1 into the function:
x(1) = 2·tri(1/4)*δ(1 - 2)

Since δ(1 - 2) is the Dirac delta function at a point different from zero (specifically, -1), its value is 0.

Therefore,
x(1) = 2·tri(1/4) * 0 = 0

b. x(-1):
To find the value of x(-1), we need to substitute t = -1 into the function:
x(-1) = 2·tri(-1/4)*δ(-1 - 2)

Again, since δ(-1 - 2) is the Dirac delta function at a point different from zero (specifically, -3), its value is 0.

Therefore,
x(-1) = 2·tri(-1/4) * 0 = 0

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The inflection point of the polynomial p(x) = [tex]x^5/20[/tex] is at x = 0. This is the only one inflection point.

To find the x-coordinates of the inflection points for the polynomial p(x) = [tex]x^5/20[/tex], we'll need to follow these steps:

1. Find the first derivative, p'(x), to determine the slope of the function.
2. Find the second derivative, p''(x), to determine the concavity of the function.
3. Set p''(x) equal to zero and solve for x to find the inflection points.

Step 1: Find the first derivative, p'(x):
p'(x) = [tex]d(x^5/20)/dx = (5x^4)/20 = x^4/4[/tex]

Step 2: Find the second derivative, p''(x):
p''(x) = [tex]d(x^4/4)/dx = (4x^3)/4 = x^3[/tex]

Step 3: Set p''(x) equal to zero and solve for x:
[tex]x^3[/tex] = 0
x = 0

There is only one inflection point, and its x-coordinate is 0.

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The percent of tools should be produced by factories B and C so that a tool picked at random in the market will have a probability of being non defective equal to 96% = 0.96 are 5% and 95% respectively.

We have three factories produce the same tool and supply it to the market. Let's consider three events defined as, A = event for tools produced by factory A

B = event for tools produced by factory B

C = event for tools produced by factory C and N be the count that tools produced by all factories is not defective.

The probability that the tools produced by factory A for the market, P( A) = 30%

= 0.30

The probability that the tools produced by factories B and C for the market, P( B and C) = 70% = 0.70

The Probability that tools are non- defective and that are produced in factory A, P( N/A) = 98%

= 0.98

The Probability that tools are non- defective and that are produced in factory B, P( N/B) = 95%

= 0.95

The Probability that tools are non- defective and that are produced in factory C, P( N/C) = 97% = 0.97

Now, since only three factories supply to the whole market, then by probability law, P(A) + P(B) + P( C) = 1

=> 0.3 + P(B) + P( C) = 1

=> P(B) = 0.7 - P(C) --(1)

We have to determine percent of tools should be produced by factories B and C that is P(C) and P(B) when probability of non defective, P(N) is 96% = 0.96. From the law of total probability law, P(N) is written by, P( N) = P( N/A) P(A) + P( N/B) P(B) + P( N/C) P( C)

=> 0.96 = 0.98 × 0.3 + 0.95 × ( 0.7 - P(C) ) + 0.97 × P(C)

=> 0.96 = 0.98 × 0.3 + 0.95 × 0.7 - 0.95 P(C) + 0.97 × P(C)

=> 0.96 = 0.98 × 0.3 + 0.95 × 0.7 - 0.95 P(C) + 0.97 × P(C)

=> 0.96 = 0.294 + 0.665 + 0.02 × P(C)

=> 0.96 = 0.959 + 0.02 × P(C)

=> 0.02 × P(C) = 0.96 - 0.959

=> 0.02 × P(C) = 0.001

=> P(C) = 0.05 = 5%

from equation (1), P(B) = 1 - P(C)

=> P( B) = 1 - 0.05 = 0.95

Hence, required percentage is 95%.

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Make the amount of money they have £100 because this makes the question easier.

Colin gets 3/10 of the money, so Colin will get £30.

After Colin has taken his share £70 will be left over.

The ratio give is 3 : 2. So 3 + 2 is equal to 5.

The amount of money left over is then divided by the ratio added in this case its 70/5.

70/5 gives us an answer of 14 .

This means that each share is equal to £14.

Emma gets the ratio of 3 so we do 3 x 14 which gives us he answer of  £42.

And if we do 3 x 2 we get the answer of £28.

We then know Dave gets £26 pounds from the £100 at the start.

26/100 converted to a percentage is 26%.

The kind of geometric transformation shown by the line of music is: Reflection

A transformation is a mathematical manipulation that moves a geometric shape or function from one space to another.

Now, a set of image transformations where the geometry of image is changed without altering its actual pixel values are commonly referred to as “Geometric transformations"

Looking at the music note, we can see that the note didn't change in size but looks to be a mirror image of its' previous notes.

Since it is a mirror image, the obvious transformation will be a reflection

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The particular solution is [tex]3x^(-1) - 1/27 + 3(9)^(-2)[/tex] based on integration.

To find a particular solution to given equatio we need to integrate twice. First, we integrate with respect to x to get [tex]-3x^(-2)[/tex].

Then, we integrate again with respect to x to get 3x^(-1) + C1, where C1 is a constant of integration.

Next, we use the initial condition 6′ 9 to solve for C1. Taking the derivative of [tex]3x^(-1) + C1[/tex], we get [tex]-3x^(-2)[/tex]. Plugging in x = 9, we get [tex]-3(9)^(-2) = -1/27[/tex].

Therefore, [tex]-1/27 = -3(9)^(-2) + C1[/tex], and solving for C1, we get[tex]C1 = -1/27 + 3(9)^(-2)[/tex].

Thus, the particular solution is [tex]3x^(-1) - 1/27 + 3(9)^(-2)[/tex].
Hi! It seems there might be a typo in your question, making it difficult to understand the exact problem you need help with. However, I will try to address the terms "solution" and "particular."

A "solution" refers to the result or answer obtained when solving an equation, problem, or system of equations. It is the value or values that satisfy the given conditions or equations.

A "particular solution" is a specific instance of a solution, usually when there are multiple solutions or when dealing with differential equations. It is a single example of a valid answer that meets the given criteria.

If you can provide more clarification on your question, I would be happy to help you find the particular solution you're looking for!

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The solution of the differential equation dx/dt = sin(t²)×t with the initial condition x(0) = 20 is [tex]x(t) = 20 + \int_{0}^{t}tsin(t^2) dt[/tex].

In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.

To solve the given differential equation dx/dt = sin(t²)×t with the initial condition x(0) = 20 and leaving the answer as a definite integral, follow these steps:

1. Identify the given differential equation:

dx/dt = sin(t²)×t.

2. Recognize the initial condition:

x(0) = 20.

3. Integrate both sides of the equation with respect to t:

∫dx = ∫sin(t²)×t dt.

4. Apply the initial condition to determine the constant of integration:

x(0) = 20.

5. Write the final solution:

[tex]x(t) = 20 + \int_{0}^{t}tsin(t^2) dt[/tex].

So, the solution is [tex]x(t) = 20 + \int_{0}^{t}tsin(t^2) dt[/tex].

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The answer is: Test the series for convergence or divergence. [infinity] Σ (-1)n+1/3n² - converges.

To test the series Σ (-1)n+1/3n² for convergence or divergence, we can use the alternating series test. This test states that if a series alternates in sign and the absolute value of its terms decreases monotonically to zero, then the series converges.

In this case, the series Σ (-1)n+1/3n² alternates in sign and the absolute value of its terms is given by 1/3n², which decreases monotonically to zero as n increases. Therefore, we can apply the alternating series test and conclude that the series converges.

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Using the graph, we can find the following:

a) The gymnast stays 4 seconds in the air.

b) The maximum height that the gymnast reaches is 10 ft.

c) After 2 seconds the gymnast starts to descend.

d) The quadratic function that models this situation is:

y = mx + c

Quantitative data can be represented and analysed graphically. In a graph, variables representing data are drawn over a coordinate plane. Analysing the magnitude of one variable's change in light of other variables' changes became simple.

Here in the question,

a. We can see from the graph that the curve above x-axis starts from the origin (0,0) and ends at (4,0) on the x-axis.

So, the gymnast stays 4 seconds in the air.

b. As we can see from the graph that it rises and then at point (2,10) it starts to descend.

So, the maximum height that the gymnast reaches is 10 ft.

c. As we can see from the graph that it rises and then at point (2,10) it starts to descend.

So, after 2 seconds the gymnast starts to descend.

d. The quadratic function that models this situation is:

y = mx + c

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First, let's identify the coordinates of the vertices of rhombus WXYZ:
W(1,5), X(6,3), Y(1,1), and Z(4,3)

Now, we will perform a reflection over the x-axis. To do this, we simply need to negate the y-coordinate of each vertex while keeping the x-coordinate the same.

Step-by-step:

1. Reflect point W(1,5):
  The x-coordinate stays the same: 1
  Negate the y-coordinate: -5
  New coordinates for W': W'(1,-5)

2. Reflect point X(6,3):
  The x-coordinate stays the same: 6
  Negate the y-coordinate: -3
  New coordinates for X': X'(6,-3)

3. Reflect point Y(1,1):
  The x-coordinate stays the same: 1
  Negate the y-coordinate: -1
  New coordinates for Y': Y'(1,-1)

4. Reflect point Z(4,3):
  The x-coordinate stays the same: 4
  Negate the y-coordinate: -3
  New coordinates for Z': Z'(4,-3)

The coordinates of the image of rhombus WXYZ under the reflection in the x-axis are W'(1,-5), X'(6,-3), Y'(1,-1), and Z'(4,-3).

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When a regression model has an interaction term and a dummy variable in statistics and probability, there will be more than one slope and more than one intercept (D)

When there is an interaction term and a dummy variable in a regression model, we can have more than one slope and more than one intercept. The interaction term allows for different slopes for different levels of the dummy variable, while the intercepts represent the expected value of the dependent variable when the dummy variable is equal to zero for each level of the interaction term.

When a regression model has an interaction term and a dummy variable, it means that the effect of one independent variable on the dependent variable varies depending on the value of the other independent variable. In other words, the slope and intercept of the regression line will change depending on the value of the dummy variable.

More specifically, the model will have one intercept and two slopes: one for the dummy variable and one for the interaction term. As a result, the relationship between the dependent variable and the independent variables will vary depending on the value of the dummy variable, which will result in different slopes and intercepts.

Therefore, the correct answer is (d): more than one slope and more than one intercept.

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Answer:

Step-by-step explanation:

...... The problem is glitched re send the image/

$5744 is Ashley’s monthly payment.

The amount of the down payment made by Ashlee is 15% of $875,000, which is:

Down payment = 0.15 x $875,000 = $131,250

The amount that Ashlee took out on a mortgage is:

Mortgage amount = Purchase price - Down payment

= $875,000 - $131,250

= $743,750

The monthly payment on a mortgage:

[tex]M = P [ i(1 + i)^n ] / [ (1 + i)^n - 1 ][/tex]

where:

M = monthly payment

P = principal amount (mortgage amount)

i = monthly interest rate (annual interest rate / 12)

n = total number of monthly payments (amortization period x 12)

In this case, the annual interest rate is 6.95% and the term is for 5 years, so we need to first calculate the monthly interest rate and the total number of monthly payments.

Monthly interest rate = 6.95% / 12 = 0.57917%

Total number of monthly payments = 20 years x 12 = 240

Substituting these values into the formula, we get:

M = $743,750 [ 0.0057917 (1 + 0.0057917)^240 ] / [ (1 + 0.0057917)^240 - 1 ]

= $5744.002

Therefore, Ashley's monthly payment on the mortgage is $5744.

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Answer:

s = 5[tex]\sqrt{6}[/tex]

Step-by-step explanation:

using the cosine ratio in the right triangle and the exact value

cos30° = [tex]\frac{\sqrt{3} }{2}[/tex] , then

cos30° = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{s}{10\sqrt{2} }[/tex] = [tex]\frac{\sqrt{3} }{2}[/tex] ( cross- multiply )

2s = 10[tex]\sqrt{2}[/tex] × [tex]\sqrt{3}[/tex] = 10[tex]\sqrt{6}[/tex] ( divide both sides by 2 )

s = 5[tex]\sqrt{6}[/tex]

Completing the inductive step and identifying where the inductive hypothesis is used, the correct multiple choice answer is: Replacing the quantity in brackets on the left-hand side of part (c) by what it equals by virtue of the Inductive hypothesis, we have (kok+") + (x + 1)² = (x + 1)² +68+4)* (+1)(x+2) as desired.

In an induction proof, the inductive step involves assuming a statement is true for some arbitrary value, say k, and then proving it's true for the next value, k+1.

Here, the inductive hypothesis corresponds to the term (kok+"). By replacing this term on the left-hand side of part (c) with its equivalent based on the inductive hypothesis, we can show that the equation holds for the (k+1) case as well. This is crucial for proving the statement using induction, as it establishes the necessary pattern for all cases.

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The number of boxes that can fit into the crate is 7 boxes.

A cuboid has a hexahedron six-faced solid shape and the volume is determined by multiplying the length by width by height. Here; the volume of the crate is determined by finding the volume of the cuboid.

Volume of the cuboid is: 2.4 m × 1.8 m × 1.1 m

Volume of the cuboid  = 4.752 m³

To cm, volume of the cuboid  = 475.2 cm³

Now, since the cube has a length of 60 cm, then the number of boxes that will fit into the crate can be estimated by dividing the volume of the cuboid shape by the length of the cube.

Thus, the number of boxes that can fit into the crate is:

= 475.2 cm/ 60 cm

= 7. 92

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By using the Concept of Permutations,There are 59,280 different ways to choose the president, vice president, and secretary from a class of 40 students.

To determine the number of different ways people can be chosen as president, vice president, and secretary from a class of 40 students:

You can use the concept of permutations.

Step 1: Choose the president. There are 40 students in the class, so there are 40 choices for the president position.

Step 2: Choose the vice president. Since the president has been chosen, there are now 39 remaining students to choose from for the vice president position.

Step 3: Choose the secretary. After selecting the president and vice president, there are 38 remaining students to choose from for the secretary position.

Now, multiply the number of choices for each position:

40 (president) x 39 (vice president) x 38 (secretary) = 59,280 different ways to choose the president, vice president, and secretary from a class of 40 students.

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If fy (a,b) = f, (a,b) = 0, does it follow that f has a local maximum or local minimum at (a,b) then it does not follows that (a,b) is a critical point of f, and (a,b) is a candidate for a local maximum or local minimum. Therefore, the correct option is option A. No. It follows that (a,b) is a critical point of f, and (a,b) is a candidate for a local maximum or local minimum.

If fy (a,b) = f, (a,b) = 0, does it follow that f has a local maximum or local minimum at (a,b) then it does not follows that (a,b) is a critical point of f, and (a,b) is a candidate for a local maximum or local minimum.

This is because fy (a,b) = f, (a,b) = 0 indicates that the partial derivatives of f with respect to both a and b are zero at (a,b), which makes (a,b) a critical point of f. However, this does not guarantee that f has a local maximum or local minimum at (a,b), as further analysis is required to determine the nature of the critical point.

Just because both partial derivatives are zero does not guarantee that (a,b) is a local maximum or minimum. It could also be a saddle point or an inflection point. To determine whether it is a local maximum, local minimum, or neither, you would need to use the second partial derivative test or examine the nature of the function around the point (a,b).

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Answer:

{-1, 0, 1, 2, 3}

Step-by-step explanation:

. 2-6y<14

-6y < 12

y > 12/-6

y > -2.

1<21-5y

-5y > -20

y < 4.

So -2 < y < 4

and the solution set of integers is

{-1, 0, 1, 2, 3}

The volume of the cone B using scale factor k = 3/2 is equal to 54π cubic centimeters.

Volume of cone A = 16π cubic centimeters

Scale factor 'k' = 3/2

Two solids A and B are similar.

This implies ,all corresponding lengths in solid B are 3/2 times the lengths in solid A.

The volume of a cone is given by the formula

V = (1/3)πr²h,

where r is the radius and h is the height.

Volume of cone A is 16π cubic centimeters.

Let r₁ and h₁ be the radius and height of cone A and r₂ and h₂ of cone B.

⇒16π = (1/3)π(r₁²)(h₁)

Multiplying both sides by 3 and dividing by π, we get,

⇒48 =(r₁²)(h₁)

Since solid A and B are similar with a scale factor of 3/2, we have,

h₂ = (3/2)h₁ and r₂= (3/2)r₁

Using these relationships, the volume of cone B is,

Volume of cone B = (1/3)π(r₂²)(h₂)

Volume of cone B = (1/3)π[(3/2)r₁]²[(3/2)h₁]

Volume of cone B = (1/3)π(9/4)(r₁²)(3/2)(h₁)

Volume of cone B = (27/8)(1/3)π(r₁²)(h₁)

Substituting Volume of cone A = 16π, we get,

Volume of cone B = (27/8)(16π)

Volume of cone B = 54π

Therefore, the volume of cone B is 54π cubic centimeters.

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The market rate of substitution between good x and y is represented by the ratio of their prices, which is px/py. Therefore, in this case, the market rate of substitution is 20/5 = 4. However, this answer choice is not listed. The closest answer choice is b. -4, which is the negative inverse of the market rate of substitution (-1/4).
The market rate of substitution between good X and Y is represented by the marginal rate of substitution (MRS), which is the ratio of the marginal utilities of both goods. In this case, we are given income (I) = 500, the price of good X (Px) = 20, and the price of good Y (Py) = 5.

To find the MRS, we can use the formula:

MRS = - (Px / Py)

Plugging in the values, we get:

MRS = - (20 / 5)

MRS = -4

So, the market rate of substitution between good X and Y is -4 (option b).

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Kathy runs at a speed of 2 mph, and Dennis rides his bike at a speed of 7 mph.

To find the speed that Kathy is running and that Dennis is riding, the first relationship is:

K = D - 5

Then use the formula for time:

Time for Kathy = Time for Dennis

4 mi / K = 14 mi / D

4 mi / (D - 5) = 14 mi / D

4D = 14(D - 5)

4D = 14D - 70

-10D = -70

D = 7 mph

Then we can find Kathy's speed :

K = 7 - 5

K = 2 mph

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The value of x from the Intersecting chords that extend outside circle is 13

From the question, we have the following parameters that can be used in our computation:

Intersecting chords that extend outside circle

Using the theorem of intersecting chords, we have

8 * (3x - 2 + 8) = 12 * (x + 5 + 12)

This gives

8 * (3x + 6) = 12 * (x + 17)

Using a graphing tool, we have

x = 13

Hence, the value of x is 13

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