Given that tangent theta = negative 1, what is the value of secant theta, for StartFraction 3 pi Over 2 EndFraction less-than theta less-than 2 pi?
Negative StartRoot 2 EndRoot
StartRoot 2 EndRoot
0
1

By answering the above question, we may infer that Ling will thus require equation 42 feet of wallpaper border to install at the top of the three walls.

A mathematical equation links two statements and utilises the equals sign (=) to indicate equality. In algebra, an equation is a mathematical assertion that proves the equality of two mathematical expressions. For instance, in the equation 3x + 5 = 14, the equal sign separates the numbers by a gap. A mathematical formula may be used to determine how the two sentences on either side of a letter relate to one another. The logo and the particular piece of software are usually identical. like, for instance, 2x - 4 = 2.

Three walls, each 14 feet long and 12 feet tall, are being wallpapered by Ling. Just the top of the wall will be covered by the border. As a result, each wall requires a 14-foot-long wallpaper border.

The length of the border on each of the three walls must be added together to determine the overall length of wallpaper border required. So:

Whole wallpaper border length is one wall's border length times the number of walls.

Overall wallpaper border length is 14 feet multiplied by 3 walls.

42 feet is the total length of the wallpaper border.

Ling will thus require 42 feet of wallpaper border to install at the top of the three walls.

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The equation 7y = 4x - 7.

An equation is a statement that asserts the equality of two mathematical expressions. It contains an equals sign (=) and typically has variables, constants, and mathematical operations on both sides.

Equations are used in many areas of mathematics, science, and engineering to represent relationships between variables and to solve problems. In algebra, equations are often used to solve for the values of variables that make the equation true.

We have;

y = 4/7x - 1

Multiplying through by 7

7y = 4x - 7

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The solution of the given problem of unitary comes out to be the difference grows in the same proportion that we grow the minute.

The data collected from this nanosection must be multiplied by two in order to complete the task using the unitary technique. In essence, the color portions of both the unit method are either skipped or the indicated entity is set when a wanted item shows. For forty pens, a variable charge of Inr ($1.01) could have been required. It's possible that one country will have total influence over the approach taken to accomplish this.

Here,

The difference in a sustraction is equal to the remainder of the minute less the sustrancing.

The remainder between the new minuendo and the sustraendo will be greater than the previous remainder if the minuendo increases by just 10 units.

For instance, the difference is 15 if the remainder is

=> 25 – 10.

If we only increase the minimum by 10 units,

we will have

=>35 – 10 and the new difference will be 25.

In other words, the difference grows in the same proportion that we grow the minute.

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Complete question:

If in a subtraction, the minuend increases by 10 units, how much does the difference increase?

If there are 240 students in the sixth-grade class then Naomi received 336 votes.

A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0.

Bridgette and Naomi ran for sixth-grade class president.

In the election, every sixth-grade student voted for either Bridgette or Naomi.

Bridgette received 5 votes for every 7 votes Naomi received.

the first figure represents 5:7 ratio in the story.

Since Bridgette received 5 votes for every 7 votes Naomi received, Bridgette received 5/7 of the total votes and Naomi received 2/7 of the total votes.

We know that the total number of votes cast is equal to the total number of students in the class, which is 240.

Therefore, we can set up the equation:

5/7(x) + 2/7(x) = 240

Simplifying the equation, we get:

(5x + 2x)/7 = 240

7x/7 = 240

x = 240 × 7/5

x = 336

Therefore, If there are 240 students in the sixth-grade class then Naomi received 336 votes.

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The correct answer is B. 56 4/7 in². This is determined by multiplying 22/7 by the square of the radius, which is 4 in this case. The answer is equal to 56 4/7 in².

The area of a circle is calculated by the formula A=πr2, where r is the radius of the circle. Therefore, to determine the area of a circle, one must know the radius of the circle. Once the radius is known, the area of the circle can be determined by multiplying pi (π) by the radius squared (r2).

The area of a circle is equal to pi multiplied by the square of the radius of the circle. Pi is equal to 22/7, so the area of the circle can be calculated by multiplying 22/7 by the square of the radius. The correct answer is B. 56 4/7 in².

This can be calculated by multiplying 22/7 by the square of the radius, which is 4 in this case. 22/7 multiplied by 4 squared is equal to 56 4/7 in².

To summarize, the correct answer is B. 56 4/7 in2. This is determined by multiplying 22/7 by the square of the radius, which is 4 in this case. The answer is equal to 56 4/7 in².

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Answer:

Step-by-step explanation:

24 (56) *

The two real solutions of the equation are:

z=32.5/18 ± √(81+32.5z)/18

The equation 18z2-65z-72=0 can be solved by completing the square. We start by taking half of the coefficient of the squared term and squaring it:

(18/2)2 = (9)2 = 81

We then add 81 to both sides of the equation to complete the square:

18z2-65z-72+81=0+81

18z2-65z+9=81

Now, we take half of the coefficient of the z-term, subtract it from both sides of the equation, and add it in the parentheses:

18z2-(65/2)z+(65/2)z+9=81+(65/2)z

(18z-32.5)2=81+32.5z

We now have a perfect square on the left side, so we can simplify:

(18z-32.5)2=81+32.5z

18z-32.5=±√(81+32.5z)

18z=32.5±√(81+32.5z)

z=32.5/18 ± √(81+32.5z)/18

Therefore, the two real solutions of the equation are:

z=32.5/18 ± √(81+32.5z)/18

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He will have $1322.5 in 2 years.

Compound Interest is the interest calculated on the principal and the interest accumulated over the previous period. It is also the interest-based on the initial principal amount and the interest collected over the period of time.

The formula is A = P(1 + r/n)^nt

Where A = Amount compounded annually

P = Principal = $1000

r = Rate of interest = 15%

n = Number of times interest is compounded per year

t = Time in years

So, A = 1000(1 + 15%/1)^1*2

A = 1000(1 + 0.15)^2

A = 1000(1.15)^2

A = 1000(1.3225)

A = $1322.5

Therefore, the amount he will have in 2 years is $1322.5

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One way is to divide both numbers (the numerator and denominator) by 5. You will get 2/5.

Another way is to multiply both numbers (the numerator and denominator) 10. You will get 100/250.

→ the lenght = 13 square units

→ the width = 2 square units

→ the area of the rectangle = 13×2 = 26 square units

→ the area of the triangle = 26 : 2 = 13 square units

Answer:

Step-by-step explanation:

you would copy and paste another triangle on top to form a rectangle. That would make the rectangle's length 13 units and its width 2 units. the area of a rectangle is found by A=length x width which in this case is 26 units^2. A single triangle's area is 1/2 the area of the rectangle making it 26/2 which is 13.

So the area of the triangle is 13 units^2.

Answer: Your welcome!

Step-by-step explanation:

Expanded form: 7.4 x 10-4 cm

Exponential form: 7.4E-4 cm

Expanded form is when a number is written as the sum of its digits multiplied by their respective powers of 10. In this case, the number is 7.4 x 10-4, which means that 7.4 is multiplied by 10 to the power of -4.

Exponential form is when a number is written as a base number times a power of 10. In this case, the number is 7.4E-4, which means that 7.4 is multiplied by 10 to the power of -4.

The z-scores for 17, 33, and 53 are -0.82, 0.22, and 1.52, respectively.

To find the z-scores for 17, 33, and 53, we first need to calculate the mean and standard deviation of the data set.

Mean = (10 + 13 + 17 + 19 + 22 + 23 + 29 + 33 + 34 + 35 + 35 + 38 + 53 + 68)/14 = 29.57

Standard deviation = √[(10-29.57)² + (13-29.57)² + (17-29.57)² + (19-29.57)² + (22-29.57)² + (23-29.57)² + (29-29.57)² + (33-29.57)² + (34-29.57)² + (35-29.57)² + (35-29.57)² + (38-29.57)² + (53-29.57)² + (68-29.57)²]/13 = 15.37

Now we can calculate the z-scores using the formula:

z-score = (data point - mean)/standard deviation

Z-score for 17 = (17-29.57)/15.37 = -0.82

Z-score for 33 = (33-29.57)/15.37 = 0.22

Z-score for 53 = (53-29.57)/15.37 = 1.52

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The pair of functions that are not inverses of each other is (A).

To determine if two functions, g(x) and h(x), are inverses of each other, we need to check if the composition of the two functions, g(h(x)) and h(g(x)), both result in x.

A. g(x) = 2 + 9 = 11, h(x) = 1/2x - 9

g(h(x)) = g(1/2x - 9) = 2 + 9 = 11

h(g(x)) = h(11) = 1/2(11) - 9 = -3/2

Since g(h(x)) ≠ x and h(g(x)) ≠ x, the functions g(x) and h(x) are not inverses of each other.

B. g(x) = x - 1, h(x) = x + 1

g(h(x)) = g(x + 1) = (x + 1) - 1 = x

h(g(x)) = h(x - 1) = (x - 1) + 1 = x

Since g(h(x)) = x and h(g(x)) = x, the functions g(x) and h(x) are inverses of each other.

C. g(x) = 3x - 6, h(x) = 1/3x + 2

g(h(x)) = g(1/3x + 2) = 3(1/3x + 2) - 6 = x

h(g(x)) = h(3x - 6) = 1/3(3x - 6) + 2 = x

Since g(h(x)) = x and h(g(x)) = x, the functions g(x) and h(x) are inverses of each other.

D. g(x) = 3x + 4, h(x) = x - 4/3

g(h(x)) = g(x - 4/3) = 3(x - 4/3) + 4 = 3x - 4

h(g(x)) = h(3x + 4) = (3x + 4) - 4/3 = 3x + 8/3

Since g(h(x)) ≠ x and h(g(x)) ≠ x, the functions g(x) and h(x) are not inverses of each other.

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The domain of the function y = log(6 + 5x) is (-6/5, ∞).

To find the domain of the given function, y = log(6 + 5x), we need to determine the values of x for which the function is defined.

Step 1: The logarithmic function is only defined for positive values of the argument. So, we need to set the argument of the logarithm greater than zero:
6 + 5x > 0

Step 2: Solve for x:
5x > -6
x > -6/5

Step 3: The domain of the function is the set of all x values that satisfy the inequality. So, the domain is:
{x | x > -6/5}

In interval notation, the domain can be written as:
(-6/5, ∞)

Therefore, the domain of the function y = log(6 + 5x) is (-6/5, ∞).

Answer: The domain of the function y = log(6 + 5x) is (-6/5, ∞).

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The amount of cement needed is 609 cubic ft.

The amount of cement needed for the job can be calculated by finding the difference between the volume of the outer cistern and the volume of the inner cistern.

The volume of the outer cistern can be found using the formula for the volume of a cylinder, V = πr²h, where r is the radius and h is the height. The outer radius is half the outer diameter, or 10ft/2 = 5ft. The outer height is 20ft. So the volume of the outer cistern is:

V = π(5ft)²(20ft) = 1570.8 cubic ft

The volume of the inner cistern can be found using the same formula, but with the inner radius and inner height. The inner radius is the outer radius minus the thickness of the walls, or 5ft - 1ft = 4ft. The inner height is the outer height minus the thickness of the bottom, or 20ft - 1ft = 19ft. So the volume of the inner cistern is:

V = π(4ft)²(19ft) = 961.6 cubic ft

The difference between the two volumes is the amount of cement needed:

1570.8 cubic ft - 961.6 cubic ft = 609.2 cubic ft

To the nearest cubic foot, the amount of cement needed is 609 cubic ft.

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1. A person should choose one family pizza.

2. The area of two medium pizza < area of 1 large pizza.

3. The area of two medium pizza is 1413.8 square cm and the area of one large pizza is 1661.9 square cm.

What is area?

An object's area is how much space it takes up in two dimensions. It is the measurement of the quantity of unit squares that completely cover the surface of a closed figure.

I would one family pizza.

I would choose one family because the total area of two medium pizzas is less than the area of a family pizza.

This means that the two medium pizzas will have less pizza to share between the same number of people, so everyone will get a smaller slice of pizza.

To calculate the areas of the pizzas, we need to use the formula for the area of a circle -

Area of a circle = π × (radius)²

For the family pizza with a diameter of 46 cm, the radius is 23 cm. Therefore, the area of the family pizza is -

Area of family pizza = π × (23 cm)² ≈ 1661.9 square cm

For two medium pizzas with a diameter of 30 cm, the radius is 15 cm. Therefore, the area of each medium pizza is -

Area of each medium pizza = π × (15 cm)² ≈ 706.9 square cm

The total area of two medium pizzas is -

Total area of two medium pizzas = 2 × (Area of each medium pizza) ≈ 1413.8 square cm

Since the total area of two medium pizzas is less than the area of the family pizza, it means that two medium pizzas have less pizza to share between the same number of people.

Therefore, choosing one family pizza would be a better option.

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A family pizza or two medium?

* Each medium pizza is 30 cm in diameter

* The family pizza has only a diameter of 46 cm

1. What do you choose?

2. Why?

3. Please conclude your facts

According to the information the perimeter of the triangle is 69.19cm

To find the perimeter of a triangle, it is necessary to add the length of its sides. However, we do not know the length of its height, so it is necessary to apply the Pythagorean theorem to find the length of the height. In this case we must apply the following variant:

a = [tex]\sqrt{c^{2} - b^{2} }[/tex]

So we just have to replace the values and get the result.

Then we can infer that the base measures 16cm and the hypotenuse measures 29cm; the formula would be like this:

[tex]a = \sqrt{29^{2} - 16^{2} } \\a = \sqrt{841 - 256}\\a = \sqrt{585} \\a = 24.18[/tex]

According to the above, we can infer that the height of this triangle is 24.18cm. Now to find the perimeter of the figure we must add the length of its sides:

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The equation that defines "f"  for the amount of medicine in the blood stream is f(x) = 0.75ˣ * f(0).

The mathematical function known as exponential decay can be used to illustrate a quantity's progressive decline over time. The quantity that is lost or decays in each unit of time is proportionate to the amount that is still present, which is defined by a constant relative rate of change. Exponential decay is frequently seen in physical, chemical, and biological systems, where it explains the ageing of a population or the deterioration of radioactive isotopes, soil minerals, drugs, or nutrients over time.

The following equation to model the situation:

[tex]f(t) = f(0) * e^{(-kt)}[/tex]

where, f(0) is the initial amount of medicine

[tex]f(t) = f(0) * e^{(-k*1)} = 0.75\\\\f(0) * e^{-k} = 0.75\\\\[/tex]

Taking ln on both sides:

-k = ln(0.75 / f(0))

k = -ln(0.75 / f(0))

Substituting this value of "k" back into the equation for "f(x)", we get:

f(t) = f(0) * (0.75 / f(0))ˣ

Simplifying this equation further, we get:

f(t) = 0.75ˣ * f(0)

Therefore, the equation that defines "f" is f(x) = 0.75ˣ * f(0).

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The complete question is:

The probability that a randomly selected student is female or a physics major would be 0.65.

Probability refers to a possibility that deals with the occurrence of random events.

The probability of all the events occurring needs to be 1.

The probability that a randomly selected student is female or a physics major can be calculated:

P(PM or F) = P(PM) + P(F) - P(PMF)

Where P(PM or F) = Probability of student selected is Physics Major or Female needed to find.

P(PM) = Probability of student selected is Physics Major

P(PM) = Number of Physics Major / Total number of students in the Physics class

P(PM)  = 14 / 40 = 0.35

P(F) = Probability of student selected is Female = Number of female students in the Physics class / Total number students in the Physics class = 18 / 40 = 0.45

P(PMF) = Probability of student selected is Physics Major and Female  = 6 / 40 = 0.15

Substituting the values into equation (1);

P(PM or F) = 0.35 + 0.45 - 0.15

P(PM or F) = 0.65

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Based on the information in the table, we can infer that the number that correctly replaces A is 880.

To find the number that replaces A we must analyze the information in the table. In this case we must find the difference between the distances to the destination in 1, 2, 3, 5 hours.

In this case we can subtract the distance of hour 1 from that of hour 2 to identify the difference.

So, the difference would be 60. To check it we can test with the difference between hour 2 and 3

So we can infer that the distance difference between each hour is 60 miles. Therefore, the value of A would be the value of the distance after 3 hours (940) minus 60.

So the correct option would be option B, 880.

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Yes, this is exponential growth. Exponential growth occurs when the rate of increase is proportional to the current amount. In this case, the retail revenue from shopping on the Internet is projected to grow at a rate of 56% per year, meaning that the amount of growth each year is 56% of the current amount. This is an example of exponential growth because the rate of growth is proportional to the current amount.

To further illustrate this point, let's say that the retail revenue from shopping on the Internet in year 1 is $100. In year 2, it would grow by 56% to $156 ($100 + ($100 * 0.56)). In year 3, it would grow by 56% again to $243.36 ($156 + ($156 * 0.56)). As you can see, the amount of growth each year is proportional to the current amount, which is the definition of exponential growth.

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For the inequality y ≤ -2/3x + 6 the graph is plotted with the points (0, 6) and (9, 0).

What is an inequality?

In Algebra, an inequality is a mathematical statement that uses the inequality symbol to illustrate the relationship between two expressions. An inequality symbol has non-equal expressions on both sides. It indicates that the phrase on the left should be bigger or smaller than the expression on the right, or vice versa.

The inequality y ≤ -2/3x + 6 is already in slope-intercept form (y = mx + b), where the slope is -2/3 and the y-intercept is 6.

To find some possible solutions for this inequality, we can choose any value of x and then solve for y.

Let's choose x = 0 -

y ≤ -2/3(0) + 6

y ≤ 6

So when x = 0, any y value less than or equal to 6 will satisfy the inequality.

This gives us the solution set -

{(x, y) | y ≤ 6}

Now choose a value of y to find another point that satisfies the inequality. Let's choose y = 0 -

0 ≤ -2/3x + 6

2/3x ≤ 6

x ≤ (6 × 2)/3

x ≤ 9

So when y = 0, any x value less than or equal to 0 will satisfy the inequality.

Therefore, some possible solutions for the inequality y ≤ -2/3x + 6 are -

(0, 6) and (9, 0).

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4312 liters of water need to be filled to calculate the volume of water occupying 70% volume of the tank.

What is volume ?

Volume is the amount of space occupied by an object or a substance. It is a measure of the amount of three-dimensional space an object or a substance occupies. The volume of an object or substance is typically measured in cubic units, such as cubic meters (m³), cubic centimeters (cm³), or cubic feet (ft³).

A) To calculate the area of glass for the tank without lid, we need to find the total surface area of the rectangular box.

The front and back sides have dimensions 1.1 m x 7 dm = 1.1 m x 0.7 m = 0.77 m² each.

The top and bottom sides have dimensions 1.1 m x 8 m = 8.8 m² each.

The left and right sides have dimensions 7 dm x 8 m = 0.7 m x 8 m = 5.6 m² each.

Therefore, the total surface area of glass for the tank without lid is:

0.77 m² + 0.77 m² + 8.8 m² + 8.8 m² + 5.6 m² + 5.6 m² = 30.6 m²

B) To calculate the volume of water occupying 70% of the tank, we first need to calculate the volume of the tank.

The volume of a rectangular box is found by multiplying its length, width, and height:

Volume = length x width x height

We have:

length = 1.1 m

width = 8 m

height = 7 dm = 0.7 m

Volume = 1.1 m x 8 m x 0.7 m = 6.16 m³

Next, we need to find 70% of the volume of the tank:

70% of 6.16 m³ = 0.7 x 6.16 m³ = 4.312 m³

Finally, we need to convert cubic meters to liters, using the conversion factor given:

1 m³ = 1000 L

Therefore,

4.312 m³ x 1000 L/m³ = 4312 L

Therefore, 4312 liters of water need to be filled to calculate the volume of water occupying 70% volume of the tank.

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Earthquake in Turkey was 1000 times much stronger, compared to a magnitude 5​ earthquake

The Richter scale (also known as the Richter magnitude scale) assigns  magnitude numbers to quantify the energy released by an earthquake. Developed by Charles Richter in the 1930s, the Richter scale is a base 10 logarithmic scale that defines magnitude as the logarithm of the ratio of  seismic wave amplitudes to smaller amplitudes.

As measured by seismographs, an earthquake measuring 5.0 on the Richter scale has 10 times more shaking amplitude than a 4.0 earthquake.

Given,

Magnitude 6 is 10² = 100 times greater than magnitude 4.

Magnitude 4 is 10 times greater than magnitude 4.

By the above data, 1 digit greater magnitude means 10 times powerful.

Earthquake on turkey is of 8 magnitude

∵ Difference of 8 magnitude and 5 magnitude = 8 - 5 = 3

∴ 8 magnitude earthquake is 1000 times greater.

Hence, 1000 times much stronger was earthquake in Turkey, compared to a magnitude 5​ earthquake.

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The basic subtraction of rational expression simplified is (7)/(12d).

To subtract these rational expressions, we need to find a common denominator. The least common denominator (LCD) of 4d and 6d is 12d. We can then rewrite the expressions with the LCD as the denominator:

(3)/(4d) = (3 * 3)/(4d * 3) = (9)/(12d)
(1)/(6d) = (1 * 2)/(6d * 2) = (2)/(12d)

Now we can subtract the numerators and keep the same denominator:

(9)/(12d) - (2)/(12d) = (9 - 2)/(12d) = (7)/(12d)

We can simplify this fraction by dividing both the numerator and denominator by their greatest common factor (GCF), which is 1:

(7)/(12d) = (7/1)/(12d/1) = (7)/(12d)

So the final answer is (7)/(12d).

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a. The inverse of a linear transformation T₁,  which is obtained is:

T₁⁻¹(x,y) = (37x - 9y)/2 , (-8x + 2y)/2

b. The inverse of T₂ does not exist.

c. Entering the values of T₁ into the equations gives:  

d. Entering the values of T₂ into the equations gives:  

a. To find the inverse of a linear transformation T₁ , we need to solve the system of equations:

2x + 9y = a
8x + 37y = b

We can use the determinant of the matrix associated with this system to find the inverse:

[tex]\left[\begin{array}{cc}2&9\\8&37\end{array}\right][/tex]

The determinant Δ is:

Δ = (2)(37) - (9)(8)

Δ = 74 - 72

Δ =  2

The inverse of T₁ is:

T₁⁻¹(a,b) = (1/2)(|37 -9| |a|) = (37a - 9b)/2 , (-8a + 2b)/2

T₁⁻¹(x,y) = (37x - 9y)/2 , (-8x + 2y)/2

b. To find the inverse of a linear transformation T₂, we need to solve the system of equations:

x + z = a
x + y = b
y + z = c

We can use the determinant of the matrix associated with this system to find the inverse:

[tex]\left[\begin{array}{ccc}1&0&1\\1&1&0\\0&1&1\end{array}\right][/tex]

The determinant Δ is:

Δ =  (1)(1)(1) + (0)(0)(1) + (1)(1)(0) - (1)(0)(0) - (1)(1)(1) - (0)(1)(1)

Δ = 1 - 1

Δ = 0

Since the determinant is 0, the inverse of T₂ does not exist.

c. To find x and y given T₁(x,y) = (5,-1), we can plug in the values into the equations for T₁:

2x + 9y = 5
8x + 37y = -1

We can use substitution to solve for x and y. From the first equation, we can solve for x:

x = (5 - 9y)/2

Plugging this into the second equation:

8(5 - 9y)/2 + 37y = -1

Simplifying:

20 - 36y + 37y = -2

y = -22

Plugging this back into the first equation to solve for x:

x = (5 - 9(-22))/2

x = 101/2

d. To find x, y, and z given T₂(x,y,z) = (5,-3,-1), we can plug in the values into the equations for T₂:

x + z = 5
x + y = -3
y + z = -1

We can use substitution to solve for x, y, and z. From the first equation, we can solve for x:

x = 5 - z

Plugging this into the second equation:

5 - z + y = -3

Simplifying:

y = -8 + z

Plugging this back into the third equation:

-8 + z + z = -1

2z = 7

z = 7/2

Plugging this back into the equations to solve for x and y:

x = 5 - 7/2

x = 3/2

y = -8 + 7/2

y = -9/2

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To make the trinomial y2 - 18y a perfect square, we need to add a constant that is equal to (18/2)^2 = 81.

This is because a perfect square trinomial is in the form (x + a)^2 = x^2 + 2ax + a^2. In this case, x = y and 2a = -18, so a = -9 and a^2 = 81. So, the trinomial becomes y2 - 18y + 81, which is a perfect square.

To factor the trinomial, we can use the formula (x + a)^2 = x^2 + 2ax + a^2, where x = y and a = -9. So, the trinomial y2 - 18y + 81 can be factored as (y - 9)^2.

Therefore, the proper constant to add to the binomial y2 - 18y to make it a perfect square trinomial is 81, and the factored form of the trinomial is (y - 9)^2.

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The matrix \( A \) is not diagonalizable as an element of \( M_{3 \times 3}(\mathbb{R}) \), because there does not exist a basis of \( \mathbb{R}^{3} \) consisting of eigenvectors of \( A \).

The matrix \(A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -1 & 1\end{array}\right] \) is diagonalizable as an element of \( M_{3 \times 3}(\mathbb{R}) \) if and only if there exists a basis of \( \mathbb{R}^{3} \) consisting of eigenvectors of \( A \).

To determine if \( A \) is diagonalizable, we need to find the eigenvalues and eigenvectors of \( A \).

The characteristic polynomial of \( A \) is given by:

\(\det(A-\lambda I)=\left|\begin{array}{ccc}1-\lambda & 0 & 0 \\ 0 & 1-\lambda & 1 \\ 0 & -1 & 1-\lambda\end{array}\right|=(1-\lambda)^{3} \)

The eigenvalues of \( A \) are the roots of the characteristic polynomial, which are \( \lambda=1 \) with multiplicity 3.

To find the eigenvectors of \( A \), we need to solve the equation \( (A-\lambda I)x=0 \) for each eigenvalue.

For \( \lambda=1 \), we have:

\( (A-1I)x=0 \)

\(\left[\begin{array}{ccc}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0\end{array}\right]x=0 \)

The solution to this equation is the eigenspace of \( \lambda=1 \), which is spanned by the eigenvector \( x=\left[\begin{array}{c}1 \\ 0 \\ 0\end{array}\right] \).

Since the eigenspace of \( \lambda=1 \) has dimension 1, there is only one linearly independent eigenvector for this eigenvalue. Therefore, the matrix \( A \) is not diagonalizable as an element of \( M_{3 \times 3}(\mathbb{R}) \), because there does not exist a basis of \( \mathbb{R}^{3} \) consisting of eigenvectors of \( A \).

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