The Venn diagram here shows the cardinality of each set. Use this to find the cardinality of the given set.
n(A)=

Answer:

B

Step-by-step explanation:

It's like a piece of paper, fold it in half

Answer:

The original price of petrol is R12.58 per litre and the new price is R13.28 per litre. To find the percentage increase in the price, we can use the following formula:

Percentage increase = ((new price - old price) / old price) x 100%

Substituting the values, we get:

Percentage increase = ((13.28 - 12.58) / 12.58) x 100%

Percentage increase = (0.70 / 12.58) x 100%

Percentage increase = 0.0557 x 100%

Percentage increase = 5.57%

Therefore, the percentage increase in the price of petrol is 5.57%.

Step-by-step explanation:

The mean and standard deviation for 30 multiple-choice questions for the number of correct answers is Mean(μ) = 6 and Standard deviation(σ) = 2.19.

What is mean?

A collection of values are averaged to form the mean. The total points were divided by the total scores. When population samples are tiny, the mean is sensitive to extreme scores. For example, if two students in a class of 20 earned significantly higher than the rest, the mean will be skewed higher than the other students' scores might suggest. When using means, larger sample sizes are preferable.

Define standard deviation.

In statistics, a measure of variability known as the standard deviation ( standard deviation ) is frequently used. It demonstrates how different things are from the norm. (mean). When the SD is low, the data tend to be close to the mean, while when it is high, the data are dispersed over a wide variety of values.

Given: number of questions(n) = 30, p = [tex]\frac{1}{5}[/tex], q = [tex]\frac{4}{5}[/tex]

Mean μ = n*p

            = 30  *[tex]\frac{1}{5}[/tex]

         μ = 6

Standard deviation σ = [tex]\sqrt{n*p*q}[/tex]

                                   = [tex]\sqrt{30*\frac{1}{5}*\frac{4}{5}}[/tex]

                                   = [tex]\sqrt{4.8}[/tex]

                                σ = 2.19

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[tex]\bold{Solution:}[/tex]

[tex]\Delta[/tex][tex]DE[/tex][tex]F[/tex] congruent to [tex]\Delta[/tex][tex]JKI[/tex]

[tex]\bold{FD=JI} \text{(corresponding angles of congruent triangles)}[/tex]

[tex]z + 22 = 3z[/tex]

[tex]\text{or,} \ z-3 z= -22[/tex]

[tex]\text{or,} \ -2z = -22[/tex]

[tex]\text{or,} \ z = \bold{11}[/tex]

[tex]\bold{EF=KI} \text{(corresponding angles of congruent triangles)}[/tex]

[tex]5y+13=6y[/tex]

[tex]\bold{y=13}[/tex]

Answer:

the two numbers are 15 and 35, and their ratio is 3:7, and their difference is 20.

Step-by-step explanation:

The product of expected value of xy is  p^2.

If x and y are i.i.d. Bernoulli(p) random variables, then the probability mass function of each variable is:

P(x=1) = p

P(x=0) = 1-p

P(y=1) = p

P(y=0) = 1-p

To find the expected value of the product xy, we can use the definition of expected value:

E[xy] = Σ Σ xy P(x,y)

where the summation is over all possible values of x and y.

Since x and y can only take on the values of 0 or 1, there are only four possible outcomes:

x=0, y=0

x=0, y=1

x=1, y=0

x=1, y=1

We can calculate the probability of each outcome using the probability mass function:

P(x=0, y=0) = P(x=0)P(y=0) = (1-p)(1-p)

P(x=0, y=1) = P(x=0)P(y=1) = (1-p)p

P(x=1, y=0) = P(x=1)P(y=0) = p(1-p)

P(x=1, y=1) = P(x=1)P(y=1) = p^2

Therefore, we have:

E[xy] = (0)(0)(1-p)(1-p) + (0)(1)(1-p)p + (1)(0)p(1-p) + (1)(1)p^2

E[xy] = 0 + 0 + 0 + p^2

E[xy] = p^2

So the expected value of xy is p^2.

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____The given question is incomplete, the complete question is given below:

g suppose that x and y are i.i.d. bernoulli(p) random variables (i.e., with the same value of p, 0. find the product of expected value of xy.

The minors and cofactors of the given matrix are:

|882 270 -72| |-490 194 198| |78 -44 -72|

To find the minors and cofactors of the given matrix, we first need to understand what a minor and a cofactor are.

The minor of an element a_ij is the determinant of the submatrix obtained by deleting the i-th row and j-th column of the original matrix.

The cofactor of an element a_ij is the minor of a_ij multiplied by (-1)^(i+j).

Using this definition, we can find the minors and cofactors of the given matrix as follows:

Minor of a_11:

|-7 9 5 -1|

|-5 -9 -7 9|

|-6 -3 -7 9|

Minor = (-7)(-9)(-7) + (9)(-7)(-6) + (5)(-5)(-3) + (-1)(-6)(9) = 882

Cofactor of a_11:

Cofactor = (-1)^(1+1) * 882 = 882

Minor of a_12:

|-6 -3 -7 9|

|-5 -9 -7 9|

|-5 -1 -9 5|

Minor = (-6)(-9)(-9) + (-3)(-7)(5) + (-7)(-5)(-1) + (9)(-5)(-5) = -270

Cofactor of a_12:

Cofactor = (-1)^(1+2) * (-270) = 270

Minor of a_13:

|-6 -6 -3|

|-5 -9 -7|

|-5 -1 -9|

Minor = (-6)(-9)(-9) + (-6)(-7)(-1) + (-3)(-5)(-1) + (-9)(-5)(-6) = -72

Cofactor of a_13:

Cofactor = (-1)^(1+3) * (-72) = 72

Minor of a_21:

|-5 -9 -7 9|

|-6 -3 -7 9|

|-5 -1 -9 5|

Minor = (-5)(-7)(-9) + (-9)(-5)(-5) + (-7)(-5)(-1) + (9)(-1)(-5) = 490

Cofactor of a_21:

Cofactor = (-1)^(2+1) * 490 = -490

Minor of a_22:

|-6 -6 -3|

|-5 -1 -9|

|-5 -1 -9|

Minor = (-6)(-9)(-1) + (-6)(-1)(-5) + (-3)(-5)(-1) + (-9)(-5)(-5) = 194

Cofactor of a_22

Cofactor = (-1)^(2+2) * 194 = 194

Minor of a_23:

|-6 -6 -3|

|-5 -9 -7|

|-5 -1 -9|

Minor = (-6)(-9)(-7) + (-6)(-7)(-1) + (-3)(-5)(-1) + (-7)(-5)(-6) = 198

Cofactor of a_23:

Cofactor = (-1)^(2+3) * (-198) = 198

Minor of a_31:

|-5 -9 -7|

|-6 -3 -7|

|-5 -1 -9|

Minor = (-5)(-3)(-9) + (-9)(-5)(-1) + (-7)(-6)(-1) + (-6)(-1)(-7) = -78

Cofactor of a_31:

Cofactor = (-1)^(3+1) * (-78) = 78

Minor of a_32:

|-6 -6 -3|

|-5 -1 -9|

|-5 -1 -9|

Minor = (-6)(-9)(-1) + (-6)(-1)(-5) + (-3)(-5)(-1) + (-1)(-5)(-5) = 44

Cofactor of a_32:

Cofactor = (-1)^(3+2) * 44 = -44

Minor of a_33:

|-6 -6 -3|

|-5 -9 -7|

|-6 -3 -7|

Minor = (-6)(-9)(-7) + (-6)(-7)(-3) + (-3)(-6)(-1) + (-7)(-5)(-6) = -72

Cofactor of a_33:

Cofactor = (-1)^(3+3) * (-72) = -72

Therefore, the minors and cofactors are:

|882 270 -72| |-490 194 198| |78 -44 -72|

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Answer: 1979.2 meters

Step-by-step explanation:

Diameter of wheel is 63 cm. Circumference is 63π = 197.92 cm. 1000 revolutions of 197.92 = 197920 cm = 1979.2 meters.

let's reword it

what is the arc of a circle whose central angle is 100 revolutions and has a radius of 63÷2?

let's keep in mind that radius is half the diameter, and a revolution is 2π radians or 360°, so hmmm 100 revolutions will just be 360*100 = 36000°.

[tex]\textit{arc's length}\\\\ s = \cfrac{\theta \pi r}{180} ~~ \begin{cases} r=radius\\ \theta =\stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ \theta =36000\\ r=\frac{63}{2} \end{cases}\implies s=\cfrac{(36000)\pi (\frac{63}{2})}{180}\implies s=\cfrac{(36000)(\frac{22}{7}) (\frac{63}{2})}{180} \\\\\\ s=200(\frac{22}{7}) (\frac{63}{2})\implies s=(200)(99)\implies \boxed{s=19800}~metres[/tex]

Answer:

80 %

Step-by-step explanation:

FRACTIONS

X/100 = 25,390/32,000

CROSS MULTIPLICATION

X * 32,000 = 32,000X

25,390*100 = 2,539,000

DIVIDE

2,539,000/32,000 = 32000X/32000- CANCEL

X = 79.34375 % Round - X = 80%

The negative of the gradient (-grad f(x)) points in the direction of the maximum decrease of f from x.

The gradient of a scalar-valued function is a vector that points in the direction of the maximum increase of the function.

In other words, if we consider a point in the domain of the function and take the gradient at that point, the direction of the gradient vector indicates the order in which the function increases the most from that point. Conversely, the negative gradient points in the direction of the maximum decrease of the function.

Specifically, let f be a scalar-valued function of n variables [tex](f: R^n - > R),[/tex]and let x be a point in the domain of f. The gradient of f at x is defined as the vector:

[tex]grad f(x) = (∂f/∂x1, ∂f/∂x2, ..., ∂f/∂xn)[/tex]

where ∂f/∂xi denotes the partial derivative of f with respect to xi evaluated at x, the direction of the gradient vector grad f(x) at x is the direction in which f increases the most from x, and the magnitude of the gradient vector is the rate of change of f at x in that direction.

The negative of the gradient (-grad f(x)) points in the direction of the maximum decrease of f from x.

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Answer: 2:1

Step-by-step explanation:

a) The total selling price of 20 mobile sets is Rs. 3,00,000. Therefore, the selling price of each mobile set is:

Selling price of each mobile set = Total selling price ÷ Number of mobile sets sold

= 3,00,000 ÷ 20

= Rs. 15,000

Therefore, the selling price of each mobile set is Rs. 15,000.

b) The cost of each mobile set is given as Rs. 12,500. The total cost of 20 mobile sets is:

Total cost = Cost of each mobile set × Number of mobile sets sold

= 12,500 × 20

= Rs. 2,50,000

The profit made by the shopkeeper is the difference between the selling price and the cost price of the mobile sets:

Profit = Selling price - Cost price

= 3,00,000 - 2,50,000

= Rs. 50,000

Therefore, the profit made by the shopkeeper is Rs. 50,000.

The profit percent is given by:

Profit percent = (Profit ÷ Cost price) × 100

= (50,000 ÷ 2,50,000) × 100

= 20%

Therefore, the shopkeeper made a profit of 20%.

c) The ratio of the length to the breadth of each mobile set is:

Length:breadth = 12:6

= 2:1

Therefore, the ratio of the length to the breadth of each mobile set is 2:1.

Answer:

Step-by-step explanation:

a) To find the selling price of each mobile set, we need to divide the total amount sold by the number of mobile sets sold:

Selling price of each mobile set = Total amount sold / Number of mobile sets sold

Selling price of each mobile set = Rs 3,00,000 / 20

Selling price of each mobile set = Rs 15,000

Therefore, the selling price of each mobile set is Rs 15,000.

b) The cost price of each mobile set is Rs 12,500.

Profit or loss percent = [(Selling price - Cost price) / Cost price] x 100

Profit or loss percent = [(15,000 - 12,500) / 12,500] x 100

Profit or loss percent = 20%

Therefore, the shopkeeper has made a profit of 20%.

c) The ratio of the length to the breadth of the mobile set is:

Length : Breadth = 12 cm : 6 cm

Simplifying the ratio by dividing both sides by 6:

Length : Breadth = 2 : 1

Therefore, the ratio of the length to the breadth of the mobile set is 2 : 1.

Answer: June uses 192 jellybeans for every dozen (12) plastic eggs she fills, which means she uses 192/12 = <<192/12=16>>16 jellybeans per plastic egg.

To find how many plastic eggs can be filled with 1,152 jellybeans, we can divide 1,152 by the number of jellybeans per egg:

Number of eggs = 1,152 ÷ 16 = <<1152/16=72>>72

Therefore, June can fill 72 plastic eggs with 1,152 jellybeans.

Step-by-step explanation:

Answer:

Step-by-step explanation:

Answer:

4 combination if order doesn't matter, 8 if order matter (for example is not the same 1x24 or 24x1)

Step-by-step explanation:

1x24
2x12
3x8
4x6

IF order matter also:
6x4
8x3
12x2
24x1

To solve this equation, begin by expanding the parentheses.

y = 4x² + 40x + 100 + 16

Then, collect like terms on the left hand side.

y = 4x² + 40x + 116

Finally, rearrange the equation to get y on the left side.

4x² + 40x - (y - 116) = 0

The final answer will be a quadratic equation.

If we total up the dots plot for 3, 4, and 5 pets, we find that 3 people have 2 pets at home, 10 individuals have at least 3 pets at home.

The fact that dogs are the most common pet in the world shouldn't be shocking. There is a reason why there are tens of millions of dogs living in the United States alone, which is why some people say that dogs are a man's greatest friend. Around the world, at least one dog is kept in one-third of all households.

A fully domesticated animal kept constitutes a "household pet." a pet kept by you for personal company, like a dog, cat, reptile, bird, or mouse. Any kind of horse, cow, pig, sheep, goat, chicken, turkey, other captive fur-bearing animal is not considered a household pet, nor is any animal that is typically kept for food or profit.

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The total number of people with pets at home is 11, which is the sum of the heights of the columns.

What is equation?

A math equation is a method that links two claims and represents equivalence using the equals sign (=). An equation is a mathematical statement that establishes the equivalence of two mathematical expressions in algebra.

Based on the given dot plot, we can answer the following questions:

How many people have 2 pets at home?

Answer: Two people have 2 pets at home, as indicated by the two dots in the second column.

How many people have at least 3 pets at home?

Answer: Six people have at least 3 pets at home, as indicated by the dots in the third column and beyond.

How many more people have 2 pets than 5 pets?

Answer: There are no dots in the last column, which represents 5 pets. Therefore, the difference between the number of people with 2 pets and those with 5 pets is 2 - 0 = 2.

How many people have less than 3 pets at home?

Answer: Three people have less than 3 pets at home, as indicated by the dots in the first two columns.

Therefore, the total number of people with pets at home is 11, which is the sum of the heights of the columns.

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The measure of angle BAC is 55°, which is closest to option B (50°).

The ratio of the length of the side directly opposite an acute angle to the side directly adjacent to the angle is known as the tangent in trigonometry. Only triangles with straight angles can have this.

Let's give the angles shown in the diagram the following labels:

Angle ACD = 55°

Angle ABD = 35°

Angle BCD = 90°

To determine the size of angle ABC, we can use the knowledge that a triangle's total angles equal 180°. Because the straight line formed by angles ABD and BCD, we have:

[tex]Angle ABC = 180° - Angles ABD and BCD.[/tex]

[tex]Angle ABC = 180° - 35° - 90°Angle ABC = 55°[/tex]

Given that triangle ABC has two angles, we can use the knowledge that a triangle's total of angles equals 180° to determine the size of angle BAC:

[tex]Angle BAC = 180° - Angle ABC - Angle ACBAngle BAC = 180° - 55° - 70°Angle BAC = 55°[/tex]

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It is most similar to option B (50°) when the angle BAC is 55°.

What is a tangent angle?

The tangent in trigonometry is the length of the side directly opposite an acute angle divided by the length of the side directly next to the angle.

This property can only be found in triangles with straight angles.

Let's give the angles shown in the diagram the following labels:

Angle ACD = 55°

Angle ABD = 35°

Angle BCD = 90°

We can use the fact that a triangle's total number of angles is 180° to calculate the size of angle ABC. due to the fact that the straight line created by angles ABD and BCD

Triangle ABC has two angles, so we can use the fact that a triangle's sum of angles is 180° to calculate the size of angle BAC.

Therefore, the BAC measurement is 55°, which is closest to option B's 50°.C is 55°, which is closest to option B (50°).

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The best description of the research design for this study is a controlled experiment.

What is a controlled experiment?

A controlled experiment is a type of scientific investigation in which an independent variable is manipulated and the dependent variable is measured while controlling for other variables that may affect the outcome.

Finding the best description of the research design for this study :

In this research study, the researchers are manipulating the environment in which the dogs are kept. Two dogs are kept away from human beings during the day and freed at night, while the other two dogs are kept in an environment where they regularly engage with human beings at both the day and the evening. By doing this, the researchers can observe and measure the behavior of the dogs in both groups and determine if there is a difference in their behavior towards trespassers.

This research design is a controlled experiment because the researchers are manipulating one variable, which is the environment in which the dogs are kept, while keeping other variables constant. This allows them to establish cause-and-effect relationships between the independent variable (environment) and the dependent variable (dog behavior).

In conclusion, the research design for this study is a controlled experiment because it involves manipulating the environment in which the dogs are kept and measuring their behavior while controlling for other variables that may affect the outcome.

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Answer:

a. For the function:

f(x) = { sin x/3, if x ≤ π

{ x√3/2π, if x > π

I. To find lim f(x) as x approaches π from the negative side, we need to evaluate f(x) for values of x that are slightly less than π. In this case, since sin(x/3) is a continuous function, we can simply evaluate it at x = π:

lim f(x) as x approaches π- = f(π-) = sin(π/3) = √3/2

II. To find lim f(x) as x approaches π from the positive side, we need to evaluate f(x) for values of x that are slightly greater than π. In this case, we can simply evaluate the other part of the piecewise function at x = π:

lim f(x) as x approaches π+ = f(π+) = π√3/2π = √3/2

III. To find lim f(x) as x approaches π, we need to check whether the left-hand and right-hand limits are equal. In this case, since both the left- and right-hand limits exist and are equal, we have:

lim f(x) as x approaches π = √3/2

b. For the function:

f(x) = (x^2 - 36)/√(x^2 - 12x + 36)

I. To find lim f(x) as x approaches 6 from the negative side, we need to evaluate f(x) for values of x that are slightly less than 6. In this case, we can substitute x = 6 - h, where h is a positive number approaching zero, to get:

lim f(x) as x approaches 6- = lim f(6 - h) as h approaches 0

Substituting x = 6 - h into the function, we get:

f(6 - h) = [(6 - h)^2 - 36]/√[(6 - h)^2 - 12(6 - h) + 36]

= [h^2 - 12h]/√[h^2]

Simplifying the numerator and denominator separately, we get:

f(6 - h) = h(h - 12)/|h|

Since h approaches 0 from the positive side, we have:

lim f(6 - h) as h approaches 0+ = lim h(h - 12)/h as h approaches 0+ = lim (h - 12) as h approaches 0+ = -12

II. To find lim f(x) as x approaches 6 from the positive side, we need to evaluate f(x) for values of x that are slightly greater than 6. In this case, we can substitute x = 6 + h, where h is a positive number approaching zero, to get:

lim f(x) as x approaches 6+ = lim f(6 + h) as h approaches 0

Substituting x = 6 + h into the function, we get:

f(6 + h) = [(6 + h)^2 - 36]/√[(6 + h)^2 - 12(6 + h) + 36]

= [h^2 + 12h]/√[h^2]

Simplifying the numerator and denominator separately, we get:

f(6 + h) = h(h + 12)/|h|

Since h approaches 0 from the positive side, we have:

lim f(6 + h) as h approaches 0+ = lim h(h +

Step-by-step explanation:

An explanation without explaining why that was right is called

dogmatic explanation.

An explanation without a reason why it is true is called a dogmatic statement.

We have,

Dogmatic statement:

This is a statement that is presented as true without providing any evidence or logical reasoning to support it.

It is often based on personal beliefs, opinions, or assumptions rather than objective facts or evidence.

Dogmatic statements can be problematic in academic or scientific contexts, where claims must be supported by evidence and logical reasoning.

It is important to be critical and skeptical of dogmatic statements and to seek out evidence and logical reasoning before accepting any claim as true.

Thus,

An explanation without explaining why that was right is called

dogmatic ex[planation.

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According to the given information, Mary's total annual premium is $574 (rounded to the nearest dollar).

In mathematics, multiplication is an arithmetic operation that combines two or more numbers to produce a product. It is represented by the symbol "×" or "*", or by placing the numbers next to each other with no symbol between them.

Mary is 21 years old, so according to the rating factor table, her rating factor is 1.22 for a female.

For liability insurance, Mary has chosen the 50/100/25 coverage, which means $50,000 for bodily injury per person, $100,000 for bodily injury per accident, and $25,000 for property damage per accident. The premium for this coverage is $385.

For collision and comprehensive insurance, Mary has chosen a $500 deductible, so her premiums are $102 for collision and $87 for comprehensive.

To find the total annual premium, we add up the premiums for liability insurance and collision/comprehensive insurance:

Total premium = Liability premium + Collision premium + Comprehensive premium

Total premium = $385 + $102 + $87

Total premium = $574

Therefore, according to the given information, Mary's total annual premium is $574 (rounded to the nearest dollar).

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The probability of spinning a slice that is not yellow is equal to the total number of non-yellow slices divided by the total number of slices.

Total number of non-yellow slices = 9 (3 red + 5 blue)

Total number of slices = 12

Therefore, the probability of spinning a slice that is not yellow is 9/12 or 75%.

If there are 95 families to feed then each family will receive 11 cans of assorted tins.

Assorted tins typically refer to cans or containers of food items that come in different varieties or flavors.

To calculate the number of cans each family will receive, we need to first determine the total number of cans in the 19 boxes.

19 boxes × 55 cans per box = 1045 cans

Now we can divide the total number of cans by the number of families to determine the number of cans each family will receive:

1045 cans ÷ 95 families = 11 cans per family (rounded to the nearest whole number)

Therefore, each family will receive 11 cans of assorted tins.

For example, a box of assorted tins may contain cans of different types of vegetables, fruits, soups, or other food items. The specific assortment may vary depending on the context and the supplier. In the context of the given question, the 19 boxes of assorted tins likely contain cans of different types of food that will be packed into food parcels for the poor.

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The required simplified forms are 3x - 1, 6x , 2x² - 10x , a + 2b , 15x - 8 , x² + 2x + 6 , x² - y² , 6x² - 22x.

the definition of an equation is a mathematical statement that demonstrates that two mathematical expressions are equal. For instance, the equation 3x + 5 = 14 consists of the two expressions 3x + 5 and 14, which are separated by the 'equal' symbol.

Simplified forms of the equation are

[tex]$c: 3(x-2)+5=3x-6+5=3x-1$[/tex]

[tex]$f: 2x(3-x)+x^2=6x-x^2+x^2=6x$[/tex]

[tex]$i: 7x^2-5x(x+2)=7x^2-5x^2-10x=2x^2-10x$[/tex]

[tex]$c: 2a-(a-2b)=2a-a+2b=a+2b$[/tex]

f: [tex]$3x-4(2-3x)=3x-8+12x=15x-8$[/tex]

[tex]$x(x+4)-2(x-3)=x^2+4x-2x+6=x^2+2x+6$[/tex]

[tex]$x(x+y)-y(x+y)=(x-y)(x+y)=x^2-y^2$[/tex]

[tex]$o: 4x(x-3)-2x(5-x)=4x^2-12x-10x+2x^2=6x^2-22x$[/tex]

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The parabola is passing through (0, 0) and (9, 0).

The locus of a moving point is the location that maintains a constant distance between a stationary point and a predetermined line.

The directrix is a non-moving line, while the focus is a non-moving point.

As you can see, the quadratic variable is not, which modifies the figure's orientation.

This parabola doesn't actually represent a function because it will open horizontally, specifically toward negative infinity, as the graphic linked illustrates.

The vertical line test will reveal that the vertical line in question intercepts the figure in two places, proving that it is not a function.

According to our question-

The equation of the parabola is given below.

y = (3/2)x² - 6x

Then the factor of the equation will be

y = 3x (x / 2 - 3)

Then the zeroes of the function will be

y = 0

x = 0, 9

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Answer:157

Step-by-step explanation:

it just is =]

A replacement value is a value that can be substituted for a variable in an equation or expression in order to solve for that variable or simplify the expression.

An expression is a combination of numbers, variables, and/or operators that represents a mathematical relationship or calculation. An expression can be as simple as a single number or variable, or it can be more complex and involve multiple operations and variables.

A variable is a symbol or letter that represents a quantity or value that can change or vary in different contexts.

In the given question,

A replacement value is a value that can be substituted for a variable in an equation or expression in order to solve for that variable or simplify the expression.

For example, suppose you have the equation 2x + 3 = 7, and you want to solve for x.

To do so, you can use a replacement value.

You can subtract 3 from both sides of the equation to isolate the term with x, which gives you:

2x = 4

Now, you can divide both sides of the equation by 2 to solve for x:

x = 2

In this case, the replacement value was the number 2, which was substituted for the variable x in the equation in order to solve for x.

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10.5% is better because 20 dollers is good but it not gonna increase like 10.5%

The statement "T is injective iff T preserves linear independence" means that T maps distinct vectors in V to distinct vectors in W and that if a set of vectors in V is linearly independent, then their images under T are also linearly independent.

Now, let V and W be two vector spaces and let T: V→W be a linear transformation. We say that T is injective if and only if every distinct vector in V gets mapped to a distinct vector in W.

Now, the statement "T preserves linear independence" means that if a set of vectors in V is linearly independent, then their images under T (i.e., the vectors they get mapped to in W) are also linearly independent.

To see why this is true, suppose we have a linearly independent set of vectors v₁, v₂, ..., vₙ in V. If we assume that their images under T are linearly dependent, then there exist coefficients c₁, c₂, ..., cₙ such that

c₁Tv₁ + c₂Tv₂ + ... + cₙTvₙ = 0

But since T is linear, this is equivalent to

T(c₁v₁ + c₂v₂ + ... + cₙvₙ) = 0

which implies that c₁v₁ + c₂v₂ + ... + cₙvₙ = 0. But this contradicts the assumption that v₁, v₂, ..., vₙ are linearly independent. Therefore, their images under T must also be linearly independent.

Conversely, if T is injective, then it preserves linear independence because distinct vectors in V get mapped to distinct vectors in W. Therefore, if a set of vectors in V is linearly independent, then their images under T must also be linearly independent.

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