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Therefore, the only solution is c1 = c2 = 0, which means that the set of functions {sin²(x), 1 - cos(2x)} is linearly independent on the interval (-∞, ∞).

To check if the given set of functions is linearly independent, we need to determine if there exist constants c1 and c2, not both zero, such that:

c1sin²(x) + c2(1 - cos(2x)) = 0 for all x in (-∞, ∞)

To do this, we can look for values of x that make one of the terms equal to zero, and then solve for the other constant.

If we let x = 0, then we have:

c10 + c2(1 - cos(0)) = c2 = 0

So we know that c2 = 0, which means that the equation reduces to:

c1*sin²(x) = 0 for all x in (-∞, ∞)

This can only be true if c1 = 0, because sin²(x) is never zero for all x.

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The first problem is asking for the number of combinations which is 20, while the second problem is asking for the number of permutations which is 336.

In the first problem, the homeowners' association needs to form a committee of 3 people from a board of 6 supervisors. Here, the order in which the committee members are selected does not matter, and only the combination of individuals is important. Therefore, the problem is asking for the number of combinations, and the formula to calculate it is nCr = n! / (r!(n-r)!), where n is the total number of items (6 supervisors) and r is the number of items to be selected (3 committee members). Thus, the answer to the first problem is the number of combinations: 6C3 = 6! / (3!(6-3)!) = 6! / (3!3!) = (6x5x4) / (3x2x1) = 20.

In the second problem, the director needs to cast the male roles of Romeo, Mercutio, and Benvolio from a pool of 8 actors. Here, the order in which the actors are selected matters, as each actor will be assigned to a specific role. Therefore, the problem is asking for the number of permutations, and the formula to calculate it is nPr = n! / (n-r)!, where n is the total number of items (8 actors) and r is the number of items to be selected (3 roles). Thus, the answer to the second problem is the number of permutations: 8P3 = 8! / (8-3)! = 8! / 5! = (8x7x6) / (3x2x1) = 336.

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Answer:

slope -intercept form

y = -328.275 x + 3142.15

Step-by-step explanation:

Step(i):-

Given that the points are (2, 2485.6) and (6,1172.5)

The slope of the line

               \(m = \frac{y_{2}- y_{1} }{x_{2}-x_{1} }\)

              \(m = \frac{1172.5-2485.6 }{6-2 }\)

            \(m = - 328.275\)

Step(ii):-

Equation of the straight line passing through the Point (2,2485.6)

\(y - y_{1} = m(x-x_{1} )\)

y - 2485.6 = (-328.275)(x-2)

y - 2485.6 = -328.275x+656.55

  y =  -328.275x+656.55 + 2485.6

y = -328.275 x + 3142.15

Final answer:-

slope -intercept form

y = -328.275 x + 3142.15

The equivalent equation to the (x² - 5)² - 3x² + 15 = -2 in form of 'm' is given by option c.  m² -3m + 2 = 0.

The equation is equal to,

(x² - 5)² - 3x² + 15 = -2

let the value of m be equals to x² - 5.

Simplify the equation we have,

⇒ ( x² - 5 )² - 3x² + 15 = -2

Take '3' common factor from the 3x² + 15 so that it get convert into x² - 5 we get,

⇒ ( x² - 5 )² - 3 (x² - 5 )  = -2

Now replace x² - 5 by m to get the equivalent equation,

⇒ ( m )² - 3 (m ) = -2

⇒ m² -3m + 2 = 0

Therefore, the equivalent equation to the given equation is written as option c.  m² -3m + 2 = 0.

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The answer is (d.) (x - 2)^2 = 9, which is equivalent to the original equation when completing the square.  To solve the given quadratic equation using the completing the square method, we'll rewrite the equation in the form (x - h)² = k.

Given equation: 0 = 3x² - 12x - 15

First, let's divide by 3 to simplify the equation:

0 = x² - 4x - 5

Next, let's complete the square by adding and subtracting the square of half the coefficient of x:

0 = (x² - 4x + 4) - 5 - 4

Now, we can rewrite the left side as a perfect square:

0 = (x - 2)² - 9

Finally, add 9 to both sides to get the equation in the desired form:

(x - 2)² = 9

So the correct answer is (d.) (x - 2)² = 9.

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The probability of exactly two foreclosures is 0.09.

If one loan out of every 136 loans is foreclosed, the probability of a loan being foreclosed is 1/136. Therefore, the probability of a loan not being foreclosed is 135/136.

Since the bank approved 300 loans last month, the number of loans that can be foreclosed ranges from 0 to 300. We can use the binomial probability formula to calculate the probability of exactly 2 foreclosures in 300 loans:

P(X = 2) = (300 choose 2) * (1/136)^2 * (135/136)^298

Using a calculator, we can evaluate this expression to get:

P(X = 2) ≈ 0.091

Therefore, the probability of exactly two foreclosures is 0.09 (rounded to two decimal places).

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The standard form of the given expression is as follows:

1.4.1.) = = -2.9 × 10^-1

Scientific notation is defined as a process by which large number are bing restructured into a smaller number for easy identification.

For question 1.4.1.)

(-3.4×10-⁷ )×(8.4×10⁵)

Solving separately;

-3.4×8.4= 28.56

That is;

= 28.56 × 10-⁷×10⁵

= -2.9 × 10^(1-7+5)

= -2.9 × 10^-1

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Answer:

66 cm^2

Step-by-step explanation:

12*6=72 cm^2 and to find the missing pience of the rectangle you multiply 3*2= 6 cm^2.

72 - 6 = 66

When a positive point charge q is placed at point P, it produces an electric field at all points surrounding it. Electric fields are defined as the forces that a charged particle experiences when placed in the vicinity of other charged particles, and they are also called the Coulomb forces.

The electric field generated by the point charge is measured by the force it exerts on a small test charge placed at a distance r from the point charge.

This force is given by Coulomb's law, which states that the magnitude of the force between two charges is proportional to the product of the charges and inversely proportional to the distance between them.

The electric field is a vector quantity and is given by the force experienced by the test charge divided by the magnitude of the test charge.

It is represented by an arrow, with the direction of the arrow indicating the direction of the electric field at that point.

The electric field generated by a positive point charge is radially outward from the charge, which means that it points away from the charge in all directions.

This is because a positive charge repels other positive charges and attracts negative charges, and therefore, a test charge would be pushed away from the positive point charge and towards negative charges, resulting in a radially outward electric field.

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The solution of the given differential equation by using variable separable is \(y=\frac{x^2c}{2}-2\)

An equation involving one or more functions and their derivatives is referred to as a differential equation in mathematics. The rate of change of a function at a particular moment is determined by the function's derivatives. It is mostly employed in disciplines like biology, engineering, and physics. The study of solutions that satisfy the equations and the characteristics of the solutions is the main goal of the differential equation.

The rules to solve differential equations using variables separable,

1) \(\frac{dy}{dx}=f(x)/f(y)\)

2) separate terms with the same variable

f(y)dy=f(x)dx

3) Take integration on both sides

4) solve integration by using a suitable method

5) Then get the Solution of the differential equation.

The given differential equation is-

\(\frac{dy}{dx}=\frac{2y+4}{x}\\\\\frac{dy}{2y+4}=\frac{dx}{x}\\\\\int\frac{dy}{2y+4}=\int\frac{dx}{x}\\\\\frac{log(2y+4)}{2}=logx+logc\\log(2y+4)=2logxc\\2y+4=x^2c\\2y=x^2c-4\\y=\frac{x^2c}{2}-2\)

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Answer:

are this the answers?  

Step-by-step explanation:

The value of x is 16 degrees and the measure of the arc DFG for circles of homework 2 are 1 6.056 times radius of circle.

The property of straight angle says that, the angle of a straight line is equal to the 180 degrees.

It can also be define as the sum of all the angle made on the straight line is equal to the 180 degrees.

For the first problem, the line GE, which is the diameter of the circle has two angles over it (x-3) degree and (12x-25) degrees.

For the diameter GE the sum of these two angle will be equal to 180 degrees. Thus,

\((x-3)+(12x-25)=180\\x-3+12x-25=180\\13x-28=180\\13x=180+28\\x=\dfrac{208}{13}\\x=16^o\)

The value of mDE will be equal to (12x-25) as it is verticle angle of mGF. Thus,

\(arcDE=(12x-25)\\arcDE=(12(16)-25)\\arcDE=167^o\)

Similarly the value of mEF

\(arcEF=(x-3)\\arcEF=(16-3)\\arcEF=13^o\)

Let the radius of the circle is R. Then the measure of arc DFG is equal to the sum of areDEF and arc GF. Thus,

\(arcDFG=\dfrac{2\pi R}{2}+\dfrac{2 \pi R \angle GHF}{360}\\arcDFG=\dfrac{2\pi R}{2}+\dfrac{2 \pi R (167^o)}{360}\\arcDFG=6.056R\)

For the second problem, the angle MN and QO are equal due to the vertical angle property. Therefore,

\((10x-45)=(6x-1)\\10x-6x=-1+45\\4x=44\\x=11\)

The value of mMN is,

\(MN=(10x-45)\\MN=(10(11)-45)\\MN=65^o\)

The value of NP using the property of straight line for line segment MP can be given as,

\(NP=180-MN\\NP=180-65\\NP=115^o\)

Thus, the value of x is 16 degrees and the measure of the arc DFG for circles of homework 2 are 1 6.056 times radius of circle.

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The population in 22 years is 2176000.

The percentage is calculated by dividing the required value by the total value and multiplying by 100.

Example:

Required percentage value = a

total value = b

Percentage = a/b x 100

Example:

50% = 50/100 = 1/2

25% = 25/100 = 1/4

20% = 20/100 = 1/5

10% = 10/100 = 1/10

We  have,

Population = 400,000

Rate of increase per year = 8%

We can make an equation for the number of populations in x years.

y = 400000 \(1.08^x\)

Now,

For x = 22,

y = 400000 \(1.08^{22}\)

y = 400000 x 5.44

y = 2176000

Thus,

The population is 2176000 after 22 years.

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Answer:

\(D = 170\º\)

Step-by-step explanation:

Given

Shape: Pentagon

\(A = 87\º\)

\(B = 125\º\)

\(C = 63\º\)

\(E = 95\º\)

Required

Find D

From the sum of angles in a Pentagon; we have that

\(A + B + C + D + E = 540\)

Substitute values for A, B, C and E

\(87\º + 125\º + 63\º + D + 95\º = 540\º\)

Collect Like Terms

\(87\º + 125\º + 63\º + 95\º + D = 540\º\)

\(370\º + D = 540\º\)

Collect Like Terms

\(D = 540\º - 370\º\)

\(D = 170\º\)

Hence, from the list of given options, the correct answer is 170

There are 277,200 ways to select a committee of four Republicans, three Democrats, and two Independents from the given group.

To select a committee of four Republicans, three Democrats, and two Independents from a group of 10 distinct Republicans, 12 distinct Democrats, and 4 distinct Independents, you can use combinations.

For Republicans: C(10,4) = 10! / (4!(10-4)!) = 210 ways
For Democrats: C(12,3) = 12! / (3!(12-3)!) = 220 ways
For Independents: C(4,2) = 4! / (2!(4-2)!) = 6 ways

Now, multiply the combinations together to get the total ways:
210 (Republicans) × 220 (Democrats) × 6 (Independents) = 277,200 ways

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Answer:

k=20

Step-by-step explanation:

The constant of proportionality, k, measures the constant value of the ratio between two proportional quantities, being y and x in this case. Essentially, y/x will always equal 20, the constant of proportionality.

Answer:

2x + 4

Step-by-step explanation:

I'm guessing there are no exact numbers

home = x + 4

school = x

x + x + 4

2x + 4

The equation represented by a graph that is centered at (-4, 5) and includes the point (-2, 5) is: (x+2)²+(y-5)² = 4

What is graph ?

In mathematics, a graph is a visual representation of a set of objects where some pairs of the objects are connected by links. The objects are often represented as points or vertices, and the links are represented as lines or arcs that connect the vertices. Graphs are used to model and analyze many real-world situations, such as social networks, transportation systems, and communication networks. They are also an important tool in various fields of mathematics, such as algebra, geometry, and topology.

According to the question:
The equation represented by a graph that is centered at (-4, 5) and includes the point (-2, 5) is:

(x+2)²+(y-5)² = 4

To see why, we can use the center-radius form of the equation of a circle:

(x - h)² + (y - k)² = r²

where (h, k) is the center of the circle and r is the radius.

From the given information, we know that the center of the circle is (-4, 5) and that the circle passes through the point (-2, 5). Therefore, we can plug these values into the equation above:

(x - (-4))² + (y - 5)² = r²

(x + 4)² + (y - 5)² = r²

We now need to find the value of r, which is the radius of the circle. We know that the circle passes through the point (-2, 5), which is 2 units away from the center (-4, 5) along the x-axis. Therefore, the radius must be 2:

r² = 2²

r² = 4

Substituting r² = 4 into the equation above, we get:

(x + 4)² + (y - 5)² = 4

which matches the first option: (x+2)²+(y-5)² = 4. So, the answer is:

(x+2)²+(y-5)² = 4

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the members of k are d and 4, while the members of l are all real numbers between 4 and 5 (inclusive).

In set-builder notation, {x | P(x)} represents the set of all elements x that satisfy the property P(x).

Using this notation, we have:

k = {x | x is an element of the set {d, 4}}

l = {x | x is a real number such that x is greater than or equal to 4 and less than or equal to 5}

what is  set-builder notation?

Set-builder notation is a way of specifying a set by describing the properties that its elements must satisfy.

In set-builder notation, we use braces { } to enclose a description of the elements of a set. The description consists of a variable, followed by a vertical bar "|" symbol (read as "such that" or "where"), and then a condition that the variable must satisfy.

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The correct answer is (c) analytical. Levels of a variable can have meaning and order when the variable is analytical variable.

Analytical variables are variables that have a meaningful order or ranking, such as height or weight. For example, height is an analytical variable because people can be ranked from shortest to tallest, and each height value is meaningful in relation to other height values.

Prescriptive variables describe the desired or prescribed behavior, such as in a set of rules or guidelines. Categorical variables divide data into categories or groups, such as gender or hair color. Continuous variables have an infinite number of values and can take on any value within a given range, such as temperature or time.

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Answer:

A. I

B.R

C.R

D.R

E.I

Step-by-step explanation:

I= Irrational

R=Rational

Answer:

1) irrational

2) rational

3) rational

4) rational

5) irrational

Step-by-step explanation:

hope this helps!

if its right, plz mark me brainliest!

Answer: -24x^2+28x

Step-by-step explanation: -4x*6x-(-4x)*7 to -24x^2+28x

The solution to the system of equations is x = 3/2 and y = 6

The system of equations is given as:

1/2x = 3/4 and y = 6

The variables x and y in each of the above equations are independent

This is so because they only occur in one equation at a time

So, we have:

1/2x = 3/4

Multiply both sides by 2

x = 3/2

Hence, the solution to the system of equations is x = 3/2 and y = 6

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Answer:

The temperature rose by 5 degrees celcius between 11 am and 4 pm

Step-by-step explanation:

Here in this question, we want to find the integer which represents the temperature change from 11 am to 4 pm.

Initially, in the morning, the temperature was -2 . By 11, it had decreased by 3 degrees. What this means is that the temperature went down further by 3 degrees.

Mathematically, the temperature at 11 will be -2-3 = -5 degrees celcius

Now at 4 pm, the temperature went from -5 to 0 degrees.

The temperature change between 11 am and 4 pm will be; 0-(-5) = 0 + 5 = 5 degrees celcius

This means that the temperature rose by 5 degrees celcius between 11 am and 4 pm

The equation of the line is y = -3x + 16

The general equation of a line is expressed as;

y = mx + c

Given that;

From the diagram shown, the coordinates of the line is expressed as;

(3, 7) and (2.5, 3)

The intercept. c = -3

Substitute the values

7 = -3(3) + c

expand the bracket

7 = -9 + c

collect like terms

c = 7 + 9

c = 16

The equation of the line is then given as;

y = -3x + 16

Hence, the equation is y = -3x + 16

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Answer:

i don't know

Step-by-step explanation:

I'm doing this for no reason

The first six terms of the sequence are: 1, 3, 7, 15, 31, 63.

The first six terms of the sequence given by aₙ = 2ⁿ - 1, we substitute the values of n from 1 to 6 into the equation and calculate the corresponding terms.

For n = 1:

a₁ = 2¹ - 1 = 2 - 1 = 1

For n = 2:

a₂ = 2² - 1 = 4 - 1 = 3

For n = 3:

a₃ = 2³ - 1 = 8 - 1 = 7

For n = 4:

a₄ = 2⁴ - 1 = 16 - 1 = 15

For n = 5:

a₅ = 2⁵ - 1 = 32 - 1 = 31

For n = 6:

a₆ = 2⁶ - 1 = 64 - 1 = 63

Therefore, the first six terms of the sequence are:

1, 3, 7, 15, 31, 63.

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