Prove that for every real number if c is the root of a polynomial with rational coefficients then root ofa polynomial with integer coefficients: It may be helpful to suppose that is a solution to the polynomial equation: GnXn An-1xn-1_ +q1* + 4o Where qi € Q

The given statements are true or false are shown below, about linear system has n variables and m equations, then the augmented matrix has n rows.

First, let's write how A and C look like.

A = [C|b], where b is the constant matrix.

(a) False.

Example

[tex]\left[\begin{array}{ccc}1&0&2\\0&1&3\\\end{array}\right][/tex]

We can see that z = t and so we have infinitely many solutions but there's no row of zeros.

(b) False.

Example

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

Here; x1 = 1 and x2 = 1 is a unique solution and we have a row of zeros.

(c) True.

In the row-echelon form, the last row is either a row of zeros or a row that contains a leading 1. If the row has a leading 1, then there is a solution. Since we assume there is no solution, then the row must be a row of zeros.

(d) False.

Example

[tex]\left[\begin{array}{cc}1&3\\0&0\end{array}\right][/tex]

Here; x₁ = 1 − 3t and x2 = t. Thus, the system is consistent.

(e) True.

Suppose we have a typical equation in a system

а1x1 + A2X2 + ··· + anxn = b

Now, if b≠0 and x1 = x2 = ··· = x₂ = 0, then the system is Xn inconsistent. But, if b = 0, then we have a solution.

(f) False.

Example

[tex]\left[\begin{array}{cc}1&2&0&0\end{array}\right][/tex]

If a = 0, then it's consistent(infinitely many solutions) but if a 0, then it's inconsistent.

(g) Ture.

Since the rank would be at most 3 and this will lead to a free variable (4 columns in C and the rank is 3, so there is at leat 1 free variable). Thus, the system has more thatn one solution.

(h) True.

Because the rank is the number of leading 1's lying in different rows and A has 3 rows. Thus, the rank ≤ 3.

(i) False.

Because we could have a row of zeros in C and a leading 1 in A. In other words, a31 = a32 = A33 = A34 = 0 and c3 1. This makes the system inconsistent.

(j) True.

If the rank of C = 3, then there will be a free variable and this means the system is consistent.

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Complete question:

In each case either show that the statement is true, or give an example showing it is false. (a) If a linear system has n variables and m equations, then the augmented matrix has n rows. quations • ( *b) A consistent linear system must have infinitely many solutions. . (c) If a row operation is done to a consistent linear system, the resulting system must be consistent. (d) If a series of row operations on a linear system results in an inconsistent system, the original system is inconsistent.

99% confidence interval is that the true population proportion of adults who say national identity is strongly tied to birthplace in Country A is between 30.7% and 37.3%.

The confidence interval is a set of values that, with a given level of certainty, contains the real population parameter. The likelihood that the genuine population parameter falls inside the interval is represented by the degree of confidence. In this instance, we created 99% confidence intervals for the percentage of individuals in each demographic who claim that national identity is highly correlated with birthplace. This indicates that 99% of the intervals we build would include the genuine population percentage if the survey were to be repeated numerous times.

The confidence interval is given using the formula:

CI = p ± z*(√((p*(1-p))/n))

Using the given values for different intervals the 99% for different countries are:

For Country A we have:

p = 0.34

n = 1007

z = 2.58 (based on a 99% confidence level)

CI = 0.34 ± 2.58*(√((0.34*(1-0.34))/1007)) = 0.307 to 0.373

For Country B we have:

p = 0.22

n = 1005

z = 2.58

CI = 0.22 ± 2.58*(√((0.22*(1-0.22))/1005)) = 0.187 to 0.253

For Country C we have:

p = 0.29

n = 1016

z = 2.58

CI = 0.29 ± 2.58*(√((0.29*(1-0.29))/1016)) = 0.259 to 0.321

For Country D we have:

p = 0.5

n = 1016

z = 2.58

CI = 0.5 ± 2.58*(√((0.5*(1-0.5))/1016)) = 0.469 to 0.531

For Country E we have:

p = 0.14

n = 1000

z = 2.58

CI = 0.14 ± 2.58*(√((0.14*(1-0.14))/1000)) = 0.108 to 0.172.

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The value of d/c = 8, when the given graph represents the equation f(x) = 5.4321*(2)ˣ, and A (1, c), and B (4, d) pass through the graph.

The graph of an equation passes through all the points that satisfy the given equation. The graph shows a curve joining all those points.

In the question, we are given a graph and its equation f(x) = 5.4321*(2)ˣ.

We are given the coordinates of two points on graphs A: (1, c) and B: (4, d).

We know that any point of the form (x, y) which passes through the graph of the equation, satisfies the equation.

∴ A (1, c) and B (4, d) satisfies the equation f(x) = 5.4321*(2)ˣ.

∴ c = 5.4321*(2)¹ = 5.4321 * 2

d = 5.4321*(2)⁴ = 5.4321 * 16

∴ d/c = (5.4321*16)/(5.4321*2) = 16/2 = 8.

∴ The value of d/c = 8, when the given graph represents the equation f(x) = 5.4321*(2)ˣ, and A (1, c), and B (4, d) pass through the graph.

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Answer:

[tex]\dfrac{4096\pi}{5}\approx 2573.593\; \sf (3\;d.p.)[/tex]

Step-by-step explanation:

The shell method is a calculus technique used to find the volume of a solid revolution by decomposing the solid into cylindrical shells. The volume of each cylindrical shell is the product of the surface area of the cylinder and the thickness of the cylindrical wall. The total volume of the solid is found by integrating the volumes of all the shells over a certain interval.

The volume of the solid formed by revolving a region, R, around a vertical axis, bounded by x = a and x = b, is given by:

[tex]\displaystyle 2\pi \int^b_ar(x)h(x)\;\text{d}x[/tex]

where:

[tex]\hrulefill[/tex]

We want to find the volume of the solid formed by rotating the region bounded by y = 0, y = √x, x = 0 and x = 16 about the y-axis.

As the axis of rotation is the y-axis, r(x) = x.

Therefore, in this case:

[tex]r(x)=x[/tex]

[tex]h(x)=\sqrt{x}[/tex]

[tex]a=0[/tex]

[tex]b=16[/tex]

Set up the integral:

[tex]\displaystyle 2\pi \int^{16}_0x\sqrt{x}\;\text{d}x[/tex]

Rewrite the square root of x as x to the power of 1/2:

[tex]\displaystyle 2\pi \int^{16}_0x \cdot x^{\frac{1}{2}}\;\text{d}x[/tex]

[tex]\textsf{Apply the exponent rule:} \quad a^b \cdot a^c=a^{b+c}[/tex]

[tex]\displaystyle 2\pi \int^{16}_0x^{\frac{3}{2}}\;\text{d}x[/tex]

Integrate using the power rule (increase the power by 1, then divide by the new power):

[tex]\begin{aligned}\displaystyle 2\pi \int^{16}_0x^{\frac{3}{2}}\;\text{d}x&=2\pi \left[\dfrac{2}{5}x^{\frac{5}{2}}\right]^{16}_0\\\\&=2\pi \left[\dfrac{2}{5}(16)^{\frac{5}{2}}-\dfrac{2}{5}(0)^{\frac{5}{2}}\right]\\\\&=2 \pi \cdot \dfrac{2}{5}(16)^{\frac{5}{2}}\\\\&=\dfrac{4\pi}{5}\cdot 1024\\\\&=\dfrac{4096\pi}{5}\\\\&\approx 2573.593\; \sf (3\;d.p.)\end{aligned}[/tex]

Therefore, the volume of the solid is exactly 4096π/5 or approximately 2573.593 (3 d.p.).

[tex]\hrulefill[/tex]

[tex]\boxed{\begin{minipage}{4 cm}\underline{Power Rule of Integration}\\\\$\displaystyle \int x^n\:\text{d}x=\dfrac{x^{n+1}}{n+1}(+\;\text{C})$\\\end{minipage}}[/tex]

The profit-maximizing combination of resources for a firm is  MRPl / Pl = MRPc / Pc = 10/2 = 5/1 . The correct option is D).

The profit-maximizing combination of resources for a firm is determined by the equality of the marginal revenue product (MRP) per unit of input (labor, L, or capital, C) to the price per unit of input.

Therefore, the equation that represents the profit-maximizing combination of resources for a firm is:

MRPl / Pl = MRPc / Pc

where MRPl is the marginal revenue product of labor, Pl is the price of labor, MRPc is the marginal revenue product of capital, and Pc is the price of capital.

Among the given options, only option D satisfies the above equation. Therefore, option D represents the profit-maximizing combination of resources for a firm.

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

Which of the following equations represent the profit-maximising combination of resources for a firm?

A. MRPl / Pl = MRPc / Pc = 1

B. MRPl / Pl = MRPc / Pc = 5

C. MRPl / Pl = MRPc / Pc = 10/10 = 5/5

D. MRPl / Pl = MRPc / Pc = 10/2 = 5/1

The proof that the lines CD and XY are parallel is shown below in paragraghs

Given that

∠CAY ≅ ∠XBD

This means that the angles CAY and XBD are congruent angles

The above means that

By the corresponding angles, we have

∠BXY = ∠AYX

∠ACD = ∠BDC

By the congruent angles above, the following lines are parallel

Line AC and BX

Line AY and BD

Line CD and XY

Hence, the lines CD and XY are parallel

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Answer:

The table given provides the number of male and female students who voted for each party, but it does not give the total number of students in the survey. To find the probability of selecting a student who voted for the Republican party, we need to know the total number of students who participated in the survey.

The total number of students in the survey is:

50 + 75 + 125 + 50 = 300

The number of students who voted for the Republican party is:

75 + 50 = 125

Therefore, the probability of selecting a student who voted for the Republican party is:

125/300 = 0.4167 (rounded to four decimal places)

So, the answer is option D: 125/300

(please mark my answer as brainliest)

For the standard normal curve, the area to the left of 0.1 is the same as the area to the right of 0.9 is true . So, the correct answer is D.

The standard normal curve is a normal distribution with a mean of 0 and a standard deviation of 1. This curve is often used in statistics to model natural phenomena, and it has many important properties.

Option A is incorrect because the area under the standard normal curve between 0 and 2 is not twice the area between 0 and 1. The area under the curve increases as we move away from the mean, so the area between 0 and 2 will be greater than the area between 0 and 1.

Option B is also incorrect because the area under the standard normal curve between 0 and 2 is not half the area between -2 and 2. The area between -2 and 2 covers more of the curve than the area between 0 and 2, so the area between 0 and 2 will be smaller than half the area between -2 and 2.

Option C is incorrect because the standard normal curve does not have a fixed IQR (interquartile range). The IQR depends on the quartiles of the distribution, which can vary depending on the sample size and the distribution's shape.

Option D is the correct answer because the standard normal curve is symmetric around the mean of 0. This means that the area to the left of any point on the curve is the same as the area to the right of its negative counterpart. Therefore, the area to the left of 0.1 is equal to the area to the right of 0.9.

Therefore, Correct option is D.

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Answer:

Step-by-step explanation:

Since they are traveling in the same direction, the distance between them increases at a rate of the difference of their speeds.

The relative speed of train A with respect to train B is:

105 kmph - 87 kmph = 18 kmph

In three hours, the distance between them will be:

Distance = Speed x Time

Distance = 18 kmph x 3 hours

Distance = 54 km

Therefore, the two trains will be 54 kilometers apart after three hours.

Answer:

a)220 4 leaf clovers, 780 3 leaf clovers b)440 four leaf clovers, 1560 3 leaf clovers

Step-by-step explanation:

100:1000=1:10 22:x=1:10 x=220

220x2=440

1000-220=780

780x2=1560

A parametric equation is one where the x and y coordinates of the curve are both written as functions of another variable called a parameter; this is usually given the letter t or θ . And the value of a= 8, b= 1, c= -2 and d= 0.

Equation of this form is known as a parametric equation; it uses an independent variable known as a parameter (often represented by t) and dependent variables that are defined as continuous functions of the parameter and independent of other variables.
You require 4 independent solutions because there are 4 unknowns.  You can put two equations at each end point if you know t at each end point. (one for the x value and one for the y value).
At (8,-2), time is equal to zero as follows: 8 = a + bt = a + b(0)  a = 8 -2 = c + dt = c + d(0)  c = -2
At (9,-2), t = 1 because 9 = a + bt = 8 + b(1)  b = 1 and -2 = c + dt = -2 + d(1)  d = 0.
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Answer:

Area = 559.17 square feet

Perimeter = 94.26 ft

Step-by-step explanation:

Make sure all the units are the same and consistent.

r = radius of semi-circle

 = [tex]\frac{Diameter}{2}[/tex]

 = [tex]\frac{18}{2}[/tex] ft

 = 9 ft

Area of composite figure = Area of rectangle + Area of semi-circle:

                                          = [Length × Breadth] + [[tex]\frac{1}{2}[/tex] × (Area of circle)]
                                          = [24 ft × 18 ft] + [[tex]\frac{1}{2}[/tex] × ([tex]\pi r^{2}[/tex])]

                                          = 432 [tex]ft^{2}[/tex] + [[tex]\frac{1}{2}[/tex] × ([tex]\pi 9^{2}[/tex])] [tex]ft^{2}[/tex]

                                          = 432 + [[tex]\frac{1}{2}[/tex] × (3.14) ×(81)]

                                          = 559.17[tex]ft^{2}[/tex]

Perimeter of composite figure =

     Circumference of semi-circle + 3 outer sides of rectangle:

 = [[tex]\frac{1}{2}[/tex] × [tex]2\pi r[/tex]] + [24 + 18 + 24]

 = ( [tex]\pi r[/tex] + 66) ft

 = [(3.14)(9) + 66] ft

 = 94.26 ft

you need to match it to the percentage of the fraction for example 1/2 is 50 percentage and than you do 1/4 is 25 percent.

Answer:

Answer:

$38.08

Step-by-step explanation:

12% of 34.00 as an equation would be [tex]34*0.12[/tex], [tex]34*0.12=4.08[/tex] so we add 4.08 to the total of the lunch already. [tex]34.00+4.08=38.08[/tex]. So the answer is $38.08. Hope this helps :)

The relationship between AB and BC is given as follows:

AB > BC.

Two angles are defined as supplementary angles when the sum of their measures is of 180º.

The supplementary angles for this problem are given as follows:

By the law of sines, we have that:

AB/sin(110º) = BC/sin(70º).

As sin(110º) > sin(70º), the inequality for this problem is given as follows:

AB > BC.

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Answer:

AB>BC

Step-by-step explanation:

AI-generated answer

Based on the given information, the relationship between AB and BC depends on the measure of angle ZADB. If mZADB is 110°, we can determine the relationship as follows:

Since triangle ABD and triangle CBD share side AB, the larger the angle ZADB, the longer the side AB will be compared to side BC. Therefore, if mZADB is 110°, we can conclude that AB is greater than BC.

In summary, when mZADB is 110°, the relationship between AB and BC is:

AB > BC.

The probability of P(B U E) is 0.5984, the probability of the complement of G (i.e. Gᶜ) is 0.6818, and the probability of C given F is 0.4706

To find the probability P(B U E), we need to add the frequencies in column B and row E, but since the cell where they intersect has already been counted in both, we need to subtract it once. Therefore:

P(B U E) = (6+13+17) + (15+20+10+45) - 43 = 73

Then we divide by the total number of observations:

P(B U E) = 73/122 = 0.5984 (rounded to four decimal places)

To find the probability of the complement of G (i.e. Gᶜ), we need to subtract the frequency in the cell where G and Total intersect from the total number of observations:

P(Gᶜ) = 1 - 39/122 = 0.6818 (rounded to four decimal places)

To find the probability of C given F, we need to divide the frequency in the cell where C and F intersect by the total frequency in the row where F is located:

P(C|F) = 24/51 = 0.4706 (rounded to four decimal places)

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Complete question is attached below

Answer:

9 CDs

Step-by-step explanation:

r u d0mb? 45 divided by 5 = 5 10 15 20 25 30 35 40 45

count the numbers

BOOM ANSWER

Answer:

x²

Step-by-step explanation:

[tex]{ \tt{ \frac{ {x}^{ - 3} . {x}^{2} }{ {x}^{ - 3} } }} \\ \\ \dashrightarrow{ \tt{x {}^{( - 3 + 2 - ( - 3))} }} \\ \dashrightarrow{ \tt{ {x}^{( - 3 + 2 + 3)} }} \: \: \: \: \\ \dashrightarrow{ \boxed{ \tt{ \: \: \: \: {x}^{2} \: \: \: \: \: \: }}} \: \: \: \: [/tex]

Answer:

If the fire truck is parked 25 feet away from the high rise and the ladder on the truck reaches 100 feet, we can use the Pythagorean theorem to find out how high up the high rise the ladder can reach.

Let's assume that the height the ladder can reach is "x". Then, we can set up the following equation:

25^2 + x^2 = 100^2

Simplifying this equation, we get:

625 + x^2 = 10000

Subtracting 625 from both sides, we get:

x^2 = 9375

Taking the square root of both sides, we get:

x = sqrt(9375)

x ≈ 96.81

Therefore, the ladder on the fire truck can reach a height of approximately 96.81 feet up the high rise

1. Begin the program by including the header file <iostream> which includes basic input/output library functions as well as the <vector> library which is needed for this program.

2. Declare a vector of integer type named 'ageGroups' which will store the age groups and the corresponding membership fee and reductions.

3. Create a void function named 'calculateFee()' which will take a parameters age and activities attended.

4. In the calculateFee() function, use the switch statement for the age group and store the corresponding membership fee and reduction value in variables.

5. Use an if statement to check that the number of activities attended is non-negative.

6. Calculate the membership fee using the variables and store it in a variable named 'fee'.

7. Use an if statement to check if the fee is less than 1 and if yes, assign the fee to 1.

8. Print the fee to the user.

9. End the program.

Answer:

Step-by-step explanation:

|x-3|=x-3,if x-3≥0,or x≥3

|x-3|=-(x-3),if x-3<0 ,or x<3

The equation of the line is y = -1/2 + 13/2.

The point is (3, 5).

An equation of line is x + 2y = 5.

To determine the slope intercept form of the equation using the point and line we first determine the slope of the equation from the given line.

Convert the equation of line in slope intercept form.

x + 2y = 5

Subtract x on both side, we get

2y = -x + 5

Divide by 2 on both side, we get

y = -1/2 x + 5

On comparing with y = mx + c, where m is slope, we get

m = -1/2

Now the equation of the line is;

y - y₁ = m(x - x₁)

y - 5 = -1/2(x - 3)

Simplify the bracket

y - 5 = -1/2x + 3/2

Add 5 on both side, we get

y = -1/2x + 3/2 + 5

y = -1/2 + 13/2

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The complete question is:

Write an equation of the line containing the given point and parallel to the given line. Express your answer in the form y = mx + b.

(3, 5); x + 2y = 5

The equation of the line is ____ . (Type an equation Type your answer in slope intercept form. Use integers or fractions for any numbers in the equation. Simplify your answer)

Answer:

52

Step-by-step explanation:

10.40 TIMES 3

Answer:

9. $18  

10. 68

Step-by-step explanation:

$8 + $2 + $3 + $5= $18

104 - 36= 68

Answer:

The half-life of a substance is the amount of time it takes for half of the initial amount of the substance to decay.

We can use the following formula to calculate the half-life (t1/2) of a substance with a decay rate of r:

t1/2 = (ln 2) / r

where ln 2 is the natural logarithm of 2 (approximately 0.693).

In this case, the decay rate is 2.5% per year, or 0.025 per year. Plugging this into the formula, we get:

t1/2 = (ln 2) / 0.025

t1/2 = 27.73 years (rounded to two decimal places)

Therefore, the half-life of the substance is approximately 27.73 years.

In the context of data-flow diagrams (DFDs), a(n) option a. data-flow line shows either the source or destination of the data.

In data-flow diagrams (DFDs), an entity symbol represents an external agent or entity that interacts with the system being modeled. It could be a person, organization, or system that sends or receives data from the system being modeled. An entity symbol is represented as a rectangle with its name written inside it. It shows either the source or destination of the data in the DFD. An example of an entity symbol in a DFD could be a customer who provides orders to a company, or a supplier who delivers goods to a company. The entity symbol helps to illustrate the flow of data between external entities and the system being modeled.

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Check the picture below.

[tex]\measuredangle G=\cfrac{\stackrel{far~arc}{148}-\stackrel{near~arc}{106}}{2}\implies \measuredangle G=21^o \\\\\\ \hspace{6em}\measuredangle F=\cfrac{106}{2}\implies \measuredangle F=53^o\hspace{8em}\measuredangle G=106^o \\\\[-0.35em] ~\dotfill\\\\ ~\hfill \measuredangle FGH\textit{ is not an isosceles}~\hfill[/tex]