A small deck of four cards consists of one red card and three green cards. Draw 7 times with replacement. Assume each draw is a random selection of one card.



Let X = the number of red cards drawn



compute the variance of X. Round to 2 decimal places.



Var(X) =

The tangent to the curve at P is y = -x + (2 + π/2) and the horizontal tangent to the curve at Q is y = 2.2653.

The straight line that most closely resembles (or "clings to") a curve at a given location is known as the tangent line to the curve. It might be thought of as the limiting position of straight lines that pass between the specified point and a neighbouring curve point as the second point gets closer to the first.

Slope of a tangent to a curve at a given point is,

dy/dx

so, dy/dx = 4 + cotx - 2cosecx

dy/dx = 0 + ([tex]\frac{-1}{sin^2x}[/tex]) - 2(-cotx cosecx)

dy/dx = 2(cotx.cosecx) - 1/sin²x

At p(π/2, 0)

dy/dx = -1.

slope is -1 so equation of tangent is given by

y = mx + c

y = (-1)x + c  atp(π/2, 0)

c = 2 + π/2

So y = -x + (2 + π/2) tangent at P.

Tangent at Q is parallel to x-axis

Q (1, y) hence, its shape is O

put the point in curve Q

y = 4 + cot(1) - 2cosec(1)

y = 2.2653

So y = mx+c

y = c

Sp y = 2.2653 is horizontal tangent at point Q.

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The motion of Angela, riding on the 59 feet radius Ferris wheel indicates;

a) Please find attached the drawing represent the situation created with MS Word

b) The angle Angela swept out along the ard is about 0.751 radians

c) The measure of the angle Angela swept out in degrees is about 43.02°

The radius of a circular figure is the distance from the center of the figure to the circumference.

The specified parameters are;

Radius of the Ferris wheel = 59 feet

The distance along the arc, traveled by Angela, s = 44.3 feet

Let θ represent the angle Angela swept out along the arc, we get;

a) Please find attached the drawing of the situation created with MS Word

b)The formula for the arc length, s, of a circular motion is; s = r × θ

Where;

r = The radius of the circular motion, therefore;

θ = s/r

θ = 44.3/59 ≈ 0.751

The angle that Angela swept out, θ ≈ 0.751 radians

c) The angle swept out in degrees can be found as follows;

s = (θ/360) × 2 × π × r

Therefore;

44.3 = (θ/360) × 2 × π × 59

θ = 44.3° × 360°/(2 × π × 59) ≈ 43.02°

The angle Angela swept out is approximately 43.02°

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Answer:

The point on the graph that is closest to the point (8, 1.5) is:

[tex]\left(\sqrt[3]{4}, 2 \sqrt[3]{2}+1\right) \approx \left(1.587,3.520)[/tex]

Step-by-step explanation:

To find the point on the graph of y = x² + 1 that is closest to the point (8, 1.5), we need to find the point on the parabola that is at the shortest distance from (8, 1.5). We can use the distance formula to do this.

[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Distance Formula}\\\\$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$\\\\\\where:\\ \phantom{ww}$\bullet$ $d$ is the distance between two points. \\\phantom{ww}$\bullet$ $(x_1,y_1)$ and $(x_2,y_2)$ are the two points.\\\end{minipage}}[/tex]

Any point (x, y) on the parabola y = x² + 1 can be defined as (x, x²+1).

Therefore:

Substitute these points into the distance formula to create an equation for the distance between any point on the parabola and (8, 1.5):

[tex]d = \sqrt{(x - 8)^2 + (x^2+1 - 1.5)^2}[/tex]

Simplifying this expression for d², we get:

[tex]d = \sqrt{(x - 8)^2 + (x^2-0.5)^2}[/tex]

[tex]d^2 = (x - 8)^2 + (x^2-0.5)^2[/tex]

[tex]d^2 = x^2-16x+64 + x^4-x^2+0.25[/tex]

[tex]d^2=x^4-16x+64.25[/tex]

To find the x-coordinate that will minimize this distance, take the derivative of the expression with respect to x, set it equal to zero and solve for x:

[tex]\implies 2d \dfrac{\text{d}d}{\text{d}{x}}=4x^3-16[/tex]

[tex]\implies \dfrac{\text{d}d}{\text{d}{x}}=\dfrac{4x^3-16}{2d}[/tex]

Set it equal to zero and solve for x:

[tex]\implies \dfrac{4x^3-16}{2d}=0[/tex]

[tex]\implies 4x^3-16=0[/tex]

[tex]\implies 4x^3=16[/tex]

[tex]\implies x^3=4[/tex]

[tex]\implies x=\sqrt[3]{4}[/tex]

Finally, to find the y-coordinate of the point on the graph that is closest to the point (8, 1.5), substitute the found value of x into the equation of the parabola:

[tex]\implies y=\left(\sqrt[3]{4}\right)^2+1[/tex]

[tex]\implies y=\sqrt[3]{4^2}+1[/tex]

[tex]\implies y=\sqrt[3]{16}+1[/tex]

[tex]\implies y=\sqrt[3]{2^3 \cdot 2}+1[/tex]

[tex]\implies y=\sqrt[3]{2^3} \sqrt[3]{2}+1[/tex]

[tex]\implies y=2 \sqrt[3]{2}+1[/tex]

Therefore, the point on the graph that is closest to the point (8, 1.5) is:

[tex]\left(\sqrt[3]{4}, 2 \sqrt[3]{2}+1\right) \approx \left(1.587,3.520)[/tex]

Additional information

To find the minimum distance between the point on the graph and (8, 1.5), substitute x = ∛4 into the distance equation:

[tex]\implies d = \sqrt{(\sqrt[3]{4} - 8)^2 + ((\sqrt[3]{4})^2-0.5)^2}[/tex]

[tex]\implies d = 6.72318283...[/tex]

We obtain x 3 + 42 35 mod 49 by solving for x modulo 49. Thus, x ≡ 35 mod 56 is a solution to f(x) = 0 mod 56.

A value or values of a set of variables that satisfy a formula or system of equations are referred to as solutions in mathematics. A remedy can also refers to a process of discovering such values.

given

(A) For x = 0, 1, 2, 3, 4, we calculate f(x) modulo 5:

Thus, f(x) = 0 mod 5 for x = 2, 3.

We compute the following to see if f'(x) is not congruent to 0 modulo 5 at either x = 2 or x = 3.

f'(x) = 2x

f'(2) = 4, f'(3) = 6

Thus, xo = 2 or xo = 3 will work.

(b) We use Hensel's lemma to lift solutions from mod 5 to mod 55 and mod 56.

For mod 5 to mod 55, we start with xo = 2. Since f'(2) = 4 is invertible modulo 5, we can find a unique solution modulo 25 using Hensel's lemma. We get:

f'(2) = 4

f(2) = 5

f(2) + 4(3)(x - 2) = 0 mod 25

f'(3) = 6

f(3) = 10

f(3) Plus 6(2)(x - 3) = 0 mod 49

We obtain x 3 + 42 35 mod 49 by solving for x modulo 49. Thus, x ≡ 35 mod 56 is a solution to f(x) = 0 mod 56.

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The complete question is:-  Let f(x) = x2 + 1 € Z[X].

(a) Find an integer 0 < xo < 5 with f(x) = 0 mod 5 and f'(xo) # 0 mod 5.

(b) Use Hensel's lemma to find solutions to the congruences f(x) = 0 mod 55 and f(x) = 0 mod 56.

To show that there are infinitely many rational numbers between any two real numbers x and y, where x< y, we can use the Archimedean property of the real numbers.

The Archimedean property states that for any two positive real numbers a and b, there exists a positive integer n such that na>b. Let's choose a positive integer n such that 1/n< y-x. Then we can divide the interval(x,y) into n subintervals of equal length:

(x, y) = (x, x + (y - x)/n) ∪ (x + (y - x)/n, x + 2(y - x)/n) ∪ ... ∪ (x + (n - 1)(y - x)/n, y).

Each of these intervals  has length(y-x)/n, which is less than 1/n. therefore, there must be at least one integer k such that x+k(y-x) is a rational number. This is because the numerator k(y-x) is an integer, and the denominator n is a positive integer.Since there are n subintervals, we have found at least n different rational numbers between x and y.

However, since the choice of n was arbitrary we can choose a larger n to find even more rational numbers between x and y. Therefore, there must be infinitely many rational numbers between x and y.

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Let x and y be reals with x<y. Show that there are infinitely many rationals b such that x<b<y.

1. Select the negative square root as sine function is negative for sin 195°. 2. Select the positive square root as cosine function is positive for cos 58°. 3. Select the negative square root as tangent function is negative for tan 225°.

4. Select the negative square root as sine function is negative and cosine of 20° is positive for sin(-10°) = √(1-cos(-20°))/2.

1. Since 195° is in the third quadrant, the sine function is negative. Therefore, we should select the negative square root.

2. Since 58° is in the first quadrant, the cosine function is positive. Therefore, we should select the positive square root.

3. Since 225° is in the third quadrant, the tangent function is negative. Therefore, we should select the negative square root.

4. Since -10° is in the fourth quadrant, the sine function is negative. Also, since cosine is an even function, cos(-20°) = cos(20°), which is positive since 20° is in the first quadrant. Therefore, we should select the negative square root.

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Answer:

hey baby

Step-by-step explanation:

hi thwrw honey i love you lol

The operation we use to simplify a ratio after finding the greatest common factor (GCF) is division.

Option A is the correct answer.

An expression contains one or more terms with addition, subtraction, multiplication, and division.

We always combine the like terms in an expression when we simplify.

We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.

Example:

1 + 3x + 4y = 7 is an expression.

3 + 4 is an expression.

2 x 4 + 6 x 7 – 9 is an expression.

33 + 77 – 88 is an expression.

We have,

To simplify a ratio after finding the greatest common factor (GCF), we use division.

We divide both terms of the ratio by the GCF.

This reduces the ratio to its simplest form.

Thus,

The operation we use to simplify a ratio after finding the greatest common factor (GCF) is division.

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Answer:

375

Step-by-step explanation:

Based on the given conditions, formulate: 53 +2x = 803

Rearrange variables to the left side of the equation:

2x = 803 - 53

Calculate the sum or difference:

2x = 750

Divide both sides of the equation by the coefficient of variable:

x = 750/2

Cross out the common factor: x = 375

The solution of the graphs are as follows

first graph

second graph

The graphs consist of two sets of equations plotted, each has shade peculiar to the equation.

The solution of the graph consist of the ordered pair that fall within the parts covered by the two shades

For the first graph by the left, the solutions are

For the second graph by the left, the solutions are

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Answer:

Step-by-step explanation:

To find the current price index number using the 1975 price of 56.7 cents as the reference value, we can use the formula:

Price Index = (Current Price / Base Price) x 100

Where "Current Price" is the current cost of gasoline, and "Base Price" is the 1975 price of 56.7 cents.

Substituting the values given in the problem, we get:

Price Index = ($2.93 / $0.567) x 100

Price Index = 516.899

Therefore, the current price index number, using the 1975 price of 56.7 cents as the reference value, is 516.899.

Answer:

We are given the equation 50 - x = 38, where x represents the cost of two boxes of cookies.

To find the cost of two boxes of cookies, we need to isolate the variable x.

First, we will subtract 38 from both sides of the equation:

50 - x - 38 = 0

Simplifying:

12 - x = 0

Now, we will add x to both sides of the equation:

12 = x

Therefore, the cost of two boxes of cookies is $12.

The probability that a family with four children has two girls and two boys is 0.3734, or approximately 0.3734 rounded to four decimal places. We can solve it in the following manner.

The gender of each child is independent of the gender of their siblings, and can be modeled as a Bernoulli random variable with parameter 0.49 for female and 0.51 for male. Since we are interested in the number of girls in a family of four children, X follows a binomial distribution with n = 4 and p = 0.49.

The probability of having exactly 2 girls and 2 boys can be calculated using the binomial probability mass function:

P(X = 2) = (4 choose 2) * 0.49² * 0.51²

= 6 * 0.2401 * 0.2601

= 0.3734

Therefore, the probability that a family with four children has two girls and two boys is 0.3734, or approximately 0.3734 rounded to four decimal places.

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Add the times together:

19/100 + 4/10

Find the common denominator, which is 100 so rewrite 4/10 as 40/100

Now add:

19/100 + 40/100 = 59/100

The first car beat the third car by 59/100 seconds.

The difference of the medians as a multiple of the interquartile range is 0.5,So the correct answer is option (A) 0.5.

The median is a measure of central tendency that represents the middle value in a data set when the values are arranged in numerical order.

For example, consider the data set {3, 5, 2, 6, 1, 4}. When the values are ordered from smallest to largest, we get {1, 2, 3, 4, 5, 6}. The median in this case is the middle value, which is 3.

We can first find the medians and interquartile ranges of the two dot plots.

For Mr. Wilson's class:

Median = 12

Q1 = 10

Q3 = 14

IQR = Q3 - Q1 = 14 - 10 = 4

For Mr. Watson's class:

Median = 10

Q1 = 8

Q3 = 12

IQR = Q3 - Q1 = 12 - 8 = 4

The difference of the medians is |12 - 10| = 2. Therefore, the difference of the medians as a multiple of the interquartile range is:

$$\frac{2}{4} = \boxed{0.5}$$

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The required value is 1.5(IQR) equals 6.

A dataset is a collection of facts that relates to a particular subject. The test results of each pupil in a particular class are an illustration of a dataset. Datasets can be expressed as a table, a collection of integers in a random sequence, or by enclosing them in curly brackets.

The IQR (interquartile range) is the difference between the third quartile (Q3) and the first quartile (Q1). So, we first need to calculate IQR:

IQR = Q3 - Q1 = 16 - 12 = 4

Now we can calculate 1.5 times the IQR:

1.5(IQR) = 1.5(4) = 6

Therefore, 1.5(IQR) equals 6.

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Complete question:

The five-number summary of a data set is given below.

Minimum: 3 Q1: 12 Median: 15 Q3: 16 Maximum: 20

Which of the following equals 1.5(IQR)?

Since the p-value (0.0803) exceeds the significance threshold (0.05), the null hypothesis cannot be ruled out.

The mean in mathematics is a measurement of a collection of numerical data's central tendency. It is determined by adding up all of the values in the set and dividing the result by the total number of values. This value is frequently referred to as the average value. The mean (or mathematical mean) is calculated as follows: (Sum of Values) / Mean (number of values)

given

The null hypothesis states that the mean number of units generated during the day and night shifts is the same. The contrary hypothesis (Ha) states that more units are created on average on the night shift than on the day shift.

"day" + "night"

Bravo! Night precedes day.

b. The following method can be used to calculate the test statistic:

t = sqrt(1/n night + 1/n day) * sqrt(x night - x day)

where s p is the pooled standard deviation and x night and x day are the sample averages, n night and n day are the sample sizes, and s p is represented by:

Sqrt(((n night - 1)*s night2 + (n day - 1)*s day2) / (n night + n day - 2)) yields the value s p.

S p is equal to sqrt(((74 - 1)*35 + (68 - 1)*28) / (74 + 68 - 2)), which equals 31.88.

t = (358 - 352) / (31.88 * sqrt(1/74 + 1/68)) = 1.19

1.19 is the test result.

the p-value is 0.0803 as a result (rounded to 4 decimal places).

Since the p-value (0.0803) exceeds the significance threshold (0.05), the null hypothesis cannot be ruled out.

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The complete question is :- Clark Heter is an industrial engineer at Lyons Products. He would like to determine whether there are more units produced on the night shift than on the day shift. The mean number of units produced by a sample of 68 day-shift workers was 352. The mean number of units produced by a sample of 74 night-shift workers was 358. Assume the population standard deviation of the number of units produced is 28 on the day shift and 35 on the night shift.

Using the 0.05 significance level, is the number of units produced on the night shift larger?

a. State the null and alternate hypotheses.

O : Day/Night: H:

Day Night

b. Compute the test statistic. (Negative values should be indicated by a minus sign. Round your answers to 2 decimal places.)

c. Compute the p-value. (Round your answer to 4 decimal places.) p-value

(a) Experimental probability (5 or 8) = 0.193

(b) Theoretical probability (5 or 8) = 0.200

(c) As the number of trials increases, we expect the experimental and theoretical probabilities to become closer, though they might not be equal.

Probability is the likelihood of a desired event happening.

Experimental probability is a probability that relies mainly on a series of experiments.

Theoretical probability is the theory behind probability. To find the probability of an event, an experiment is not required. Instead, we should know about the situation to find the probability of an event occurring.

(a) From these results, the experimental probability of getting a 5 or 8 will be:

Experimental probability (5 or 8) = P(5) + P(8)

Experimental probability (5 or 8) = (15/150) + (14/150)

Experimental probability (5 or 8) = 29/150

Experimental probability (5 or 8) = 0.193

(b) The theoretical probability of getting a 5 or 8 will be:

0,1, 2, 3, 4, 5, 6, 7, 8, 9

Theoretical probability (5 or 8) = P(5) + P(8)

Theoretical probability (5 or 8) = (1/10) + (1/10)

Theoretical probability (5 or 8) = 2/10

Theoretical probability (5 or 8) = 0.200

(c) As the number of trials increases, we expect the experimental and theoretical probabilities to become closer, though they might not be equal.

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The given function rand() % 50 + 5  will generate random numbers in the range of 5 to 54 inclusive.

The code snippet you provided contains a syntax error.

It seems like there is a typo between the '%' and '5' characters.

Assume that it meant to write,

rand() % 50 + 5;

Assuming that rand() function generates a random integer between 0 and RAND_MAX

Which may vary depending on the implementation.

The expression rand() % 50 will generate a random integer between 0 and 49 inclusive.

Then, adding 5 to the result will shift the range of the generated numbers up by 5.

Producing a random integer between 5 and 54 inclusive.

Therefore, the range of the random numbers generated by the code snippet rand() % 50 + 5 will be from 5 to 54 inclusive.

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P = 2 is the answer to the equation 2(p + 4) = 12.

An expression is made up of a number, a variable, or a combination of a number, a variable, and operation symbols. Two expressions are combined into one equation by using the equal symbol. For illustration: When you add 8 and 3, you get 11.

Divide the two among the terms between the parenthesis:

2p + 8 = 12

Add 8 to both sides of the equation, then subtract 8:

2p + 8 - 8 = 12 - 8

2p = 4

multiply both sides by two:

2p/2 = 4/2 \sp = 2

p = 2 is the answer to the equation 2(p + 4) = 12 as a result.

Simply put p = 2 back into the equation and simplify to obtain p + 4:

[tex]p + 4 = 2 + 4 = 6[/tex]

Hence, p + 4 = 6.

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[tex]\begin{cases} x=\frac{2+i\sqrt{2}}{3}\implies 3x=2+i\sqrt{2}\implies 3x-2-i\sqrt{2}=0\\\\ x=\frac{2-i\sqrt{2}}{3}\implies 3x=2-i\sqrt{2}\implies 3x-2+i\sqrt{2}=0 \end{cases} \\\\\\ \stackrel{ \textit{original polynomial} }{a(3x-2-i\sqrt{2})(3x-2+i\sqrt{2})=\stackrel{ 0 }{y}} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{ \textit{difference of squares} }{[(3x-2)-(i\sqrt{2})][(3x-2)+(i\sqrt{2})]}\implies (3x-2)^2-(i\sqrt{2})^2 \\\\\\ (9x^2-12x+4)-(2i^2)\implies 9x^2-12x+4-(2(-1)) \\\\\\ 9x^2-12x+4+2\implies 9x^2-12x+6 \\\\[-0.35em] ~\dotfill\\\\ a(9x^2-12x+6)=y\hspace{5em}\stackrel{\textit{now let's make}}{a=-\frac{1}{9}} \\\\\\ -\cfrac{1}{9}(9x^2-12x+6)=y\implies \boxed{-x^2+\cfrac{4}{3}x-\cfrac{2}{3}=y}[/tex]

Answer:

its reflection

Step-by-step explanation:

a reflection is known as a flip. A reflection is a mirror image of the shape. An image will reflect through a line, known as the line of reflection. A figure is said to reflect the other figure, and then every point in a figure is equidistant from each corresponding point in another figure.

Answer:

It is Reflection. Check if it is in the list.

Answer: 27.3

Step-by-step explanation:

I took the outcomes of the Aces from the trial and found the average and the answer I got was 27.3%

Hope this helps.

A positive semi-definite operator's rayleigh quotient must have a minimum value of zero to be considered positive.

Let A be a non-positive definite positive semi-definite operator. This proves that a non-zero vector x exists such that Ax = 0. The Rayleigh quotient of A with regard to x may thus be defined as follows:

[tex]R(x) = (x^T)Ax / (x^T)x[/tex]

A is positive semidefinite, hence for each vector x, (xT)Ax >= 0 is true. However, there is a non-zero vector x such that Ax = 0 if A is not a positive definite. In this instance, the Rayleigh quotient's numerator is 0, and as a result, the Rayleigh quotient is also 0. Since there is always a non-zero vector x such that Ax = 0, we may infer that the Rayleigh quotient's lowest value for a positive semi-definite but not positive definite operator is 0.

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The cumulative distribution function (CDF) for given random variable fx(x)  is given by F(x) = 1 - e^[(-a²)(x²/2)]        x > 0,

                           F(x) = 0                                x ≤ 0.

The cumulative distribution function (CDF) F(x) for a Rayleigh random variable X is defined as,

F(x) = P(X ≤ x)

To find the CDF of X, we integrate the PDF of X over the interval [0, x],

F(x) = ∫₀ˣ a²x e^[(-a²)(x²/2)] dx

Using the substitution u = (-a²x²/2),

Simplify the integral as follows,

F(x) = ∫₀ˣ a²x e^[(-a²)(x²/2)] dx

= ∫₀^((-a²x²)/2) -e^u du (where u = (-a²x²/2) and x = √(2u/a²))

= [e^u]₀^((-a²x²)/2)

= 1 - e^[(-a²)(x²/2)]

Therefore, the CDF of X  for the Rayleigh random variable X has PDF fx (x)   is equal to,

F(x) = 1 - e^[(-a²)(x²/2)]        x > 0,

F(x) = 0                                x ≤ 0.

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The above question is incomplete, the complete question is:

For a constant parameter a > 0, a Rayleigh random variable X has PDF

fx (x)        =    a²xe^[(-a²)(x²/2)]      x > 0

                     0                              otherwise.

What is the CDF of X?

let's recall that the conjugate of any expression is simply the same pair with a different sign between, so conjugate of "a + b" is just "a - b" and so on.  That said, let's use the conjugate of the denominator

[tex]\cfrac{\sqrt{2}+\sqrt{3}}{\sqrt{2}-\sqrt{3}}\cdot \cfrac{\sqrt{2}+\sqrt{3}}{\sqrt{2}+\sqrt{3}}\implies \cfrac{(\sqrt{2}+\sqrt{3})(\sqrt{2}+\sqrt{3})}{\underset{ \textit{difference of squares} }{(\sqrt{2}-\sqrt{3})(\sqrt{2}+\sqrt{3})}}\implies \cfrac{\stackrel{ F~O~I~L }{(\sqrt{2}+\sqrt{3})(\sqrt{2}+\sqrt{3})}}{(\sqrt{2})^2-(\sqrt{3})^2} \\\\\\ \cfrac{2+2\sqrt{2}\cdot \sqrt{3}+3}{2-3}\implies \cfrac{5+2\sqrt{6}}{-1}\implies \boxed{-5-2\sqrt{6}}[/tex]

Answer: 16 Jugs of orange juice

Step-by-step explanation:

Let

J = Jugs of orange juice

P = Batches of punch

2J = 3P

Therefore to find what 1 P equals divide both sides by 3 giving:

2/3 J = 1P

Using this ratio, take it and apply it to the given question:

2/3 J = 1P

therefore:

24 x 2/3 = Needed J

= 16J

Answer:

D I believe

Step-by-step explanation:

The measure of the angles obtained using trigonometric identities are;

Trigonometric identities are mathematical equations that consists of the trigonometric functions and which are correct for the values of the angles entered into the equations.

The value of sin(2·θ) can be obtained by making use of the Pythagorean identity as follows;

cos²(θ) + sin²(θ) = 1

sin²(θ) = 1 - cos²(θ)

sin(θ) = √(1 - cos²(θ))

3·cos(θ) = √5

cos(θ) = √5/3

sin(θ) = √(1 - (√5/3)²) = 2/3

180° ≤ θ ≤ 360°, therefore, sin(θ) is negative, which indicates;

sin(θ) = -2/3

sin(2·θ) = 2·sin(θ)·cos(θ)

sin(2·θ) = 2×(-2/3) × (√5)/3 = -(4·√5)/9

The double angle formula for cosines, indicates that we get;

cos(2·θ) = cos²(θ) - sin²(θ)

Therefore;

cos(2·θ) = ((√5)/3)² - (-2/3)² = 5/9 - 4/9 = 1/9

tan(2·θ) = sin(2·θ)/cos(2·θ)

Therefore;

tan(2·θ) = ((4·√5)/9)/(1/9) = 4·√5

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[tex]\left[\begin{array}{ccc}0&0\\0&0&\\\end{array}\right][/tex] is the solution of the cayley Hamilton theorem for the matrix .

What does Cayley-Hamilton theorem mean?

Theorem of Cayley-Hamilton: Every square matrix satisfies its own characteristic equation, according to this theorem. For the stress polynomial p(), this means that the scalar polynomial p() = det(I ) also holds true.

A = [tex]\left[\begin{array}{ccc}3&1\\4&1&\\\end{array}\right][/tex]

Cayley Hamilton Theorem states that Every square matrix A must satisfy its characteristic equation | A - kI |.

So, first find characteristic equation.

     ⇒ A - kI

      [tex]\left[\begin{array}{ccc}3&1\\1&4&\\\end{array}\right][/tex]  - k[tex]\left[\begin{array}{ccc}1&0\\0&1\\\end{array}\right][/tex]

    =  [tex]\left[\begin{array}{ccc}3&1\\1&4&\\\end{array}\right] - \left[\begin{array}{ccc}k&0\\0&k&\\\end{array}\right][/tex]

   =  [tex]\left[\begin{array}{ccc}3 -k&1\\1&4-k&\\\end{array}\right][/tex]

So,

Characteristic equation is given by

  ⇒ l A - kI l = 0

  ⇒ l 3 - k    1       l

      l 1           4 - k l

 = ( 3- k )(4 - k ) - 1 = 0

 = k² - 7k + 11 =0

So, We have to show that A must satisfy

k² - 7k + 11 =0

thus

  A² - 7A + 11I =0

So, Consider

      A² - 7A + 11I =0

     [tex]\left[\begin{array}{ccc}3&1\\1&4&\\\end{array}\right] \left[\begin{array}{ccc}3&1\\1&4&\\\end{array}\right] - 7\left[\begin{array}{ccc}3&1\\1&4&\\\end{array}\right] + 11\left[\begin{array}{ccc}3&1\\1&4&\\\end{array}\right][/tex]

    [tex]\left[\begin{array}{ccc}9+1&3+4\\3+4&1+16&\\\end{array}\right][/tex] [tex]- \left[\begin{array}{ccc}21&7\\7&28&\\\end{array}\right] + \left[\begin{array}{ccc}11&0\\0&11&\\\end{array}\right][/tex]

   [tex]\left[\begin{array}{ccc}10&7\\7&17&\\\end{array}\right] + \left[\begin{array}{ccc}-10&-7\\-7&-17&\\\end{array}\right][/tex]

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

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Answer:45

Step-by-step explanation:

Answer:

A' = (1, 0)

B' = (0, -3)

C' = (3, -1)

Step-by-step explanation:

4 units right is adding 4 to the x value.

1 unit up is adding 1 to the y value.

A' = (1, 0)

B' = (0, -3)

C' = (3, -1)

Hope this helps!