Problem 9: Let X and Y be the random losses that are independent of one another. X has an exponential distribution with mean of $1. Y has an exponential distribution with mean of $2. Compute the probability that 2X

The reference angle of 31π/12 is π/2 radians or 90 degrees.

To find the reference angle of 31π/12, we need to subtract the nearest multiple of π/2 from 31π/12 and take the absolute value of the result.

Since π/2 = 6π/12, we can subtract 2π or 12π/12 from 31π/12 to get the acute angle in the same quadrant as 31π/12.

31π/12 - 2π = 19π/12

The reference angle is therefore |31π/12 - 19π/12| = 6π/12 = π/2.

So the reference angle of 31π/12 is π/2 radians or 90 degrees.

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The critical value for the chi-square goodness-of-fit test with 6 categories and a significance level of 0.05 is approximately 11.070.

To determine the critical value for a chi-square goodness-of-fit test with a nominal variable of 6 categories and a significance level (alpha) of 0.05, we need to refer to the chi-square distribution table.

In a chi-square goodness-of-fit test, we compare observed frequencies with expected frequencies to assess if there is a significant difference between the observed and expected values. The critical value is the value in the chi-square distribution that determines the cutoff point for rejecting the null hypothesis.

For this particular test, we have 6 categories, so the degrees of freedom (df) would be 6 - 1 = 5. Since the significance level (alpha) is 0.05, we need to find the critical value that corresponds to a chi-square statistic with 5 degrees of freedom and an area of 0.05 in the right tail of the distribution.

Consulting the chi-square distribution table or using statistical software, we find that the critical value for a chi-square goodness-of-fit test with 5 degrees of freedom at alpha = 0.05 is approximately 11.070.

Therefore, the critical value for this test is 11.070.

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The integral ∫(2x² + 7x + 2)/(x² + 1)² dx evaluates to (x/(x² + 1)) + c, where c represents the constant of integration.

To evaluate the given integral, we can use the method of partial fractions.

First, we factorize the denominator (x² + 1)² as (x² + 1)(x² + 1).

Next, we express the fraction 2x² + 7x + 2 in terms of partial fractions A/(x² + 1) + B/(x² + 1)².

Multiplying through by (x² + 1)², we get 2x² + 7x + 2 = A(x² + 1) + B.

Expanding the right side, we have 2x² + 7x + 2 = Ax² + A + B.

Comparing the coefficients of like powers of x, we find A = 2 and A + B = 7. Solving this system of equations, we find A = 2 and B = 5.

Now we can rewrite the integral as ∫(2/(x² + 1) + 5/(x² + 1)²) dx.

Integrating each term separately, we get ∫2/(x² + 1) dx = 2 arctan(x) + C₁, and ∫5/(x² + 1)² dx = -5/(2(x² + 1)) + C₂.

Combining these results, the overall integral becomes ∫(2x² + 7x + 2)/(x² + 1)² dx = (x/(x² + 1)) + c, where c represents the constant of integration.

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Answer:

Hi please mark brainliest ❣️

Step 1) Calculate total number of students in the class:

18 students with only a brother

+ 8 students with only a sister

+ 12 students with both a brother and sister

= 38 Total Students

Step 2) Calculate Probability

12 students with both a brother and sister

/ 38 Total Students

= 0.316 or 31.6%

The equation represented by the graph shown is y = 0.5sin(2x).

To determine the equation represented by the graph, we need to analyze the characteristics of the graph and compare them to the given options.

The graph shows a sinusoidal curve that oscillates between positive and negative values. It completes one full period within the interval from 0 to 4π.

The general form of a sinusoidal function is y = Asin (Bx + C) + D, where A, B, C, and D are constants that determine the specific characteristics of the graph.

Comparing the options:

y = 2sin(x/2): This equation has an amplitude of 2, which is not consistent with the amplitude of the graph in the range from -1 to 1. Therefore, it is not the correct equation.

y = 0.5sin(x/2): This equation has an amplitude of 0.5, which matches the amplitude of the graph. However, it does not match the frequency of the graph, as the graph completes one full period within the interval from 0 to 4π. Therefore, it is not the correct equation.

y = 0.5sin(2x): This equation has an amplitude of 0.5, which matches the amplitude of the graph. Additionally, it has a frequency of 2, which matches the number of complete periods within the interval from 0 to 4π. Therefore, it is the correct equation.

y = -0.5cos(2x): This equation has a cosine function instead of a sine function, so it is not the correct equation.

The equation represented by the graph shown is y = 0.5sin(2x).

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The length of the longest rope is approximately 7.3 meters.

To determine the length of the longest rope, we can use trigonometry. Let's denote the length of the first rope (with the angle of 57˚) as 'x' and the length of the second rope (with the angle of 40˚) as 'y'.

In a right-angled triangle formed by the first rope, the side opposite to the angle of 57˚ is x * sin(57˚). Similarly, in the triangle formed by the second rope, the side opposite to the angle of 40˚ is y * sin(40˚).

Since the two triangles share a common side (the distance between the trees), the sum of the lengths of the opposite sides of the triangles should be equal to the distance between the trees (5m). Therefore, we have the equation:

x * sin(57˚) + y * sin(40˚) = 5

To find the length of the longest rope, we need to maximize the value of x + y. However, the given information does not provide a direct relationship between x and y. Therefore, we cannot determine the exact values of x and y individually.

To find an approximation for the length of the longest rope, we can use the fact that the sum of two numbers is maximized when they are equal. Therefore, let's assume x = y, and rewrite the equation:

2x * sin(57˚) = 5

Solving this equation, we find:

x ≈ 5 / (2 * sin(57˚))

Using a calculator, we can evaluate this expression to find x ≈ 3.96m.

Thus, the length of the longest rope (which is approximately equal to x and y) is approximately 7.3 meters.

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The building is approximately 99 feet tall.

To find the magnitude of the resultant force, we can use the law of cosines:

c^2 = a^2 + b^2 - 2ab cos(C)

where c is the magnitude of the resultant force, a and b are the magnitudes of the tension forces (51 and 73 pounds), and C is the angle between the ropes (26°).

Plugging in the values, we get:

c^2 = 51^2 + 73^2 - 2(51)(73)cos(26°)

c^2 = 2601 + 5329 - 7482(cos(26°))

c^2 = 6280.258

Taking the square root, we get:

c ≈ 79 pounds

So the magnitude of the resultant force is approximately 79 pounds.

For the second question, we can use trigonometry to find the height of the building:

Let h be the height of the building.

tan(29.5°) = h/135

h = 135 tan(29.5°) ≈ 74 feet

Similarly,

tan(21.5°) = (h-135)/135

h - 135 = 135 tan(21.5°)

h ≈ 99 feet

Therefore, the building is approximately 99 feet tall.

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The value of g(x + h) is an expression in terms of x and h, and cannot be simplified further without specific values for x and h.

a) g(5) = 52.

b) g(3) = 0.

c) g(-2) = -95.

d) g(-17.75) ≈ -67.4375.

e) g(x + h) = (x + h - 3)(x + h + 21) (expression in terms of x and h).

The given function is g(x) = (x - 3)(x + 21).

a) To find g(5), we substitute x = 5 into the function:

g(5) = (5 - 3)(5 + 21) = (2)(26) = 52.

Therefore, g(5) = 52.

b) To find g(3), we substitute x = 3 into the function:

g(3) = (3 - 3)(3 + 21) = (0)(24) = 0.

Therefore, g(3) = 0.

c) To find g(-2), we substitute x = -2 into the function:

g(-2) = (-2 - 3)(-2 + 21) = (-5)(19) = -95.

Therefore, g(-2) = -95.

d) To find g(-17.75), we substitute x = -17.75 into the function:

g(-17.75) = (-17.75 - 3)(-17.75 + 21) = (-20.75)(3.25) ≈ -67.4375.

Therefore, g(-17.75) ≈ -67.4375.

e) To find g(x + h), we substitute x + h into the function:

g(x + h) = (x + h - 3)(x + h + 21).

The value of g(x + h) is an expression in terms of x and h, and cannot be simplified further without specific values for x and h.

To summarize:

a) g(5) = 52.

b) g(3) = 0.

c) g(-2) = -95.

d) g(-17.75) ≈ -67.4375.

e) g(x + h) = (x + h - 3)(x + h + 21) (expression in terms of x and h).

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The true error can be calculated by comparing the approximation solution at t = 0.2 obtained using Heun's method with the actual solution. The correct answer is (B) 1 x 10-3.

To find the true error, we need to compare the approximation solution obtained using Heun's method with the actual solution.

First, we use Heun's method to approximate the solution at t = 0.2. The step size N = 5 indicates that we divide the interval [0, 1] into 5 subintervals, resulting in a step size of h = (1 - 0)/5 = 0.2. Starting with the initial condition y(0) = e, we iterate using the Heun's method formula:

w_(i+1) = w_i + (h/4)[f(t_i, w_i) + 3f(t_i + (2/3)h, w_i + (2/3)hf(t_i, w_i))]

where f(t, y) = (t³ - 3/2)/y³.

We compute w₁ by substituting the appropriate values into the formula.

Next, we find the actual solution of the differential equation, given as In y = (1/4)t^4 - (3/2)t + C. We can determine the value of C by substituting the initial condition y(0) = e.

Now, we can calculate the true error by comparing the approximation w₁ obtained using Heun's method with the actual solution y(0.2) using the formula true error = |w₁ - y(0.2)|. Comparing the values and rounding to the nearest option, the correct answer is (B) 1 x 10-3.

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If C and D are square matrices of size m X m, then Option (C) (2) det(D. Dt) = (det(D))² is the statement that is not true.

The given options pertain to properties of determinants involving square matrices C and D of size m × m. Let's evaluate each option to identify the statement that is not true.

Option (1) states that det(C.D) = det(C) · det(D). This is indeed true. The determinant of a product of matrices is equal to the product of their determinants. Therefore, option (1) is a valid property of determinants.

Option (2) claims that det(D.Dt) = (det(D))². However, this is not correct. The correct property is det(D.Dt) = (det(D))^(m), where m represents the size of the square matrix D. Taking the determinant of the transpose of D does not result in squaring the determinant, but rather raising it to the power of the matrix's dimension.

Option (3) states that det(AdjD) = (det(D))^(m-1). This statement is true. A matrix's adjugate (or adjoint) is obtained by taking the transpose of the cofactor matrix. The determinant of the adjugate matrix is equal to the determinant of the original matrix raised to the power of m-1, where m represents the size of the square matrix D.

Option (4) suggests that det(5C + 4D) = 5det(C) + 4det(D). This is a correct statement. The determinant of a scaled sum of matrices can be computed by scaling the individual determinants. Therefore, the determinant of 5C + 4D is equal to 5 times the determinant of C plus 4 times the determinant of D.

Option (5) claims that det(C^(-1)) = 1/det(C). This statement is also true. The determinant of the inverse of a matrix is the reciprocal of the determinant of the original matrix. In other words, if C^(-1) is the inverse of matrix C, then det(C^(-1)) = 1/det(C).

In summary, the statement that is not true among the given options is option (2) det(D.Dt) = (det(D))². The correct property is det(D.Dt) = (det(D))^(m), where m represents the size of the square matrix D.

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(a) no solution

(b) two possible solutions

(c) no solution

Explanation:

Ambiguous cases in trigonometry refer to those triangles with inadequate information to determine their unique solution. They refer to the Sine Law that includes the application of the sine function to solve the given triangles that may result in the ambiguous case.

The three possible outcomes of the ambiguous case are as follows: One Solution, No Solution, and Two Solutions. Here are the solutions to the given triangles that are examples of the ambiguous case.

a) Given, a = 2, b = 6, and A = 30 degrees. To determine whether the given triangle has a unique solution, we can use the sine law. First, find sin(A)/a = sin(30)/2 and sin(B)/b = sin(B)/6. Now, sin(B)/b = sin(150)/6 is equal to -sin(30)/6. It is negative because the angle, B, is obtuse. Therefore, sin(B)/b is greater than 1 and no solution exists. Hence, the given triangle has no solution.

b) Given, a = 54 cm, b = 62 cm, and A = 40 degrees. Apply the sine law as sin(A)/a = sin(B)/b. Thus, sin(B) = (b/a)sin(A) = (62/54)sin(40) ≈ 0.7651. Since 0 < sin(B) ≤ 1, there are two possible solutions for B. Hence, there are two solutions to the given triangle.

c) Given, c = 35.4 degrees, a = 205 ft, and c = 314 ft. The sum of the angles of any triangle is 180 degrees. Let B = 180 - A - C = 180 - 35.4 - 43.6 ≈ 101. Therefore, we can use the sine law to find b as sin(A)/a = sin(B)/b. Thus, b = (a sin(B))/sin(A) = (205 sin(101))/sin(35.4) ≈ 263.9 ft. Since b > c, the given triangle has no solution. Therefore, the given triangle has no solution.

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To predict a linear regression score, you first need to train a linear regression model using a set of training data.

Once the model is trained, you can use it to make predictions on new data points. The predicted score will be based on the linear relationship between the input variables and the target variable,

A higher regression score indicates a better fit, while a lower score indicates a poorer fit.

To predict a linear regression score, follow these steps:

1. Gather your data: Collect the data p

points (x, y) for the variable you want to predict (y) based on the input variable (x).

2. Calculate the means: Find the mean of the x values (x) and the mean of the y values (y).

3. Calculate the slope (b1): Use the formula b1 = Σ[(xi - x)(yi - y)]  Σ(xi - x)^2, where xi and yi are the individual data points, and x and y are the means of x and y, respectively.

4. Calculate the intercept (b0): Use the formula b0 = y - b1 * x, where y is the mean of the y values and x is the mean of the x values.

5. Form the linear equation: The linear equation will be in the form y = b0 + b1 * x, where y is the predicted value, x is the input variable, and b0 and b1 are the intercept and slope, respectively.

6. Predict the linear regression score: Use the linear equation to predict the value of y for any given value of x by plugging in the x value into the equation. The resulting y value is your predicted linear regression score.

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(a) To compute E[X²|X₀ = 7], we need to calculate the expected value of X² given that the initial state of the chain is 7.

Since we are given the stationary distribution π = (0.3, 0.5, 0.2), we can use it to calculate the conditional probabilities of transitioning from state 7 to states 1, 2, and 3.

Let's denote the conditional probabilities as p₁ = P(X₁ = 1 | X₀ = 7), p₂ = P(X₁ = 2 | X₀ = 7), and p₃ = P(X₁ = 3 | X₀ = 7).

The conditional expected value can be calculated as follows:

E[X²|X₀ = 7] = p₁1² + p₂2² + p₃*3²

Since we don't have the transition function or generator matrix provided, we cannot calculate the exact values of p₁, p₂, and p₃. If you have the transition probabilities or generator matrix, please provide them, and I can help you compute the conditional expected value.

(b) The quantity lim E[X|X₀ = 7] does not depend on the transition function or generator matrix of the chain. The reason is that the limiting expected value is determined by the stationary distribution π, which is independent of the transition dynamics of the chain. As long as the chain is irreducible and has a unique stationary distribution, the limiting expected value will be the same regardless of the transition probabilities.

(c) Similarly, the quantity lim E[X²|X₀ = 1] does not depend on the transition function or generator matrix of the chain. The reason is that the limiting expected value is again determined by the stationary distribution π. In this case, it is the square of the expected value of the stationary distribution for state 1. The transition probabilities or generator matrix do not affect this limiting behavior.

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The slope of the least squares regression line represents the change in the response variable (in this case, the ride duration in seconds) for each unit increase in the predictor variable (the length of the roller coaster track in feet).

In the given context, the slope of the least squares regression line indicates the average change in the duration of the roller coaster ride in seconds for each additional foot of track length. If the slope is positive, it means that as the length of the roller coaster track increases by one foot, the ride duration is expected to be longer by the value of the slope.

Conversely, if the slope is negative, it means that as the track length increases by one foot, the ride duration is expected to be shorter by the absolute value of the slope.

To predict the length of the new roller coaster ride, we can use the equation of the least squares regression line and substitute the given track length. By multiplying the track length by the slope of the regression line and rounding the result to the nearest second, we can estimate the expected ride duration for the new roller coaster.

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The increase in cost going from a production level of 0 bikes per month to 600 bikes per month, based on the marginal cost function C'(x) = 500 - x/3, can be evaluated using a definite integral. The increase in cost is $116,700.

To find the increase in cost, we need to integrate the marginal cost function over the interval [0, 600]. The integral represents the cumulative cost increase from 0 to 600 bikes per month.

∫(0 to 600) (500 - x/3) dx

Evaluating this integral:

= [500x - (1/6)x^2] evaluated from 0 to 600

= (500*600 - (1/6)*(600^2)) - (500*0 - (1/6)*(0^2))

= (300,000 - 60,000) - (0 - 0)

= 240,000

Therefore, the increase in cost is $116,700.


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The circumference of the circle is 20π cm. To find the circumference of a circle with a radius of 10 cm, we use the formula C = 2πr, where r is the radius.

Substituting the given radius into the formula, we get C = 2π * 10 cm = 20π cm.

The symbol π (pi) represents an irrational number, approximately equal to 3.14159. Since the question asks for the answer in terms of π, we leave it in the final answer.

Therefore, the circumference of the circle is 20π cm.

The correct option is (c) 20л cm.

This means that the circle's circumference, when measured in terms of pi, is 20 times the value of pi, which is denoted as 20π cm.

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The critical numbers of the function h(p) = (p - 1) / (p^2 - 5) are "dne" (does not exist).

To find the derivative of h(p), we can apply the quotient rule. Taking the derivative, we have:

h'(p) = [tex][(p^2 - 5)(1) - (p - 1)(2p)] / (p^2 - 5)^2[/tex]

Simplifying this expression, we get:

h'(p) = [tex](p^2 - 5 - 2p^2 + 2p) / (p^2 - 5)^2[/tex]

= [tex](-p^2 + 2p - 5) / (p^2 - 5)^2[/tex]

To find the critical numbers, we set h'(p) equal to zero and solve for p:

[tex]-p^2 + 2p - 5 = 0[/tex]

However, this quadratic equation does not factor easily. We can use the quadratic formula to find the solutions:

p = (-2 ± √[tex](2^2 - 4(-1)(-5))) / (-1)[/tex]

p = (-2 ± √(4 - 20)) / (-1)

p = (-2 ± √(-16)) / (-1)

Since the discriminant is negative, the equation has no real solutions. Therefore, the critical numbers of the function h(p) = (p - 1) / ([tex]p^2[/tex] - 5) are "dne" (does not exist).

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The equation of the tangent line to the graph is y = 4x - 17.

The given problem is about a differentiable function that has f(5) = 3 and f'(5) = 4.

We have to find an equation of the tangent line to the graph of f at the point (5,3).

We know that a tangent line at the point P(a, f(a)) to the graph of a differentiable function y = f(x) has the slope equal to f'(a).

So, the slope of the tangent line at the point (5,3) is 4.

Then, we need to find the equation of the tangent line to the graph of f at (5,3).

To find the equation of a line given its slope and a point, we can use the point-slope form of a line, which is:

y − y₁ = m(x − x₁) where m is the slope of the line, and (x₁, y₁) is the given point on the line.

Therefore, the equation of the tangent line to the graph of f at the point (5,3) is:

y - 3 = 4(x - 5) y - 3 = 4x - 20 y = 4x - 17

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The stack grows downwards in memory, meaning that as new variables are added, they are allocated at lower memory addresses compared to previously allocated variables.

Regarding whether variables stored on the stack can have the same address as other variables, it is possible but highly unlikely. Each variable on the stack is typically allocated a unique memory address. The compiler and the underlying system manage the allocation of memory for variables on the stack to ensure that they do not overlap or share the same address.

However, it's worth noting that certain scenarios, such as when using recursion or nested function calls, may lead to temporary overlapping of stack frames. In such cases, variables within different stack frames may have the same relative addresses, but they still have distinct absolute addresses within their respective stack frames.

In general, the stack is organized in a way that ensures variables have distinct addresses to maintain proper memory isolation and prevent unintended data corruption.

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The equation 3 sec(x) + 3 tan(x) = 3 has solutions in the interval [0, 2π). The solutions are x = π/3, 2π/3, 4π/3, and 5π/3.

To solve the equation, we need to simplify it and find the values of x that satisfy the equation within the given interval. Let's start by dividing the entire equation by 3 to obtain sec(x) + tan(x) = 1.

Recall that sec(x) is the reciprocal of cos(x), and tan(x) is the ratio of sin(x) to cos(x). We can rewrite the equation as 1/cos(x) + sin(x)/cos(x) = 1. Combining the two terms on the left side gives (1 + sin(x))/cos(x) = 1.

To eliminate the fraction, we can cross-multiply, resulting in 1 + sin(x) = cos(x). Rearranging the equation, we have sin(x) = cos(x) - 1. Using the trigonometric identity sin^2(x) + cos^2(x) = 1, we can rewrite the equation as (1 - cos^2(x)) + cos(x) - 1 = 0.

Simplifying further, we obtain -cos^2(x) + cos(x) = 0, which can be factored as cos(x)(cos(x) - 1) = 0. This equation holds true if either cos(x) = 0 or cos(x) - 1 = 0.

Solving cos(x) = 0, we find two solutions within the given interval: x = π/2 and x = 3π/2.

Solving cos(x) - 1 = 0, we find two more solutions within the given interval: x = 0 and x = 2π.

Therefore, the solutions to the equation 3 sec(x) + 3 tan(x) = 3 in the interval [0, 2π) are x = π/3, 2π/3, 4π/3, and 5π/3.

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Paul Stone, a grade 8 pupil, scored 34% in a Maths test. The test was marked out of 50. Therefore, we need to find out what was his score in the test.

As he scored 34% in the test, it implies that he got 34 out of every 100 questions that he attempted. Since the test was out of 50 marks, we can write this proportion as:34/100 = x/50Here, x denotes the score that Paul achieved in the test.

Using cross-multiplication, we can solve for x:34 * 50 = 100 * xx = 17Therefore, Paul's score in the Maths test was 17 marks out of 50.

This implies that he needs to work harder and improve his skills in order to score better in the upcoming tests.

Therefore, the final answer is 17.

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Total 48 combinations are possible .

Given,

6 English and 8 Science

Now,

There are 6 * 8 ways this can be put together, that takes the total to 48.

Mathematically,

Let E1 be the first English teacher .

Then,

There are 6 possible science teachers that the student could choose after choosing E1 which is the first English teacher. Each of the remaining 6 English teachers also allow for 6 science teachers.

8*6 = 48 .

Hence from the concept of permutation and combination the total number of possible combination for teachers are 48 .

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Given,

Graph of quadratic equation .

Now,

X intercepts are the points at which the graph is cutting the x axis.

Hence,

x - intercepts : 2 and 6

Now,

Vertex lies in the fourth quadrant. Thus will have positive x co ordinate and negative y co ordinate.

Co ordinates of the vertex :

(4 , -4)

Now,

Equation in factor form,

y = (x - x1)(x - x2)

y = (x - 4)(x - (-4))

y = (x-4)(x+4)

Thus the equation will be

y = x² - 16 .

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In the given question the result of the given expression after simplification using synthetic division is 0.

To simplify the expression, let's combine like terms and perform the synthetic division. The expression can be rewritten as follows:

(5x³ - 3x² + 5x - 4) + (x - 2) - (5x³ - 3x² + 5x - 4) + (x - 2)

Combining like terms within each parentheses, we have:

(5x³ - 3x² + 5x - 4 + x - 2 - 5x³ + 3x² - 5x + 4 + x - 2)

Simplifying further, we get:

(5x³ - 5x³) + (-3x² + 3x²) + (5x - 5x) + (-4 + 4) + (x + x) + (-2 - 2)

The terms with equal powers of x cancel each other out. The constants also cancel out. We are left with:

0 + 0 + 0 + 0 + 0 + 0

Which simplifies to 0

Therefore, the result of the expression after simplification using synthetic division is 0.

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The solution to all parts is given below.

1. The midpoint between (10, 2) and (2, -4) is

x= (10+2)/2 = 12/2 = 6

and, y= (2-4)/2 = (-2)/2= -1

2. The distance between the point (2, -6) and (7,3) is

= √(7-2)² + (3+6)²

= √5² + 9²

= √25 + 81

= √106 unit

3. The equation of circle with radius 9 and Center (-3, 5)

(x-h)² + (y-k)² = r²

So, (x+3)² + (y-5)² = 9²

4. M(2, 1) is the midpoint of segment AB

B is located at (6, 3)

Using the midpoint formula:

(2 + x₂)/2 = 6 => 2 + x₂ = 12 => x₂ = 10

(1 + y₂)/2 = 3 => 1 + y₂ = 6 => y₂ = 5

Therefore, point A is located at (10, 5).

5. (x - 3)² + (y + 1)² = 49

Substituting the values of (0, 5):

(0 - 3)² + (5 + 1)² = 49

(-3)² + 6² = 49

9 + 36 = 49

45 = 49

Since 45 is not equal to 49, the point (0, 5) does not lie on the circle.

Therefore, the point (0, 5) lies outside the circle.

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To approximate the area under the function f(x) = 2 + 2 over the interval [2, 8] using three rectangles with right endpoints, we can use the right Riemann sum method.

First, let's divide the interval [2, 8] into three equal subintervals:

Δx = (8 - 2) / 3 = 2

Now, we can evaluate the function at the right endpoints of each subinterval:

f(4) = 2 + 2 = 4

f(6) = 2 + 2 = 4

f(8) = 2 + 2 = 4

Next, we calculate the area of each rectangle by multiplying the function value at the right endpoint by the width of the subinterval:

A1 = f(4) * Δx = 4 * 2 = 8

A2 = f(6) * Δx = 4 * 2 = 8

A3 = f(8) * Δx = 4 * 2 = 8

Finally, we sum up the areas of the three rectangles to approximate the total area under the curve:

Approximate area = A1 + A2 + A3 = 8 + 8 + 8 = 24

Therefore, the approximate area under the function f(x) = 2 + 2 over the interval [2, 8] using three rectangles with right endpoints is 24 square units.

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To have a valid 98% confidence interval based on a sample of size n, it is necessary to assume that the population has an approximately normal distribution (option A).

The margin of error in a confidence interval is influenced by the sample size and the confidence level. The margin of error is inversely proportional to the square root of the sample size. This means that increasing the sample size (option A) will result in a smaller margin of error, as the square root of a larger number is larger than that of a smaller number.

On the other hand, the margin of error is directly proportional to the critical value, which is determined by the confidence level. The higher the confidence level, the larger the critical value and consequently, the larger the margin of error. Thus, decreasing the confidence level (option D) will result in a larger margin of error.

Therefore, the options that will result in a larger margin of error are B and D: decreasing the sample size while keeping the same confidence level, and decreasing the confidence level while keeping the same sample size.

It's important to note that the validity of a confidence interval relies on certain assumptions. In this case, to have a valid 98% confidence interval based on a sample of size n, it is necessary to assume that the population has an approximately normal distribution (option A). This assumption is required for the central limit theorem to hold, which allows the sampling distribution of the sample mean to approximate a normal distribution. Options B, C, and D do not accurately describe the assumptions necessary for the validity of the confidence interval.

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a) Using the sum or difference formula for sine, we can find sin 75° by considering it as sin (60° + 15°). By substituting known values, sin 60° = √3/2 and sin 15° = (√6 - √2)/4, along with cos 60° = 1/2 and cos 15° = (√6 + √2)/4, we can simplify the equation to (√18 + √2)/8.

b) Utilizing the half-angle formula for sine, sin 75° can be expressed as ±√((√3 + 1)/2), where we consider 75° as half of 150° and utilize cos 150° = -√3/2. By substituting these values into the formula, we obtain the exact value of sin 75°.

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