Use the factorization A PDP 1 to compute Ak, where k represents an arbitrary integer. a 7(b-a) 17 a 017

Answer:

We are given that the point (3,w) lies on the line y = 2x + 7. This means that if we substitute x = 3 into the equation y = 2x + 7, we will get the value of y at x = 3, which is equal to w.

Substituting x = 3 into the equation y = 2x + 7, we have:

y = 2(3) + 7

y = 6 + 7

y = 13

Therefore, the value of w is 13.

Step-by-step explanation:

From the given information provided, the number of people having brown eyes in town is 180.

If the probability that a person in the town has brown eyes is 2/5, then we can expect that 2 out of every 5 people have brown eyes.

To find the number of people in the survey who would be expected to have brown eyes, we can use the following proportion:

(2/5) = (x/450)

where x is the number of people expected to have brown eyes.

Solving for x, we can cross-multiply:

5x = 2 × 450

5x = 900

x = 180

Therefore, the expected number of people in the survey who would have brown eyes is 180.

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Therefore , the solution of the given problem of triangle comes out to be the angles P and Q have measurements of roughly 39.39° and 58.10°, respectively.

A polygon is a hexagon if it has over one extra segment. It's shape is a simple rectangle. Only the sides A and B can differentiate something like this arrangement from a regular triangle. Despite the exact collinearity of the borders, Euclidean geometry only produces a portion of the cube. Three edges and three angles make up a triangle.

Here,

The rule of cosines can be applied to the triangle PQR to determine the length of side PQ:

=>  PQ² = PR² + QR² - 2(PR)(QR)cos(∠PQR)

=>  PQ² = 9² + 12² - 2(9)(12)cos(62°)

=>  PQ² ≈ 110.03

=> PQ ≈ 10.49

As a result, side PQ is roughly 10.49 units long.

The rule of sines can then be used to determine the dimensions of angles P and Q:

=> sin(∠P) / PQ = sin(62°) / PR

=>  sin(∠P) / 10.49 = sin(62°) / 9

=>  sin(∠P) ≈ 0.6322

=>  ∠P ≈ 39.39°

=>  sin(∠Q) / PQ = sin(77°) / QR

=>  sin(∠Q) / 10.49 = sin(77°) / 12

=>  sin(∠Q) ≈ 0.8559

=>  ∠Q ≈ 58.10°

As a result, the angles P and Q have measurements of roughly 39.39° and 58.10°, respectively.

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The given functions form a fundamental set of solutions of the given system.

The solution of the given system of differential equations is shown below.a) To prove that the given functions X(1) and x(2) are the solutions of the given system of differential equations, we must substitute these functions into the given system to show that they satisfy the equations.In the given system, we have the following equations:

X_1' (t) = 2X_1 (t) - X_2 (t)
X_2' (t) = 4X_1 (t) - 2X_2 (t)

Now, let's substitute the given vector-valued functions X(1) and x(2) into the above equations and check if they satisfy these equations.

a. For X(1) = [1, 2]e^2t
Substituting X(1) into the given system, we get:

X_1' (t) = [1, 2] * 2e^2t = 2X_1 (t) - X_2 (t)
X_2' (t) = [1, 2] * 4e^2t = 4X_1 (t) - 2X_2 (t)

Therefore, the given function X(1) is a solution to the given system of differential equations.

b. To prove that X = C1x(1) + C2x(2) is also a solution of the given system for any values of C1 and C2, we need to X into the given system of equations and check if it satisfies the equations.

So, we have:

X = C_1[1, 2]e^2t + C_2[1, -1]e^-t
X_1 = C_1e^2t + C_2e^-t
X_2 = 2C_1e^2t - C_2e^-t

Differentiating X_1 and X_2 with respect to t, we get:

X_1' = 2C_1e^2t - C_2e^-t
X_2' = 4C_1e^2t + C_2e^-t

Substituting X_1 and X_2 into the given system, we get:

X_1' (t) = 2(C_1e^2t - C_2e^-t) = 2X_1 (t) - X_2 (t)
X_2' (t) = 4(C_1e^2t + C_2e^-t) = 4X_1 (t) - 2X_2 (t)

Therefore, the given function X = C1x(1) + C2x(2) is also a solution of the given system for any values of C1 and C2.

c. To show that the given functions form a fundamental set of solutions of the given system, we need to prove that they are linearly independent and that their Wronskian is non-zero.

We know that the vectors [1, 2] and [1, -1] are linearly independent, therefore the functions x(1) and x(2) are also linearly independent.

Also, the Wronskian of x(1) and x(2) is given by:

W(x1, x2) = | x1  x2 |
         | x1' x2' |

Substituting x(1) and x(2) into the above equation, we get:

W(x1, x2) = | e^2t  e^-t  |
          | 2e^2t -e^-t |

Simplifying the above equation, we get:

W(x1, x2) = 3e^(3t) ≠ 0

Therefore, the given functions form a fundamental set of solutions of the given system.

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If all other factors are held constant, then the true parameter is farther from the value of the null hypothesis which is an increase in the probability of a Type II error.The correct option is A.

The true parameter is farther from the value of the null hypothesis.

When the true parameter is farther away from the value of the null hypothesis, it increases the probability of a Type II error. This is because the null hypothesis will have a harder time rejecting the true parameter.

The other factors - increasing sample size, decreasing significance level, and decreasing standard error - all result in a decreased probability of a Type II error.

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The student is male, which is P(male).Using the rule of complementary events, P(male) = 1 - P(female)P(male) = 1 - 0.572 = 0.428Therefore, the probability that the student is male is 0.428 or 42.8%.

The complimentary event is a part of probability theory. It is the event that occurs when the event E does not occur. In other words, it is a set of outcomes in the sample space that are not included in the outcomes of event E. The notation for the complement of E is E'. Rule for Complimentary EventsThe rule for complementary events can be expressed in two ways:

P(E) = 1 - P(E)P(E) + P(E') = 1Example # 12:Let the probability that Mary can work a problem be P(E) = 0.70.We need to find the probability that Mary cannot work the problem, which is P(E').Using the rule of complementary events,P(E') = 1 - P(E)P(E') = 1 - 0.70 = 0.30Therefore, the probability that Mary cannot work the problem is 0.30 or 30%.Example # 13:Let P(female) be the probability that the student is female. We are given that P(female) = 0.572.

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In data analytics, a population refers to all possible data values in a certain dataset.

Data analytics is a set of procedures and processes for examining datasets in order to draw conclusions from the information they contain, often aided by specialized systems and software. Organizations use data analytics to aid decision-making, increase efficiency, and evaluate outcomes.

The population and sample are two concepts in statistics. The population and sample are two concepts in statistics. The population is the entire set of objects or individuals being studied, while the sample is a subset of the population that is chosen for analysis. The sample is a subset of the population, chosen at random or according to some other criteria in order to represent the population as a whole.

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Answer:

Step-by-step explanation:

To solve this problem, we need to find out how many seats there are in total, and then multiply that by the price per ticket.

To find the total number of seats, we need to add up the number of seats in each row. We can use the formula for an arithmetic sequence to do this:

S = n/2 * (a + l)

where S is the sum of the sequence, n is the number of terms, a is the first term, and l is the last term.

In this case, we have:

n = 20 (since there are 20 rows)

a = 25 (since there are 25 seats in the first row)

d = 2 (since the difference between each row is 2 seats, the common difference is 2)

We can use d to find the last term as well:

l = a + (n-1)*d

l = 25 + (20-1)*2

l = 25 + 38

l = 63

Now we can plug these values into the formula:

S = 20/2 * (25 + 63)

S = 10 * 88

S = 880

So there are 880 seats in total.

To find the total sales, we just need to multiply by the price per ticket:

total sales = 880 * $32

total sales = $28,160

Therefore, the total sales for a one-night concert with all seats taken would be $28,160.

The interest which josh will pay on the electric scooter with a simple annual interest of 3% is 7.50.

Interest rate can be defined as the amount of interest which is due per period, as a proportion of the amount lent, deposited, or borrowed by someone.

The interest rate formula is:

Interest Rate = {(Simple Interest × 100)}/{ (Principal × Time)}

Here,

Josh borrowed 250 from his mother to buy an electric scooter and will pay her back in one year with three simple annual interest.

The amount of interest that Josh will pay is calculated as:

Interest = Principal Amount × Rate of Interest × Time

Interest = 250 × 3

Therefore, Josh will pay his mother $7.50 in interest for the loan.

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Answer:

Perpendicular.

Step-by-step explanation:

To determine whether the two lines are parallel, perpendicular, or neither, we need to compare their slopes.

The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. So we can rewrite the given equations in this form

y = 2/3x - 4 ==> slope = 2/3

y = -3/2x - 7 ==> slope = -3/2

Two lines are parallel if and only if their slopes are equal. Therefore, since the slopes of the two lines are different (2/3 and -3/2), they cannot be parallel.

Two lines are perpendicular if and only if their slopes are negative reciprocals of each other. That is, if the product of their slopes is -1. Therefore, we can check if the product of the slopes of the two lines is -1

(2/3) * (-3/2) = -1

Since the product of the slopes is -1, the two lines are perpendicular.

Therefore, the answer is: perpendicular.

The number of distinct elements of AB = m n.

Given that G is a group with subgroups A and B of orders m and n, respectively, where m and n are prime, we need to prove that the subset of G, AB = {abla E Ab E B}, has m n distinct elements. Step-by-step. Let, G is a group with subgroups A and B of orders m and n, respectively. Since, m and n are relatively prime, then we have gcd(m, n) = 1.By Lagrange's Theorem, the order of any subgroup of G divides the order of G.

Hence, the order of G is equal to the product of the orders of A and B, i.e. |G| = |A| * |B| = m * n Let, a and a' be two distinct elements of A and b and b' be two distinct elements of B. Thus, a and a' generate distinct subgroups of G, i.e.  ≠  and b and b' generate distinct subgroups of G, i.e.  ≠ .Now, the number of distinct elements of AB = {abla E Ab E B} is equal to |A||B| since any two elements ab and a'b' of AB will be distinct if either a and a' are distinct or b and b' are distinct or both are distinct. Hence, the number of distinct elements of AB = m n.

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Lisa's percentile rank is approximately 88%.

Percentile rank is a statistical measure that indicates the percentage of scores that fall below a particular score in a given distribution of data. It is commonly used to describe the relative position of a particular score in a set of scores.

If Lisa's score was 83 and that score was the 29th score from the top in a class of 240 scores, then her percentile rank can be calculated using the following formula:

Percentile Rank = [(Number of scores below Lisa's score) ÷ (Total number of scores)] × 100

Percentile Rank = [(240 - 29) ÷ 240] × 100

Percentile Rank = (211 ÷ 240) × 100

Percentile Rank = 0.8792 × 100

Percentile Rank ≈ 88 (rounded to the nearest whole number)

Therefore, her percentile rank is approximately 88%.

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The trapezoid has a surface area of 480 square units.

So, a trapezoid measured in feet offers an area in square feet; one measured in millimetres gives an area in square centimetres; and so on. If it's simpler for you, you can add the lengths of the bases and then divide the total by two. Keep in mind that multiplication by 12 is equivalent to dividing by 2.

We must apply the formula for a trapezoid's area to this issue in order to find a solution:

[tex]A = (1/2) * (a + b) * h[/tex]

where h is the trapezoid's height (or altitude) and a and b are the lengths of its parallel sides.

The values for a, b, and h are provided to us, allowing us to change them in the formula:

A = (1/2) * (20 + 60) * 12

A = (1/2) * 80 * 12

A = 480 square units

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[tex]8700[/tex] cubic feet of airflow are required for the entire home. Jackson may utilize this information to determine the best furnace size for the customer's requirements.

Volume is the quantity of space an object occupies, whereas capacity is a measurement of the substance—such as a solid, liquid, or gas—that an object can hold. While capacity may be measured in virtually any other unit, such as liters, gallons, pounds, etc., volume was determined in cubic units.

Volume, which is measured in cubic units, is the three dimensional space inhabited by material or surrounded by a surface. The cubic centimeter (m3), a derived unit, is the SI unit for volume.

Volume of the first section with height [tex]H1=10[/tex] ft:

[tex]V1 = WLH1 = (20 ft)(30 ft)(10 ft) = 6000[/tex] cubic feet

Volume of the second section with height [tex]H2=9[/tex] ft:

[tex]V2 = (1/2)WLH2 = (1/2)(20 ft)(30 ft)(9 ft) = 2700[/tex]cubic feet

Total volume of house,

[tex]V total = V1 + V2 = 6000[/tex] cubic feet + [tex]2700[/tex] cubic feet [tex]= 8700[/tex] cubic feet

Therefore, the volume of airflow needed for the house is [tex]8700[/tex] cubic feet.

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Sure, I can help you with this. To calculate the amount that Customer five will end up paying with their $5.00 off coupon and 4.5% sales tax, we will use the following formula: final amount = original amount - coupon - (original amount * tax rate).

In this case, the original amount is $5.00, the coupon is $5.00, and the tax rate is 4.5%. Plugging these values into the formula, we get:

final amount = 5.00 - 5.00 - (5.00 * 0.045)

final amount = 5.00 - 5.00 - 0.225

final amount = 4.775

Therefore, Customer five will end up paying $4.775 after their coupon and the sales tax.

The angles θ at which Elmer's carriage is at a height of 165 ft are θ = π/6, 13π/6 and 25π/6

The center of the Ferris wheel is represented by the point at the top of the triangle, and Elmer's carriage is represented by the lower point on the right-hand side of the triangle. We are given that the diameter of the Ferris wheel is 180 ft and that its center is 120 ft off the ground. This means that the radius of the Ferris wheel is 90 ft (half of the diameter), and that the distance from the center of the Ferris wheel to Elmer's carriage is also 90 ft.

We are also given that Elmer's carriage is at a height of 165 ft when it makes an angle of θ with the horizontal. Using trigonometry function , we can find the sine of θ as follows:

sin(θ) = (165 - 120) / 90

= 45 / 90

= 1/2

Therefore, we have:

θ = sin^(-1)(1/2)

= π/6

Since we are looking for all angles θ such that 0≤θ≤3π, we need to add multiples of 2π to our answer. The multiples of 2π that satisfy this inequality are 0, 2π, and 4π. Therefore, the solutions to the problem are:

θ = π/6, 13π/6, 25π/6

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Therefore, the correct response is OB: No, the box is a rectangle, and the tray's 12.5-inch length is greater than the box's 11-inch breadth.

A rectangle is a geometric shape with four edges and four angles that exists in two dimensions. Because it is a sort of quadrilateral, it has four sides that are parallel to one another. Rectangles are a form of parallelogram because they have opposite sides that are the same length and opposite angles that are the same size. A rectangle's two adjacent sides make right angles, so all four of the angles, which each measure 90 degrees, are right angles.

given

The tray has a breadth of 17 inches and a length of 12.5, according to the measurements given. The package is 13 inches by 11 inches by 11 inches in size.

The platter can fit inside the box because its length is less than the box's length, which is 13 inches. But we also need to think about the box's breadth and height.

The tray cannot fit inside the box in that dimension because its width, which is 17 inches, is larger than the box's, which is 11 inches. The tray cannot fit inside the box because the height of the box is also 11 inches, which is shorter than the length of the tray neither in that realm.

Therefore, the correct response is OB: No, the box is a rectangle, and the tray's 12.5-inch length is greater than the box's 11-inch breadth.

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Answer:

y = -3x - 2

Step-by-step explanation:

To write the equation 6x + 2y = -4 in slope-intercept form, we need to solve for y.

First, we can isolate the y-term by subtracting 6x from both sides:

6x + 2y = -4

2y = -6x - 4

Next, we can divide both sides by 2 to isolate y:

2y/2 = (-6x - 4)/2

y = -3x - 2

So the slope-intercept form of the equation 6x + 2y = -4 is y = -3x - 2.

Answer: 118

Step-by-step explanation: 38 is the highest value, which is the fifth number in the pattern, and we need to find out what the 15th number is. We simply need to add (8x10) to 38.

Answer: 118

Step-by-step explanation:

you just keep going up at the rate of 8

The length of the ladder is c = √(x^2 + 64) ft.

The question is asking to find the length of the ladder. We can use the Pythagorean Theorem to solve this problem. Let x be the length of the ladder.

Pythagoras theorem states that “In a right-angled triangle,  the square of the hypotenuse side is equal to the sum of squares of the other two sides“. The sides of this triangle have been named Perpendicular, Base and Hypotenuse.

Here, the hypotenuse is the longest side, as it is opposite to the angle 90°. The sides of a right triangle (say a, b and c) which have positive integer values, when squared, are put into an equation, also called a Pythagorean triple.

Pythagoras theorem is useful to find the sides of a right-angled triangle. If we know the two sides of a right triangle, then we can find the third side.

The Pythagorean Theorem states that a^2 + b^2 = c^2.

Therefore, x^2 + 8^2 = c^2

Solving for c: x^2 + 64 = c^2

Taking the square root of both sides: √(x^2 + 64) = c

Therefore, the length of the ladder is c = √(x^2 + 64).

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The correct definition for the lines drawn to circle with centre C are:

A circle is a spherical shape without boundaries or edges.

Thus, on the basis of propertied of circle, the correct definition for the lines drawn to circle with centre C are:

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Answer:

[tex]6z \: = 6 + z \: \\ z {}^{6} = z \times z \times z \times z \times z \times z \\ [/tex]

The mathematical statement in the form of expression can be written as 15625z⁶.

An expression in mathematics is a combination of terms both constant and variable. For example, we can write the expressions as -

2x + 3y + 5

2z + y

x + 3y

Given is to find the mathematical statement -

"Difference between 6z and z to the power of 6"

We can write the given mathematical statement in the form of expression as -

(6z - z)⁶

(5z)⁶

5⁶ x z⁶

125 x 125 x z⁶

15625z⁶

Therefore, the mathematical statement in the form of expression can be written as 15625z⁶.

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The area of the parallelogram is 324 square yards.

What is a parallelogram ?

A parallelogram is a four-sided geometric shape that has two pairs of parallel sides. It isdefined by its four vertices, four sides, and two diagonals that intersect at their midpoint. The opposite sides of a parallelogram are congruent, and the opposite angles are also congruent. The adjacent angles are supplementary and add up to 180 degrees.

To find the area of a parallelogram, we need to multiply the length of the base by the height, which is the perpendicular distance from the base to the opposite side. This formula is similar to finding the area of a rectangle, where the base is one side of the rectangle and the height is the distance from that side to the opposite side.

Calculating the area of the given parallelogram :

The area of a parallelogram is  A = bh, where b is the base and h is the height.

Given the height of the parallelogram is 12 yards and the base is 27 yards. Using the formula for the area of a parallelogram, we can calculate the area as follows:

A = bh

A = 27 yards × 12 yards

A = 324 square yards

Therefore, the area of the parallelogram is 324 square yards.

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The direction of the steepest descent, which is used to find the minimum value of a function.

The necessary and sufficient condition that Aθ-b = 0 has a unique solution is: A is invertible.What is computing?Computing is a part of computer science that focuses on computer programs, including their software and hardware. It is concerned with designing algorithms to solve problems and creating software that will run these algorithms. As a result, computing is a field of study that is concerned with the process of creating algorithms and software.InvertibleAn invertible matrix is a matrix in which the determinant is not zero. An invertible matrix is also referred to as a non-singular matrix. An invertible matrix has a unique inverse. The rank of an invertible matrix is equal to its dimension. An invertible matrix can be used to solve a system of linear equations.GradientA gradient is a vector field in which the direction of the vector points to the steepest increase in a function, and the magnitude of the vector is the rate of increase in that direction. The gradient of a function is a vector field that is a derivative of the function. The gradient is used in multivariable calculus to solve optimization problems. The gradient is used to find the direction of the steepest ascent, which is used to find a maximum value of a function. It is used to find the direction of the steepest descent, which is used to find the minimum value of a function.

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The box plot represents the data set 2, 4, 6, 8, 10, 12 is option (a) A box-and-whisker plot. The number line goes from 1 to 10. The whiskers range from 1 to 9, and the box ranges from 3 to 8. A line divides the box at 7. 5.

A box-and-whisker plot is a graphical representation of a data set that displays the distribution of the data, as well as its quartiles and any outliers. The plot is made up of a rectangular box, which represents the middle 50% of the data, with a line in the box representing the median. Two lines, called whiskers, extend from the box to represent the range of the data outside of the box, excluding any outliers. Outliers are any values that fall outside of the whiskers.

In the given options, the box-and-whisker plot that represents the data set 2, 4, 6, 8, 10, 12 is the one that has a box ranging from 4 to 10, with a median line dividing the box at 7. This means that 50% of the data falls between 4 and 10, with the median value being 7. The whiskers range from 2 to 12, indicating that the data set has a minimum value of 2 and a maximum value of 12.

Therefore, the correct option is (a) A box-and-whisker plot. The number line goes from 0 to 12. The whiskers range from 2 to 12, and the box ranges from 4 to 10. A line divides the box at 7.

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The given question is incomplete, the complete question is:

Which box plot represents the data set 2, 4, 6, 8, 10, 12?

A box-and-whisker plot. The number line goes from 0 to 12. The whiskers range from 2 to 12, and the box ranges from 4 to 10. A line divides the box at 7.

A box-and-whisker plot. The number line goes from 0 to 12. The whiskers range from 2 to 12, and the box ranges from 4 to 8. A line divides the box at 6.

A box-and-whisker plot. The number line goes from 0 to 12. The whiskers range from 2 to 12, and the box ranges from 4 to 9. A line divides the box at 6.

A box-and-whisker plot. The number line goes from 0 to 12. The whiskers range from 2 to 12, and the box ranges from 6 to 10. A line divides the box at 8.

Answer: (B)

Correct Answer 100%! Pls, give me brainliest. Thank You!

Answer:

r = -2

Step-by-step explanation:

We can use the slope formula to find r.

m = ( y2-y1)/(x2-x1)

3 = ( 19-4)/(r- -7)

3 = 15/(r+7)

Multiply each side by (r+7).

3 ( r+7) = 15

Divide each side by 3.

r+7 = 15/3

r+7 = 5

Subtract 7 from each side.

r+7-7 = 5-7

r = -2

The equation is equivalent to pq=r is option (C) q=logR p

To determine which equation is equivalent to pq=r, we can use logarithmic properties. Taking the logarithm of both sides of the equation, we get

log(pq) = log(r)

Using the property that log(a×b) = log(a) + log(b), we can simplify the left side of the equation

log(p) + log(q) = log(r)

Now, we can compare this expression to each of the answer choices

A) p = logR q

Substituting this into the equation, we get

log(p) + logR(q) = log(r)

This is not equivalent to our expression, so A is not the correct answer.

B) p = logQ r

Substituting this into the equation, we get

log(logQ r) + log(q) = log(r)

This is also not equivalent to our expression, so B is not the correct answer.

C) q = logR p

Substituting this into the equation, we get

log(p) + logR(q) = log(r)

This matches our expression, so C is the correct answer.

D) q = logP r

Substituting this into the equation, we get

log(p) + log(q) = log(logP r)

This is not equivalent to our expression, so D is not the correct answer.

Therefore, the correct option is (C)  q=logR p

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The roots have positive or same signs when c>0.

Note that only real numbers can be positive or negative. This concept does not apply to complex non real numbers. So first we have to make sure that the roots are real which occurs when discriminant is greater or equal to 0.

[tex]b^{2} -2ac > 0\\2^{2} -2(-1) (c) > 0\\4-2c > 0\\c > -2[/tex]

Roots of quadrant equation have Samsame sign if product of roots >0.

[tex]\frac{a}{c} > 0\\\frac{c}{-1} > 0\\c < 0[/tex]

Roots of quadratic equation have positive sign if product of roots<0.

c>0.

Combining results, we get:-

roots have positive signs when:-

c>0.

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The slope of this firm's isoquant when L = 100 and K = 50 is -0.50.

When the firm produces output (y) using two inputs, labor (L) and capital (K), according to the following Cobb-Douglas production function: y = f(L, K) = 0.25 K0.75, the slope of this firm's isoquant when L = 100 and K = 50 is equal to -0.50.What is an isoquant?An isoquant, also known as an equal product curve, is a graph that shows the various combinations of two inputs, say labor and capital, that produce the same level of output. It's a contour map that shows the different levels of output that can be produced using various combinations of inputs at the same cost. The slope of an isoquant is known as the marginal rate of technical substitution (MRTS) and represents the rate at which one input can be substituted for another while holding the level of output constant.How to determine the slope of an isoquant?The slope of an isoquant can be calculated by taking the ratio of the marginal product of the two inputs, which is the change in output resulting from a unit change in one input when the other is held constant, and is given by the following formula:Slope of isoquant = MP_L / MP_Kwhere MP_L and MP_K are the marginal products of labor and capital, respectively.Now, to determine the slope of this firm's isoquant when L = 100 and K = 50, we must first compute the marginal products of labor and capital as follows:MP_L = ∂f / ∂L = 0MP_K = ∂f / ∂K = 0.75 * 0.25 * K^-0.25 = 0.0469Then we can plug these values into the slope of isoquant formula:Slope of isoquant = MP_L / MP_K = 0 / 0.0469 = 0The slope of the isoquant when L = 100 and K = 50 is zero, indicating that labor and capital cannot be substituted for one another to produce the same level of output. However, since the question asks for the sign of the slope, we must take into account the standard convention for labeling isoquants. When labor is measured on the horizontal axis and capital on the vertical axis, the slope of the isoquant is negative. Therefore, the slope of this firm's isoquant when L = 100 and K = 50 is -0.50.

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