Macy of New York sold LeeCo. of Chicago office equipment with a $6,300 list price. Sale terms were 3/10, n/30 FOB New York. Macy agreed to prepay the $40 freight. LeeCo. pays the invoice within the discount period. What does LeeCo. pay Macy?

Answer:

Step-by-step explanation:

We can use the formula for compound interest:

A = P(1 + r/n)^(nt)

where A is the final amount, P is the principal (initial amount), r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time (in years).

In this case, we know that P = $10,000, r = 6.5% = 0.065, and we want to find t when A = $29,500. We also know that the interest is compounded annually, so n = 1.

Substituting these values into the formula, we get:

$29,500 = $10,000(1 + 0.065/1)^(1t)

Dividing both sides by $10,000, we get:

2.95 = (1.065)^t

Taking the natural logarithm of both sides, we get:

ln(2.95) = ln(1.065)^t

Using the property of logarithms that ln(a^b) = b ln(a), we can rewrite the right side as:

ln(2.95) = t ln(1.065)

Dividing both sides by ln(1.065), we get:

t = ln(2.95)/ln(1.065) ≈ 18.2

Therefore, the person must leave the money in the bank for about 18.2 years to reach $29,500. To the nearest tenth of a year, the answer is 18.2 years.

Answer:

Through observation, a=2, b=1 or vice versa .

Step-by-step explanation:

If a=2 and b=1

a² +b²=?

2² +1² =4+1=5

a²+b²=5

On Wednesday, there is a higher (0.7395) likelihood that both managers will be accessible than on Monday (0.7216).

The study of random occurrences and their results falls under the category of probability, which is a subfield of mathematics. A number between 0 and 1, where 0 denotes an impossibility and 1 denotes a certainty, is used to express the likelihood of an event happening. By dividing the number of favourable outcomes by the total number of potential outcomes, the probability of an occurrence is determined. It is used in many fields, including statistics, finance, engineering, and science, to generate predictions and inform decision-making.

given

We must calculate the product of the odds of each manager being available on a given day in order to determine the day with the highest likelihood that both managers will be available.

The likelihood that both managers will be accessible on Monday is 0.82 x 0.88, or 0.7216.

The likelihood that both managers will be accessible on Wednesday is 0.87 x 0.85 = 0.7395.

Therefore, the solution is option C: Wednesday. On Wednesday, there is a higher (0.7395) likelihood that both managers will be accessible than on Monday (0.7216).

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Step-by-step explanation:

B) 20:23

C) 6,0375

hope this helps

Answer168 Bighas of land.

Step-by-step explanation:

Since the machine can dig a land of five bighas in 14 hours, put it into a ration 5:14 5(land it can dig) 14(amount of hours it takes.) Have X as the time to dig 60 bighas, 60(amount of land) : x (time). The equation would be 60/x=5/14 ===> 168 bughas of land.

Answer:

Step-by-step explanation:

8.35 times 105 is 876.75

1.6 times 10 is 16

  Hope this helps

Step-by-step explanation:

Here is ONE way

sum of exterior angles of a polygon = 360 °

  there are 5 , se each EXTERIOR angle = 360/5 = 72 °

     the EXT angles are SUPPLEMENTARY with the interior angles

          ext + int = 180°

            72 + int = 180°      INTERIOR angle = 108 °

Answer:

Step-by-step explanation:

D is it

The value of a is 2.

the vertex form of the graph : y = 2 ( x + 1 )² - 9,

the standard form of the graph : y = 2x² + 4x - 7,

the factored form of the graph : y = 2 ( x + 1) ( x + 1 ) - 9.

What is the quadratic equation?

The solutions to the quadratic equation are the values of the unknown variable x, which satisfy the equation. These solutions are called roots or zeros of quadratic equations. The roots of any polynomial are the solutions for the given equation.

The equation of the parabola in vertex form:

The vertex v is at v ( -1, -9 ) this is equivalent to v ( h, k )

the equation of parabola in vertex form is y = a ( x- h )² + k

y = a ( x - (-1) )² + (-9)

substituting point ( -3, -5 ) from the graph we have:

-5 = a ( -3 + 1 )² - 9

-5 = 2a - 9

a = 2

substituting the known values to y = a ( x- h ) + k

y = 2 ( x + 1 )² - 9

the equation of the parabola in factored form

y = 2 ( x + 1) ( x + 1 ) - 9

the equation of the parabola in standard form

y = 2 ( x + 1) ( x + 1 ) - 9

y = 2 ( x² + 2x + 1 ) - 9

y = 2x² + 4x - 7

Hence, the value of a is 2

the vertex form of the graph : y = 2 ( x + 1 )² - 9

the standard form of the graph : y = 2x² + 4x - 7

the factored form of the graph : y = 2 ( x + 1) ( x + 1 ) - 9

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Therefore, A, B, C, and F are the proper responses as the congruence theories or postulates based on the data.

Having three straight sides and three angles where they intersect, a triangle is a closed, two-dimensional shape. It is one of the fundamental geometric shapes and has a number of characteristics that can be used to study and resolve issues that pertain to it. The triangle inequality theory states that the sum of a triangle's interior angles is always 180 degrees, and that the longest side is always the side across from the largest angle. Triangles can be used to solve a wide range of mathematical issues in a variety of disciplines and can be categorised based on the length of their sides and the measurement of their angles.

given

We can use the following congruence theories or postulates based on the data in the diagram:

A. ASA

B. AAS

C. LL (corresponding angles hypothesis)

F. SAS

Therefore, A, B, C, and F are the proper responses as the congruence theories or postulates based on the data.

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Answer:

x = 21

Step-by-step explanation:

Given the angle measurements of two angles, we can deduce that this is a 30-60-90 triangle. The formula for side lengths of this type of triangle is given by s → s√3 → 2s with s being the shortest side (the side opposite the 30° angle).

We are looking for the side length represented by s[tex]\sqrt{3}[/tex]. We can see that the hypotenuse is represented by 2s, so...

2s = 24

s = 12

The side length of the shortest side is 12. Therefore, we can say that

x = s  [tex]\sqrt{3}[/tex]

x = 12 * [tex]\sqrt{3}[/tex]

x = 20.785

The question asks for our answer to the nearest whole number, so

x = 21

Here's the completed table and the graph:

x  h(x)

-90°  1.0

-75°  -0.5

-60°  -1.0

-45° -0.0

-30° 1.0

-15° 0.5

0° 1.0

15° 0.5

30° -0.0

45° -1.0

60° -0.5

75° 1.0

90° 1.0

In mathematics, a function is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output. Functions are often represented as a set of ordered pairs, where the first element of each pair is an input and the second element is the corresponding output. Functions are a fundamental concept in many areas of mathematics and have many real-world applications, including in science, engineering, and economics.

Here,

To calculate the values of h(c) in the table, we plug in the given values of x into the function h(c) = cos 2x and evaluate. For example, to find h(c) when x = -75°:

h(c) = cos 2x

h(c) = cos 2(-75°) (substitute -75° for x)

h(c) = cos (-150°) (simplify using the double angle identity)

h(c) = -0.5 (evaluate using the unit circle or a calculator)

We repeat this process for each value of x to fill out the table.

To graph the function h(x) = cos 2x, we plot each point from the table on the given system of axes. The x-axis represents the angle x in degrees, and the y-axis represents the value of h(x) = cos 2x. We then connect the points with a smooth curve to obtain the graph.

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The values for a and b such that f(t) = a + logb(t) are,

a = log(1/log(49))

b = 49

Therefor

We can start by simplifying the expression for f(t)

f(t) = log49(t7)

f(t) = log(t7)/log(49) [Using the change of base formula]

Now we can rewrite f(t) as:

f(t) = log(t)/log(b) + a [Using the logarithmic properties log(ab) = log(a) + log(b)]

Comparing this expression with f(t) = a + logb(t), we can see that:

a = log(1/log(49)) [since log(b) = 1/log(49)]

b = t

Therefore, we have

f(t) = log(t7)/log(49) = log(t)/log(b) + log(1/log(49))

= log(t)/log(49) + log(1/log(49))

= log(1/log(49)) + log(t)/log(49)

So, a = log(1/log(49)) and b = 49.

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The 1,262 students who received a merit scholarship, the amount they received varied per student and totaled an average of $3,458 ($454). That amount is 78.2% of the full tuition cost of $4,400.

Merit is a term used to describe the quality of something or someone that makes them worthy of recognition or respect. It is an indicator of worthiness and is usually based on a person's ability, effort, or accomplishments. Merit is often used when evaluating an individual or a group for a promotion, hiring, or award. Merit is subjective, as different people have different standards for what merits recognition.

Therefore, 21.8% of the students who received a merit scholarship did not receive enough to cover full tuition. Therefore, the percentage of students who received a merit scholarship and did not receive enough to cover full tuition is 21%.

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The correct sampling technique for the given case is C. Simple random sampling

Each person in a population has an equal probability of being chosen to be a member of specific sample when using simple random sampling as a statistical sampling technique. With this technique, a random sample is picked at random from a wider population, giving each person a fair chance of being chosen. This indicates that there is no bias in selection process and that any person or object in overall population has same likelihood of getting chosen.

Simple random sampling is frequently employed in research investigations because it is an easy-to-use and objective sampling technique. Given that 100 names were randomly chosen from a list of all registered voters, simple random sampling is appropriate sampling technique in this particular case.

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Answer:

Step-by-step explanation:

To find the number of students who studied in the library or cafeteria (including both), we need to add the number of students who studied in the library and the number of students who studied in the cafeteria, but we need to subtract the number of students who studied in both the library and cafeteria to avoid counting them twice.

So, the number of students who studied in library or cafeteria is:

170 + 135 - 76 = 229

Therefore, 229 students studied in the library or cafeteria (including both).

Answer :

Step-by-step explanation to problem:

2/3 * 3 = 2

we do 2/3 times 3 because $3 is for 1 pound and here we only need 2/3 of a pound

$2

Correct Answer = D

The statements below describe the graph of y > mx, where m > 0

The inequality is given as:

y > mx

Where m > 0

The inequality symbol implies that:

The condition m > 0 implies that:

Next, we assume that:

m = 5

So, we have:

y > 5x

See attachment for the graph of y > 5x

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This can be simplified to 19/100, which is equivalent to 6/31 or approximately 0.19 or 19%. The answer is: 6 over 31.

what is probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 indicates that an event is impossible, and 1 indicates that an event is certain. Probability can be expressed as a fraction, decimal, or percentage.

The question asks for the experimental probability of drawing a 3, 5, or 7 from a deck of cards, given the frequency table provided.

The frequency table tells us how many times each card was drawn out of a total of 100 draws.

To calculate the frequency of drawing a 3, 5, or 7, we add the frequencies of these cards: 4 (for 3) + 6 (for 5) + 6 (for 7) = 16. Note that we do not include the frequencies of any other cards.

To calculate the probability of drawing a 3, 5, or 7, we divide the frequency of these cards (16) by the total number of draws (100): 16/100 = 0.16. This means that in our experiment, we drew a 3, 5, or 7 approximately 16% of the time.

The total number of draws is 100, and the frequency of cards 3, 5, and 7 is 7+6+6=19.

Therefore, the experimental probability of drawing a 3, 5, or 7 is:

19/100 = 0.19

This can be simplified to 19/100, which is equivalent to 6/31 or approximately 0.19 or 19%.

Therefore, the answer is: 6 over 31.

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The position of the mass at any time t is given by u(t) = -1/2 cos(2√2 t) + sin(2√2 t) + 1/2 sin(4t) - 1/2 cos(4t), and the first four times at which the velocity of the mass is zero, excluding t = 0, are t = 0.0567, 0.283, 0.510, and 0.736.

To solve this problem, we need to use Newton's second law of motion which states that the force acting on an object is equal to its mass times its acceleration. In this case, the force acting on the mass is the sum of the force due to gravity and the external force, and the acceleration is given by the second derivative of the displacement.

Using Hooke's law, the force due to the spring is proportional to the displacement u of the mass from its equilibrium position, with a proportionality constant k

F_spring = -k u

where k is the spring constant. Since the mass is pulled down 6 in. from its equilibrium position, the initial displacement is u(0) = -0.5 ft. The spring stretches 24 in. under a weight of 8 lb, so the spring constant is

k = F_spring / u = (8 lb) / (2 ft) = 4 lb/ft

The force due to gravity is

F_gravity = m g

where m is the mass of the object and g is the acceleration due to gravity. The mass is given in pounds, so we need to convert it to slugs:

m = 8 lb / (32.2 ft/s^2) = 0.248 slugs

Thus, the force due to gravity is

F_gravity = (0.248 slugs) (32 ft/s^2) = 7.94 lb

The equation of motion for the displacement u(t) is

m u''(t) = F_gravity + F_spring + F_external

where F_external = 4 sin 4t lb is the external force. Substituting the expressions for these forces and dividing by m, we get

u''(t) + 8 u(t) = 2 sin 4t

This is a homogeneous linear second-order differential equation with constant coefficients, which can be solved by finding the roots of the characteristic equation:

r^2 + 8 = 0

The roots are r = ±2i√2. The general solution of the differential equation is then

u(t) = c1 cos(2√2 t) + c2 sin(2√2 t) + u_p(t)

where u_p(t) is a particular solution of the nonhomogeneous equation. Since the right-hand side of the equation is a sinusoidal function with frequency 4/π, we can guess a particular solution of the form

u_p(t) = A sin(4t) + B cos(4t)

Substituting this into the equation and equating coefficients, we get

16 A - 16 B + 8 A = 2, or A = 1/2

-16 A - 16 B + 8 B = 0, or B = -1/2

Therefore, the particular solution is

u_p(t) = 1/2 sin(4t) - 1/2 cos(4t)

The general solution is then

u(t) = c1 cos(2√2 t) + c2 sin(2√2 t) + 1/2 sin(4t) - 1/2 cos(4t)

We can use the initial conditions u(0) = -0.5 and u'(0) = 0 to determine the values of c1 and c2

u(0) = c1 + 1/2 = -0.5, or c1 = -1/2

u'(0) = -2√2 c1 + 2√2 c2 - 2 = 0, or c2 = 1

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The given question is incomplete, the complete question is:

A mass that weighs 8 lb stretches a spring 24 in. The system is acted on by an external force of 4 sin 4t. If the mass is pulled down 6 in. and then released, determine the position of the mass at any time t. (Use g = 32 ft/s2 for the acceleration due to gravity. Let u(t), measured positive downward, denote the displacement in feet of the mass from its equilibrium position at time t seconds.)

u(t) = 1/2sin(4t)+1/2cos(4t)-2tcos(4)

Determine the first four times at which the velocity of the mass is zero. (Exclude t = 0 as trivial.)

If the probability distribution is f(x) = 2(1/3)ˣ for x = 1,2,3,... then the moment generating function is Mₓ(t) = [tex]\frac{2e^{t} }{3-e^{t} }[/tex].

The moment generating function (MGF) of a random variable is defined as a mathematical function that is used to determine expected values of the random variable.

It is a way to characterize the entire probability distribution of the random variable by generating a sequence of moments from the function.

The probability distribution is  f(x) = 2(1/3)ˣ for x = 1,2,3,... ;

The MGF will be ⇒ Mₓ(t) = [tex]\[ \sum_{x=1}^{\infty} e^{tx} .2(\frac{1}{3})^{x}[/tex]

⇒ [tex]2 \[ \sum_{x=1}^{\infty}(\frac{e^{t} }{3} )^{x}[/tex] ;

This expression [tex]\[ \sum_{x=1}^{\infty}(\frac{e^{t} }{3} )^{x}[/tex] represents an infinite geometric series.

So, [tex]\[ \sum_{x=1}^{\infty}(\frac{e^{t} }{3} )^{x}[/tex] = [tex]\frac{e^{t}/3 }{1-e^{t}/3 }[/tex] = [tex]\frac{e^{t} }{3-e^{t} }[/tex]

We get, Mₓ(t) = [tex]\frac{2e^{t} }{3-e^{t} }[/tex].

Therefore, the Moment generating Function for the given Probability Distribution is Mₓ(t) = [tex]\frac{2e^{t} }{3-e^{t} }[/tex].

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The given question is incomplete, the complete question is

Find the moment-generating function of the discrete random variable X that has the probability distribution f(x) = 2(1/3)ˣ for x = 1,2,3,....

For a limit expression to be solved using L'Hospital's rule, the expression must be in the form of 0/0 or infinity/infinity, and the derivatives of the numerator and denominator must exist and approach a finite limit or infinity/negative infinity at the point of evaluation.

L'Hospital's rule applies to limit expressions of the form 0/0 or infinity/infinity. Specifically, if we have a limit expression of form f(x)/g(x), where f(x) and g(x) both approach 0 or infinity as x approaches a certain value, then L'Hospital's rule may be applied to find the limit of the expression.

It is important to note that L'Hospital's rule can only be applied if the limit of the derivative of f(x) divided by the derivative of g(x) exists and is finite, or if the limit of the derivative of f(x) divided by the derivative of g(x) approaches infinity or negative infinity.

Therefore, for a limit expression to be solved using L'Hospital's rule, the expression must be in the form of 0/0 or infinity/infinity, and the derivatives of the numerator and denominator must exist and approach a finite limit or infinity/negative infinity at the point of evaluation.

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By using rational zeros theorem, we find that there are no real zeros of the polynomial function f(x) = x^3 - x^2 - 37x - 35, so we cannot factor f(x) over the real numbers.

To find the real zeros of the polynomial function f(x) = x^3 - x^2 - 37x - 35, we can use the rational zeros theorem, which states that any rational zeros of the function must have the form p/q, where p is a factor of the constant term (-35) and q is a factor of the leading coefficient (1).

The possible rational zeros of f are therefore ±1, ±5, ±7, ±35. We can then test each of these values using synthetic division or long division to see if they are zeros of the function. After testing all of the possible rational zeros, we find that none of them are actually zeros of the function.

Therefore, we can conclude that there are no real zeros of the function f(x) = x^3 - x^2 - 37x - 35.

However, we could factor it into linear and quadratic factors with complex coefficients using the complex zeros of f(x). But since the problem only asks for factoring over the real numbers, we can conclude that the factored form of f(x) is:

f(x) = x^3 - x^2 - 37x - 35 (cannot be factored over the real numbers)

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The required number of boxes that the truck contain is 70 small boxes and 55 large boxes.

In mathematics, the operation and interpretation of a function to make it simple or easier to grasp is known as simplifying, and the process is known as simplification.

Given that, Number of large boxes weigh 45 pounds each, and the small boxes weigh 35 pounds each. There are 125 boxes in all, the truck is carrying a total of 4925 pounds

Let the number of large boxes will be l and the number of small boxed be s,

According to the question,

l + s = 125

l = 125 - s - - - - - (1)

Again,

45l + 35s = 4925

put 1 in the above equation,

45[125 - s] + 35s = 4925

5625 - 45s + 35s = 4925

10s = 700

s = 70

Now,

l = 125 - 70

l = 55

Thus, the required number of boxes that the truck contain is 70 small boxes and 55 large boxes.

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For Victor to have enough money to purchase the new iPhone 14 Pro Max, he will need to work for roughly 12 weeks.

An expression is made up of a number, a variable, or a combination of a number, a variable, and operation symbols. An equation is created by joining two expressions with an equal sign. For illustration: When you add 8 and 3, you get 11.

a) Here is the equation for Victor's savings account:

V(w) = 50w + 500

Slope (m) = 50

Y-intercept (b) = 500

b) The slope (m = 50) represents the rate at which Victor is saving money per week. For every week that he works as a tutor, he saves $50. The y-intercept (b = 500) represents the initial amount of money that Victor had saved up before he started working as a tutor.

(c)To find out how many weeks Victor will have to work to buy the new iPhone 14 Pro Max, we need to set up an equation where the amount of money he has saved up (V(w)) is equal to the price of the phone ($1099). Therefore, we can write:

50w + 500 = 1099

Subtracting 500 from both sides, we get:

50w = 599

Dividing both sides by 50, we get:

w = 11.98

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Solving the system of equations, it is found that a viable time that the trains are side by side is given by:

15 seconds

A system of equations is when two or more variables are related, and equations are built to find the values of each variable.

In this problem, the equations are:

[tex]y=14t^2[/tex]

[tex]y=130t+1200[/tex]

They will be side by side when:

[tex]14t^2=130t+1200[/tex]

[tex]14t^2-130t-1200=0[/tex]

It's solution is a quadratic function with coefficients a = 14, b = -130, c = -1200, hence:

[tex]\Delta=(-130)^2-4(14)(-1200)=84100[/tex]

[tex]x_1=\dfrac{130+\sqrt{84100} }{28}=1.5[/tex]

[tex]x_2=\dfrac{130-\sqrt{84100} }{28}=-5.71[/tex]

Time is a non-negative measure, hence 15 seconds is correct.

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Answer:

Answer:

The width of the frame is 10 inches.

5x7 = 35

80-35 = 45

45/7 = 6.4

5 + 6.4 = 11.4

11.4 rounded down = 11

11 - 5 = 10

Width of the frame = 10 inches

Sally will supply 3 quilts at $500.00 per quilt according to the market supply schedule she constructed.

The market supply schedule only takes into account the price per quilt and the quantity supplied per month. Other factors such as production costs, availability of raw materials, competition, and changes in technology are not taken into account in this supply schedule.

a. If Sally purchases a new sewing machine that helps her produce quilts in less time, her production efficiency will increase. This will enable her to produce more quilts in a shorter period of time, which will result in an increase in the supply of quilts.

b. If Sally anticipates an increase in prices for quilts in three months, she may increase the quantity supplied per month in order to take advantage of the higher prices. This may result in a temporary increase in the supply of quilts.

Lastly, c. If there are twice as many quilt suppliers as there were one year ago, Sally will likely face increased competition. This may result in a decrease in sales for Sally and a decrease in the price of quilts. In order to remain competitive, Sally may need to reduce the quantity supplied per month or find ways to differentiate her quilts from those of her competitors.

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See full question below

Part One

Sally Thomas is an entrepreneur. She makes hand-made quilts and sells them at craft shows and at her home studio. Below is a market supply schedule she constructed. Answer the questions that appear below it.

Price per quilt

Quantity Supplied Per Month

$300.00

1

$400.00

2

$500.00

3

$600.00

4

$700.00

5

1. How many quilts will Sally supply at $500.00 per quilt?

3

2. Are any other factors besides price taken into account with the market supply schedule she has constructed?

Supply

3. Several situations are listed below that Sally has encountered in her business. Explain how each of these situations might alter the supply of quilts. a. Sally purchases a new sewing machine that helps her produce quilts in less time.

Sally will be able to produce more quilts in less time, increasing the supply of quilts.

b. Sally anticipates an increase in prices for quilts in three months.

Sally will increase the quantity supplied per month for three months.

c. There are twice as many quilt suppliers as there were one year ago.

Sally will likely experience a decrease in sales which in turn will result in a decrease of supply and price.

a. Coefficient of .x³in (1 + .x)⁶ is 20. b. Coefficient of .x³ in (2.x - 3)⁶ is -540. c. Coefficient of .x³ in (x + 1)²⁰ + (x - 1)²⁰ is zero. d. Coefficient of .x³ y³ in (.x + 1)⁴ is 4.

a. To find the coefficient of .x³ in the expansion of (1 + .x)⁶, we can use the binomial theorem. The coefficient of .x³ is given by the expression 6C₃(1)³(.x)³, where 6C₃ is the number of ways to choose three items from a set of six items. Evaluating this expression gives us:

6C₃(1)³(.x)³ = (6!)/(3!3!)(1³)(.x)³ = 20(.x)³

Therefore, the coefficient of .x³ in (1 + .x)⁶ is 20.

b. Similarly, we can use the binomial theorem to find the coefficient of .x³ in the expansion of (2.x - 3)⁶. The coefficient of .x³ is given by the expression 6C₃(2.x)³(-3)⁶, which simplifies to:

6C₃(2³)(.x)³(-3)⁶ = 6C₃(8)(.x)³(729)

The value of 6C₃ is 20, so we have:

20(8)(.x)³(729) = 116,640(.x)³

Therefore, the coefficient of .x³ in (2.x - 3)⁶ is -116,640.

c. In the expansion of (x + 1)²⁰ and (x - 1)²⁰, each term will be of the form (xⁿ)(1)⁽²⁰⁻ⁿ⁾ and (xⁿ)(-1)⁽²⁰⁻ⁿ⁾, respectively, where n is an even integer between 0 and 20. Since the powers of x are even, the only term that can contain .x³ is the term where n = 6, which is (x⁶)(1⁽¹⁴⁾) and (x⁶)(-1⁽¹⁴⁾) in the two expansions. Adding these two terms together gives us:

(x⁶)(1⁽¹⁴⁾ - 1⁽¹⁴⁾) = 0

Therefore, the coefficient of .x³ in (x + 1)²⁰ + (x - 1)²⁰ is zero.

d. To find the coefficient of .x³ y³ in (.x + 1)⁴, we can use the binomial theorem once again. The coefficient of .x³ y³ is given by the expression 4C₁(1³)(.x)³(y)¹, which simplifies to:

4C₁(1³)(.x)³(y)¹ = 4(.x)³(y)

Therefore, the coefficient of .x³ y³ in (.x + 1)⁴ is 4.

Learn more about Binomial Theorem here: brainly.com/question/30100273

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