Binomial expansion

(a) Expand (1+e^x)+(e^3x) in ascending powers of x to the term x^2.

(b) If the coefficient of x^2 in the expansion [tex]\frac{(1+e^{x}-e^{3x})(1-3x)^{n}}{e^{3x} }[/tex] is [tex]\frac{433}{2}[/tex] , find n.

The probability that the actor will get his lines straight for the first time on the fourth attempt is 0.0864, rounded to three decimal places.

Probability can be defined as the ratio of favorable outcomes to the total number of events.

Here,
The probability that the actor will get his lines straight on any one take is 0.40. This means that the probability of him not getting his lines straight on any one take is 1 - 0.40 = 0.60.

P(getting lines straight on 4th attempt)
= P(not getting lines straight on 1st, 2nd, or 3rd attempt) x P(getting lines straight on 4th attempt)

P(not getting lines straight on any one attempt) = 0.60

P(not getting lines straight on 1st, 2nd, or 3rd attempt) = 0.60 x 0.60 x 0.60 = 0.216

P(getting lines straight on 4th attempt) = 0.40

P(getting lines straight for the first time on the fourth attempt) = P(not getting lines straight on 1st, 2nd, or 3rd attempt) x P(getting lines straight on 4th attempt)
= 0.216 x 0.40 = 0.0864

Therefore, the probability that the actor will get his lines straight for the first time on the fourth attempt is 0.0864, rounded to three decimal places.

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The slope of the line that passes through the points (0, - 4) and (30, -9) is -1/6.

In Mathematics, the slope of any straight line can be determined by using this mathematical equation;

Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)

Slope (m) = rise/run

Based on the information provided, we can logically deduce the following data points on the line:

Substituting the given data points into the slope formula, we have the following;

Slope, m = (-9 + 4)/(30 - 0)

Slope, m = -5/30

Slope, m = -1/6.

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The values typically reported when presenting the results of a t-test are: mean, standard deviation, and degrees of freedom (df). Therefore, the correct answer is b.

The mean, standard deviation and degrees of freedom (df) are often shown together with the t-value and p-value when the outcomes of a t-test are reported.

The central tendency and variability of the sample data under analysis are revealed by the mean and standard deviation. Determine the critical values for the t-distribution by using the degrees of freedom (df), which stands for the number of independent observations utilised in the t-test.

These statistics, along with the t-value and p-value, offer crucial details regarding the strength and significance of the relationship or difference under study in the t-test.

Therefore, the correct answer is b. mean, standard deviation and degrees of freedom.

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Brand₂ has the highest impact on the price of a smartphone, increasing the price by $80 for each unit increase in that variable.

Regression analysis is a statistical method that helps us understand the relationship between two or more variables.

Based on the regression equation, we can determine which brand increases the price of a smartphone the most. The regression equation is as follows:

Price = 1100 + 50 (Brand₁) + 80 (Brand₂) + 30 (Brand₃) + 20 (Battery Life) + 10 (Camera Resolution)

In this equation, the intercept (1100) represents the average price of a smartphone with all other variables set to zero. The coefficients for each variable represent the change in the price of a smartphone for a one-unit increase in that variable, holding all other variables constant.

Brand₁ and Brand₂ have a lower impact on the price, increasing it by $50 and $30, respectively, for each unit increase in those variables. Battery life and camera resolution have the smallest impact on the price, increasing it by $20 and $10, respectively, for each unit increase in those variables.

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Complete Question:

Can brand, battery life, and camera resolution affect a smartphone's price? Use α= o.05 to perform a regression analysis for the Smartphones01BC dataset and answer the following questions. Solve this problem using Excel and Mega Stat, and write your complete answers in the Response Box so I know that you can interpret the output.

Based on your regression equation, which brand increases the price of a smartphone the most? Use the statistics from your output to explain your answer.

Answer: 2.25 per gallon

Step-by-step explanation:

4.50 divided by 2 = 2.25

Using arithmetic operation Jude's typing rate is 182 characters per minute, which is option C.

What is arithmetic operation?

A subject of mathematics known as arithmetic operations deals with the study and use of numbers in all other branches of mathematics. Basic operations including addition, subtraction, multiplication, and division are included.

To find the typing rate in characters per minute, we need use arithmetic operation of division.

Divide the number of characters by the number of minutes -

Typing rate = Number of characters / Number of minutes

In this case, Jude types 1,274 characters in 7 minutes, so her typing rate is -

Typing rate = 1274 / 7 = 182

Therefore, the typing rate is 182 characters/minute.

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The two points that would lie on the graph of function g are (-2,-12) and (1,30). The solution has been obtained by using functions.

What is a function?

An association between a number of inputs and a single output for each input is known as a function. A function is often written as f(x), where x stands for the input. A function is typically expressed as y = f(x).

We are given that f(x) passes through the points (-2,-4) and (1,10).

So, it means that

For x = -2,  f(-2) = -4

Similarly,

For x = 1, f(1) = 10

We are given g(x) = 3f(x)

So, on substituting the obtained values of f(x) in the g(x) function, we get

For x = -2,

⇒g(-2) = 3f(-2)

⇒g(-2) = 3(-4)

⇒g(-2) = -12

Similarly,

For x = 1,  

⇒g(1) = 3f(1)

⇒g(1) = 3(10)

⇒g(1) = 30

Hence, the two points that would lie on the graph are (-2,-12) and (1,30).

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Question: An exponential function, f, passes through the points (-2,-4) and (1,10). Which two points would lie on the graph of function g if g(x)=3f(x)?

The pair of figures is similar by SSS

The complete question is added as an attachment

From the question, we have the following parameters that can be used in our computation:

Triangles BDC and FGC

When the side lengths of the triangle are compared, we have:

Side lengths of FGC are 5/3 multiplied by the side lengths of BDC

This means that the triangles are similar

And the similarity statement is SSS

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52 students passed only one subject

Let's use the following variables to represent the number of students who passed each subject:

P(Ga): the number of students who passed Ga

P(Fante): the number of students who passed Fante

From the problem statement, we know that:

P(Ga) = P(Fante) + 6, since the number of students who passed Ga is 6 more than the number who passed Fante

P(Ga ∩ Fante) = 8, since 8 students passed both subjects

P(Ga ∪ Fante) = 60, since every student passed at least one subject

We want to find the number of students who passed only one subject, which is equal to the total number of students who passed Ga minus the number of students who passed both subjects, plus the number of students who passed Fante but not Ga. In symbols:

P(Ga) - P(Ga ∩ Fante) + (P(Fante) - P(Ga ∩ Fante))

Substituting the values we know:

(P(Fante) + 6) - 8 + (P(Fante) - 8)

Simplifying:

2P(Fante) - 10

We also know that P(Ga) + P(Fante) - P(Ga ∩ Fante) = P(Ga ∪ Fante) = 60.

Substituting the values we know:

P(Ga) + P(Fante) - 8 = 60

P(Ga) + P(Fante) = 68

Substituting P(Ga) = P(Fante) + 6:

2P(Fante) + 6 = 68

2P(Fante) = 62

P(Fante) = 31

Therefore, the number of students who passed only one subject is:

2P(Fante) - 10 = 2(31) - 10 = 52

Therefore, 52 students passed only one subject.

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Answer:

m<20

Step-by-step explanation:

We can solve this equation by setting B2 = [c d], so that AB2 = [c+d 1] = [0 1]. This gives us the system of equations c + d = 0 and c = 1. Solving this system, we have c = 1, where I is the 2x2 identity matrix

Let A be the given 2x2 matrix A = [1 2 ; 1 1]. To find a 3x2 matrix B such that AB = I, where I is the 2x2 identity matrix, we can use the hint provided: If B1 and B2 are the columns of B, then ABj = Ij. This means that for each column j of B, we need to solve the equation ABj = Ij. For j = 1, we have AB1 = I1 = [1 0]. We can solve this equation by setting B1 = [a b], so that AB1 = [a+b 1] = [1 0]. This gives us the system of equations a + b = 1 and a = 0. Solving this system, we have a = 0 and b = 1, so B1 = [0 1].For j = 2, we have AB2 = I2 = [0 1]. We can solve this equation by setting B2 = [c d], so that AB2 = [c+d 1] = [0 1]. This gives us the system of equations c + d = 0 and c = 1. Solving this system, we have c = 1.

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Answer:

360-137+x+y

Step-by-step explanation:

The relation:

{x = y : x, y ε A}.

And this relation is a function.

A relation between a collection of inputs and outputs is known as a function. A function is a connection between inputs in which each input is connected to precisely one output. Each function has a range, co-domain, and domain.

Given:

A={2,4,6} and B={135}

R={(2,2), (4,4), (6,6)} is a relation on A.

That means, each value of set A connects to the same point.

The relation:

{x = y : x, y ε A}

Hence, {x = y : x, y ε A}.

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The options that represent an amount and price of gas that Pedro purchased are 19 gallons of gas for $52.25, 16 gallons of gas for $44. So options C and D are correct.

To calculate the amount of gas and the total price that Pedro purchased, we need to use the formula:

Price = Cost per gallon * Number of gallons

Dividing the total price by the cost per gallon will give us the number of gallons purchased. We can then compare this with the given options to see which one(s) match.

Using the given cost per gallon of $2.75, we can check each option:

a) 13.75 gallons of gas for $41.25

Price = $2.75/gallon * 13.75 gallons = $37.81

This option does not match the total price of $41.25, so it is not correct.

b) 22 gallons of gas for $71.5

Price = $2.75/gallon * 22 gallons = $60.50

This option does not match the total price of $71.5, so it is not correct.

c) 19 gallons of gas for $52.25

Price = $2.75/gallon * 19 gallons = $52.25

This option matches the total price of $52.25, so it is correct.

d) 16 gallons of gas for $44

Price = $2.75/gallon * 16 gallons = $44

This option matches the total price of $44, so it is correct.

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Complete question is:

Pedro fills up his truck with gas he paid $2.75 a gallon which of these represents an amount and price of gas that Pedro purchased select all that apply.

a) 13.75 gallons of gas for $41.25

b) 22 gallons of gas for $71.5

c) 19 gallons of gas for $52.25

d) 16 gallons of gas for $44

Answer: 55+p

Step-by-step explanation: Since p is in unlike terms as 55, you can just write that as is an the expression to represent the dog's weight.

The two major properties of similar shapes include the following:

In Mathematics, two (2) geometric shapes are said to be similar when the ratio of their corresponding side lengths are equal and their corresponding angles are congruent.

In Geometry, the scale factor of a geometric figure can be calculated by dividing the dimension of the image (new figure) by the dimension of the pre-image (original figure):

This ultimately implies that, two (2) similar geometric shapes can have different sizes depending on the scale factor that is applied because it would either enlarge (increase) or reduce (decrease) them.

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There are two solutions to the system of linear equations depicted in the graph: one at (0,4) and one at (-3,0). There is only one distinct solution to the system of equations where the two lines intersect at the points (-3,0) and (0,4).

If the slopes are the same but the y-intercepts vary, the system has no solution. If the slopes are different, the system only has one solution. The system has an endless number of solutions if the slopes and y-intercepts match.

There are therefore countless possible answers. A different kind of linear equation system is a system with two parallel lines represented by the equations is referred to be an inconsistent system. The lines have distinct y-intercepts but the same slope.

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The percent will the fraction decrease should be 80%.

Important information:

In the case when its numerator is decreased by 85% and its denominator is decreased by 25%

Calculation of percentage:

= (100-85) ÷ (100 - 25)

=  15%÷ 75%

= 20%

Now the decrease in percentage should be

= 100% - 20%

= 80%

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The correct answer is 80%

3

Step-by-step explanation:

35 times 3.sub

Answer: To convert a Senary (base 6) number to decimal, we can use the following formula:

Decimal = (5 * 6^4) + (4 * 6^3) + (7 * 6^2) + (2 * 6^1) + (5 * 6^0)

= (5 * 1296) + (4 * 216) + (7 * 36) + (2 * 6) + (5 * 1)

= 6480 + 864 + 252 + 12 + 5

= 7603

So the Senary (base 6) number 54724 is equivalent to the decimal number 7603.

Step-by-step explanation:

The value of variables x and y are,

⇒ x = 80°

⇒ y = 130°

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Since, A parallelogram is forming here,

Then, Opposite angels are going to be equal.

Thus, We get;

⇒ y = (x + 50)°

Now, As you know that sum of all angles of a quadrilateral equals 360°

Hence, We get;

2 {(x + 50) + (x - 30)} = 360°

(x + 50) + (x - 30) = 180°

2x + 20 = 180

2x = 160

x = 80°

So, We get;

⇒ y = (x+50)

      = 80 +50

      = 130°

And,

⇒ x = 80°

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The expression [tex]m^{7} * n^{4}[/tex] divided by [tex]m^{4} * n^{3}[/tex] simplifies to [tex]m^{3} * n[/tex], meaning m raised to the power of 3 times n. This is found by subtracting the exponents in the denominator from the exponents in the numerator.

To simplify the expression [tex]\frac{m^{7} * n^{4}}{m^{4} * n^{3}}[/tex] , we need to divide the exponents of m and n. To do this, we subtract the exponents of m and n in the denominator from the exponents of m and n in the numerator.

Starting with m, we have:

[tex]\frac{m^{7}}{m^{4}}[/tex] = [tex]m^{(7-4)}[/tex] = [tex]m^{3}[/tex]

This means that we divided the exponent of m in the numerator (7) by the exponent of m in the denominator (4), and the result is m raised to the power of 3.

Next, we do the same thing with n:

[tex]\frac{n^{4} }{n^{3} }[/tex] = [tex]n^{(4-3)}[/tex] = [tex]n^{1}[/tex] = n

This means that we divided the exponent of n in the numerator (4) by the exponent of n in the denominator (3), and the result is n raised to the power of 1, which is just n.

Finally, we can combine the results:

[tex]m^{3}*n[/tex] = [tex]m^{3}*n[/tex]

So, the simplified form of the expression [tex]\frac {m^{7 }* n^{4}} { m^{4} * n^{3}}[/tex] is [tex]m^{3}*n[/tex].

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Answer:

m^3n

Step-by-step explanation:

m3n

Answer:

2.57 square meters

Step-by-step explanation:

r = Radius of circle

Area of shaded region = Area of semi-circle:

[tex]= \frac{1}{2}(\pi r^{2})[/tex]

= [tex]\frac{\pi}{2} r^{2}[/tex]

= [tex]\frac{\pi }{2} (1.28m)^{2}[/tex]

= 2.57 [tex]m^{2}[/tex] (Rounded to 2 decimal places)

The value of the functions at 9m is

Generally, In mathematics, a function is a rule that maps one set of values (the domain) to another set of values (the range). The values in the range are determined by applying the function to the values in the domain.

Functions can take many forms and be expressed in different ways, including:

Linear functions: a function that produces a straight line when graphed. It has the form f(x) = mx + b, where m is the slope of the line and b is the y-intercept.

Quadratic functions: a function that produces a parabola when graphed. It has the form f(x) = ax^2 + bx + c, where a, b, and c are constants.

Therefore, this function

- f(x) = 4x

g(x) = 9x¹/2

for

F(x)=4x

F(9)=4*9

f(9)=36

for

g(x) = 9x¹/2

g(9)=9√9

=27

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Answer:

98 km

Step-by-step explanation:

To find the area of the figure;

Total Area = Area A + Area B + Area C

Area A = 14 * 2 = 28 km2

Area B = 11 * 2 = 22 km2

Area C = 12 * 4 = 48 km2

Total Area = 98 km2

Sum of the given geometric sequence is -100.

A geometric sequence is the special type of sequence in which the ratio of every two successive terms is constant.

Sum of each infinite geometric sequence

-50, -25, -12.5, ...

The infinite sum of geometric sequence is given by,

[tex]s_{\infty} =\frac{a}{1-r}[/tex]

Where a is the first term in the sequence and r is the common ratio.

[tex]s_{\infty} =\frac{-50}{1-(-0.5)}[/tex]

      = -100.

Hence, sum of the given geometric sequence is -100.

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Answer:

7 cents

Step-by-step explanation:

0.072222 rounded to the nearest cent is 0.07

The correct relationship of the three principles is: The well-ordering principle for the integers is equivalent to strong mathematical induction, and neither is equivalent to weak mathematical induction.

The well-ordering principle for the integers states that every non-empty set of non-negative integers has a least element. Strong mathematical induction is a proof technique that involves proving that a statement is true for a base case, and then assuming that it is true for all cases up to a certain value, and then proving that it must also be true for the next case.

This requires assuming the truth of all cases up to a certain value, which is where the well-ordering principle comes in. Weak mathematical induction is a proof technique that involves proving that a statement is true for a base case, and then assuming that it is true for a particular case, and then proving that it must also be true for the next case.

Weak mathematical induction does not require assuming the truth of all cases up to a certain value, and is therefore weaker than strong mathematical induction.

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