In the first game with a matrix of [-1 1 2 5], the maximin strategy for the row player is Row 2, and the minimax strategy for the column player is Column 1. In the second game with a matrix of [2 5 4 6 -2 -4], the maximin strategy for the row player is Row 1, and the minimax strategy for the column player is Column 2.
To determine the maximin and minimax strategies for a two-person, zero-sum matrix game, we need to analyze the payoffs in the game matrix.
Game Matrix: [−1 1 2 5]
Row Player's Strategies: Row 1, Row 2
Column Player's Strategies: Column 1, Column 2
Let's start by finding the maximin strategy for the Row Player:
For Row 1, the minimum payoff is -1.
For Row 2, the minimum payoff is 2.
Since the Row Player wants to maximize their minimum payoff, they will choose Row 2 as their maximin strategy.
Next, let's determine the minimax strategy for the Column Player:
For Column 1, the maximum payoff is 2.
For Column 2, the maximum payoff is 5.
Since the Column Player wants to minimize the maximum payoff of the Row Player, they will choose Column 1 as their minimax strategy.
Therefore, the maximin strategy for the Row Player is to play Row 2, and the minimax strategy for the Column Player is to play Column 1.
Game Matrix: [2 5 4 6 −2 −4]
Row Player's Strategies: Row 1, Row 2
Column Player's Strategies: Column 1, Column 2, Column 3
Let's find the maximin strategy for the Row Player:
For Row 1, the minimum payoff is 2.
For Row 2, the minimum payoff is -4.
The Row Player will choose Row 1 as their maximin strategy since it yields the higher minimum payoff.
Next, let's determine the minimax strategy for the Column Player:
For Column 1, the maximum payoff is 4.
For Column 2, the maximum payoff is 6.
For Column 3, the maximum payoff is -2.
The Column Player will choose Column 2 as their minimax strategy since it yields the lower maximum payoff for the Row Player.
Therefore, the maximin strategy for the Row Player is to play Row 1, and the minimax strategy for the Column Player is to play Column 2.
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Find the partial fraction decomposition for the rational expression. AX+B CX+D + (x+ 9) (5x + 76) メ+9 5x2 +76 4x - 7 17. Sketch the graph of the equation by transforming it to rectangular coordina
The partial fraction decomposition for the rational expression is (0x - 7/76) / (x + 9) + (Cx + 4/9) / (5x² + 76)
The sketch of the graph of the equation is illustrated below.
To find the partial fraction decomposition of the rational expression (4x - 7) / [(x + 9) (5x² + 76)], we need to express it as a sum of simpler fractions. In this case, we have a quadratic term in the denominator, so we need to decompose it into partial fractions of the form:
(4x - 7) / [(x + 9) (5x² + 76)] = (Ax + B) / (x + 9) + (Cx + D) / (5x² + 76)
To determine the values of A, B, C, and D, we need to find a common denominator for the right side and then equate the numerators. Multiplying both sides of the equation by [(x + 9) (5x² + 76)] gives us:
(4x - 7) = (Ax + B) (5x² + 76) + (Cx + D) (x + 9)
Expanding and collecting like terms, we get:
4x - 7 = (5A) x³ + (9C + 5B) x² + (76A + 9D) x + 76B
By equating coefficients of corresponding powers of x, we can form a system of equations to solve for A, B, C, and D. Equating the coefficients of x^3, we have 5A = 0, which gives A = 0. Equating the coefficients of x², we have 9C + 5B = 0. Equating the coefficients of x, we have 76A + 9D = 4, which gives D = 4/9. Finally, equating the constant terms, we have 76B = -7, which gives B = -7/76.
Substituting the values of A, B, C, and D back into the partial fraction decomposition equation, we have:
(4x - 7) / [(x + 9) (5x² + 76)] = (0x - 7/76) / (x + 9) + (Cx + 4/9) / (5x² + 76)
To sketch the graph of the equation (4x - 7) / [(x + 9) (5x² + 76)], we can transform it into rectangular coordinates by plotting points and connecting them.
The graph will consist of two parts: the line defined by (0x - 7/76) / (x + 9) and a curve defined by (Cx + 4/9) / (5x² + 76). The line will have a y-intercept at -7/76 and approach zero as x approaches negative infinity. The curve will vary depending on the value of C, which we have not determined yet.
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Complete Question:
Find the partial fraction decomposition for the rational expression.
(4x - 7) / [(x + 9) (5x² + 76) ] = (Ax + B) / ( x + 9) + (Cx + D) / (5x² + 76)
Sketch the graph of the equation by transforming it to rectangular coordinates.
A wildlife conservation group is designing a monitoring study of wallaby behaviour in a remote Queensland national park. The group has decided to study several regions in the park, the boundary of which form squares with side lengths W km and areas X km². A statistician has decided to choose the regions such that the region area, X, is a uniformly distributed random variable on the interval 1 < x < a such that X - U (1, a).
The statistician has deduced that W = vX is a random variable that describes the side length of the regions. He has also deduced that W has the cumulative distribution function Fw(w) = b/2 (w^2 - 1). The value of b and the range of W depends on a.
a, Show that b = 2/a-1
(b) The group choose the maximum allowable region area, a, such that the average region area is equal to 5 km? What is the average region side length, E(W)? (c) The monthly monitoring cost comprises a base rate of $500 plus $50 per km². i. Write an expression for the monitoring cost, C, in terms of the region area, X. ii. Find the average monitoring cost. iii. Find the variance of the monitoring cost.
a) b = 2/(a-1).
b) the average region side length, E(W), is 5v km.
c) the average monitoring cost is $750.
Var(C) = $50² * (16/3) = $40000/3
The variance of the monitoring cost is $40000/3.
What is the average?
This is the arithmetic mean and is calculated by adding a group of numbers and then dividing by the count of those numbers. For example, the average of 2, 3, 3, 5, 7, and 10 is 30 divided by 6, which is 5.
(a) To find the value of b in terms of a, we need to calculate Fw(w) using the given cumulative distribution function (CDF) and then compare it with the given equation X - U(1, a).
Given: Fw(w) = b/2 (w² - 1)
To find Fw(w), we differentiate the CDF with respect to w:
fw(w) = d/dw (Fw(w))
= d/dw (b/2 (w² - 1))
= b/2 (2w)
= bw
Now, we equate fw(w) to the density function of X - U(1, a):
bw = 1/(a-1) [Since X - U(1, a) is a uniformly distributed random variable on the interval 1 < x < a]
Comparing the coefficients of w on both sides of the equation, we have:
b = 1/(a-1)
Therefore, b = 2/(a-1).
(b) The average region area is given as 5 km². We can find the value of a using the equation for the average:
E(X) = (1/2) * (1 + a)
Given E(X) = 5, we can solve for a:
5 = (1/2) * (1 + a)
10 = 1 + a
a = 9
The maximum allowable region area, a, is 9 km².
To find the average region side length, E(W), we substitute the value of a into the expression W = vX:
E(W) = E(vX) = v * E(X) = v * (1/2) * (1 + a) = v * (1/2) * (1 + 9) = 5v km
Therefore, the average region side length, E(W), is 5v km.
(c) i. The monitoring cost, C, is given by the expression:
C = $500 + $50 * X
ii. To find the average monitoring cost, E(C), we need to find E(X) and substitute it into the expression for C:
E(C) = $500 + $50 * E(X) = $500 + $50 * 5 = $750
Therefore, the average monitoring cost is $750.
iii. To find the variance of the monitoring cost, Var(C), we can use the fact that Var(aX) = a² * Var(X) for a constant "a" and a random variable "X". In this case, "a" is the cost per km², $50.
Var(C) = Var($500 + $50 * X) = $50^2 * Var(X)
Since X is uniformly distributed on the interval 1 < x < 9, the variance of X is given by:
Var(X) = (9 - 1)² / 12 = 8² / 12 = 64 / 12 = 16/3
Therefore, Var(C) = $50² * (16/3) = $40000/3
The variance of the monitoring cost is $40000/3.
Hence,
a) b = 2/(a-1).
b) the average region side length, E(W), is 5v km.
c) the average monitoring cost is $750.
Var(C) = $50² * (16/3) = $40000/3
The variance of the monitoring cost is $40000/3.
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allowed
The data below is going to be plotted on a line graph.
a) What is the maximum value of time we need to plot?
b) What is the maximum value of distance we need to plot?
Time since start Distance from start
(minutes)
(km)
0
0
5
6
10
9
15
7
20
8
25
5
30
4
Q maths
For the line graph, we need to plot time values ranging from 0 to 30 minutes and distance values ranging from 0 to 9 km.
To determine the maximum values for time and distance from the given data, we need to identify the highest values in the respective columns.
a) Maximum value of time we need to plot:
Looking at the "Time since start" column, we can see that the highest value is 30 minutes.
Therefore, the maximum value of time we need to plot is 30 minutes.
b) Maximum value of distance we need to plot:
Examining the "Distance from start" column, we can observe that the highest value is 9 km.
Thus, the maximum value of distance we need to plot is 9 km.
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How many rounds of golf do those physicians who play golf play per year? A survey of 12 physicians revealed the following numbers: 7, 41, 16, 4, 32, 38, 21, 15, 19, 25, 12, 52 Estimate with 90% confidence the mean number of rounds played per year by physicians, assuming that the population is normally distributed with a standard deviation of 8. Note: For each confidence interval, enter your answer in the form (LCL, UCL). You must include the parentheses and the comma between the confidence limits. Confidence Interval =
The 90% confidence interval for the mean number of rounds played per year by physicians, assuming a normal distribution with a standard deviation of 8, is (15.15, 34.15).
To estimate the mean number of rounds played per year by physicians with a 90% confidence interval, we can use the formula:
CI = X ± Z * (σ / √n)
Where:
CI is the confidence interval
X is the sample mean
Z is the critical value for the desired confidence level (90% in this case)
σ is the population standard deviation
n is the sample size
Given:
Sample size (n) = 12
Sample mean (X) = (7 + 41 + 16 + 4 + 32 + 38 + 21 + 15 + 19 + 25 + 12 + 52) / 12 = 23.25
Population standard deviation (σ) = 8
Critical value (Z) for a 90% confidence level is 1.645 (obtained from a standard normal distribution table)
Plugging in the values into the formula, we have:
CI = 23.25 ± 1.645 * (8 / √12)
CI = 23.25 ± 1.645 * 2.3094
CI = 23.25 ± 3.7983
CI ≈ (15.15, 34.15)
Therefore, with 90% confidence, we can estimate that the mean number of rounds played per year by physicians is between 15.15 and 34.15.
This means that we are 90% confident that the true population mean falls within this range based on the given sample.
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an object oscillates as it moves along the x-axis. Its displacement varies with time according to the equation x=4 cos(pi*t+Pi/4) where t=time in seconds and x=displacement in meters. What is the displacement between t=0 and t=1 second??
The displacement of the object between t=0 and t=1 second is 5.66 m.
What is the displacement?The displacement of the object between t=0 and t=1 second is calculated as follows;
The given equation of the object's motion;
x = 4 cos (πt + π/4)
where;
x is the object's displacementat a time, t = 0 second, the displacement of the object is calculated as;
x = 4 cos (πt + π/4)
x = 4 cos (0 + π/4)
x = 4 cos (π/4)
x = 2.83 m
at time t = 1 second, the displacement of the object is calculated as;
x = 4 cos (π + π/4)
x = 4 cos (5π/4)
x = -2.83 m
The displacement of the object between the time given;
x = 2.83 m - ( - 2.83 m )
x = 5.66 m
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The approximation of J 2 1 xin (x +1/2) dx using two points Gaussian quadrature formula is: O 1.06589 O 2.8191 O 4.08176 O 3.0323
The approximation of ∫0^1 J21xin(x + 1/2)dx using two points Gaussian quadrature formula is 1.5324 (approx). Hence, the correct option is O 1.06589.
To approximate the given integral using two points Gaussian quadrature formula, we use the following formula:∫a^bf(x)dx≈[(b−a)/2]∑i=1^2wi*f[(b−a)/2*xi+(b+a)/2]Here, f(x) = J21xin(x + 1/2), a = 0, b = 1, w1 = w2 = 1, x1 = -√(1/3) and x2 = √(1/3)We have to calculate ∫0^1 J21xin(x + 1/2)dx≈[(1−0)/2]∑i=1^2wi*f[(1−0)/2*xi+(1+0)/2]Putting the values of weights and abscissae, we have∫0^1 J21xin(x + 1/2)dx ≈ [1/2]{f(-√(1/3)) + f(√(1/3))}≈ [1/2]{J21xi(-√(1/3) + 1/2) + J21xi(√(1/3) + 1/2)}Putting x1 = -√(1/3) and x2 = √(1/3), we get∫0^1 J21xin(x + 1/2)dx ≈ [1/2]{J21xi(1/6 - √(1/3)) + J21xi(1/6 + √(1/3))}≈ [1/2]{1.40628 + 1.65847}≈ [1/2]*3.06475≈ 1.53238 ≈ 1.5324 (correct to 4 decimal places)Therefore, the approximation of ∫0^1 J21xin(x + 1/2)dx using two points Gaussian quadrature formula is 1.5324 (approx). Hence, the correct option is O 1.06589.
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Catherine Ndlovu, a school senior contemplates her future immediately after graduation She thinks there is a 25% change that she will join Boston School and teach English in South Africa for the next few years. Alternatively, she believes that there is a 35% change that she will enroll in a full-time Law School program in the Madagascar. The probability that she does not choose either of these options is 0.00
The probability that Catherine chooses either option A or option B is 1 (or 100%).
Let's denote the events as follows:
A: Catherine joins Boston School and teaches English in South Africa.
B: Catherine enrolls in a full-time Law School program in Madagascar.
C: Catherine does not choose either of these options.
We are given the following probabilities:
P(A) = 0.25 (25% chance of joining Boston School)
P(B) = 0.35 (35% chance of enrolling in Law School)
P(C) = 0.00 (0% chance of not choosing either option)
To determine the probability that Catherine does not choose either of these options, we can use the fact that the sum of probabilities for all possible outcomes must equal 1:
P(A) + P(B) + P(C) = 1
Since P(C) = 0.00, we can rewrite the equation as:
P(A) + P(B) + 0 = 1
From this equation, we can solve for P(A) + P(B):
P(A) + P(B) = 1 - P(C) = 1 - 0 = 1
Therefore, the probability that Catherine chooses either option A or option B is 1 (or 100%).
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Suppose that someone wants to earn $4,259 in 9 years in an account that as an annual rate of 3.2% compounded quarterly. How much should be invested? (round up to 2 decimal places)
To earn $4,259 in 9 years with an annual interest rate of 3.2% compounded quarterly, one should invest approximately $3,066.79.
To determine the amount that should be invested, we can use the formula for compound interest:
[tex]A = P(1 + r/n)^{(nt)[/tex]
Where:
A is the future value of the investment ($4,259),
P is the principal amount (the amount to be invested),
r is the annual interest rate (3.2% or 0.032),
n is the number of times the interest is compounded per year (quarterly, so 4),
and t is the number of years (9).
Plugging in the given values, we can rearrange the formula to solve for P:
P = A / (1 + r/n)^(nt)
Substituting the values, we have:
P = $[tex]4,259 / (1 + 0.032/4)^{(4*9)[/tex]
P = $[tex]4,259 / (1.008)^{(36)[/tex]
P ≈ $3,066.79
Therefore, approximately $3,066.79 should be invested to earn $4,259 in 9 years with an annual interest rate of 3.2% compounded quarterly.
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Find the p-values for the following critical values: (Assume one sided hypothesis) a. 2.03 b. 1.50 c. 1.20 d. 2.76 7. Find the p-values for the following critical values: (Assume two sided hypothesis) a. 2.03 b. 1.50 c. 1.40 d. 2.26
The p-value for 2.03 is 0.0212.
The p-value for 1.50 is 0.0668.
The p-value for 1.20 is 0.1151.
The p-value for 2.76 is 0.0029.
To calculate the p-values for the given critical values, we need to refer to a standard normal distribution graph or a z-table. This table provides the probabilities associated with different z-scores (standardized scores). The z-score represents the number of standard deviations a data point is away from the mean.
a. Critical value: 2.03
To find the p-value for the critical value of 2.03, we look at the standard normal distribution graph or z-table. Locate the value of 2.03 on the graph and find the corresponding area under the curve. This means that if the null hypothesis is true, there is a 0.0212 probability of obtaining a test statistic as extreme as 2.03 or more extreme.
b. Critical value: 1.50
Similarly, we locate the value of 1.50 on the standard normal distribution graph and find the corresponding area. This implies that if the null hypothesis is true, there is a 0.0668 probability of obtaining a test statistic as extreme as 1.50 or more extreme.
c. Critical value: 1.20
Again, we locate the value of 1.20 on the standard normal distribution graph and find the corresponding area. This means that if the null hypothesis is true, there is a 0.1151 probability of obtaining a test statistic as extreme as 1.20 or more extreme.
d. Critical value: 2.76
Locating the value of 2.76 on the standard normal distribution graph, we find the corresponding area. This indicates that if the null hypothesis is true, there is a 0.0029 probability of obtaining a test statistic as extreme as 2.76 or more extreme.
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In a particular very small region, the consumer price index, C, depends on the current value of gross regional domestic expenditure E, number of people living in poverty P, and the average number of household members in a family F, by the following formula: e-Ep C = 100 + F If it is known that the gross regional domestic expenditure is decreasing at a rate of PHP 50 per year, and the number of people living in poverty and the average number of household members in a family are increasing at 3 and 1 per year, resp., how fast does the consumer price index change per year at the moment when E = 1,000, P = 200, and F = 5? (5 points)
The consumer price index (C) changes at a rate of 8,001 per year at the moment when E = 1,000, P = 200, and F = 5.
To find how fast the consumer price index (C) changes per year at the moment when E = 1,000, P = 200, and F = 5, we need to calculate the derivative of C with respect to time and evaluate it at those specific values.
Given the formula for C in terms of E, P, and F:
C = 100 + F - E * P
We can differentiate C with respect to time (t) using the chain rule:
dC/dt = dC/dE * dE/dt + dC/dP * dP/dt + dC/dF * dF/dt
Given the rates of change:
dE/dt = -50 (gross regional domestic expenditure is decreasing at a rate of PHP 50 per year)
dP/dt = 3 (number of people living in poverty is increasing at a rate of 3 per year)
dF/dt = 1 (average number of household members in a family is increasing at a rate of 1 per year)
We need to calculate the partial derivatives dC/dE, dC/dP, and dC/dF:
dC/dE = -P
dC/dP = -E
dC/dF = 1
Now, we can substitute the values E = 1,000, P = 200, F = 5, and the calculated partial derivatives into the derivative equation:
dC/dt = (-P) * (-50) + (-E) * 3 + 1 * 1
= 50P - 3E + 1
Substituting the given values E = 1,000 and P = 200:
dC/dt = 50(200) - 3(1,000) + 1
= 10,000 - 3,000 + 1
= 8,001
Therefore, the consumer price index (C) changes at a rate of 8,001 per year at the moment when E = 1,000, P = 200, and F = 5.
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Watch the last percentages video on multistep percentage problems.
a. You have an item that costs $40 that is 25% off. At the register they take an additional 30% off because of a daily coupon. What do you pay?
b. Some people might assume that 25% off followed by 30% off is 55% off. Why might people think that? How would you convince them it's not? What is the overall percentage when you take 25% off followed by 30% off?
a. You will pay $24.b. People might think that 25% off followed by 30% off is 55% off because they are adding the two percentages together.
However, this is not correct. When you take multiple discounts, you need to multiply the percentages together. In this case, the overall percentage is 42.5%.To calculate the final price, we can use the following formula:
Final price = Original price * (1 - Discount 1) * (1 - Discount 2)
Where:
The final price is the price you will pay
The original price is the price of the item
Discount 1 is the first discount
Discount 2 is the second discount
In this case, the original price is $40, the first discount is 25%, and the second discount is 30%.
Final price = $40 * (1 - 0.25) * (1 - 0.30)
Final price = $24
As you can see, the final price is $24. This is because the overall percentage is 42.5%.
People might think that 25% off followed by 30% off is 55% off because they are adding the two percentages together. However, this is not correct. When you take multiple discounts, you need to multiply the percentages together. In this case, the overall percentage is 42.5%.
To convince people that 25% off followed by 30% off is not 55% off, you can use the following analogy. Imagine that you have a $100 bill and you take 25% off. This means that you will have $75 left. Now, imagine that you take another 30% off of the $75. This means that you will have $52.50 left. As you can see, the overall percentage is not 55%. It is 42.5%.
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Consider the statement: For any integers a, b and d * 0, if d divides a and d divides b, then d divides a + b. (a) [2 marks] Write this statement in symbols as a universal conditional statement. (b) [3 marks] For this statement, give, in symbols, the: (i) Contrapositive (ii) Converse (iii) Negation
(a) The statement in symbols: ∀a, b, d ∈ Z, (d ≠ 0) → ((d divides a) ∧ (d divides b)) → (d divides (a + b)).
(b) Symbols for contrapositive: ∀a, b, d ∈ Z, (d ≠ 0) → (¬(d divides (a + b))) → (¬((d divides a) ∧ (d divides b))).
Symbols for converse: ∀a, b, d ∈ Z, (d ≠ 0) → ((d divides a + b) → (d divides a) ∧ (d divides b)).
Symbols for negation: ∃a, b, d ∈ Z, (d ≠ 0) ∧ ((d divides a) ∧ (d divides b)) ∧ ¬(d divides (a + b)).
(a) The statement can be written in symbols as: ∀a, b, d ∈ Z, (d ≠ 0) → ((d divides a) ∧ (d divides b)) → (d divides (a + b)).
(b) The symbols for the contrapositive, converse, and negation of the statement are as follows:
(i) Contrapositive:
∀a, b, d ∈ Z, (d ≠ 0) → (¬(d divides (a + b))) → (¬((d divides a) ∧ (d divides b)))
(ii) Converse:
∀a, b, d ∈ Z, (d ≠ 0) → ((d divides a + b) → (d divides a) ∧ (d divides b))
(iii) Negation:
∃a, b, d ∈ Z, (d ≠ 0) ∧ ((d divides a) ∧ (d divides b)) ∧ ¬(d divides (a + b))
Note: The symbols "∀" represents the universal quantifier "for all", "∃" represents the existential quantifier "there exists", "∈" represents "belongs to", "Z" represents the set of integers, "¬" represents "not", and "∧" represents "and".
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. Which of the following functions defined on (-inf,inf) has an inverse? (a) f(x) = x^2 (b) f(x) = |x| (c) f(x) = x^3 (d) f(x) = e^x (e) None of the above
The only function that has an inverse on the domain of (-∞, ∞) is given by option d. f(x) = eˣ which is equal to ln(x).
To determine which of the given functions has an inverse,
check if each function satisfies the criteria for having an inverse.
f(x) = x²
This function does not have an inverse on the entire domain of (-∞, ∞) because it fails the horizontal line test.
It fails the test because different values of x can produce the same output, violating the one-to-one correspondence required for an inverse.
f(x) = |x|,
This function also does not have an inverse on the entire domain of (-∞, ∞) since it fails the horizontal line test for the same reason as function (a).
Different values of x produce the same output, making it non-invertible.
f(x) = x³
Similar to the previous functions, this function fails the horizontal line test and does not have an inverse on the entire domain of (-∞, ∞).
Different x-values can produce the same output, so it is not one-to-one.
f(x) = eˣ
The exponential function f(x) = eˣ does have an inverse.
It is called the natural logarithm function, denoted as ln(x).
The inverse function of f(x) = eˣ is g(x) = ln(x), defined on the positive real numbers (0, ∞).
However, it is important to note that the domain of f(x) = eˣ is (−∞, ∞), while the domain of its inverse, g(x) = ln(x), is (0, ∞).
Therefore, based on the above analysis option (d) f(x) = eˣ is the only function that has an inverse on the given domain.
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you may need to use the appropriate appendix table or technology to answer this question. z is a standard normal random variable. find p(−1.86 ≤ z ≤ 1.5). a. 0.0314 b. 0.0982 c. 0.9018 d. 0.9332
The correct option is (c).
To find the probability P(−1.86 ≤ z ≤ 1.5), where z is a standard normal random variable, we need to use the standard normal distribution table or a technology tool.
Using a standard normal distribution table, we look up the z-values −1.86 and 1.5. The table provides the area under the standard normal curve up to those z-values.
From the table, we find that the area to the left of z = −1.86 is 0.0314 and the area to the left of z = 1.5 is 0.9332.
To find the probability between −1.86 and 1.5, we subtract the smaller area from the larger area:
P(−1.86 ≤ z ≤ 1.5) = 0.9332 - 0.0314 = 0.9018
Therefore, the correct answer is c) 0.9018.
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An auditorium has 79 rows of seats. The first row contains 60 seats. As you move to the rear of the auditorium, each row has 3 more seats than the previous row. How many seats are in row 24? How many seats are in the auditorium? Question 10 Find the infinite sum, if it exists for this series: (-2) + (0.5) + (-0.125) +
9. Row 24 of the auditorium has 129 seats.
10. The sum of the infinite series (-2) + (0.5) + (-0.125) + ... is -4/3.
To find the number of seats in row 24 of the auditorium, we can use the given information that each row has 3 more seats than the previous row. Starting from the first row with 60 seats, we can determine the number of seats in row 24 by adding 3 seats for each subsequent row:
Number of seats in row 24 = Number of seats in the first row + (Number of rows - 1) * 3
= 60 + (24 - 1) * 3
= 60 + 23 * 3
= 60 + 69
= 129
Therefore, row 24 of the auditorium has 129 seats.
To find the total number of seats in the auditorium, we need to sum up the number of seats in each row. Since each row has 3 more seats than the previous row, we can use an arithmetic progression to find the sum.
The sum of an arithmetic progression can be calculated using the formula:
Sum = (n/2) * (first term + last term)
where n is the number of terms in the progression.
In this case, the number of terms is 79 (number of rows), the first term is 60 (number of seats in the first row), and the last term can be calculated as:
Last term = Number of seats in the first row + (Number of rows - 1) * 3
= 60 + (79 - 1) * 3
= 60 + 78 * 3
= 60 + 234
= 294
Now we can calculate the total number of seats in the auditorium:
Total number of seats = (79/2) * (60 + 294)
= 39.5 * 354
= 14,013
Therefore, the auditorium has a total of 14,013 seats.
For Question 10:
The given series is: (-2) + (0.5) + (-0.125) + ...
We notice that each term is obtained by multiplying the previous term by (-0.5). This indicates a geometric series.
To find the sum of the infinite geometric series, we can use the formula for the sum of an infinite geometric series:
Sum = a / (1 - r)
where "a" is the first term and "r" is the common ratio.
In this case, the first term (a) is -2 and the common ratio (r) is -0.5.
Sum = (-2) / (1 - (-0.5))
= (-2) / (1 + 0.5)
= (-2) / (1.5)
= -4/3
Therefore, the sum of the infinite series (-2) + (0.5) + (-0.125) + ... is -4/3.
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Find a point c satisfying the conclusion of the Mean Value Theorem for the following function and interval. f(x) = x-7, (1,5) C=
The point c = 3 satisfies the conclusion of the Mean Value Theorem for the function f(x) = x - 7 on the interval (1,5).
To apply the Mean Value Theorem, we need to verify that f(x) is continuous on [1,5] and differentiable on (1,5).
Since f(x) = x - 7 is a polynomial function, it is continuous and differentiable for all real numbers. Therefore, f(x) is also continuous on [1,5] and differentiable on (1,5).
Now, we can use the Mean Value Theorem to find a point c in (1,5) such that:
f'(c) = (f(5) - f(1))/(5 - 1)
First, let's evaluate f(5) and f(1):
f(5) = 5 - 7 = -2
f(1) = 1 - 7 = -6
Next, let's calculate the derivative of f(x):
f'(x) = d/dx (x - 7) = 1
So, we have:
1 = (-2 - (-6))/(5 - 1)
Simplifying the right-hand side, we get:
1 = 1
Therefore, there exists at least one point c in (1,5) such that f'(c) = 1. To find the value of c, we can solve for c as follows:
f'(c) = 1
d/dx (x - 7) = 1
1 = 1
Since the derivative is constant, it is equal to 1 for all values of x. Thus, we can choose any value of c in (1,5), for example, c = 3.
Therefore, the point c = 3 satisfies the conclusion of the Mean Value Theorem for the function f(x) = x - 7 on the interval (1,5).
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x² Given that f(x): = 2+x* a) Use the definition of derivative to find f'(x). b) find the slope of tangent line to the f(x) at x=2. c) Find the tangent line at (2,1).
The derivative of f(x) = 2 + x² is f'(x) = 2x, the slope of the tangent line to f(x) at x = 2 is 4 and the equation of the tangent line at the point (2, 1) is y = 4x - 7.
To find the derivative of the function f(x) = 2 + x² using the definition of derivative
we need to compute the limit of the difference quotient as it approaches zero.
The definition of derivative is given by:
f'(x) = lim (h -> 0) [f(x + h) - f(x)] / h
Substituting f(x) = 2 + x² into the definition, we have:
f'(x) = lim (h -> 0) [2 + (x + h)² - (2 + x²)] / h
f'(x) = lim (h -> 0) [2 + x² + 2xh + h² - 2 - x²] / h
f'(x) = lim (h -> 0) 2x + h
Taking the limit as h approaches zero, the term h goes to zero, and we are left with:
f'(x) = 2x
Therefore, the derivative of f(x) = 2 + x² is f'(x) = 2x.
b) To find the slope of the tangent line to the function f(x) at x = 2, we can evaluate the derivative at that point.
Substituting x = 2 into the derivative f'(x) = 2x, we have:
f'(2) = 2(2) = 4
The slope of the tangent line to f(x) at x = 2 is 4.
c) To find the equation of the tangent line at the point (2, 1), we can use the point-slope form of a linear equation, where the slope is 4 (as found in part b) and the point (2, 1) lies on the line.
The point-slope form is given by:
y - y₁ = m(x - x₁)
Substituting the values, we have:
y - 1 = 4(x - 2)
y - 1 = 4x - 8
y = 4x - 7
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Given that f(x) = |x| and g(x) = 9x + 2, calculate (a) f o g(x) = (b) g o f(x) = (c) f o f(x) = (d) g o g(x) =
To calculate the compositions of the given functions:
(a) f o g(x):
To find f o g(x), we substitute g(x) into f(x):
f o g(x) = f(g(x))
Since g(x) = 9x + 2, we substitute it into f(x):
f o g(x) = f(9x + 2)
Now, we evaluate f(9x + 2) by substituting (9x + 2) into |x|:
f o g(x) = |9x + 2|
(b) g o f(x):
To find g o f(x), we substitute f(x) into g(x):
g o f(x) = g(f(x))
Since f(x) = |x|, we substitute it into g(x):
g o f(x) = g(|x|)
Now, we evaluate g(|x|) by substituting |x| into 9x + 2:
g o f(x) = 9|x| + 2
(c) f o f(x):
To find f o f(x), we substitute f(x) into f(x):
f o f(x) = f(f(x))
Since f(x) = |x|, we substitute it into f(x):
f o f(x) = f(|x|)
Now, we evaluate f(|x|) by substituting |x| into |x|:
f o f(x) = ||x|| = |x|
(d) g o g(x):
To find g o g(x), we substitute g(x) into g(x):
g o g(x) = g(g(x))
Since g(x) = 9x + 2, we substitute it into g(x):
g o g(x) = g(9x + 2)
Now, we evaluate g(9x + 2) by substituting (9x + 2) into 9x + 2:
g o g(x) = 9(9x + 2) + 2 = 81x + 20
Therefore, the calculations are as follows:
(a) f o g(x) = |9x + 2|
(b) g o f(x) = 9|x| + 2
(c) f o f(x) = |x|
(d) g o g(x) = 81x + 20
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a) Show that the triangle with vertices D(-2, 5), E(-4, 1), and F(2, 3) is a right triangle.
b) Verify that the midpoint of the hypotenuse of △DEF is equidistant from all three vertices.
(a) The triangle with vertices D(-2, 5), E(-4, 1), and F(2, 3) is a right triangle, as the slopes of the sides DE and EF are negative reciprocals.
(b) The midpoint of the hypotenuse of △DEF, which is (-1, 2), is equidistant from all three vertices, as the distances from the midpoint to each vertex are equal.
(a) To show that △DEF is a right triangle, we can calculate the slopes of two sides and check their relationship. The slope of DE is (1 - 5) / (-4 - (-2)) = -4 / -2 = 2. The slope of EF is (3 - 1) / (2 - (-4)) = 2 / 6 = 1/3. Since these slopes are negative reciprocals (2 (1/3) = 2/3), the sides DE and EF are perpendicular, indicating that △DEF is a right triangle.
(b) To verify that the midpoint of the hypotenuse of △DEF is equidistant from all three vertices, we can calculate the distances from the midpoint to each vertex and compare them. The midpoint of DE is [(2 + (-4)) / 2, (3 + 1) / 2] = (-1, 2).
Distance from (-1, 2) to D(-2, 5) = √[[tex](-2 - (-1))^2 + (5 - 2)^2[/tex]] = √[[tex]1^2 + 3^2[/tex]] = √10.
Distance from (-1, 2) to E(-4, 1) = √[[tex](-4 - (-1))^2 + (1 - 2)^2[/tex]] = √[[tex]3^2 + (-1)^2[/tex]] = √10.
Distance from (-1, 2) to F(2, 3) = √[[tex](2 - (-1))^2 + (3 - 2)^2[/tex]] = √[[tex]3^2 + 1^2[/tex]] = √10.
As the distances from the midpoint (-1, 2) to each vertex are equal (√10), it verifies that the midpoint is equidistant from all three vertices of the triangle △DEF.
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Find the approximate value I of the integral ∫^3_0 f(x)dx using the Trapezoidal rule with n= 3, where f(x) = x(1+e^x). If ∫^3_0 f(x)dx-1 = -3/12 f"(c) where 0 < c <3, estimate the error |∫^3_0 f(x)dx - I|
The approximate value of the integral ∫^3_0 f(x)dx using the Trapezoidal rule with n = 3 is 52.057. The estimated error |∫^3_0 f(x)dx - I| is less than or equal to 13.673.
To approximate the value of the integral ∫^3_0 f(x)dx using the Trapezoidal rule with n = 3, we first divide the interval [0, 3] into n subintervals of equal width. In this case, with n = 3, we have h = (3 - 0) / 3 = 1.
Next, we evaluate the function f(x) at the endpoints and midpoints of each subinterval:
f(0) = 0(1 + e^0) = 0(1 + 1) = 0
f(1) = 1(1 + e^1) = 1(1 + 2.718) ≈ 4.718
f(2) = 2(1 + e^2) = 2(1 + 7.389) ≈ 16.778
f(3) = 3(1 + e^3) = 3(1 + 20.086) ≈ 63.258
Using the Trapezoidal rule formula, the approximation of the integral is:
I ≈ (h/2) * [f(0) + 2f(1) + 2f(2) + f(3)]
≈ (1/2) * [0 + 2(4.718) + 2(16.778) + 63.258]
≈ 52.057
To estimate the error |∫^3_0 f(x)dx - I|, we can use the error formula for the Trapezoidal rule:
Error = - (h^3 / 12) * f''(c), where 0 < c < 3.
The second derivative of f(x) is:
f''(x) = 2e^x + x(e^x) = e^x(2 + x)
To find the maximum value of f''(x) on the interval [0, 3], we can evaluate it at the endpoints:
f''(0) = e^0(2 + 0) = 2
f''(3) = e^3(2 + 3) ≈ 164.076
Since f''(x) is continuous on the interval [0, 3], the maximum value must occur at some point within the interval.
Therefore, |∫^3_0 f(x)dx - I| ≤ (1^3 / 12) * 164.076
= 13.673
The estimated error is approximately 13.673.
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The mean price of houses in the US is $383,500. A real estate agent believes the mean price of houses in a local neighborhood is less than the national mean. The agent takes a random sample of 30 houses and finds the mean price to be $295,089 with a standard deviation of $156,321. Do the data provide convincing evidence at the Alpha = 0.05 level that the mean price of the houses in the area is less than $383,500?
What are the test statistic and P-value for this significance test?
Find the t-table here and the z-table here.
t = 3.10 and 0.001 < P-value < 0.0025
z = 3.10 and 0.001 < P-value < 0.0025
t = –3.10 and 0.001 < P-value < 0.0025
z = –3.10 and 0.001 < P-value < 0.0025
The test statsistic and the p value are t = –3.10 and 0.001 < P-value < 0.0025
How to solve the test statisticTo determine the test statistic and P-value for this significance test, let's proceed with the calculations.
Given information:
Sample mean (x) = $295,089
Population mean (μ₀) = $383,500
Standard deviation (σ) = $156,321
Sample size (n) = 30
Alpha level (α) = 0.05
First, let's calculate the test statistic (t-statistic) using the formula:
t = (x - μ₀) / (σ / √n)
Substituting the values:
t = ($295,089 - $383,500) / ($156,321 / √30)
t ≈ (-88311) / (28514.87 / 5.477)
t ≈ -3.45
So the calculated t-statistic is approximately -3.45.
To find the P-value for this t-statistic, we need to refer to the t-distribution table. The degrees of freedom (df) for this test is n - 1 = 30 - 1 = 29.
Looking up the absolute value of the t-statistic (-3.45) and the degrees of freedom (df = 29) in the t-distribution table, we find that the P-value is between 0.001 and 0.0025.
Therefore, the correct answer is:
t = –3.10 and 0.001 < P-value < 0.0025
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Given a population of travel expense vouchers for a university in academic year, indicate what the sampling distribution for samples of 20 would consist of. Choose the correct answer below.
A. The sampling distribution is a representative collection of 20 samples, each containing 20 vouchers, selected with replacement.
B. The sampling distribution is the average result from all possible samples of 20 vouchers.
C. The sampling distribution is the distribution of the results for all possible samples of 20 vouchers.
D. The sampling distribution is a representative collection of 20 samples, each containing 20 vouchers, selected without replacement.
C. The sampling distribution is the distribution of the results for all possible samples of 20 vouchers.
In more detail, a sampling distribution represents the distribution of a statistic (in this case, the results of the travel expense vouchers) across all possible samples of a specific size (in this case, 20). It provides information about the variability and characteristics of the statistic when repeatedly sampling from the population. Each sample is obtained by randomly selecting 20 vouchers from the population.
The sampling distribution is constructed by calculating the desired statistic (e.g., mean, standard deviation) for each sample and organizing these values into a distribution. In this case, the sampling distribution would consist of the results (e.g., average travel expenses) for all possible samples of 20 vouchers. It allows us to examine the overall pattern, central tendency, and spread of the statistic across the samples.
Option A suggests sampling with replacement, where vouchers are selected and then returned to the population before the next selection. Option D suggests sampling without replacement, where vouchers are selected and not returned, resulting in a different distribution. Option B refers to the average result from all possible samples, but does not capture the full distribution of the results. Therefore, option C accurately represents the concept of the sampling distribution for samples of 20 vouchers.
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(a) Differentiate the following function implicitly. y? + cos y = x6 + 3xy x (b) Differentiate the following function from first principles. f(x) = x3
The implicit differentiation of y? + cos y = x^6 + 3xyx yields (dy/dx)^2 - sin(y) * dy/dx = 6x^5 + 3y + 3xy * dy/dx. The first principles differentiation of f(x) = x^3 involves expanding [(x + h)^3 - x^3] / h and simplifying to find f'(x) = 3x^2.
To differentiate the function implicitly, we take the derivative of both sides with respect to x, applying the chain rule and power rule. The result is (dy/dx)^2 - sin(y) * dy/dx = 6x^5 + 3y + 3xy * dy/dx.
To differentiate the function from first principles, we use the definition of the derivative. Simplifying [(x + h)^3 - x^3] / h and taking the limit as h approaches 0, we obtain the derivative f'(x) = 3x^2.
(a) In order to differentiate the function implicitly, we consider the derivative of each term on both sides of the equation with respect to x. We apply the chain rule to differentiate the terms involving y, and the power rule to differentiate the terms involving x. Combining these derivatives, we obtain the differentiated equation.
(b) To differentiate the function f(x) = x^3 from first principles, we apply the definition of the derivative: [f(x + h) - f(x)] / h. Expanding the numerator, we simplify the expression and eliminate the terms that vanish as h approaches 0. The resulting expression represents the derivative of f(x) with respect to x, which is 3x^2.
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T/F?
a) There is no proper non-trivial ideal in any Field
b A local Ring is a Ring with only One Maximal ideal c) The union of two ideals is an ideal d Every non-zero element in an Integral Domain is a unit
True. In an integral domain, not every non-zero element is a unit. Units are elements that have a multiplicative inverse, and not all elements in an integral domain possess this property.
a) In fact, every field has two trivial ideals, which are the zero ideal and the whole field itself.
b) A local ring is defined as a ring that has a unique maximal ideal.
c) The union of two ideals is not always an ideal. For example, consider the ideals (2) and (3) in the ring Z (the integers). The union of these two ideals is {2, 3}, which is not closed under addition and therefore not an ideal.
d) A unit in an integral domain is an element that has an inverse. Not every non-zero element in an integral domain is a unit. For example, in the ring Z (the integers), the only units are 1 and -1.
A field has no proper non-trivial ideals because its only ideals are the zero ideal and the entire field itself.
A local ring is defined as a ring with a unique maximal ideal, which means it has only one maximal ideal. The union of two ideals is not necessarily an ideal, as it may not be closed under subtraction or multiplication by elements of the ring. In an integral domain, not every non-zero element is a unit. Units are elements that have a multiplicative inverse, and not all elements in an integral domain possess this property.
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A new young executive is perplexed at the number of interruptions that occur due to employee relations. She has decided to track the number of interruptions that occur during each hour of her day. Over the last month, she has determined that between 0 and 3 interruptions occur during any given hour of her day. The data is shown below.
Number of Interruptions in 1 hour
Probability
0 interruption
0.5
1 interruptions
0.3
2 interruptions
0.1
3 interruptions
0.1
On average, she should expect 0.8 interruptions per hour. ?
The expected value for the average number of interruptions per hour based for the given of data of number of interruptions is equal to 0.8.
To determine the average number of interruptions per hour,
Calculate the expected value of the number of interruptions using the given probabilities.
The expected value or mean of a discrete random variable is,
Calculated by multiplying each possible value by its corresponding probability and summing them up.
Here, the number of interruptions can take values 0, 1, 2, or 3, with the corresponding probabilities given.
Expected Value (μ)
= (0 × 0.5) + (1 × 0.3) + (2 × 0.1) + (3 × 0.1)
= 0 + 0.3 + 0.2 + 0.3
= 0.8
Therefore, the expected value for the average number of interruptions per hour based on the given data is 0.8.
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Where does the plus minus come from (+-) ???
pls I need help
the idea of the ± in an even root, well, square root is an even root, however 4th, 6th and so on are also even roots and this applies to all even roots, is that, even root of some number say "x" is "y", that means that if we squared "y", we'd get "x", but but but if we square the negative of "y", we'd also get the same "x", so either the positive or negative version will really give us "x", that's a bit mumbled, let's put it this way
[tex]\sqrt{16}=4\implies 16=4^2\qquad \textit{well, to be honest}\qquad 16=(-4)^2\qquad too \\\\\\ \textit{how do we know }\text{\LARGE 16}\textit{ came from }(+4)^2 ~~ or ~~ (-4)^2 ~~ ?\quad \textit{ we really don't know} \\\\\\ \textit{so we } incl ude \textit{ both and say }16=(\pm 4)^2\implies \sqrt{16}=\pm 4\implies \mp\sqrt{16}=4[/tex]
so the even root could have come from either the negative or positive version of the same value, because once the power is even, any negatives will turn to positives.
Work Problem 2 (15 Points) (a) Sketch a graph of a function f such that :
f' > 0 and f" < 0,for x< 3 f' < 0 and f" > 0, for X > 3. (b) Does the graph of the function f in Part (a) have an inflection point? Explain.
(a) By considering a simple piecewise-defined function, we can sketch a graph of a function f satisfying the given conditions.
(b) Yes, the graph of the function f in part (a) has an inflection point.
(a) To sketch a graph of a function f satisfying the given conditions, we can start by considering a simple piecewise-defined function.
Let's define the function as follows:
For x < 3: f(x) = -(x - 3)²
For x > 3: f(x) = (x - 3)²
Let's analyze the derivatives of this function:
For x < 3:
f'(x) = -2(x - 3)
f"(x) = -2
For x > 3:
f'(x) = 2(x - 3)
f"(x) = 2
(b) Yes, the graph of the function f in part (a) has an inflection point. An inflection point occurs where the concavity of the function changes. In this case, since the second derivative f" changes sign at x = 3 (from negative to positive), there is an inflection point at x = 3.
At x = 3, the graph transitions from a concave down (negative concavity) to a concave up (positive concavity). This change in concavity indicates the presence of an inflection point.
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Divide (15) base 16 by (011) base 2. Express the
answer in base 10.
To divide (15) base 16 by (011) base 2, we first need to convert the numbers to the same base. Let's convert (15) base 16 to base 2:
(15) base 16 = (1111) base 2.
Now we can perform the division:
1111
011 ) 1111
0110
-----
101
011
-----
100
The quotient is (101) base 2, which is equivalent to (5) base 10. Therefore, when dividing (15) base 16 by (011) base 2, the result is (5) base 10.
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Assume that the Monroes want to invest $10,000. They decide to invest $7,000 in Portfolio A with the remainder in the S&P 500. Changes in the S&P 500 account for 25% of the returns for Portfolio A. If Portfolio A has a standard deviation of 20% and the S&P 500 has a standard deviation of 11.5%, what is the standard deviation of the combined $10,000 portfolio?
The combined $10,000 portfolio's standard deviation is roughly 0.335, or 33.5% (rounded to the closest percentage).
To calculate the standard deviation of the combined $10,000 portfolio, we need to consider the correlation between Portfolio A and the S&P 500. The formula for the standard deviation of a portfolio that consists of two assets is:
[tex]\sigma_{\text{total}} = \sqrt{w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho \sigma_1 \sigma_2}[/tex]
where:
- [tex]\begin{equation}\sigma_\text{total}[/tex] is the standard deviation of the combined portfolio.
- [tex]$w_1$[/tex] and [tex]$w_2$[/tex] are the weights of Portfolio A and the S&P 500, respectively.
- [tex]$\sigma_1$[/tex] and [tex]$\sigma_2$[/tex] are the standard deviations of Portfolio A and the S&P 500, respectively.
- ρ is the correlation coefficient between Portfolio A and the S&P 500.
Given:
- Portfolio A has a standard deviation of 20% ([tex]$\sigma_1$[/tex] = 0.20).
- The S&P 500 has a standard deviation of 11.5% ([tex]$\sigma_2$[/tex] = 0.115).
- Changes in the S&P 500 account for 25% of the returns for Portfolio A (ρ = 0.25).
- The Monroes invest $7,000 in Portfolio A and the remainder ($10,000 - $7,000 = $3,000) in the S&P 500.
First, let's calculate the weights of each investment:
[tex]The weight of Portfolio A (w_1) is calculated as:\[w_1 = \frac{7,000}{10,000} = 0.7\]\\The weight of the S\&P 500 (w_2) is calculated as:\[w_2 = \frac{3,000}{10,000} = 0.3\][/tex]
Now, we can substitute the given values into the standard deviation formula:
[tex]\sigma_{\text{total}} = \sqrt{0.7^2 \times 0.20^2 + 0.3^2 \times 0.115^2 + 2 \times 0.7 \times 0.3 \times 0.25 \times 0.20 \times 0.115}[/tex]
Calculating this expression:
[tex]\begin{equation}\sigma_\text{total}[/tex] = sqrt(0.098 + 0.00303875 + 0.00994275) = sqrt(0.1119815) ≈ 0.335 (rounded to three decimal places).
Therefore, the standard deviation of the combined $10,000 portfolio is approximately 0.335, or 33.5% (rounded to the nearest percentage).
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I already saw the responses to this question but I want another
way. Please don't copy and past it! Please show all work.
(10) 8. Determine if 0010 belongs to each of the following regular sets: a. 0(01)0 b. (000) (10) C. (00) 1 (00) d. (001)*0* e. 00(11) (01)*
If 0010 belongs to each of the following regular sets
a. 0(01)0: 0010 belongs.
b. (000) (10): 0010 does not belong.
c. (00) 1 (00): 0010 does not belong.
d. (001)*0*: 0010 belongs.
e. 00(11) (01)*: 0010 does not belong.
To determine if 0010 belongs to each of the following regular sets, we will analyze the patterns and rules of each set.
a. 0(01)0: This set consists of strings that start and end with 0, with the sequence 01 in between. Since 0010 starts with 0, has 01 in the middle, and ends with 0, it belongs to this set.
b. (000) (10): This set consists of strings that have three consecutive 0's followed by 10. Since 0010 does not have three consecutive 0's, it does not belong to this set.
c. (00) 1 (00): This set consists of strings that have two 0's followed by 1 and then two more 0's. Since 0010 has two 0's followed by 1 and then only one more 0, it does not belong to this set.
d. (001)*0*: This set consists of strings that have any number of occurrences of 001 followed by any number of 0's. Since 0010 starts with 001 and is followed by 0, it belongs to this set.
e. 00(11) (01)*: This set consists of strings that start with 00, followed by 11, and then have any number of occurrences of 01. Since 0010 starts with 00, is followed by 11, and does not have any occurrence of 01, it does not belong to this set.
In summary:
a. 0(01)0: 0010 belongs.
b. (000) (10): 0010 does not belong.
c. (00) 1 (00): 0010 does not belong.
d. (001)*0*: 0010 belongs.
e. 00(11) (01)*: 0010 does not belong.
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