it due in 5 min help

True. After testing H0: p = 0.33 versus HA: p < 0.33 at α = 0.05, with a sample proportion of 0.20 and a sample size of n = 100, we do not reject H0.

True. After conducting a hypothesis test with the given parameters, if the p-value is greater than the significance level (α = 0.05), we do not reject the null hypothesis (H0: p = 0.33). This means that there is not enough evidence to support the alternative hypothesis (HA: p < 0.33) and we conclude that the proportion is not significantly less than 0.33 based on the sample data.

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The probability of drawing 2 pennies and 2 dimes in any order is 0.0706 or approximately 7.06%.

To find the probability of drawing 2 pennies and 2 dimes in any order, we first need to calculate the total number of ways to select 4 coins without replacement from the given purse.
The total number of coins in the purse is 4 + 5 + 8 = 17. So, the total number of ways to select 4 coins without replacement is:
17 choose 4 = 17C4 = (17!)/(4!13!) = 2380
Next, we need to calculate the number of ways to select 2 pennies and 2 dimes in any order.
The number of ways to select 2 pennies from 4 is:
4 choose 2 = 4C2 = (4!)/(2!2!) = 6
Similarly, the number of ways to select 2 dimes from 8 is:
8 choose 2 = 8C2 = (8!)/(2!6!) = 28
To get the total number of ways to select 2 pennies and 2 dimes in any order, we need to multiply the number of ways to select 2 pennies by the number of ways to select 2 dimes:
6 * 28 = 168
Finally, we can calculate the probability of drawing 2 pennies and 2 dimes in any order by dividing the number of ways to select 2 pennies and 2 dimes by the total number of ways to select 4 coins without replacement:
168/2380 = 0.0706 or approximately 7.06%
Therefore, the probability of drawing 2 pennies and 2 dimes in any order is 0.0706 or approximately 7.06%.

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To express the integrand as a rational function, we can make the substitution u = 1 + x^2. Then, du/dx = 2x, so dx = du/(2x).

You are asked to evaluate the integral of a function that can be expressed as a rational function using substitution. The given integrand is:

∫((Tc-4)^(-1/3) dTc)

Step 1: Make a substitution to express the integrand as a rational function
Let's perform a substitution:
u = Tc - 4

Then, differentiate both sides with respect to Tc:
du/dTc = d(Tc - 4)/dTc = 1

Now, solve for dTc:
dTc = du

Now, substitute the expressions for u and dTc into the original integrand:
∫(u^(-1/3) du)

Step 2: Evaluate the integral
Now, integrate the new expression:
∫(u^(-1/3) du) = (3/2)u^(2/3) + C

Step 3: Substitute back to the original variable, Tc
Now, substitute back the original variable, Tc, using u = Tc - 4:
(3/2)(Tc - 4)^(2/3) + C

Therefore, the integral of the given expression is:
(3/2)(Tc - 4)^(2/3) + C

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The correct answer is D. To solve the division problem 42 ÷ 7, we need to find how many times 7 can go into 42.

This is the same as asking "what is the quotient of 42 and 7?" So, the number sentence that can be used to find 42 ÷ 7 = ? is D. which is 7 ÷ 42 = _. To solve for the quotient, we divide 42 by 7, which gives us the answer of 6. So, 42 ÷ 7 = 6.

Answer A, 42 x _ = 7, is not a valid number sentence to find the answer to 42 ÷ 7, because we cannot multiply any number by 42 to get 7.

Answer B, 7 x _ 42 = 42, is also not a valid number sentence to find the answer to 42 ÷ 7, because this equation is equivalent to 7 = 1, which is not true.

Answer C, 42 - 7 = _, is not a valid number sentence to find the answer to 42 ÷ 7, because subtracting 7 from 42 does not give us the quotient of 42 and 7.

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The indefinite integral is: (2x + 5)/2 - (5/2) * ln|2x + 5| + C

To evaluate the indefinite integral, ∫(2x)/(2x+5) dx, we can use a change of variables, also known as substitution. Let's set:

u = 2x + 5

Now, differentiate u with respect to x:

du/dx = 2

So, dx = du/2

Substitute u and dx in the original integral:

∫(2x)/(u) * (du/2) = ∫(u - 5)/(u) * (du/2)

Now, split the fraction:

∫(u/u - 5/u) * (du/2) = ∫(1 - 5/u) * (du/2)

Now, integrate with respect to u:

(1/2) * ∫(1 - 5/u) du = (1/2) * (u - 5 * ln|u|) + C

Now, substitute back the original variable, x:

(1/2) * ((2x + 5) - 5 * ln|2x + 5|) + C

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The correction factor that should be used to compute the standard error is 2.326. When constructing a confidence interval for the mean, we use the formula:

CI = X ± tα/2 (SE)

Where X is the sample mean, tα/2 is the critical value from the t-distribution with (n-1) degrees of freedom, and SE is the standard error of the mean. Since the population size is finite and the sample size is less than 10% of the population size, we need to use a correction factor to adjust for the bias in the standard error calculation. The correction factor is calculated as:

CF = sqrt((N-n)/(N-1))

Where N is the population size and n is the sample size. In this case, the correction factor is:

CF = sqrt((1000-100)/(1000-1)) = 0.993

Therefore, the corrected standard error is:

SE* = SE * CF = SE * 0.993

To find the critical value, we need to look up the t-score in a t-table with (n-1) degrees of freedom and a confidence level of 98%. This gives us a critical value of 2.326. Therefore, the final formula for the confidence interval is:

CI = X ± 2.326(SE*)

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We have shown that if a, b, c, and m are integers such that m ≥ 2, c > 0, and a ≡ b (mod m), then ac ≡ bc (mod mc) holds true.

To show that if a, b, c, and m are integers such that m ≥ 2, c > 0, and a ≡ b (mod m), then ac ≡ bc (mod mc), we can use the definition of congruence modulo m.

According to the definition of congruence modulo m, we have:

a ≡ b (mod m) --> a - b is divisible by m, or m divides (a - b).

Since a ≡ b (mod m), we can write a - b = km for some integer k.

Now, we want to show that ac ≡ bc (mod mc), which means that ac - bc is divisible by mc, or mc divides (ac - bc).

We can start by expanding ac and bc:

ac = (a - b + b)c = (km + b)c,

bc = (km + b)c,

where we have used the fact that a - b = km.

Now, we can subtract bc from ac:

ac - bc = (km + b)c - (km + b)c,

ac - bc = (km + b - km - b)c,

ac - bc = 0,

which means that ac ≡ bc (mod mc), since mc divides (ac - bc).

Therefore, we have shown that if a, b, c, and m are integers such that m ≥ 2, c > 0, and a ≡ b (mod m), then ac ≡ bc (mod mc) holds true.

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The exponential function P(x) undergoes a transformation of -I(x). The correct options are A, D, E, and G

Since a graph of two functions is shown P(x) and I(x) from observation each of the graphs is an opposite replica of the other.
When,
⇒-I(x)⇒P(x)

Following the above trasformation, all the properties will also get changed in an opposite manner such as intercept, slope, and so on.

Thus, the exponential function P(x) undergoes a transformation of -I(x). Option D is correct.

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Answer:

6x - 4 = 8x + 12

Step-by-step explanation:

Translate a small part of the sentence into an expression until you translate the entire sentence.

"Four less than six times a number is the same as eight times the number, x, increased by 12."

6x

"Four less than six times a number is the same as eight times the number, x, increased by 12."

6x - 4

"Four less than six times a number is the same as eight times the number, x, increased by 12."

6x - 4 =

"Four less than six times a number is the same as eight times the number, x, increased by 12."

6x - 4 = 8x

"Four less than six times a number is the same as eight times the number, x, increased by 12."

6x - 4 = 8x + 12

Answer: 343p

Step-by-step explanation: did the test

The 9th derivative of f(x) evaluated at x=0 is approximately 0.4394.

We start by writing the Maclaurin series for the cosine function:

[tex]cos(x) = Σ (-1)^n * x^(2n) / (2n)![/tex]

We can then rewrite the given function as:

[tex]f(x) = cos^2(x) - 1\\f(x) = [Σ (-1)^n * x^(2n) / (2n)!]^2 - 1[/tex]

Expanding the square and simplifying, we get:

[tex]f(x) = Σ (-1)^n * x^(4n) / [(2n)!]^2 - 1[/tex]

To find the 9th derivative of f(x) evaluated at x=0, we need to differentiate the function 9 times with respect to x. Each differentiation will reduce the power of x by 4, and we will be left with a term of the form [tex]x^0 = 1[/tex] when we evaluate the function at x=0. The terms with negative powers of x will disappear.

The 9th derivative of f(x) is:

[tex]f^(9)(x) = Σ (-1)^n * (4n)! / [(2n)!]^2 * x^(4n-36)[/tex]

Evaluating this expression at x=0, we get:

[tex]f^(9)(0) = (-1)^9 * (49)! / [(29)!]^2\\f^(9)(0) = 362880 / (2^18)\\f^(9)(0) = 0.4394...[/tex]

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In each case, the dimension of the subspace of P_3 spanned by the given sets can be found by determining the linear independence of the vectors within each set.

1. S = span{x, x - 2, x^2 + 2}
The given vectors are linearly independent, as there is no scalar multiple or linear combination of the first two vectors that can result in the third vector. Therefore, dim(S) = 3.

2. S = span{x, x - 2, x^2 + 2, x^2 - 2}
The first three vectors are linearly independent, as previously determined. The fourth vector, x^2 - 2, can be obtained by subtracting the second vector (x - 2) from the third vector (x^2 + 2), making the fourth vector linearly dependent on the other vectors. Thus, dim(S) = 3.

3. S = span{x^2, x^2 - x - 2, x + 2}
These vectors are linearly independent, as there is no scalar multiple or linear combination of any two vectors that can result in the third vector. Therefore, dim(S) = 3.

4. S = span{3x, x - 3}
The given vectors are linearly independent since neither vector is a scalar multiple of the other. Therefore, dim(S) = 2.

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The series we are considering is: Σ (from n=1 to infinity) [1/(9 + e^(-n))] is divergent.

This is an infinite series of positive terms, so we can apply the Comparison Test. To use the Comparison Test, we need to find a series that we know is convergent or divergent and that dominates our given series term by term. In this case, we can compare it to the series:

Σ (from n=1 to infinity) [1/e^n]

This is a geometric series with a common ratio of 1/e (which is less than 1), and thus, it converges. Since e^(-n) is always positive, 9 + e^(-n) > 9, and therefore:

1/(9 + e^(-n)) < 1/9 * 1/e^n

Since the series on the right is convergent and our given series is smaller term by term, by the Comparison Test, our given series is also convergent.

To find the sum of the given series, we can't directly use the geometric series formula because of the added 9 in the denominator. However, we can rewrite the series as a sum of two separate series:

Σ (from n=1 to infinity) [1/9] + Σ (from n=1 to infinity) [1/e^n]

The first series is a geometric series with a common ratio of 1, and thus, it diverges. The second series is convergent, as previously mentioned, but since one of the series diverges, the sum of the two series also diverges.

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Complete question:

Determine whether the series is convergent or divergent. Sigma_n=1^infinity 1/9 + e^-n convergent divergent If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.)

No, growing up south of Denver and enjoying country music are related in some way.

To apply this concept to the given data, let's define two events:

A: "Growing up south of Denver"

B: "Enjoying country music"

We can calculate the probability of event A happening by dividing the number of high school seniors who grew up south of Denver by the total number of high school seniors in the study:

P(A) = 570/2000 = 0.285

Similarly, we can calculate the probability of event B happening by dividing the number of high school seniors who enjoy country music by the total number of high school seniors in the study:

P(B) = 1120/2000 = 0.56

Now, if events A and B are independent, then the probability of both events happening together should be equal to the product of the individual probabilities:

P(A and B) = P(A) x P(B)

We can calculate the probability of high school seniors growing up south of Denver AND enjoying country music by dividing the number of high school seniors who meet both criteria (386) by the total number of high school seniors in the study:

P(A and B) = 386/2000 = 0.193

So, if events A and B are independent, we should have:

P(A and B) = P(A) x P(B)

0.193 = 0.285 x 0.56

0.193 = 0.1596

0.193 is not equal to 0.1596, we can conclude that events A and B are not independent.

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We know the least possible value of a positive integer when given the greatest common divisor (GCD) and the least common multiple (LCM) of two numbers. Let's represent the two numbers as a and b, and the GCD and LCM as gcd(a, b) and lcm(a, b), respectively.

1. First, recall that the greatest common divisor (GCD) of two positive integers is the largest number that divides both of them without leaving a remainder.

2. Next, remember that the least common multiple (LCM) of two positive integers is the smallest multiple that both numbers evenly divide into.

3. There's a relationship between the GCD and LCM of two numbers, expressed as: a * b = gcd(a, b) * lcm(a, b).

Now, to find the least possible value of one of the integers (let's say "a"), we need to consider a scenario where gcd(a, b) = 1, because when the GCD is 1, the numbers are relatively prime and don't share any common factors other than 1. In this case:

a * b = lcm(a, b)

Since we want to minimize the value of "a", we can set a = 1. Therefore, in this case:

1 * b = lcm(1, b)

So, the least possible value of a positive integer in this context is 1.

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Toni invested in pounds= $1350

rate of compound interest=4%

Here, we have,

Toni invests money into an account which pays a fixed rate of compound interest each year. The total value, £V, of her investment after t years is given by the formula:  

V = 1350 x 1.04^t

Answer questions a & b

a - How much money did Toni invest in pounds

b - What rate of compound interest is paid each year

the formula for calculating compound interest is given by :

V = P (1 + r)^t

P = Present value  -= amount invested = 1350

R = interest rate  = 4%

N = number of years  = t

we have,

V = 1350 x 1.04^t

P= 1350 is the initial amount invested

1+r= 1.04  (subtract 1 from both sides)

r = 0.04

multiply it by 100 to get percentage

r= 4%

Hence, Toni invested in pounds= $1350

rate of compound interest=4%

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The arrangement of the number from least to greatest is: [tex]-0.35, -0.7, 1/5, 0.95[/tex].

We will start by identifying the smallest number which is -0.35. Next, we move on to the next smallest number which is -7/10.

It is important to note that negative numbers come before positive numbers when ordering from least to greatest, so, the next number in the set is 1/5, which is larger than -7/10 but smaller than 0.95

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The side of the square is the same length as the edge of the cube, which is "a," the area of the AA1C1C square is a².

To find the area of the AA1C1C square, we first need to understand the geometry of a cube. A cube is a three-dimensional shape with six faces that are all congruent squares. Each of the edges of the cube has the same length, which we are given as "a."

To find the area of the AA1C1C square, we need to know the length of one of its sides. Since the edge AA1 and the side of the square that it shares are the same length, we can use "a" as the length of the side of the square. Therefore, the area of the AA1C1C square can be found by squaring the length of one of its sides. In this case, the length of the side is "a," so we can write:

Area of AA1C1C square = (length of side)² = a²

So, the area of the AA1C1C square on a cube with an edge length of "a" is a².

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The correct answer is A. Animal is a class with a constructor that takes one parameter of the String type, Mammal is its subclass that has no constructors defined.

In Java, when you create an object using the "new" keyword, the constructor of that object's class is called. In the given statement, we are creating an object of the Animal class and initializing it with a Mammal object that has the name "Elephant".

Option A is the correct answer because it states that the Animal class has a constructor that takes a String parameter and Mammal is its subclass that has no constructors defined. Therefore, the Animal class constructor will be called and it will accept the "Elephant" parameter without any errors.

Option B is incorrect because it states that Mammal is a class with a constructor that takes a String parameter, and Animal is its subclass. This is not possible as a subclass cannot have a constructor that its superclass does not have.

Option C is incorrect because it states that an Animal is an interface, and Mammal implements Animal. An interface does not have a constructor, so this option is not possible.

Option D is incorrect because it states that Animal has a public static data field String Mammal. This does not make sense as a data field is not the same as a constructor.

In summary, the statement Animal a = new Mammal("Elephant") will compile with no errors if the Animal class has a constructor that takes a String parameter, and Mammal is its subclass that has no constructors defined. Therefore the correct option is A

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The hexadecimal instruction 2888006416 is associated with the MIPS32 command "d. sub $v0, $t9, $t8".

In MIPS32 assembly language, "sub" is a command used for subtraction. In this particular instruction, the command is subtracting the value stored in register $t8 from the value stored in register $t9 and storing the result in register $v0.

It's important to note that hexadecimal instructions are machine code instructions that are represented in hexadecimal format for ease of reading. They are not typically used by programmers directly. Instead, programmers write code in assembly language and then use an assembler to translate it into machine code.

In summary, the MIPS32 command associated with the hexadecimal instruction 2888006416 is "sub $v0, $t9, $t8", which subtracts the value stored in register $t8 from the value stored in register $t9 and stores the result in register $v0.

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The particular solution is: y(x) = -9e^(-0.6x) To solve the separable differential equation dy/dx = -0.6y, we need to separate the variables by dividing both sides by y and dx:

1/y dy = -0.6 dx

Then, we integrate both sides:

ln|y| = -0.6x + C

where C is the constant of integration. To find the particular solution satisfying the initial condition y(0) = -9, we substitute x = 0 and y = -9 into the above equation:

ln|-9| = -0.6(0) + C

ln 9 = C

Therefore, the particular solution is:

ln|y| = -0.6x + ln 9

|y| = e^(ln 9) e^(-0.6x)

|y| = 9 e^(-0.6x)

Since y(0) = -9, we take the negative sign for the absolute value of y:

y(x) = -9 e^(-0.6x)

Therefore, the particular solution satisfying the initial condition y(0) = -9 is y(x) = -9 e^(-0.6x).

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The mathematics teacher should use the data from the clicker quiz to identify the areas where her students have misconceptions or errors in their understanding of mathematical concepts.

She can then adjust her lesson plans to focus on these areas and provide additional instruction or resources to help her students improve their understanding. By analyzing the data and using it to inform her teaching strategies, the teacher can help her students develop better mathematical proficiency and achieve better results on future assessments.

Based on the given scenario, if the mathematics teacher's goal is to improve her students' mathematical proficiency, her best use of the data from the two-question clicker quiz would be to:

1. Identify areas of misunderstanding and error patterns: By analyzing the students' responses, the teacher can pinpoint specific concepts or problem-solving strategies that are causing difficulties.

2. Tailor instruction accordingly: Once the teacher has identified areas of weakness, she can adapt her lessons and teaching methods to address these issues more effectively, ensuring that students receive targeted support to improve their understanding.

3. Monitor progress over time: Regularly collecting and analyzing data from the quizzes allows the teacher to track the progress of her students and determine if her instructional adjustments are resulting in improved mathematical proficiency.

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The mathematics teacher can use the data from the clicker quiz to identify the areas where her students are struggling the most and focus her teaching on those topics.

She can also use the data to provide individualized feedback to each student, addressing their specific misconceptions and errors. By analyzing the patterns in the data, the teacher can modify her teaching strategies and methods to better suit the learning needs of her students.

In short, the data from the clicker quiz can be used to inform and improve the teacher's instruction and enhance her students' mathematical proficiency.

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Answer:

Ordered pairs are equal if and only if their corresponding elements are equal. That is, (a, b) and (c, d) are equal if and only if a = c and b = d.

Examples:

(2, 4) and (2, 4) are equal ordered pairs since they have the same first and second elements.

(0, -3) and (0, -3) are equal ordered pairs since they have the same first and second elements.

(1, 8) and (1, 8) are equal ordered pairs since they have the same first and second elements.

The area of the surface obtained by rotating the curve y = 5 x + 1, 0 ≤ x ≤ 10 about the x-axis is  2 * π * (260 * sqrt(26)).

To find the area of the surface obtained by rotating the curve y = 5x + 1, 0 ≤ x ≤ 10 about the x-axis, we can use the Surface of Revolution formula, which is given by:

Area = 2 * π * ∫[y * sqrt(1 + (dy/dx)^2)] dx, from a to b

where y is the function y = 5x + 1, dy/dx is its derivative, and a and b are the limits of integration (0 and 10, respectively).

Step 1: Find the derivative of y = 5x + 1 with respect to x:
dy/dx = 5 (since the derivative of 5x is 5 and the derivative of 1 is 0)

Step 2: Calculate 1 + (dy/dx)^2:
1 + (5)^2 = 1 + 25 = 26

Step 3: Calculate y * sqrt(1 + (dy/dx)^2):
(5x + 1) * sqrt(26)

Step 4: Set up the integral with limits of integration from 0 to 10:
Area = 2 * π * ∫[(5x + 1) * sqrt(26)] dx, from 0 to 10

Step 5: Integrate the function with respect to x:
Area = 2 * π * [((5x^2)/2 + x) * sqrt(26)] evaluated from 0 to 10

Step 6: Evaluate the integral at the limits of integration:
Area = 2 * π * [((5(10)^2)/2 + 10) * sqrt(26) - ((5(0)^2)/2 + 0) * sqrt(26)]

Step 7: Simplify and calculate the area:
Area = 2 * π * [((250) + 10) * sqrt(26)] = 2 * π * (260 * sqrt(26))

Thus, the area of the surface obtained by rotating the curve y = 5x + 1, 0 ≤ x ≤ 10 about the x-axis is 2 * π * (260 * sqrt(26)).

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If a car uses 1.5 gallons of gas every 30 miles, it means that it uses 0.05 gallons of gas per mile. Therefore, if you have 6 gallons of gas, you can drive for:

6 gallons / 0.05 gallons per mile = 120 miles
So you can drive for 120 miles with 6 gallons of gas.
To find out how many miles can be driven with 6 gallons of gas, you can use the given ratio of 1.5 gallons for every 30 miles.

First, find out how many times 1.5 gallons fits into 6 gallons:
6 gallons / 1.5 gallons = 4
Now, multiply this number (4) by the 30 miles per 1.5 gallons:
4 * 30 miles = 120 miles

So, with 6 gallons of gas, you can drive 120 miles.

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(a) Lower quartile of the discounts is £15.75 and the upper quartile of the discounts is £45.75.

(b) The interquartile range of the discounts is £30.

a) To find the upper and lower quartiles of the discounts, we first need to arrange the discounts in order from lowest to highest. Since each item can be discounted by up to £60, the possible discounts are between £0 and £60.

Assuming the discounts are evenly distributed between £0 and £60, we can use the formula for finding quartiles:

Lower Quartile (Q1) = (n + 1)/4-th term

Upper Quartile (Q3) = 3(n + 1)/4-th term

where n is the number of data points, which in this case is 80.

Lower Quartile (Q1):

Q1 = (n + 1)/4-th term

Q1 = (80 + 1)/4-th term

Q1 = 20.25-th term

Since we can't have a fractional term, we round up to the 21st term.

The 21st term in the ordered list of discounts would be:

21st term = (21/80) x £60

21st term = £15.75

So the lower quartile of the discounts is £15.75.

Upper Quartile (Q3):

Q3 = 3(n + 1)/4-th term

Q3 = 3(80 + 1)/4-th term

Q3 = 60.75-th term

Again, we round up to the 61st term.

The 61st term in the ordered list of discounts would be:

61st term = (61/80) x £60

61st term = £45.75

So the upper quartile of the discounts is £45.75.

b) The interquartile range (IQR) is the difference between the upper and lower quartiles:

IQR = Q3 - Q1

IQR = £45.75 - £15.75

IQR = £30

So the interquartile range of the discounts is £30.

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Answer:

[tex]m = \frac{ - 6 - ( - 4)}{ - 2 - ( - 5)} = - \frac{2}{3} [/tex]

The basis of U is  {v1, v2}.  {v1, v2, v3, v4, v5} is a basis of R5 by verifying that these vectors are linearly independent and span R5. A basis of [tex]U^\perp[/tex] is {w1, w2}, where w1 = (-3,1,0,0,0) and w2 = (0,0,-7,1,0)

(a) To find a basis of U, we need to find linearly independent vectors that span U. We can start by setting x2 = 1 and x4 = 1 and solving for the other variables. This gives us two vectors in U:

v1 = (3,1,7,0,0)

v2 = (0,0,0,1,0)

We can check that these vectors are linearly independent by setting [tex]a1v1 + a2v2 = 0[/tex] and solving for a1 and a2. This gives us a1 = a2 = 0, so the vectors are linearly independent. Therefore, {v1, v2} is a basis of U.

(b) To extend the basis {v1, v2} of U to a basis of R5, we need to find three more linearly independent vectors that are not in U. We can choose:

v3 = (1,0,0,0,0)

v4 = (0,1,0,0,0)

v5 = (0,0,1,0,0)

We can check that {v1, v2, v3, v4, v5} is a basis of R5 by verifying that these vectors are linearly independent and span R5.

(c) To find a subspace W of R5 such that R5 = U direct sum W, we can choose W to be the orthogonal complement of U. We can find a basis of W by finding a basis of [tex]U^\perp[/tex], where

[tex]U^\perp = {(x1,x2,x3,x4,x5)[/tex] belongs to R5: x1 = -3x2, x3 = -7x4

A basis of [tex]U^\perp[/tex] is {w1, w2}, where

w1 = (-3,1,0,0,0)

w2 = (0,0,-7,1,0)

We can verify that U and W are orthogonal complements by checking that any vector in R5 can be written as a unique sum of a vector in U and a vector in W, and that U and W are orthogonal (i.e., the dot product of any vector in U with any vector in W is zero).

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The domain and the range of the function are given as follows:

Hence the domain and the range of the function are given as follows:

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