Help please it would mean a lot

The value of  [tex]\sum _{n=1}^{400}\:\left(2n-1\right)[/tex] is 16000.

Sequence is an enumerated collection of objects in which repetitions are allowed and order matters.

Given equation is  [tex]\sum _{n=1}^{400}\:\left(2n-1\right)[/tex]

[tex]\sum \:a_n+b_n=\sum \:a_n+\sum \:b_n[/tex]

By Applying the sum rule

[tex]=\sum \:_{n=1}^{400}2n-\sum \:_{n=1}^{400}1[/tex]

=2×2(400+1)/2 -400

=2×200×401−400

=16000

Hence, the value of  [tex]\sum _{n=1}^{400}\:\left(2n-1\right)[/tex] is 16000.

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The vector that is perpendicular to the tangent vector of the curve at (0, 1, 0) is <1, 0, -1>, or option c

The space curve given is r(t) = sin t i + e^t j + ln(t+1) k. To find the tangent vector of the curve at a given point, we need to take the derivative of the curve with respect to t. The derivative of the curve is r'(t) = cos t i + e^t j + (1/(t+1)) k.

At the point (0, 1, 0), t = 0. So, the tangent vector at this point is r'(0) = cos 0 i + e^0 j + (1/(0+1)) k = <1, 1, 1>.

Now, we need to find which of the given vectors is perpendicular to the tangent vector. Two vectors are perpendicular if their dot product is equal to 0. So, we need to find the dot product of the tangent vector with each of the given vectors and see which one is equal to 0.

a. <1, 1, 1> • <0, 1, 0> = 0 + 1 + 0 = 1 (not perpendicular)
b. <1, 1, 1> • <-1, 0, -1> = -1 + 0 - 1 = -2 (not perpendicular)
c. <1, 1, 1> • <1, 0, -1> = 1 + 0 - 1 = 0 (perpendicular)
d. <1, 1, 1> • <1, -1, 1> = 1 - 1 + 1 = 1 (not perpendicular)
e. <1, 1, 1> • <1, 0, 1> = 1 + 0 + 1 = 2 (not perpendicular)

Therefore, the vector that is perpendicular to the tangent vector of the curve at (0, 1, 0) is <1, 0, -1>, or option c.

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The minimum value of the objective function is 16 and it occurs at x = 1 and y = 5.

The given objective function is 9x+3y. The given constraints are y≥−2x+11, y≤−x+10, y≤−3, x+6y≥−4, and x+4.

To minimize this objective function, we need to set up the Lagrangian function:

L(x,y,λ1,λ2,λ3) = 9x+3y - λ1(y+2x-11) - λ2(y-x+10) - λ3(y+3) - λ4(x+6y-4) - λ5(x+4)

We can then find the critical points of this function by taking the partial derivatives of the Lagrangian with respect to x, y, λ1, λ2, and λ3, and setting them equal to zero:

∂L/∂x = 9 - λ2 + 6λ4 = 0
∂L/∂y = 3 - λ1 - λ2 - λ3 - 6λ4 = 0
∂L/∂λ1 = -(y+2x-11) = 0
∂L/∂λ2 = -(y-x+10) = 0
∂L/∂λ3 = -(y+3) = 0
∂L/∂λ4 = -(x+6y-4) = 0
∂L/∂λ5 = -(x+4) = 0

Solving these equations, we get the solution x = 1, y = 5, λ1 = 4, λ2 = -3, λ3 = 8, λ4 = -3, and λ5 = -1.

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Answer:

Step-by-step explanation:

The value of [tex]x[/tex] that satisfies the equation is [tex]x = 6[/tex].

How can we solve an equation with a fraction and a quadratic expression?

To solve an equation with a fraction and a quadratic expression, we can first try to simplify the fractions by finding a common denominator. Then, we can move all the terms to one side of the equation and simplify it into a quadratic equation. Finally, we can solve the quadratic equation using factoring or the quadratic formula.

Find the value of [tex]x[/tex]:

The given equation is:

[tex]\frac{50}{4x-12}+\frac{x-4}{x^2+x-12}=\frac{x}{2}[/tex]

The first step is to find the LCM of the denominators. Factoring the denominator [tex]x^2+x-12[/tex], we get:

[tex]x^2+x-12=(x+4)(x-3)[/tex]

So the LCM of the denominators is [tex](4x-12)(x+4)(x-3)[/tex]. Multiplying both sides of the equation by this LCM, we get:

[tex]50(x+4)(x-3)+(x-4)(4x-12)=\frac{x}{2}(4x-12)(x+4)(x-3)[/tex]

Expanding and simplifying both sides, we get:

[tex]6x^3-26x^2-13x+56=0[/tex]

Factoring out [tex](x-4)[/tex], we get:

[tex](x-4)(6x^2-10x-14)=0[/tex]

Solving for [tex]x[/tex] using the quadratic formula, we get:

[tex]x=\frac{5\pm\sqrt{29}}{3} \text{ or } x=4[/tex]

However, we need to check if any of these solutions make the denominator zero, which would be undefined. Checking each of the possible solutions, we find that [tex]x=4[/tex] is the only solution that does not make any of the denominators zero.

Therefore, the solution is [tex]x=4[/tex].

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The scale factor of the preimage to image is 3. Then the correct option is C.

Dilation is the process of increasing the size of an item without affecting its form. Depending on the scale factor, the object's size can be raised or lowered. There is no effect of dilation on the angle.

The picture rectangle has points on coordinates (0, 0), (0, 8), (9, 8), and (9, 0). The pre-image has points (0, 0), (0, 2.5), (3, 2.5), and (3, 0).

The scale factor is given as,

SF = 9 / 3

SF = 3

The scale factor of the preimage to image is 3. Then the correct option is C.

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The complete question is given below.

What is the scale factor in the dilation?

On a coordinate plane, the image rectangle has points (0, 0), (0, 8), (9, 8), and (8, 0). The pre-image has points (0, 0), (0, 2.5), (3, 2.5), and (3, 0).

a) One-sixth

b) One-third

c) 3

d) 6

Answer:

Step-by-step explanation: C  !!!!!! / 3

Edge 2023

1/6

1/3

3   <--------------- correct

6

This as \( 5.4 \times 10^{-10} \) meters, or \( 5.4 \times 10^{-9} \) centimeters

The diameter of a certain atom is about \( 0.54 \) nanometers, which can be rewritten in scientific notation as \( 5.4 \times 10^{-1} \) nanometers. This is because scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. In this case, we move the decimal point one place to the left and multiply by \( 10^{1} \) to get the number in scientific notation. Since nano- means "billionth," we can also write this as \( 5.4 \times 10^{-10} \) meters, or \( 5.4 \times 10^{-9} \) centimeters.

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The ordered pair that is a member of both equations is (-27/7, -11/7).

To find the ordered pair that is a member of both equations, we need to solve the system of equations. We can do this by using the elimination method.

First, we will multiply the first equation by 2 and the second equation by -4 to eliminate the x variable:

-4x - 8y = 28
-20y - 8x = 16

Next, we will add the two equations together:

-28y = 44

Now we can solve for y:

y = -44/28
y = -11/7

Now we can plug this value of y back into one of the original equations to solve for x:

-2x - 4(-11/7) = 14
-2x + 44/7 = 14
-2x = 14 - 44/7
-2x = 54/7
x = -27/7

Therefore, the ordered pair that is a member of both equations is (-27/7, -11/7).

Answer: The ordered pair that is a member of both equations is (-27/7, -11/7).

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Option B, On a coordinate plane, the locations of the points (9, -5) and (9, 5) related reflection across the x-axis.

The two locations (9, -5) and (9, 5) have the same x-coordinate, but their y-coordinates are different. We need to see how the coordinates of two points change as they undergo various transformations to understand their relationship.

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The required cost of the cans of weedkiller needed to cover lawn are  £ 47.4.

The formula A = 12 (a Plus b) h is used to calculate the area of a trapezoid, where a and b are the bases (parallel sides) and h is the height (the angle between the bases).

We have;

Dimensions of Trapezoid; a = 20, b = 12, Height = h

So;

[tex]$Area = \frac{(a+b)h}{2}$[/tex]

For h, Using Pythagoras theorem;

17² = 8² + h²

h = 15 m

Then,

[tex]$Area = \frac{(12+20)15}{2}$[/tex]

Area = 240 Sq meter

Then,

100  Sq meter = £19.75

For 240  Sq meter

= £19.75(2.4)

= £ 47.4

Thus, required cost is  £ 47.4.

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Answer:

To simplify the expression, we can combine the like terms.

(3x + 1/2) + (7x - 4 1/2)

= 3x + 7x + 1/2 - 4 1/2 (distributing the addition)

= 10x - 4 (combining the like terms)

Therefore, (3x + 1/2) + (7x - 4 1/2) simplifies to 10x - 4.

The first ship sails on a bearing of 83° at 30 knots while the second on a bearing of 173° at 20 knots. After two hours, the distance between the ships is 72 NM

The two ships are travelling on different bearings and at different speeds. To calculate the distance between them after two hours, we need to use the formula:

Distance = Speed x Time

The distance the first ship has travelled after two hours:

Distance 1st ship = 30 knots x 2 hours = 60 nautical miles (NM)

The distance the second ship has travelled after two hours:

Distance 2nd ship = 20 knots x 2 hours = 40 nautical miles (NM)

The angle difference = 173° -  83°  = 90°

Since the port, the first ship, and the second ship form a right triangle, we can find the total distance between the two ships using the Pythagorean Theorem.

Distance² = (60 NM)² + (40 NM)²

Distance = √ (3600 + 1600) = √ 5200 = 72 NM


Therefore, after two hours, the two ships will be 72 NM apart.

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The probability that the person chosen at random is a boy is 8/13 and the probability that they are between 120 cm and 130 cm tall is 6/10.

Polygon is a closed figure that consists of three or more straight lines. It is a two-dimensional shape with straight sides and angles. Polygons are used to construct many shapes in geometry such as triangles, rectangles, squares and many more. Polygons are used in many fields such as computer graphics, navigation, engineering and architecture.

The frequency polygon show that there are more boys than girls in the class. This can be seen from the higher peak on the boys side of the graph. The total number of boys is 8 and the total number of girls is 5. This gives us a probability of 8/13 that the person chosen at random is a boy.

To work out the probability that the person is between 120 cm and 130 cm tall, we must first count the number of individuals in the class in this height range. From the graph, we can see that there are 6 boys and 4 girls in this range. This gives us a probability of 6/10 that the person chosen at random is between 120 cm and 130 cm tall.

In conclusion, the probability that the person chosen at random is a boy is 8/13 and the probability that they are between 120 cm and 130 cm tall is 6/10.

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Step-by-step explanation:

a) To form a committee of 8 with 3 prefects, we need to choose 3 prefects from the 5 available prefects, and 5 non-prefects from the remaining 7 boys. We can do this by using the combination formula:

Number of ways = (Number of ways to choose 3 prefects) × (Number of ways to choose 5 non-prefects)

Number of ways to choose 3 prefects from 5 = C(5, 3) = 10

Number of ways to choose 5 non-prefects from 7 = C(7, 5) = 21

Therefore, the total number of committees of 8 with 3 prefects that can be formed is:

Number of ways = 10 × 21 = 210

b) To form a committee of 8 with at least 3 prefects, we need to consider two cases: one where we choose exactly 3 prefects, and one where we choose all 5 prefects. We can calculate the number of ways for each case using the combination formula:

Number of ways to choose exactly 3 prefects and 5 non-prefects = C(5, 3) × C(7, 5) = 210

Number of ways to choose all 5 prefects and 3 non-prefects = C(5, 5) × C(7, 3) = 35

Therefore, the total number of committees of 8 with at least 3 prefects that can be formed is:

Number of ways = (Number of ways to choose exactly 3 prefects and 5 non-prefects) + (Number of ways to choose all 5 prefects and 3 non-prefects)

Number of ways = 210 + 35 = 245

The equation of exponential function=y = 0.054 × [tex]1.028^{x}[/tex]

As the name implies, exponents are utilised in exponential functions. But remember that a variable serves as the exponent instead of a constant as the basis of an exponential function.

A function is a power function rather than an exponential function if its base is a variable and its exponent is a constant.

If the graph of exponential function passes through the points (1960,19.005) and (2010,19.548), then the coordinates of these points satisfy the equation, y = [tex]ab^{x}[/tex]

Now, 19.005 = a × [tex]b^{1960}[/tex]

⇒  [tex]b^{1960}[/tex] = 19.005/a

Similarly, 19.548 = a × [tex]b^{2010}[/tex]

⇒ [tex]b^{2010}[/tex] = 19.548/a

⇒  [tex]b^{1960}[/tex] × [tex]b^{50}[/tex] = 19.548/a

⇒ [tex]b^{50}[/tex] = 19.548/a × a/19.005

         = 1.028

Putting this value to find a,

we get a = 0.054

So, the equation of exponential function=

y = 0.054 × [tex]1.028^{x}[/tex]

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Answer:

This is an odd function

Step-by-step explanation:

Given a function f(x),

if f(-x) = f(x) then the f(x) is even
If f(-x) = -f(x0 then the function is odd

If neither of the above is true then the function is neither odd nor even

We are given
[tex]f(x) = 4x^3 - x^5 + 5x\\f(-x) = 4(-x)^3 -(x)^5 + 5(-x)\\\\(-x)^3 = - x^3\\(-x)^5 = -x^5\\\\\\\therefore\\f(-x) = 4(-x^3) - (-x^5) + 5 (-x)\\\\= -4x^3 + x^5 - 5x\\\\= -(4x^3 -x^5 + 5x)\\\\= -f(x)[/tex]

So the function is odd

Answer:

See image attached for complete answer

Step-by-step explanation:

The attached image shows the calculation of probabilities for each branch

I believe it is self-explanatory

Since P(bus to shops) = 0.4, P(walk to shops ) = 1- 0.4 = 0.6

Since P(bus from shops) = 0.7, P(walk from shops) = 1- 0.7 = 0.3

The probabilities at the leaves of the tree(rightmost entries) show the combined probabilities of to and from

For example if she takes a bus to the shops and takes a bus from the shops, the combined probability = 0.4 x 0.7 = 0.28

If she walks to the shops and takes a bus from the shops the combined probability = 0.6 x 0.7 = 0.42

The other two probabilities can be computed the same way

The probability that she walks at least one way
= P(walking to shop) or P(walking from shop)

= sum of the probabilities in the yellow shaded boxes

= 0.12 + 0.42 + 0.18

= 0.72

The area enclosed is 3 unit²

the line y=0 simply denotes the x axis. Now, the straight line y= (1/2)x+1 can be written as (x/(-2))+(y/1) = 1. So, this line touches x axis at (-2,0) and y axis at (0,1). the other line y= -(1/4)x+1 can be written as (x/4)+(y/1)=1. So, this line touches y axis at (4,0) and y axis at (0,1). so, it is evident that the triangle that got forms have the vertices of (4,0), (-2,0) and (0,1), the line joining (4,0) and (-2,0) being the base of the triangle.

so, the base of the triangle is 4+2= 6 unit and height of the triangle is 1 unit.

area of the triangle is 1/2* base* height= 1/2*6*1 = 3 unit^2.

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The result obtained  when x^(4)-6x^(2)-x-19 is divided by x-3  using long division method is    x3 - 18x2 - 9x - 57 + 19/x-3.

The question is, "Use the long division method to find the result when x^(4)-6x^(2)-x-19 is divided by x-3".

To solve this problem using long division, we need to set up the division like this:

    x4 - 6x2 - x - 19

    ___________________      (x-3)

                      0

                           -3x3 - 18x2 - 9x - 57

Now, divide the first term of the numerator by the first term of the denominator, x4/x = x3, and write it above the division. Next, multiply the result x3 by the denominator (x-3) to get -3x3.

Now subtract -3x3 from the numerator (x4 - 6x2 - x - 19) to get -6x2 - x - 19. Divide this by the denominator (x-3) to get -18x2 - 9x - 57, and write it below the division. Now multiply the result -18x2 by the denominator (x-3) to get -54x2 - 27x - 171.

Finally, subtract -54x2 - 27x - 171 from -6x2 - x - 19 to get the remainder, -19.

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a) The length of DP is,

DP = 11 Units

b) The lengths are,

RY = 12

YN = 24

YT = 10

BT = 30

IY = 10

And, IE = 15

TN = 25

And, IN  = 50

A triangle is a three sided polygon, which has three vertices and three angles which has the sum 180 degrees.

Given that;

Triangle are shown in figure.

a) We have;

AM = MP

Hence,

12x - 4 = 4x + 12

12x - 4x = 4 + 12

8x = 16

x = 2

Hence, Lenght of DP is,

DP = 7x - 3

DP = 7×2 - 3

DP = 14 - 3

DP = 11

b) We know that;

RY : YN = 1 : 2

Here, RN = 36

Hence, We get;

36 = x + 2x

3x = 36

x = 12

Hence,

RY = x = 12

YN = 2x = 2 x 12 = 24

And, If BY = 20;

YT = 20/2

YT = 10

And, BT = 20 + 10 = 30

If EY = 5;

IY = 2 × 5 = 10

And, IE = 5 + 10 = 15

If IT = 25

Then, TN = 25

And, IN = 25 + 25 = 50

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(a) The average rate of change of f on [−3,1] is:
f(1)-f(-3)/(1-(-3)) = (1+3)/(1-(-3)) - ((-3)+3)/(-3-(-3)) = (4/4) - (0/6) = 1 - 0 = 1

(b) The average rate of change of f on [x,x+h] is:
f(x+h)-f(x)/(x+h-x) = (x+h+3)/(x+h-8) - (x+3)/(x-8) = (x+h+3)(x-8) - (x+3)(x+h-8)/(x+h-8)(x-8) = (x^2-5x-8h-11)/(x^2-8x-8h+64)

The average rate of change of a function is the slope of the line that passes through two points on the graph of the function. It is calculated by the difference in the y-values of the two points divided by the difference in the x-values of the two points.

The average rate of change of f on [−3,1] is:
f(1)-f(-3)/(1-(-3)) = (1+3)/(1-(-3)) - ((-3)+3)/(-3-(-3)) = (4/4) - (0/6) = 1 - 0 = 1

The average rate of change of f on [x,x+h] is:
f(x+h)-f(x)/(x+h-x) = (x+h+3)/(x+h-8) - (x+3)/(x-8) = (x+h+3)(x-8) - (x+3)(x+h-8)/(x+h-8)(x-8) = (x^2-5x-8h-11)/(x^2-8x-8h+64)

Therefore, the average rate of change of f on [x,x+h] is (x^2-5x-8h-11)/(x^2-8x-8h+64).

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3

x

+

7

(

3

x

+

4

)

3

x

+

7

×

3

x

+

7

×

4

3

x

+

21

x

+

28

24

x

+

28

Answer:

Step-by-step explanation:

45 points for signing up.

2.5 points for each visit [tex]=2.5v[/tex]

Needs to total 75 points.

So [tex]45+2.5v=75[/tex] is the solution.

This is shown by option [tex](D)[/tex].

Michael will need to pay a total of R276,000 over the 60 months for the Mex tractor.

First, let's calculate the total amount he will need to finance:

Total amount to finance = R160,000 + 0.15 x R160,000 (VAT at 15%) = R184,000

Next, let's calculate the total interest he will need to pay over the 60 months:

Total interest = (principal x rate x time) / 100

where

Total interest = (R184,000 x 10% x 5) / 100 = R92,000

Therefore, the total amount Michael will need to pay over the 60 months is the sum of the total amount financed and the total interest:

Total amount to pay = Total amount to finance + Total interest = R184,000 + R92,000 = R276,000

So, Michael will need to pay a total of R276,000 over the 60 months for the Mex tractor.

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The exact values are:
cosα = -√7/3
sin2α = 8√7/9

cos2α = -7/9
tan2αcos2α/sin2α = 1

If sinα = −4/3, π⩽α⩽2π/3, we can find the exact value of cosα, sin2α, cos2α, and tan2αcos2α/sin2α by using the trigonometric identities.

cosα = √(1 - sin²α) = √(1 - (-4/3)²) = √(1 - 16/9) = √(-7/9) = -√7/3

sin2α = 2sinαcosα = 2(-4/3)(-√7/3) = 8√7/9

cos2α = 1 - sin²α = 1 - (-4/3)² = 1 - 16/9 = -7/9

tan2αcos2α/sin2α = (sin2α/cos2α)(cos2α/sin2α) = 1

Therefore, the exact values are:
cosα = -√7/3
sin2α = 8√7/9
cos2α = -7/9
tan2αcos2α/sin2α = 1

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Ordered pair (0, 3), (1, 48) and many more lies on the graph of the function.

What are Functions?

A function in math is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range) with the property that each input is related to exactly one output. It can be represented using an equation, graph, or table of values.

The exponential function [tex]f(x) = 3(4)^(2x)[/tex] can be evaluated for any value of x to get the corresponding value of y. To find ordered pairs that lie on the graph of this function, we can substitute different values of x into the function and calculate the corresponding values of y.

For example, if we substitute x = 0 into the function, we get:

[tex]f(0) = 3(4)^(2(0)) = 3(4)^0 = 3(1) = 3[/tex]

So the ordered pair (0, 3) lies on the graph of the function.

Similarly, if we substitute x = 1 into the function, we get:

[tex]f(1) = 3(4)^(2(1)) = 3(4)^2 = 3(16) = 48[/tex]

So the ordered pair (1, 48) also lies on the graph of the function.

We can repeat this process for other values of x to find more ordered pairs that lie on the graph of the function.

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When we square the two sides, we obtain: Z = square (82) , YZ thus has a length of about 9.06 units.

The connection among all three opposites of a right triangle is a key idea in geometry known as Pythagoras' theorem. According to this rule, the square of the hypotenuse's length—the side that faces the right angle—in a right triangle is the product of the squares of the dimensions of the other two sides. You can write this as follows: [tex]a^2 + b^2 = c^2[/tex] where a and b are the sizes of the other two right triangle sides and c has been the angle of the hypotenuse. The theorem is named after Pythagoras, an ancient Greek mathematician who is credited with finding it and creating its mathematical proof.

given

We may apply the Pythagorean theorem to determine the length of YZ.

Triangle VYZ is a right-angled triangle with the legs VY and VZ and the hypotenuse YZ, as may be seen. From the information provided, we can determine the lengths of VY and VZ.

VY = 35 - 36 = -1 (we can see from the diagram that V is to the left of Y, hence VY is negative) (we can see from the diagram that V is to the left of Y, so VY is negative)

VZ = 44 - 35 = 9 (we can see from the graphic that Z is below V, hence VZ is positive) (we can see from the diagram that Z is below V, so VZ is positive)

Using the Pythagorean theorem, we have the following:

[tex]VY^2 + VZ^2 = YZ^2\\YZ^2 = (-1) (-1)^2 + 9^2 \\YZ^2 = 1 + 81 \\YZ^2 = 82[/tex]

When we square the two sides, we obtain: Z = square (82) , YZ thus has a length of about 9.06 units.

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