A polar graph is shown.

If two lines have non-zero slopes and the same y-intercept, and their sum of slopes is 0, then the sum of the x-coordinates of their x-intercepts is -2 times the y-intercept divided by one of the slopes.

Let's consider two lines L1 and L2 with equations:

L1: y = m1x + b
L2: y = m2x + b

Here, m1 and m2 are the non-zero slopes, and b is the same y-intercept for both lines. We know that the sum of their slopes is 0, which means:

m1 + m2 = 0
m2 = -m1

Now, let's find the x-intercepts of both lines. To find the x-intercept, we need to set y = 0 in the equations:

For L1:
0 = m1x1 + b
x1 = -b / m1

For L2:
0 = m2x2 + b
x2 = -b / m2

As m2 = -m1, we can rewrite the x2 equation as:
x2 = -b / (-m1)

Now, let's find the sum of the x-coordinates of their x-intercepts (x1 + x2):

x1 + x2 = (-b / m1) + (-b / (-m1))
x1 + x2 = (-b / m1) + (b / m1)
x1 + x2 = (-b + b) / m1
x1 + x2 = 0

So, the sum of the x-coordinates of the x-intercepts of the two lines is 0.

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The currency exchange , If we Rounded it up to  2 decimal places , it will be  $313.95.

Firstly,  we will convert the amount given  in pounds to euros:

sin 1 pounds = 1.12 euro

And €1 is =$1.22

Therefore,

£230 x €1.12 divide by £1 = €257.60

Then, let convert the amount in euros  to dollars:

€257.60 x $1.22 divided €1 = $313.95

So, we can say  Aishah will  get $313.95 when  she converts  230 pounds  into dollars. If we Rounded it up to  2 decimal places , it will be  $313.95.

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The property of vectors that is incorrect is "Oa· (b + c) = a · b + a.c = (a + b ) + c = a +(b + c)". The correct property is "Oa· (b + c) = Oa·b + Oa·c".

The incorrect property of vectors among the given options is:

Oa·b = axb

This property is incorrect because the dot product (a·b) and cross product (axb) of two vectors are different operations with different results. The dot product is a scalar value, while the cross product is another vector that is orthogonal to the given vectors. The correct properties of vectors in your question are:

1. a x b = -b x a (cross product)
2. a · (b + c) = a · b + a · c (dot product distributive property)
3. (a + b) + c = a + (b + c) (vector addition associativity)

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Using the given terms, we'll apply Stokes' theorem to find the flux of the vector field F across the surface S.

Stokes' theorem states that the flux of the curl of a vector field F across a surface S is equal to the circulation of F around the boundary of S. Mathematically, it's expressed as:

∮_C F·dr = ∬_S curl(F)·dS

Given the vector field F = li + 3j + 1k, we first need to find the curl of F. Curl(F) is given by the determinant of the following matrix:

| i  j  k  |
| ∂/∂x  ∂/∂y  ∂/∂z |
| l  3  1 |

Curl(F) = i(∂(1)/∂y - ∂(3)/∂z) - j(∂(1)/∂x - ∂(l)/∂z) + k(∂(3)/∂x - ∂(l)/∂y)
Curl(F) = -j(0 - 0) + k(0 - 0) = 0

Since the curl of F is 0, the flux of the vector field F across the surface S is also 0. Therefore, by using Stokes' theorem, we have found that the flux of the vector field F across the surface S in the first octant is 0.

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Answer:

it c

Step-by-step explanation:

it well retun but it L to q but P in the way so that answer wod be c

The function is increasing on the interval (pi/6, 5pi/6), and it is decreasing on the intervals:

[0, pi/6) and (5pi/6, 27).

In interval notation, we can write:

The function is increasing on (pi/6, 5pi/6).
The function is decreasing on [0, pi/6) and (5pi/6, 27).

To find the intervals of increase and decrease for the function f(x) = -4cos(x) - 2x on [0, 2π), we first need to find its derivative.

Step 1: Find the derivative of f(x).
f'(x) = derivative of (-4cos(x) - 2x)
f'(x) = 4sin(x) - 2

Step 2: Identify critical points by setting the derivative equal to zero.
4sin(x) - 2 = 0

Step 3: Solve for x.
sin(x) = 1/2
x = π/6, 5π/6 (since these values are within the interval [0, 2π))

Step 4: Determine intervals of increase and decrease.
We will now test intervals around the critical points to determine where the function is increasing and decreasing.

Test interval 1: (0, π/6)
f'(π/12) = 4sin(π/12) - 2 > 0
Therefore, f(x) is increasing on (0, π/6).

Test interval 2: (π/6, 5π/6)
f'(π/2) = 4sin(π/2) - 2 < 0
Therefore, f(x) is decreasing on (π/6, 5π/6).

Test interval 3: (5π/6, 2π)
f'(3π/2) = 4sin(3π/2) - 2 > 0

To determine the intervals of increase and decrease, we need to test the sign of f'(x) in each sub-interval.
In the interval [0, pi/6), f'(x) is negative since sin(x) is less than 1/2. Therefore, f(x) is decreasing on this interval.
In the interval (pi/6, 5pi/6), f'(x) is positive since sin(x) is greater than 1/2. Therefore, f(x) is increasing on this interval.
In the interval (5pi/6, 27), f'(x) is negative again since sin(x) is less than 1/2. Therefore, f(x) is decreasing on this interval.

Therefore, the function is increasing on the interval (pi/6, 5pi/6), and it is decreasing on the intervals [0, pi/6) and

(5pi/6, 27).

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The conclusion of the Mean Value Theorem is satisfied for two values of c: [tex]c = \sqrt{(5/6)}[/tex] and [tex]c = -\sqrt{(5/6)}[/tex]. The limit of the given expression as x approaches infinity is 0.

1. To verify that the function [tex]g(x) = x^3 + x - 1[/tex] satisfies the hypotheses of the Mean Value Theorem on the interval [0, 2], we need to check two conditions: continuity and differentiability.

Firstly, g(x) is continuous on [0, 2] since it is a polynomial function. Secondly, g(x) is differentiable on (0, 2) since its derivative[tex]g'(x) = 3x^{2} + 1[/tex]is also a polynomial function and is defined for all x in the interval (0, 2).

Now, by the Mean Value Theorem, there exists a number c in (0, 2) such that [tex]g'(c) = [g(2) - g(0)]/(2 - 0)[/tex]. Therefore, we can find the value of c by solving the equation:

[tex]g'(c) = [g(2) - g(0)]/(2 - 0)[/tex]

3c² + 1 = (8 - 1)/(2)

3c² + 1 = 7/2

3c² = 5/2

c² = 5/6

[tex]c = \pm \sqrt{(5/6)}[/tex]

Hence, the conclusion of the Mean Value Theorem is satisfied for two values of c: [tex]c = \sqrt{(5/6)}[/tex] and [tex]c = -\sqrt{(5/6)}[/tex].

2. To evaluate [tex]\lim_{x \to \infty} (ln x)^3/x^2[/tex], we can use L'Hopital's Rule. Applying the rule once, we get:

[tex]\lim_{x \to \infty} (ln x)^3/x^2 = \lim_{x \to \infty} 3(ln x)^2/x[/tex]

[tex]= \lim_{x \to \infty} 6ln \;x/x[/tex]

[tex]= \lim_{x \to \infty} 6/x = 0[/tex]

Therefore, the limit of the given expression as x approaches infinity is 0.

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The number of people who are not looking for work will be 376,800.

The unemployment rate in a city is 5.8%. There are 23,200 people who are unemployed and looking for work.

The total number of people is calculated as,

⇒ 23,200 / 0.058

⇒ 400,000

The number of people who are not looking for work will be given as,

⇒ 400,000 x (1 - 0.058)

⇒ 376,800

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A schematic diagram that uses symbols to represent the parts of a system is called a schematic or a circuit diagram.

A schematic diagram is a type of diagram that represents a system or process using symbols, lines, and other graphical elements. The purpose of a schematic diagram is to convey information about a system or process in a clear and concise manner, making it easier to understand and analyze.

Schematic diagrams are commonly used in fields such as electrical engineering, mechanical engineering, and process engineering. They may be used to depict a wide range of systems or processes, including electronic circuits, hydraulic systems, HVAC systems, and manufacturing processes, among others.

In a schematic diagram, symbols are used to represent various components or elements of the system or process being depicted. For example, in an electrical circuit schematic, symbols might be used to represent resistors, capacitors, diodes, and other electronic components. Lines and other graphical elements are used to show the connections and interactions between the different components.

Schematic diagrams are an important tool in the design, analysis, and troubleshooting of systems and processes. They allow engineers and other professionals to quickly and easily understand the workings of complex systems, identify potential problems, and develop solutions.

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The quotient is x^2 - 5x - 5and remainder is -1 using synthetic division for x." – + 723 – 7.x2 + 5x – 11 2 - 1

To find the quotient and remainder using synthetic division for the given problem, we'll follow these steps:

Given polynomial: x^3 - 7x^2 + 5x - 11
Divisor: x - 2

1. Write down the coefficients of the given polynomial: (1, -7, 5, -11)

2. Write the constant term of the divisor with the opposite sign: (2)

3. Perform synthetic division:

```
   ____________________________
2 |  1    -7     5     -11
       |      2    -10   10
   ____________________________
       1    -5    -5    -1
```

4.The result gives us the coefficients of the quotient and the remainder.

The quotient is x^2 - 5x - 5, and the remainder is -1.

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The value 6 is a counterexample to this conditional statement. So, correct option is C.

The statement "If a number is divisible by three, then it is odd" is a conditional statement that can be written in the form of "If p, then q", where p represents "a number is divisible by three" and q represents "it is odd". To disprove a conditional statement, we need a counterexample where p is true and q is false.

Option C) 6 is a counterexample to this conditional statement since it is divisible by three but it is not odd. Therefore, option C) is the correct answer.

Option A) 1 is not a counterexample as it is not divisible by three and is odd.

Option B) 3 is true for both p and q, and is not a counterexample.

Option D) 9 is not a counterexample as it is divisible by three and is odd.

So, correct option is C.

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Roberto's z-score is approximately 2.29. This means that his score is about 2.29 standard deviations above the mean.

A z-score (also known as a standard score) is a measure of how many standard deviations a data point is away from the mean of a distribution. It is used to standardize data so that we can compare values from different distributions.

For example, if a student's score on a test is 80 and the mean score is 75 with a standard deviation of 5, the z-score for the student's score would be:

z = (80 - 75) / 5

z = 1

This means that the student's score is one standard deviation above the mean.

To find Roberto's z-score, we can use the formula:

(x - μ) / σ

where x is Roberto's score, μ is the mean of the test, and σ is the standard deviation of the test.

We are given that the mean was 69 and the standard deviation was 7. Roberto scored 85. So we can plug in these values into the formula and solve for z:

z = (85 - 69) / 7

z = 16 / 7

z ≈ 2.29

Therefore, Roberto's z-score is approximately 2.29.

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A figure has a surface area of 496 m². If the dimensions are doubled by half, the surface area of the new figure is 124 m².

Firstly, we will assume it being a rectangle then calculate the new area using the formula and then we will put the values of original figure into new figure.

Assume we're working with a rectangle. We know that the area equals the length (l) multiplied by the width (w).

A = l x w

If we divide the dimensions in half, we get A = (1 / 2)l x (1 / 2)w.

A = (1 / 4) × (l  x w)

As a result, the new surface area would be one-quarter of the original:

[tex]A_{original}[/tex] = 496 m²

[tex]A_{new}[/tex] = (1/4) × [tex]A_{original}[/tex]

[tex]A_{new}[/tex] = (1 / 4) × (496)

[tex]A_{new}[/tex] = 124 m²

As a result, the new area would be 124 m².

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Answer: equal to 90 degrees.

Step-by-step explanation:

The angle formed by the radius of a circle and a tangent line to the circle is always a right angle, which means it is equal to 90 degrees. This is a well-known property of tangents to circles.

Answer: (8.5, -2.5)

Step-by-step explanation:

The midpoint formula tells us the midpoint of two points...

[tex]M =( \frac{x1 + x2}{2} ,\frac{y1 + y2}{2})[/tex]

And hopefully this makes sense; we are averaging the x and y values to find their "middle"!

plugging in, you'd get the midpoint to be (8.5, -2.5)

The derivative dz/dt can be found using the chain rule. By differentiating each term with respect to t and applying the chain rule, we can calculate dz/dt as follows:

[tex]dz/dt = 2sin(t)cos(t) + 4e^tcos(t) + 4e^tsin(t) + 4e^t + 4sin(t)e^t.[/tex]

By applying the chain rule, we can find dz/dt as follows: differentiate z with respect to x, then multiply it by dx/dt, and finally differentiate z with respect to y and multiply it by dy/dt.

The function z = x² + y² + xy can be rewritten as z = (sin(t))² + (4e^t)² + (sin(t))[tex](4e^t)[/tex].

To find dz/dt, we need to find the partial derivatives of z with respect to x and y and multiply them by dx/dt and dy/dt, respectively.

The partial derivative of z with respect to x is (2x + y), and the partial derivative of z with respect to y is (2y + x).

Next, we differentiate x = sin(t) with respect to t, giving us dx/dt = cos(t).

Similarly, differentiating[tex]y = 4e^t[/tex] with respect to t yields [tex]dy/dt = 4e^t.[/tex]

Now we can apply the chain rule:

dz/dt = (2x + y) * dx/dt + (2y + x) * dy/dt

Substituting the expressions for x, y, dx/dt, and dy/dt:

[tex]dz/dt = (2sin(t) + 4e^t) * cos(t) + (2(4e^t) + sin(t)) * (4e^t)[/tex]

Simplifying this expression will yield the final result.

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The surface area of the composite figure is 416 in².

We have,

From the figure,

We have 10 surfaces.

Now,

There are 4 pairs of surfaces and 2 different surfaces.

1 pair is in square shape.

3 pairs in a rectangle shape.

Now,

Square shape surface area.

= 3² + 3²

= 9 + 9

= 18 in²

Rectangular surface area.

= (6 x 8) + (6 x 8) + (6 x 11) + (6 x 11) + (3 x 11) + (3 x 11)

= 56 + 56 + 66 + 66 + 33 + 33

= 310 in²

And,

Two different Surfaces area.

Both are in rectangular shape.

= (11 x 3) + (11 x (8 - 3))

= 33 + (11 x 5)

= 33 + 55

= 88 in²

Thus,

The surface area of the composite figure.

= 18 + 310 + 88

= 416 in²

Thus,

The surface area of the composite figure is 416 in².

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Answer: To fill the whole bucket, it will take 64 seconds so the remaining time is 40 seconds

Step-by-step explanation: As we are given 3/8 th part of the bucket is filled in 24 seconds. So by simply applying the unitary method we can say -

3/8 th part -----> 24 seconds

To fill the whole bucket multiply both sides by 8/3 in order to make the 1 unit of the bucket on the L.H.S, we get

1 bucket ----> 64 seconds.

The remaining times as it already passes 24 seconds and 3/8 th part of the bucket is filled, 64-24 seconds i.e 40 seconds is remaining in which bucket is full.

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An expression [tex](16x^4)^{\frac{1}{2} }[/tex] is equivalent to [tex](64x^6)^{\frac{1}{3} }[/tex]

The correct answer is an option (B)

Consider an expression,

[tex](16x^4)^{\frac{1}{2} }[/tex]

we know that the rule of exponents.

[tex](ab)^m=a^mb^m[/tex]

So we get,[tex](16x^4)^{\frac{1}{2} } = (16)^\frac{1}{2} \times (x^4)^{\frac{1}{2}[/tex]

Also, [tex](a^m)^n=a^{m\times n}[/tex]

Thus,

[tex](x^4)^{\frac{1}{2} }\\\\=x^{4\times\frac{1}{2}}\\\\=x^2[/tex]

We know that the n-th root of any number 'm' is represented as [tex]m^{\frac{1}{n} }[/tex]

So, the 1/2 power of 16 means the square of 16.

And [tex]\sqrt{16} =4[/tex]

If we look at the option (B) then we can observe that the (1/3) power of 64 is nothing but the cube root of 64 which is 4 and

[tex](x^6)^{\frac{1}{3}}\\\\=x^{6\times \frac{1}{3}}\\\\=x^2[/tex]

Therefore, the correct answer is an option (B)

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Height of  Ball 1: h₁(t) = −16t² + 16, Ball 2: h₂(t) = −16t² + 64. Ball 1 reaches the ground in 2 sec. Ball 2 reaches the ground in 3 sec. The correct answer is option (c)

To understand why this is the correct answer, let's first understand what the given information represents. Two identical rubber balls are dropped from different heights, and we are asked to find their respective height functions. The height function gives the height of the ball at any given time during its descent.

We know that the height function of a ball dropped from a height h₀ is given by h(t) = −16t² + h₀, where t is the time in seconds since the ball was dropped.

Using this formula, we can find the height functions for the two balls:

For the first ball dropped from a height of 16 feet, the height function is h₁(t) = −16t² + 16.

For the second ball dropped from a height of 64 feet, the height function is h₂(t) = −16t² + 64.

Now, we need to determine when each ball will reach the ground. We can do this by setting h(t) = 0 and solving for t. When h(t) = 0, the ball has hit the ground.

For ball 1: 0 = −16t² + 16, which gives t = 2. Therefore, ball 1 reaches the ground in 2 seconds.

For ball 2: 0 = −16t² + 64, which gives t = 3. Therefore, ball 2 reaches the ground in 3 seconds.

Comparing their graphs, we can see that both balls follow the same shape but start at different heights. Ball 2 starts at a higher point on the y-axis (64 ft) and takes longer to hit the ground. Ball 1 starts at a lower point (16 ft) and hits the ground sooner. This is because the greater the initial height, the longer it takes for the ball to reach the ground.

The correct answer is option (c)

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To determine the factors that should be included in the function to satisfy the given conditions, we need to analyze the end behavior of the function. An end behavior refers to the behavior of a function's graph as x approaches positive or negative infinity.

1. Increasing as x approaches negative infinity: This indicates that the graph should rise as we move to the left. This is associated with an odd-degree function with a positive leading coefficient. An example of such a function is f(x) = x^3.

2. Decreasing as x approaches positive infinity: This also indicates that the graph should fall as we move to the right. This is consistent with an odd-degree function with a positive leading coefficient, like f(x) = x^3.

To ensure that the function meets both conditions, we should include a term with an odd exponent and a positive coefficient. This will make the function increase as x approaches negative infinity and decrease as x approaches positive infinity. Some examples include x^3, 5x^5, and 7x^7.

In conclusion, include a term with an odd exponent and a positive coefficient to satisfy the given conditions. This will result in the graph of the function rising to the left (as x approaches negative infinity) and falling to the right (as x approaches positive infinity).

Which factor should be included in the function below so that the graph of the function is increasing as x approaches negative infinity and decreasing as x approaches positive infinity? Select all that apply.

f(x)=(x+4)(x−5)

-0.8

-3x

(x+5)

-(2x+1)

(3x^2+5)

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The particular solution of the differential equation is: y = e^(5x+ln(7)) y = 7e^(5x) This is the function that satisfies the given differential equation and initial condition.

To find the particular solution of the given differential equation with the initial condition, we need to follow these steps:

1. Write down the differential equation:
dy/dx * y * cos(x) = 5 * cos(x)

2. Separate variables:
(dy/dx) = 5/y * cos(x)

3. Integrate both sides with respect to x:
∫(dy/y) = ∫(5*cos(x) dx)

4. Evaluate the integrals:
ln|y| = 5 * sin(x) + C

5. Solve for y:
y = e^(5 * sin(x) + C)

6. Apply the initial condition y(0) = 7:
7 = e^(5 * sin(0) + C)

7. Solve for C:
7 = e^C => C = ln(7)

8. Substitute C back into the solution:
y(x) = e^(5 * sin(x) + ln(7))

So the particular solution of the given differential equation is:
y(x) = e^(5 * sin(x) + ln(7))

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Since matrix A is a 3x7 matrix and it has three pivot columns, it means that there are three leading ones in the row-reduced echelon form of A, which implies that the row-reduced echelon form of A has three nonzero rows. Thus, the rank of matrix A is 3.

(a) Col A- R3: The column space of A is spanned by the columns containing the pivot entries in the row-reduced echelon form of A.

Since there are three pivot columns, it means that the column space of A has dimension 3. Therefore, Col A- R3 = {0}, which means that the only linear combination of the columns of A that gives the zero vector is the trivial one.

(b) Nul A- R42: The null space of A is t solutions to the homogeneous equation Ax = 0. Since A has rank 3, the nullity of A is 7 - 3 = 4.

It follows that Nul A- R42 is the set of all solutions to the homogeneous equation Ax = 0 that can be written as a linear combination of four linearly independent vectors. Since the nullity of A is 4, it means that Nul A- R42 has dimension 4.

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The stability properties of the equilibrium at the origin for all values of a in the system of differential equations dx/dt = Ax depend on the eigenvalues of matrix A.

To determine the stability properties, first find the eigenvalues of matrix A by solving the characteristic equation, det(A - λI) = 0, where λ represents the eigenvalues and I is the identity matrix. Once you obtain the eigenvalues, analyze their real parts:

1. If all real parts are negative, the equilibrium is asymptotically stable.
2. If any real part is positive, the equilibrium is unstable.
3. If all real parts are non-positive, and there are no repeated eigenvalues with zero real parts, the equilibrium is stable.

By examining the eigenvalues, you can determine the stability properties for all values of the real-valued constant a.

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Points A and B are on side YZ of rectangle WXYZ such that WA and WB trisect ZWX. If BY = 3 and AZ = 6, then the area of rectangle WXYZ is approximately 218.23 square units.

Given that,

Points A and B are on side YZ of the rectangle WXYZ.

WA and WB trisect ZWX.

BY = 3 and AZ = 6.

From figure:

tan A = 6/y   ......(i)

And tan 2A = y/6  .....(ii)

Now tan(A) x tan(2A) = (6/y) x (y/6)

tan(A) x tan(2A) = 1

Expanding tan(2A):

[tex]tan(A) \times \dfrac{2 tan A}{1- tan^2 A} = 1\\\\3 tan^2 A = 1\\\\tanA = \dfrac{1}{\sqrt{3}} ....(iii)[/tex]

From equation (i) and (iii):

y = 6√3

Now again from the figure,

tanA = y/(x-3)

(x-3) = y/tanA

x  = 3 + 6√3√3                  from (iii)

x = 21

Therefore,

Area of the rectangle = xy,

Area of rectangle = (21)(6√3)

Area of the rectangle ≈ 218.23 square units

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If a is a subset of b, then the cardinality of a (|a|) is less than or equal to the cardinality of b (|b|).

By definition, if a is a subset of b, it means that every element in a is also an element of b. In other words, a is contained within b. The cardinality of a set refers to the number of elements it contains. Therefore, if a is a subset of b, it implies that the number of elements in a (|a|) cannot exceed the number of elements in b (|b|). In fact, |a| could be equal to |b| if a and b have the same number of elements. Hence, if a ⊂ b, it follows that |a| ≤ |b|.

To show that if a and b are sets and a ⊂ b, then |a| ≤ |b|, we need to show that there exists an injective function from a to b.

Let f(a) = a, for all a in set a. Since a is a subset of b, every element in a is also an element in b. Therefore, f(a) is a function from a to b.

To show that f is injective, suppose that f(a) = f(a'). Then, by the definition of f, we have a = a'. Therefore, f is injective.

Since we have found an injective function from a to b, by the definition of cardinality, we have |a| ≤ |b|.

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The determinant matrix found in the first step is the matrix A with the greatest common divisor, which is 5, The determinant of the original matrix A is: -1000.

The determinant of the matrix A can be found by factoring out the greatest common divisor, which is 5, and then using the fact that |cA| = cⁿ|A|.

Thus, the determinant of the matrix after factoring out the greatest common divisor is:

|A'| = 5|5 4 2 -1|

Using the fact that |cA| = cⁿ|A|, we have:

|A'| = 5⁴|1 4/5 2/5 -1/5|

Evaluating the determinant of the matrix A' gives:

|A'| = 5⁴((1)(-2/5)-(4/5)(-1/5)-(2/5)(4/5)-(1/5)(1)) = -200

|A| = 5(-200) = -1000.

The first step is to factor out the greatest common divisor, which is 5, from the rows and columns of the matrix. This results in a new matrix A' with elements that are integers. Next, we use the fact that |cA| = cⁿ|A|, where c is a scalar and n is the size of the matrix, to simplify the determinant of A'. We evaluate the determinant of A' using the formula for a 4x4 matrix and then multiply the result by 5⁴ to obtain the determinant of the original matrix A.

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The graph of the function has a vertical asymptote at x = -2 and a horizontal asymptote at y = -1. The graph is a hyperbola that opens downwards and has its center at (-2, -1).

Writing the function in the form y= a/x-h+k, rearrange as follows:

y = a / (x - h) + k

The graph is a hyperbola with a vertical asymptote at x = h and a horizontal asymptote at y = k.

The value of "a" determines the shape of the hyperbola. If a is +, the hyperbola opens upwards, and if a is -, it opens downwards.

The point (h, k) is the center of the hyperbola.

Transforming the graph of y = 1/x into the given function, apply the following transformations:

Horizontal shift: shift the graph to the right by 2 units, so h = -2.

Vertical shift: shift the graph downwards by 6 units, so k = -6.

Vertical stretch: stretch the graph vertically by a factor of -1, so a = -1.

Therefore, the function y = -1/(x+2) - 6 is the transformed function.

To solve y = (-x-2)/(x+6), simplify:

y = (-x-2)/(x+6)

y = (-1(x+2))/(x+6)

y = (-1(x+2))/((x+2)+4)

y = -1/(x+2) - 4/(x+2)

y = -1/(x+2) - 4x/(x+2)(x+2)

This expression is in the form y = a/(x-h) + k, where:

- a = -4

- h = -2

- k = -1

Therefore, the graph of the function has a vertical asymptote at x = -2 and a horizontal asymptote at y = -1. The graph is a hyperbola that opens downwards and has its center at (-2, -1).

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