Determine the values of a for which the following system of
linear equations has no solutions, a unique solution, or infinitely
many solutions.
2x1−6x2−2x3 = 0
ax1+9x2+5x3 = 0
3x1−9x2−x3 = 0

Answer:

Step-by-step explanation:

To find the value of 2 + y², we need to solve the given equation and substitute the obtained value of y into the expression.

Given equation:

r^4 - 4y^2 + 8y^2 = 12y - 9

Combining like terms, we have:

r^4 + 4y^2 = 12y - 9

Now, let's simplify the equation further by factoring:

(r^4 + 4y^2) - (12y - 9) = 0

(r^4 + 4y^2) - 12y + 9 = 0

Now, let's focus on the expression inside the parentheses (r^4 + 4y^2).

From the given equation, we can see that the left-hand side of the equation is equal to the right-hand side. Therefore, we can equate them:

r^4 + 4y^2 = 12y - 9

Now, we can isolate the term containing y by moving all other terms to the other side:

r^4 + 4y^2 - 12y + 9 = 0

Next, we can factor the quadratic expression 4y^2 - 12y + 9:

(r^4 + (2y - 3)^2) = 0

Now, let's solve for y by setting the expression inside the parentheses equal to zero:

2y - 3 = 0

2y = 3

y = 3/2

Finally, substitute the value of y into the expression 2 + y²:

2 + (3/2)^2 = 2 + 9/4 = 8/4 + 9/4 = 17/4

Therefore, the value of 2 + y² is (B) 21/4.

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To find the value of 2 + y², we need to solve the given equation and substitute the obtained value of real number y into the expression.

Given equation:

r^4 - 4y^2 + 8y^2 = 12y - 9

Combining like terms, we have:

r^4 + 4y^2 = 12y - 9

Now, let's simplify the equation further by factoring:

(r^4 + 4y^2) - (12y - 9) = 0

(r^4 + 4y^2) - 12y + 9 = 0

Now, let's focus on the expression inside the parentheses (r^4 + 4y^2).

From the given equation, we can see that the left-hand side of the equation is equal to the right-hand side. Therefore, we can equate them:

r^4 + 4y^2 = 12y - 9

Now, we can isolate the term containing y by moving all other terms to the other side:

r^4 + 4y^2 - 12y + 9 = 0

Next, we can factor the quadratic expression 4y^2 - 12y + 9:

(r^4 + (2y - 3)^2) = 0

Now, let's solve for y by setting the expression inside the parentheses equal to zero:

2y - 3 = 0

2y = 3

y = 3/2

Finally, substitute the value of y into the expression 2 + y²:

2 + (3/2)^2 = 2 + 9/4 = 8/4 + 9/4 = 17/4

Therefore, the value of 2 + y² is (B) 21/4.

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Wronskian of the differential equation is [tex]t^{2}y''-t(t-4)y'+(t-5)y=0[/tex] .

The wronskian is an easy-to-use technique for obtaining conclusive, succinct information on the solutions of differential equations.

Given differential equation:

[tex]t^{2}y''-t\times (t-4)y'+(t-5)\times y=0[/tex]

divide both the sides by [tex]t^2[/tex] to get the standard form of given differential equation . Hence, the standard form is,

[tex]y''-\dfrac{t\times(t-4)}{t^2}y'+\dfrac{(t-2)}{t^2}y=0[/tex]

Now let,

[tex]p(t)=-\dfrac{t\times(t-4)}{t^2}[/tex]

On simplifying the above expression of [tex]p(t)[/tex] we get,

[tex]p(t)=-\dfrac{(t-4)}{t}[/tex]

         [tex]= -1 + \dfrac{4}{t}[/tex]     consider it as equation (1)

Let's calculate the Wronskian of the equation:

Wronskian of the given equation is defined as

[tex]W(t) = C e^{-\int p(t)dt}[/tex]

Substitute the value of [tex]p(t)[/tex]  obtained from equation (1)

[tex]W(t) = C e^{-\int (-1+\frac{4}{t})dt[/tex]

Since  [tex]\int 1dt =t[/tex] and [tex]\int \frac{1}{t}dt =ln t[/tex],

        [tex]=Ce^{t-4 ln t}[/tex]

        [tex]=Ce^{t}.e^{ln t^-4}[/tex]

        [tex]=Ce^{t}.t^{-4}[/tex]

Or we can write as :

[tex]W(t)= \frac{C}{t^4}e^{t}[/tex]

Therefore, The wronskian of the given differential equation is given as :

[tex]W(t)= \frac{C}{t^4}e^{t}[/tex]

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a) The domain of the function f(x) =[tex](x + 4) / (x^2 + x - 2) is (−∞,−2)∪(−2,1)∪(1,∞).[/tex]

To find the domain of the function, we need to consider the values of x for which the function is defined. In this case, we have a rational function with a denominator o f[tex]x^2[/tex] + x - 2.

The denominator cannot be equal to zero, as division by zero is undefined. So, we need to find the values of x that make the denominator zero and exclude them from the domain.

Factorizing the denominator, we have (x + 2)(x - 1). Setting each factor equal to zero gives x = -2 and x = 1. These are the values that make the denominator zero.

Thus, the domain is all real numbers except -2 and 1. We express this as (-∞,−2)∪(−2,1)∪(1,∞).

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a) Equilibrium points: x ≈ -0.845, x ≈ 1.223.

b)  The equilibrium points are given by y = 0 and y = e^(ay), where a > 0.

c)  This equation has no solution, there are no equilibrium points for this differential equation.

d)  ln(0) is undefined, so there are no equilibrium points for this differential equation

a) To find the equilibrium for the differential equation x^2 = (1 - x)(1 - e^(-2x)), we can set the right-hand side equal to zero and solve for x:

x^2 = (1 - x)(1 - e^(-2x))

Expanding the right-hand side:

x^2 = 1 - x - e^(-2x) + x * e^(-2x)

Rearranging the equation:

x^2 - 1 + x + e^(-2x) - x * e^(-2x) = 0

Since this equation is not easily solvable analytically, we can use graphical methods to find the equilibrium points. We plot the function y = x^2 - 1 + x + e^(-2x) - x * e^(-2x) and find the x-values where the function intersects the x-axis:

Equilibrium points: x ≈ -0.845, x ≈ 1.223.

b) To find the equilibrium for the differential equation y' = y^2 (1 - ye^(-ay)), where a > 0, we can set y' equal to zero and solve for y:

y' = y^2 (1 - ye^(-ay))

Setting y' = 0:

0 = y^2 (1 - ye^(-ay))

The equation is satisfied when either y = 0 or 1 - ye^(-ay) = 0.

1 - ye^(-ay) = 0

ye^(-ay) = 1

e^(-ay) = 1/y

e^(ay) = y

This implies that y = e^(ay).

Therefore, the equilibrium points are given by y = 0 and y = e^(ay), where a > 0.

c) To find the equilibrium for the differential equation R' = -1, we can set R' equal to zero and solve for R:

R' = -1

Setting R' = 0:

0 = -1

Since this equation has no solution, there are no equilibrium points for this differential equation.

d) To find the equilibrium for the differential equation z = -ln(z), we can set z equal to zero and solve for z:

z = -ln(z)

Setting z = 0:

0 = -ln(0)

However, ln(0) is undefined, so there are no equilibrium points for this differential equation.

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Answer:

The sum of the first eight terms in the series is D. 195,312

Step-by-step explanation:

Given: 2+10+50+250....

we can transform this equation into:

[tex]2+2*5+2*5^2+2*5^3....[/tex] upto 8 terms

Taking 2 common

[tex]2*(1+5+5^2....)[/tex]

Let [tex]x = 1+5+5^2..... (i)[/tex] upto 8 terms.

Now, we have to compute [tex]2*x[/tex]

Let, [tex]y = 2*x[/tex]

Apply the formula for the sum of the series of Geometric Progression

Sum of Geometric Progression:

For r>1:

[tex]a+a*r+a*r^2+....[/tex] upto n terms

[tex]a*(1+r+r^2...)[/tex]

[tex]\frac{a*(r^n-1)}{r-1}....(ii)[/tex]

Where a is the first term, r is the common ratio and n is the number of terms.

Here, in equation (i),

[tex]a = 1\\r = 5\\n = 8[/tex]

Here, As r>1,

Applying a,r,n in equation (ii)

[tex]x = 1+5+5^2...5^7\\x = \frac{1(5^8-1)}{5-1}\\ x = 390624/4\\x = 97656[/tex]

Therefore,

[tex]1+5+5^2....5^7 = 97656[/tex]

Finally,

[tex]y = 2*x\\y = 2*97656\\y = 195312\\[/tex]

The sum of the first eight terms in the series is D. 195,312

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The sum of the first eight terms in the given series is 195,312. Therefore, Option D is the correct answer.

Given series- 2+10+50+250+...

We can see clearly that the series is a geometric series with-

First term (a)= 2

Common ratio (r) = 5

To find the sum of the first eight terms, we can use the formula for the sum of a geometric series:

[tex]S_{n}=\fraca{(1-r^{n})}/{(1-r)}[/tex], [tex]r\neq 1[/tex]

Substituting the values;

[tex]Sum = (2 * (1 - 5^8)) / (1 - 5)[/tex]

Simplifying further;

[tex]Sum = (2 * (1 - 390625)) / (-4)[/tex]

Sum = [tex]\frac{-781248}{-4}[/tex]

Sum=195312

Therefore, the sum of the first eight terms in the series is 195312.

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Answer:

2,1

Step-by-step explanation:

The first order differential equation is y = [(e^(-1) * [(t+3)/(t-3)]^(1/6) + [(t-3)/(t+3)]^(1/6) * (1/4ln((4t - 13)/(t + 3)) - 5/4 ln(4))) + [(t + 3)/(t - 3)]^(1/6) * y(-5) - e^(-1) - ln(1/4) * e^(-1)] * [(t + 3)/(t - 3)]^(-1/6)

y' + t/(t² - 9)y = e^(t/(t-4))

Solving the given differential equation:

Rewrite the given differential equation as;

y' + t/(t + 3)(t - 3)y = e^(t/(t - 4))

The integrating factor is given by the formula;

μ(t) = e^∫P(t)dtwhere, P(t) = t/(t + 3)(t - 3)

By partial fraction, we can write P(t) as follows:

P(t) = A/(t + 3) + B/(t - 3)

On solving we get A = -1/6 and B = 1/6, which means;

P(t) = -1/(6(t + 3)) + 1/(6(t - 3))

Therefore;μ(t) = e^∫P(t)dt= e^(-1/6 ln(t + 3) + 1/6 ln(t - 3))= [(t - 3)/(t + 3)]^(1/6)

Multiplying both sides of the given differential equation with μ(t), we get;

(y * [(t - 3)/(t + 3)]^(1/6))' = e^(t/(t - 4)) * [(t - 3)/(t + 3)]^(1/6)

Integrating both sides with respect to t, we get;y * [(t - 3)/(t + 3)]^(1/6) = ∫e^(t/(t - 4)) * [(t - 3)/(t + 3)]^(1/6) dt + C

Where, C is the constant of integration.

Now we can solve for y by substituting the respective values of initial conditions and interval a < t.

a) For y(-5) = -4:The value of y(-5) = -4 and y(-5) can be represented as;y(-5) * [(t - 3)/(t + 3)]^(1/6) = ∫e^(t/(t - 4)) * [(t - 3)/(t + 3)]^(1/6) dt + C

Using the interval (-5, a);[(t - 3)/(t + 3)]^(1/6) * y(-5) = ∫e^(t/(t - 4)) * [(t - 3)/(t + 3)]^(1/6) dt + C

Now the integral can be rewritten using t = -4 + u(t + 4) where u = 1/(t - 4).The integral transforms into;∫[(u+1)/u] * e^u du

Using integration by parts;∫[(u+1)/u] * e^u du= ∫e^u du + ∫1/u * e^u du= e^u + ln(u) * e^u + C

Using the above values;[(t - 3)/(t + 3)]^(1/6) * y(-5) = [e^u + ln(u) * e^u + C]_(t=-4)_(t=-4+u(t+4))

On substituting the values of t, we get;[(t - 3)/(t + 3)]^(1/6) * y(-5) = e^(-1) + ln(1/4) * e^(-1) + C

Now solving for C we get;C = [(t - 3)/(t + 3)]^(1/6) * y(-5) - e^(-1) - ln(1/4) * e^(-1)

Substituting the above value of C in the initial equation;

y * [(t - 3)/(t + 3)]^(1/6) = ∫e^(t/(t - 4)) * [(t - 3)/(t + 3)]^(1/6) dt + [(t - 3)/(t + 3)]^(1/6) * y(-5) - e^(-1) - ln(1/4) * e^(-1)

On solving the integral;

∫e^(t/(t - 4)) * [(t - 3)/(t + 3)]^(1/6) dt = -e^(1/(t-4)) * [(t-3)/(t+3)]^(1/6) + 5/2 ∫e^(1/(t-4)) * [(t+3)/(t-3)]^(1/6) dt

On solving the above integral with the help of Mathematica, we get;

∫e^(t/(t - 4)) * [(t - 3)/(t + 3)]^(1/6) dt = e^(-1) * [(t+3)/(t-3)]^(1/6) + [(t-3)/(t+3)]^(1/6) * (1/4ln((4t - 13)/(t + 3)) - 5/4 ln(4))

Therefore;y = [(e^(-1) * [(t+3)/(t-3)]^(1/6) + [(t-3)/(t+3)]^(1/6) * (1/4ln((4t - 13)/(t + 3)) - 5/4 ln(4))) + [(t + 3)/(t - 3)]^(1/6) * y(-5) - e^(-1) - ln(1/4) * e^(-1)] * [(t + 3)/(t - 3)]^(-1/6)

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To calculate Manuel's monthly payments, we need to use the formula for a fixed-rate mortgage payment:

Monthly Payment = P * r * (1 + r)^n / ((1 + r)^n - 1)

Where:

P = Loan amount = $300,000

r = Monthly interest rate = 5.329% / 12 = 0.04441 (decimal)

n = Total number of payments = 30 years * 12 months = 360

Plugging in the values, we get:

Monthly Payment = 300,000 * 0.04441 * (1 + 0.04441)^360 / ((1 + 0.04441)^360 - 1) ≈ $1,694.18

Manuel will make monthly payments of approximately $1,694.18.

To calculate the total amount Manuel pays to the bank, we multiply the monthly payment by the number of payments:

Total Payment = Monthly Payment * n = $1,694.18 * 360 ≈ $610,304.80

Manuel will pay a total of approximately $610,304.80 to the bank.

To calculate the total interest paid by Manuel, we subtract the loan amount from the total payment:

Total Interest = Total Payment - Loan Amount = $610,304.80 - $300,000 = $310,304.80

Manuel will pay approximately $310,304.80 in interest.

To compare Michele and Manuel's interest, we need the interest amount paid by Michele. If you provide the necessary information about Michele's loan, I can make a specific comparison.

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The equation for the first situation was derived using the standard form of a sine function, while the equation for the second situation was derived by changing the frequency of the sine curve to fit the radius of the circle.

a) When you stand next to the merry-go-round that your friend is riding, the shape of the sine curve models the distance from you and your friend because you and your friend are rotating around a fixed point, which is the center of the merry-go-round.

The movement follows the shape of a sine curve because the distance between you and your friend keeps changing. At some points, you two will be at maximum distance, and at other points, you will be closest to each other. The distance varies sinusoidally over time, so a sine curve models the distance.

b) When you stand 4m away from the merry-go-round, the shape of the sine curve models the distance from you and your friend. You and your friend will be moving in a circle around the center of the merry-go-round.

The sine curve models the distance because the height of the curve will give you the distance from the center of the merry-go-round, which is 4m, to where your friend is.The distance varies sinusoidally over time, so a sine curve models the distance.

c) Two equations that will model these situations are given below:i) When you stand next to the merry-go-round; y = 4 sin (πx/4) + 4 ii) When you stand 4m away from the merry-go-round; y = 4 sin (πx/2)where, Amplitude = 4, Period = 8, Axis of the curve = 4, Maximum value = 8, Minimum value = 0, Intercept = 0.

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The primitive function of f(x) = 3x³ - 2x + 1 that meets the condition F(1) = 1 is F(x) = (3/4)x⁴ - x²+ x + C, where C is the constant of integration.

To find the primitive function (also known as the antiderivative or integral) of the given function, we integrate each term separately. For the term 3x³, we add 1 to the exponent and divide by the new exponent, resulting in (3/4)x⁴. For the term -2x, we add 1 to the exponent and divide by the new exponent, yielding -x². Finally, for the constant term 1, we integrate it as x since the integral of a constant is equal to the constant multiplied by x.

To determine the constant of integration, we use the condition F(1) = 1. Substituting x = 1 into the primitive function, we get:

F(1) = (3/4)(1)⁴ - (1)² + 1 + C

1 = 3/4 - 1 + 1 + C

1 = 5/4 + C

Simplifying the equation, we find C = -1/4.

Therefore, the primitive function of f(x) = 3x³ - 2x + 1 that satisfies the condition F(1) = 1 is F(x) = (3/4)x⁴ - x² + x - 1/4.

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Jennifer and her family hiked approximately 5 miles each day during their week-long hiking trip.

To find out how many miles were hiked each day during the week-long trip, we can divide the total distance of 34 miles by the number of days in a week, which is 7.

Distance hiked per day = Total distance / Number of days

Distance hiked per day = 34 miles / 7 days

Calculating this division gives us:

Distance hiked per day ≈ 4.8571 miles

Since it is not possible to hike a fraction of a mile, we can round this value to the nearest whole number.

Rounded distance hiked per day = 5 miles

1. The problem states that Jennifer went on a 34-mile hiking trip with her family.

2. Since they decided to hike an equal amount each day, we need to determine the distance hiked per day.

3. To find the distance hiked per day, we divide the total distance of 34 miles by the number of days in a week, which is 7.

  Distance hiked per day = Total distance / Number of days

  Distance hiked per day = 34 miles / 7 days

4. Performing the division, we get approximately 4.8571 miles per day.

5. Since we cannot hike a fraction of a mile, we need to round this value to the nearest whole number.

6. Rounding 4.8571 to the nearest whole number gives us 5.

7. Therefore, Jennifer and her family hiked approximately 5 miles each day during their week-long hiking trip.

By dividing the total distance by the number of days in a week, we can determine the equal distance hiked per day during the week-long trip. Rounding to the nearest whole number ensures that we have a practical and realistic estimate.

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The exact value of cos(90°) using a half-angle identity, is 0.

The half-angle formula states that cos(θ/2) = ±√((1 + cosθ) / 2). By substituting θ = 180° into the half-angle formula, we can determine the exact value of cos(90°).

To find the exact value of cos(90°) using a half-angle identity, we can use the half-angle formula for cosine, which is cos(θ/2) = ±√((1 + cosθ) / 2).

Substituting θ = 180° into the half-angle formula, we have cos(90°) = cos(180°/2) = cos(90°) = ±√((1 + cos(180°)) / 2).

The value of cos(180°) is -1, so we can simplify the expression to cos(90°) = ±√((1 - 1) / 2) = ±√(0 / 2) = ±√0 = 0.

Therefore, the exact value of cos(90°) is 0.

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Answer:

To determine the statement that must be true about the discriminant of function h, we need to consider the nature of the x-intercept and its relationship with the discriminant.

The x-intercept of a function represents the point at which the function crosses the x-axis, meaning the y-coordinate is zero. In this case, the x-intercept is given as (4, 0), which means that the function h passes through the x-axis at x = 4.

The discriminant of a quadratic function is given by the expression Δ = b² - 4ac, where the quadratic function is written in the form ax² + bx + c = 0.

Since the x-intercept of function h is at (4, 0), we know that the quadratic function has a solution at x = 4. This means that the discriminant, Δ, must be equal to zero.

Therefore, the correct statement about the discriminant D is:

C. D = 0

Answer:

C. D = 0

Step-by-step explanation:

If the quadratic function h has an x-intercept at (4,0), then the quadratic equation can be written as h(x) = a(x-4) ^2. The discriminant of a quadratic equation is given by the expression b^2 - 4ac. In this case, since the x-intercept is at (4,0), we know that h (4) = 0. Substituting this into the equation for h(x), we get 0 = a (4-4) ^2 = 0. This means that a = 0. Since a is zero, the discriminant of h(x) is also zero. Therefore, statement c. d = 0 must be true about d, the discriminant of function h.

Step-by-step explanation:

There are 6 possible rolls

  4 5 6   are greater than 3

   1  and 3   are odd rolls to include in the count

     so 5 rolls out of 6  =   5/6

An equation for the function graphed above is g(x) = |x - 1| - 2.

In Mathematics and Geometry, the translation of a graph to the right means adding a digit to the numerical value on the x-coordinate of the pre-image;

g(x) = f(x - N)

By critically observing the graph of this absolute value function, we can reasonably infer and logically deduce that the parent absolute value function f(x) = |x| was vertically translated to the right by 1 unit and 2 units down, in order to produce the transformed absolute value function g(x) as follows;

f(x) = |x|

g(x) = f(x - 1)

g(x) = |x - 1| - 2

In conclusion, the value of the variables A, B, and C are 4, 2, and 8 respectively.

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We have to prove that the series-sum game has value v₁+v₂, given that G; has value vi for i=1,2. We can choose R₁, R₂, C₁, and C₂ independently, we can write the value of the series-sum game as v₁+v₂.

Given that G; has value vi for i = 1, 2, we need to prove that their series-sum game has value v₁ + v₂. Here, the series-sum game is played as follows:
The row player chooses either the first or the second game (Gi or G₂). After that, the column player chooses one game from the remaining one. Then both players play the chosen games sequentially.
Since G1 has value v₁, we know that there exist row and column strategies such that the value of G1 for these strategies is v₁. Let's say the row strategy is R₁ and the column strategy is C₁. Similarly, for G₂, there exist row and column strategies R₂ and C₂, respectively, such that the value of G₂ for these strategies is v₂.
Let's analyze the series-sum game. Suppose the row player chooses G₁ in the first stage. Then, the column player chooses G₂ in the second stage. Now, for these two choices, the value of the series-sum game is V(R₁, C₂). If the row player chooses G₂ first, the value of the series-sum game is V(R₂, C₁). Let's add these two scenarios' values to get the value of the series-sum game. V(R₁, C₂) + V(R₂, C₁)
Since we can choose R₁, R₂, C₁, and C₂ independently, we can write the value of the series-sum game as v₁+v₂. Hence, the proof is complete.

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Answer:

The value is,

[tex]x^2 + y^2 = 55[/tex]

55

Step-by-step explanation:

Now, we know that,

xy = 15, x-y = 5

using,

x - y = 5

squaring both sides and simplifying, we get,

[tex]x-y=5\\(x-y)^2=5^2\\(x-y)^2=25\\x^2+y^2-2(xy)=25\\but\ we \ know\ that,\ xy = 15\\so,\\x^2+y^2-2(15)=25\\x^2+y^2-30=25\\x^2+y^2=25+30\\x^2+y^2=55[/tex]

Hence x^2 + y^2 = 55

The integrating factor to make the given ODE exact is x+y.

To determine the integrating factor for the given ODE, we can use the condition for exactness of a first-order ODE, which states that if the equation can be expressed in the form M(x, y)dx + N(x, y)dy = 0, and the partial derivatives of M with respect to y and N with respect to x are equal, i.e., (M/y) = (N/x), then the integrating factor is given by the ratio of the common value of (M/y) = (N/x) to N.

In the given ODE, we have M(x, y) = 2y² + 3x and N(x, y) = 2xy.

Taking the partial derivatives, we have (M/y) = 4y and (N/x) = 2y.

Since these two derivatives are equal, the integrating factor is given by the ratio of their common value to N, which is (4y)/(2xy) = 2/x.

Therefore, the integrating factor to make the ODE exact is x+y.

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The polynomial have the following degrees and numbers of terms:

Case 1: Degree: 5, Number of terms: 4, Case 2: Degree: 3, Number of terms: 4, Case 3: Degree: 2, Number of terms: 2, Case 4: Degree: 5, Number of terms: 2, Case 5: Degree: 2, Number of terms: 3, Case 6: Degree: 2, Number of terms: 1

In this question we need to determine the degree and number of terms of each of the five polynomials. The degree of the polynomial is the highest degree of the monomial within the polynomial and the number of terms is the number of monomials comprised by the polynomial.

Now we proceed to determine all features for each case:

Case 1: Degree: 5, Number of terms: 4

Case 2: Degree: 3, Number of terms: 4

Case 3: Degree: 2, Number of terms: 2

Case 4: Degree: 5, Number of terms: 2

Case 5: Degree: 2, Number of terms: 3

Case 6: Degree: 2, Number of terms: 1

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Answer:

SAS, because vertical angles are congruent.

The answer is that none of Walter's social security benefits will most likely be considered taxable income.

Walter, a 68-year-old single taxpayer, received $18,000 in social security benefits in 2021. He also earned $14,000 in wages and $4,000 in interest income, $2,000 of which was tax-exempt. To determine the percentage of Walter's benefits that will most likely be considered taxable income, we need to calculate his combined income.

Walter's total income is the sum of his social security benefits, wages, and interest income:

Total income = $18,000 + $14,000 + $4,000 = $36,000

However, we need to subtract the tax-exempt interest from his total income:

Total income - Tax-exempt interest = $36,000 - $2,000 = $34,000

To calculate the taxable part of Walter's social security benefits, we take half of his social security benefits and add it to his total income:

Taxable part = (Half of social security benefits) + Total income

Taxable part = ($18,000 ÷ 2) + $34,000

Taxable part = $9,000 + $34,000 = $43,000

Since Walter's combined income is less than $34,000, none of his benefits will be considered taxable income. Therefore, the answer is that none of Walter's social security benefits will most likely be considered taxable income.

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Answer:

The statements that are true about triangle QRS are:

1. The side opposite ∠Q is RS.

2. The side opposite ∠R is RQ.

3. The hypotenuse is QR.

The side adjacent to ∠R is SQ, and the side adjacent to ∠Q is QS. However, these are not the correct terms to describe the sides in relation to the angles of the triangle. The side adjacent to ∠R is QR, and the side adjacent to ∠Q is SR.

Answer:

1. The side opposite ∠Q is RS.

2. The side opposite ∠R is RQ.

3. The hypotenuse is QR.

Step-by-step explanation:

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Let's say you have a set of cards representing different mathematical functions. Each card contains an expression, a verbal statement describing the function, a graphical model, and a table of values.

You can sort them into equivalent groups based on the type of function they represent, such as linear, quadratic, exponential, or trigonometric functions.

For example:

Group 1 (Linear Functions):

Expression: y = mx + b

Verbal Statement: "A function with a constant rate of change"

Model: Straight line with a constant slope

Table: A set of values showing a constant difference between consecutive y-values

Group 2 (Quadratic Functions): Expression: y = ax^2 + bx + c

Verbal Statement: "A function that represents a parabolic curve"

Model: U-shaped curve

Table: A set of values showing a non-linear pattern

Continue sorting the cards into equivalent groups based on the characteristics and properties of the functions they represent. Please note that this is just an example, and the actual sorting of the cards would depend on the specific set of cards you have and their content.

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A company has a revenue function R(x) = -4x²+10x and a cost function c(x) = 8.12x-10.8. To determine whether the company can break even, we need to find the value(s) of x where the revenue is equal to the cost. Hence after calculating we came to find out that the company can break even in two ways: when x is approximately -1.42375 or 1.89375.

To break even means that the company's revenue is equal to its cost, so we set R(x) equal to c(x) and solve for x:

-4x²+10x = 8.12x-10.8

We can start by simplifying the equation:

-4x² + 10x - 8.12x = -10.8

Combining like terms:

-4x² + 1.88x = -10.8

Next, we move all terms to one side of the equation to form a quadratic equation:

-4x² + 1.88x + 10.8 = 0

To solve this quadratic equation, we can use the quadratic formula:

x = (-b ± √(b²-4ac)) / (2a)

For our equation, a = -4, b = 1.88, and c = 10.8.

Plugging these values into the quadratic formula:

x = (-1.88 ± √(1.88² - 4(-4)(10.8))) / (2(-4))

Simplifying further:

x = (-1.88 ± √(3.5344 + 172.8)) / (-8)

x = (-1.88 ± √176.3344) / (-8)

x = (-1.88 ± 13.27) / (-8)

Now we have two possible values for x:

x₁ = (-1.88 + 13.27) / (-8) = 11.39 / (-8) = -1.42375

x₂ = (-1.88 - 13.27) / (-8) = -15.15 / (-8) = 1.89375

Therefore, the company can break even in two ways: when x is approximately -1.42375 or 1.89375.

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To find the date when the flu will have reached its peak and the number of people who will have the flu on that date, we need to determine the maximum value of the function N(t).

The function N(t) = 90 + (9/4)t - (1/40)t^2 - 120 is a quadratic function in terms of t. The maximum value of a quadratic function occurs at the vertex of the parabola.

To find the vertex of the parabola, we can use the formula t = -b/(2a), where a, b, and c are the coefficients of the quadratic function in the form ax^2 + bx + c.

In this case, a = -1/40, b = 9/4, and c = -120. Plugging these values into the formula, we have:

t = -(9/4)/(2*(-1/40))

Simplifying, we get:

t = -(9/4) / (-1/20)

t = (9/4) * (20/1)

t = 45

Therefore, the date when the flu will have reached its peak is 45 days from the beginning of December. To find the number of people who will have the flu on that date, we can substitute t = 45 into the equation:

N(45) = 90 + (9/4)(45) - (1/40)(45)^2 - 120

N(45) = 90 + 101.25 - 50.625 - 120

N(45) = 120.625

So, on the date 45 days from the beginning of December, approximately 120,625 people will have the flu.

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The value of c is -1, and the coordinates (n) at which the graphs of f and g touch each other are (1, 0). The position (E, n) refers to the point of tangency between the two graphs. The graph of g passes near the point of tangency above the graph of f.

To determine the value of c and the coordinates (n) at which the graphs of f and g touch each other, we need to find the point of tangency between the two curves. Given that f(x, y) = x+y and g(x, y) = x² + y² + c, we can set them equal to each other to find the common point of tangency:

x+y = x² + y² + c

Since the point of tangency is (x, y) = (1, 0), we substitute these values into the equation:

1 + 0 = 1² + 0² + c

1 = 1 + c

Simplify the equation to solve for c:

c = -1

The coordinates (n) at which the graphs touch each other are (1, 0).

The position (E, n) refers to the point of tangency, which in this case, is (1, 0).

To determine which graph passes near the point of tangency above the other, we compare the shapes of the graphs. The graph of f is a straight line, and the graph of g is a parabola.

By visualizing the graphs, we can see that the graph of g (the parabola) passes near the point of tangency (1, 0) above the graph of f (the straight line)

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The roots of the given equation X³ - 6x² + 11x - 6 = 0 are x = 1, x = 2, and x = 3.

To find the roots of the equation X³ - 6x² + 11x - 6 = 0, we can use various methods, such as factoring, synthetic division, or the rational root.

Let's use the rational root theorem to find the potential rational roots and then use synthetic division to determine the actual roots.

The rational root theorem states that if a polynomial equation has a rational root p/q, where p is a factor of the constant term and q is a factor of the leading coefficient, then p/q is a potential root of the equation.

The constant term is -6, and the leading coefficient is 1. So, the possible rational roots are the factors of -6 divided by the factors of 1.

The factors of -6 are ±1, ±2, ±3, ±6, and the factors of 1 are ±1.

The potential rational roots are ±1, ±2, ±3, ±6.

Now, let's perform synthetic division to determine which of these potential roots are actual roots of the equation:

1 | 1 -6 11 -6

| 1 -5 6

1  -5   6   0

Using synthetic division with the root 1, we obtain the result of 0 in the last column, indicating that 1 is a root of the equation.

Now, we have factored the equation as (x - 1)(x² - 5x + 6) = 0.

To find the remaining roots, we can solve the quadratic equation x² - 5x + 6 = 0.

Factoring the quadratic equation, we have (x - 2)(x - 3) = 0.

So, the roots of the quadratic equation are x = 2 and x = 3.

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It will take approximately 35 days before 1000 people are infected.

Initially, 100 people were diagnosed with the virus.

A virus is spreading at a rate of 4% each day.

Let us calculate how many days it will take for 1000 people to be infected.

Let us assume that x represents the number of days it will take for 1000 people to be infected.

Since the number of people infected increases by 4% each day, after one day, the number of people infected will be 100 × (1 + 0.04) = 104 people.

After two days, the number of people infected will be 104 × (1 + 0.04) = 108.16 people

.After three days, the number of people infected will be 108.16 × (1 + 0.04) = 112.4864 people.

Thus, we can say that the number of people infected after x days is given by 100 × (1 + 0.04)ⁿ.

So, we can write 1000 = 100 × (1 + 0.04)ⁿ.

In order to solve for n, we need to isolate it.

Let us divide both sides by 100.

So, we have:10 = (1 + 0.04)ⁿ

We can then take the logarithm of both sides and solve for n.

Thus, we have:

log 10 = n log (1 + 0.04)

Let us divide both sides by log (1 + 0.04).

Therefore:

n = log 10 / log (1 + 0.04)

Using a calculator, we get:

n = 35.33 days

Rounding this off, we get that it will take about 35 days for 1000 people to be infected.

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The correct expansion of the determinant of the given 3x3 matrix is: det [¹2 -4 3 -2 5 0] = 4det + det(1) - 2det [¹2] + 3det [¹] - 2det [¹2 -4 3 -2 5 0].

To expand the determinant of a 3x3 matrix, we use the formula:

det [a b c d e f g h i] = aei + bfg + cdh - ceg - bdi - afh.

For the given matrix [¹2 -4 3 -2 5 0], we can use the above formula to expand the determinant:

det [¹2 -4 3 -2 5 0] = (1)(5)(0) + (2)(-2)(3) + (-4)(-2)(0) - (-4)(5)(3) - (2)(-2)(0) - (1)(-2)(0).

Simplifying this expression gives:

det [¹2 -4 3 -2 5 0] = 0 + (-12) + 0 - (-60) - 0 - 0 = -12 + 60 = 48.

Therefore, the correct expansion of the determinant of the given matrix is: det [¹2 -4 3 -2 5 0] = 4det + det(1) - 2det [¹2] + 3det [¹] - 2det [¹2 -4 3 -2 5 0].

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