Which of the following is the Laplace transformation L {f(t)} if f(t)=(sin(-4t) 2 ? 32 33 +645 O None of them s2 $4 + 3252 +256 $+32 S + 64s ) 16 54 + 3252 +256

The required answer is  -

a. leaves a remainder of n² - 3n or (n - 6)(n + 7) ,the possible remainders are 0, 1, 4, 9, 5, 3

b .every n € Z that 121 | n² + 16n +20.

Explanation:-

(a) Remainder when n² + 16n + 20 is divided by 11:Let us first find out the value of n² + 16n + 20,n² + 16n + 20 = (n + 10)(n + 6) + 4As n and 11 are co-prime, by Fermat's Little Theorem, n¹⁰  ≡  1(mod 11)So, (n + 10)(n + 6) ≡ (n + 1)(n + 7) (mod 11)Hence, n² + 16n + 20 ≡ (n + 1)(n + 7) + 4 ≡ n² + 8n + 11 ≡ n² - 3n (mod 11).Therefore, when n² + 16n + 20 is divided by 11, it leaves a remainder of n² - 3n or (n - 6)(n + 7).

Thus, the possible remainders are 0, 1, 4, 9, 5, 3

(b) Prove that 121 | n² + 16n + 20 for every n ∈ Z . n ∈ Z, thenn² + 16n + 20 = (n + 8)² - 44(n + 8) + 324 = (n + 8 - 22)(n + 8 + 14) + 324= (n - 14)(n + 22) + 324.We know that 121 = 11² | (n - 14)(n + 22),

Therefore, 11 | (n - 14) or 11 | (n + 22)So, there exist k and l ∈ Z such that n - 14 = 11k or n + 22 = 11lWe have, n + 22 = n - 14 + 36Hence, n - 14 ≡ n + 22 (mod 11)If 11 | (n - 14), then 11 | (n + 22), if 11 | (n + 22), then 11 | (n - 14).Therefore, 121 | n² + 16n + 20. Thus, proved.

To know about Fermat's Little Theorem. To click the link .

https://brainly.com/question/30761350.

#SPJ11

The given quadratic expression is: $n^2 + 16n + 20$.

(a)We have to determine the possible remainders when this expression is divided by 11.

Let's solve for the remainder:$(n^2 + 16n + 20) \div 11$ $= \frac{(n + 8)^2 - 44}{11}$

We can write the given expression as $n^2 + 16n + 20 = (n + 8)^2 - 44$

Therefore,$\begin{aligned} (n^2 + 16n + 20) \div 11 & = \frac{(n + 8)^2 - 44}{11} \\ & = \frac{(n + 8)^2}{11} - 4 \end{aligned}$

For all possible values of $n$,

let's calculate the remainder of $\frac{(n + 8)^2}{11}$.

Let $a$ be the remainder when $n$ is divided by 11; then, $(n + 8) \div 11$ has a quotient of $q$ and a remainder of $r$. Therefore, $n + 8 = 11q + r$ $ \Right arrow n = 11q + r - 8$$\begin{aligned} n^2 & = (11q + r - 8)^2 \\ & = 121q^2 + r^2 + 64 + 22qr - 16q - 16r \end{aligned}$Let's now replace $n^2$

in the given expression:$(n^2 + 16n + 20) \div 11 = \frac{121q^2 + r^2 + 22qr - 16q + 6r + 64}{11}$

We have to find the remainders when $121q^2 + r^2 + 22qr - 16q + 6r + 64$ is divided by 11.

We observe that $121q^2$ is always divisible by 11, so we can ignore it.

We must find the remainders of $r^2 + 22qr + 6r + 64 - 16q$. $\begin{aligned} r^2 + 22qr + 6r + 64 - 16q & = r^2 + 2 \cdot 11qr + 11qr + 6r + 64 - 16q \\ & = (r + 11q)^2 + 6r - 16q + 64 \\ & = (r + 11q)^2 - 11(2q - r - 3) \end{aligned}$

The remainder is $r_0 = 11(2q - r - 3)$.

This remainder can take any value between $-10$ and $10$.

(b)We have to show that $121 | n^2 + 16n + 20$ for all integers $n$.We proved earlier that $n^2 + 16n + 20 = (n + 8)^2 - 44$.

We can restate this as $121 | (n + 8)^2 - 44$, which implies that $(n + 8)^2 - 44 = 121k$, where $k \in \mathbb{Z}$.Now we need to show that $k = \frac{n^2 + 16n + 20 - 121k}{121}$ is an integer.

Let's substitute $(n + 8)^2 - 44 = 121k$ into the given expression:$\begin{aligned} k & = \frac{(n + 8)^2 - 44}{121} \\ & = \frac{n^2 + 16n + 20 - 121k}{121} \\ & = \frac{n^2 + 16n + 20}{121} - k \end{aligned}$Thus, $k = \frac{n^2 + 16n + 20 - 121k}{121}$ is an integer for all $n \in \math bb{Z}$,

implying that $121 | n^2 + 16n + 20$ for all $n \in \math bb{Z}$.

Hence, we have proved that $121 | n^2 + 16n + 20$ for all $n \in \math bb{Z}$ using the method of mathematical induction.

Know more about quadratic expression:

https://brainly.com/question/10025464

#SPJ11

Answer:

[tex]y=2x-3[/tex]

Step-by-step explanation:

This is just algebraic manipulation. In order to solve for y, you need to isolate it. Start this by moving the 2x from the left side of the equation. You can do this by subtracting 2x from both sides and you should end up with:

[tex]-y=-2x+3[/tex]

After this, you still have a negative y, which means you just need to divide both sides of the equation by -1 to get rid of the negative. That should reverse the signs of all the variables in the equation, making it look like:

[tex]y=2x-3[/tex]

Answer:

Length = x + 3 meters

Step-by-step explanation:

Expression for the area of the rectangle = [tex]x^2 - 9 = (x + 3)(x - 3) m[/tex]

Expression for width of rectangle = ([tex]x - 3[/tex]) m

Area of a rectangle = [tex]Length \times Width[/tex]

⇒ Expression for length of rectangle = [tex]\frac{Area}{Width} = \frac{(x + 3)(x - 3)}{(x - 3)} = (x + 3) m[/tex]

The first three nonzero terms of two linearly independent solutions about x = 0 can be obtained by Taylor expanding the solutions in terms of the exponent r and truncating the series to the desired order.

To determine if x = 0 is a regular singular point of the differential equation xy'' + y = 0, we substitute y = x^r into the equation and solve for the exponent r. Differentiating y twice with respect to x, we have y'' = r(r - 1)x^(r - 2). Substituting these expressions into the differential equation, we get [tex]x(x^r)(r(r - 1)x^(r - 2)) + x^r = 0[/tex]. Simplifying, we obtain r(r - 1) + 1 = 0, which yields r^2 - r + 1 = 0. Solving this quadratic equation, we find that the exponents at the singular point x = 0 are complex and given by r = (1 ± i√3)/2.

To find the first three nonzero terms of two linearly independent solutions about x = 0, we can use the Taylor series expansion. Let's consider the solution y1(x) corresponding to the exponent r = (1 + i√3)/2. Expanding y1(x) as a series around x = 0, we have y1(x) =[tex]x^r = x^((1 +[/tex]i√3)/2) = x^(1/2) *[tex]x^(i√3/2[/tex]). Using the binomial series expansion and Euler's formula, we can write [tex]x^(1/2) and x^(i√3/2)[/tex] as infinite series.

Learn more about binomial series here:

https://brainly.com/question/29592813

#SPJ11

Answer:

x = 18

y = 27

Step-by-step explanation:

Answer:

x = 18

y = 27

Step-by-step explanation:

Answer:

Hi! The answer to your question is D. {(0,0),(0,1),(1,2)(1,3)}

Step-by-step explanation:

☆*: .｡..｡.:*☆☆*: .｡..｡.:*☆☆*: .｡..｡.:*☆☆*: .｡..｡.:*☆

☁Brainliest is greatly appreciated!☁

Hope this helps!!

- Brooklynn Deka

Answer: z= -0.5

Step-by-step explanation: mean m=120 and deviation s=10.

Z = (x-m)/s= (115-120)/10

Answer: z=−1.1

Step-by-step explanation:

z=\frac{x-\mu}{\sigma}

z=

σ

x−μ

​

z-score formula

z=\frac{104-115}{10}

z=

10

104−115

​

Plug in values

z=\frac{-11}{10}

z=

10

−11

​

Subtract

z=-1.1

z=−1.1

Divide

Answer:

22.8906 sq in

Step-by-step explanation:

SA of a sphere = 4 π [tex]r^{2}[/tex]

r = 2.7 / 2 = 1.35

4 x 3.14 x [tex]1.35^{2}[/tex] = 22.8906

Answer:

C

Step-by-step explanation:

y=mx+b

3 = -2(3) + b

3 = -6 + b

9 = b

y = -2x + 9

2x + y = 9

Answer: C.

Step-by-step explanation: To find the volume of something, you multiply the length, width, and height together. Since they have already multiplied the length and width together to form the base, multiply 7 by 12 (base times height) to get 84 cubic inches.

The holomorphic function f whose real part is u(x, y) = e^-2xy sin(x² - y²) is given by f(z) = e^(-z²)sin(z²).

This function is holomorphic because it satisfies the Cauchy-Riemann equations. The Cauchy-Riemann equations relate the partial derivatives of the real and imaginary parts of a holomorphic function with respect to the variables x and y.

In this case, the real part of f is u(x, y) = e^-2xy sin(x² - y²), and the imaginary part of f is v(x, y) = e^-2xy cos(x² - y²). By computing the partial derivatives of u and v with respect to x and y and checking that they satisfy the Cauchy-Riemann equations, we can verify that f is indeed holomorphic.

Know more about holomorphic function here:

https://brainly.com/question/30858417

#SPJ11

Answer:

My brain...

Step-by-step explanation:

Answer:

i believe its A on plato

Step-by-step explanation:

Answer. 2, 3, 5, 7 , 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

Step-by-step explanation:

There are 21 prime number between 100 and 200.

A number system is defined as a system of writing to express numbers.

A prime number is a whole number greater than 1 whose only factors are 1 and itself.

The prime numbers between 100 and 200 are

101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199.

One hundred one, one hundred three, one hundred seven, one hundred nine, One hundred thirteen, one hundred twenty seven, One hundred thirty one, one hundred thirty seven, one hundred thirty nine, one hundred forty nine, one hundred fifty one, one hundred fifty seven, one hundred sixty three,  one hundred sixty seven, one hundred seventy three, one hundred seventy  nine, one hundred eighty one, one hundred ninty one, one hundred ninty three, one hundred ninty seven, one hundred ninty nine.

There are no numbers  between 100 and 200 have 11 as a prime factor.

Hence there are 21 prime number between 100 and 200.

To learn more on Number system click:

https://brainly.com/question/22046046

#SPJ2

Answer:

.

Step-by-step explanation:

a) The equation for the tangent at x = 0 is y - 2 = -42x.

(b) The equation for the tangent at x = 1 is y + 38 = -38x + 38.

(c) The equation for the tangent at x = 2 is y + 62 = -34x + 68.

(d) When x  = 1,the value of f(x) is -38.  This is the y-  coordinate of the point on the tangent line at x = 1.

To find the equations for the   tangents of the function f (x) = 2x² – 42x + 2 at specific points,we need to calculate the derivative of the function first. Let's differentiate f(x) with respect to x .

f(x ) =   2x² – 42x+ 2

f'(x) = d/  dx (2x²) – d/dx (42x)+ d/dx (2)

f'(x) = 4x – 42

So

(a) Tangent at x = 0  -

To find the equation for the tangent at x  = 0 we need to evaluate f'(x) at x = 0.

f'(0) = 4(0) – 42

f'(0) = -42

Since the slope of the tangent is -42 at x = 0, the equation of the tangent can be written in point-slope form using the point (0, f(0)).

Using f(0) = 2(0)² – 42(0) + 2 = 2, the equation for the tangent at x = 0 is -

y - 2 = -42(x - 0)

y - 2 = -42x

(b) Tangent at x = 1:

To find the equation for   the tangent at x = 1,we need to evaluate f'(x) at x = 1.

f'(1  ) =4(1) – 42

f'(1) = -38

Using f(1) = 2(1)² – 42(1) + 2 = -38, the equation for the tangent at x = 1 is  -

y + 38 = -38(x - 1)

y + 38 = -38x + 38

(c) Tangent at x = 2

To   find the equation for the tangent at x = 2,we need to evaluate f'(x) at x = 2.

f'(2) = 4(2) – 42

f'(2) = -34

Using f(2) = 2(2)^2 – 42(2) + 2 = -62, the equation for the tangent at x = 2 is:

y + 62 = -34(x - 2)

y + 62 = -34x + 68

(d)  To determine what is special about f when x = 1, we can evaluate the function at x = 1

f(1) = 2(1)² – 42(1) + 2

f(1) = -38

When x = 1,the value of f(x) is -38. This is the y  -coordinate of the point on the tangent line at x = 1. Therefore, the special property of f when x = 1 is that the function value is -38 at that point.

Learn more about tangent:
https://brainly.com/question/4470346
#SPJ4

Answer:

50

Step-by-step explanation:

100÷1/2=50 so your answer would be 50

If 19,000=19% Then 100,000=100%

100,000-900=99,100

Answer: $99,100 was his salary last year

Answer:

9

plz mark as brainliest

Answer:

9

Step-by-step explanation:

[tex]\sqrt{81} =9[/tex]

9 x 9 = 81

Answer:

area = 84 in²

Step-by-step explanation:

area = (9x12) - (6x8x0.5) = 84 in²

Answer:

72.25663

Step-by-step explanation:

C=2πr=2·π·11.5≈72.25663

a) The correlation coefficient (r) is approximately 0.300, indicating a weak positive linear relationship between the ages of Best Actresses and Best Actors.

b) Based on the correlation coefficient (r), there is a weak positive linear correlation between the ages of Best Actresses and Best Actors, suggesting that as the ages of Best Actresses increase, the ages of Best Actors also tend to increase, but the relationship is not very strong.

To find the correlation coefficient (r), we can use the given ages of Best Actresses and Best Actors. The correlation coefficient measures the strength and direction of the linear relationship between two variables. Using a calculator or statistical software, we calculate the correlation coefficient to be approximately 0.300.

Based on the correlation coefficient (r) of approximately 0.300, there is a weak positive linear correlation between the ages of Best Actresses and Best Actors. This means that there is a tendency for the ages of Best Actresses and Best Actors to increase together, but the relationship is not very strong. The correlation coefficient ranges from -1 to +1, where 0 indicates no linear correlation, 1 indicates a strong positive linear correlation, and -1 indicates a strong negative linear correlation. In this case, the value of 0.300 suggests a weak positive linear relationship, indicating that as the ages of Best Actresses increase, the ages of Best Actors also tend to increase, albeit not strongly.

Learn more about Correlation

brainly.com/question/28898177

#SPJ11

Answer:

D. 2992 cubic inches of water

Step-by-step explanation:

Given data

Base Area= 187in^2

Height of 16 inches

The expression for the volume of a rectangular shaped tank is

V= Area*Height

substiute

V= 187*16

V=2992 in^3

Hence the volume is 2992 cubic inches of water

The statement is false. The appropriate differential equation for the quantity Q of toxin in the tank is not given by dQ/dt = r (c/Vo).

In a tank problem with equal inflow and outflow rates and a constant input concentration c of a toxic substance, the appropriate differential equation for the quantity Q of toxin in the tank is dQ/dt = r(c - Q/Vo), where r is the inflow/outflow rate and Vo is the total volume of the mixture in the tank.

The term (c - Q/Vo) represents the difference between the input concentration and the concentration in the tank, scaled by the volume Vo. This equation accounts for the fact that the concentration in the tank changes over time due to the inflow of fresh mixture with concentration c and the outflow of the mixture with concentration Q/Vo.

Therefore, the correct differential equation is dQ/dt = r(c - Q/Vo), which reflects the balance between the inflow and outflow of the toxic substance and its accumulation or depletion in the tank.

Learn more about concentration here:

https://brainly.com/question/3045247

#SPJ11

Answer:

3x+4y=3

Step-by-step explanation:

Plug this in desmos graphing calculator.

The complex number Z = (20 + 120i) can be expressed in polar form as Z = 2√370(cos(1.405) + isin(1.405)).

To express the complex number Z = (20 + 120i) in polar form, we need to find its magnitude (r) and argument (θ).

The magnitude of a complex number Z = a + bi is given by the formula:

|r| = √(a^2 + b^2)

In this case, a = 20 and b = 120.

Therefore, the magnitude of Z is:

|r| = √(20^2 + 120^2) = √(400 + 14400) = √14800 = 2√370.

The argument (θ) of a complex number Z = a + bi is given by the formula:

θ = arctan(b/a)

In this case, a = 20 and b = 120. Therefore, the argument of Z is:

θ = arctan(120/20) = arctan(6) ≈ 1.405 radians.

Now we can express Z in polar form as Z = r(cosθ + isinθ), where r is the magnitude and θ is the argument:

Z = 2√370(cos(1.405) + isin(1.405)).

To know more about polar form refer here:

https://brainly.com/question/11741181#

#SPJ11