19. J and ZK are supplementary. The measure of ZJ is (9x) and the measure of ZK is 45°. What is the value of x?​

The discriminant of the function [tex]f(x,y)[/tex] cannot be defined, as it does not have any quadratic terms. However, we can compute the Hessian matrix of [tex]f(x,y)[/tex], which can be used to analyze the critical points and extrema of the function.

The discriminant of a function [tex]f(x,y)[/tex] is typically defined as the expression [tex]B^2 - 4AC[/tex], where A, B, and C are coefficients of the quadratic terms in the function. However, in this case, the function [tex]f(x,y)[/tex] does not have any quadratic terms, so it is not clear how to define the discriminant.

If you meant to ask for the Hessian matrix of f, which is a square matrix of second-order partial derivatives of f, then we can compute it as follows:

[tex]f(x,y) = \frac{5x^2 y^2}{4x^2 + 10y^2}[/tex]

Taking partial derivatives with respect to x and y, we get:

[tex]f(x,y) = \frac{25xy^2}{2(2x^2+5y^2)^2}[/tex]

[tex]$f(x,y) = \frac{25yx^2}{2(5x^2+2y^2)^2}$[/tex]

Taking partial derivatives of these functions with respect to x and y again, we get:

[tex]$f_{xx}(x,y) = \frac{25y(15y^2-8x^2)}{2(2x^2+5y^2)^3}$[/tex]

[tex]$f_{yy}(x,y) = \frac{25x(15x^2-8y^2)}{2(5x^2+2y^2)^3}$[/tex]

[tex]$f_{xy}(x,y) = \frac{25xy(20x^2-20y^2)}{2(2x^2+5y^2)^2(5x^2+2y^2)^2}$[/tex]

The Hessian matrix of [tex]f(x,y)[/tex] is then:

[tex]$H_f(x,y) = \left| \begin{matrix} f_{xx}(x,y) & f_{xy}(x,y) \ f_{xy}(x,y) & f_{yy}(x,y) \end{matrix} \right|$[/tex]

[tex]$\mathbf{Hf}(x,y) = \begin{pmatrix}\frac{25y(15y^2-8x^2)}{2(2x^2+5y^2)^3} & \frac{25xy(20x^2-20y^2)}{2(2x^2+5y^2)^2(5x^2+2y^2)^2} \\frac{25xy(20x^2-20y^2)}{2(2x^2+5y^2)^2(5x^2+2y^2)^2} & \frac{25x(15x^2-8y^2)}{2(5x^2+2y^2)^3}\end{pmatrix}$[/tex]

[tex]$\left| \frac{25xy(20x^2-20y^2)}{2(2x^2+5y^2)^2(5x^2+2y^2)^2} \cdot \frac{25x(15x^2-8y^2)}{2(5x^2+2y^2)^3} \right|$[/tex]

Simplifying this expression is possible, but it is quite lengthy and not particularly insightful. Instead, we can make some observations about the Hessian matrix:

The Hessian matrix is symmetric, since [tex]fxy(x,y) = fyx(x,y)[/tex]

The Hessian matrix is continuous and has continuous first-order partial derivatives everywhere, except at the origin (0,0), where the denominator of the expression for f is zero.

Therefore, we can conclude that the discriminant of [tex]f(x,y)[/tex] does not exist at the origin, since the Hessian matrix is not defined there. Outside of the origin, the Hessian matrix is well-defined and can be used to analyze the critical points and extrema of the function [tex]f(x,y)[/tex].

To learn more about Hessian matrix

https://brainly.com/question/31850779

#SPJ4

The characteristic polynomial of the given matrix is λ(λ - 3055).

The characteristic polynomial of a matrix is obtained by taking the determinant of the matrix subtracted by a scalar multiplied by the identity matrix. In this case, the given matrix is a 2x2 matrix. Therefore, the characteristic polynomial can be obtained by:

det(a - λI) =

| 3055 - λ    -5 |

|   -1    0 - λ |

= (3055 - λ) * (-λ) - (-5 * -1)

= λ^2 - 3055λ

= λ(λ - 3055)

Therefore, the characteristic polynomial of the given matrix is λ(λ - 3055).

to learn more about polynomial click here:

brainly.com/question/11154053

#SPJ11

There are 14.25 liters in 15 quarts based on the expression and data given.

To find the number of liters in 15 quarts, we can use the conversion factor given in the table for quarts to liters. The table states that 1 quart is equal to 0.95 liters.

To convert 15 quarts to liters, we can set up the following expression:

Number of liters = (Number of quarts) × (Conversion factor)

In this case:
Number of liters = 15 quarts × 0.95 liters/quart

Now, you can simply multiply 15 by 0.95 to find the number of liters:
Number of liters = 15 × 0.95 = 14.25 liters

So, there are 14.25 liters in 15 quarts.

Learn more about liters here:

https://brainly.com/question/5827510

#SPJ11

To find the area of the region that lies inside both curves, we need to determine the points where the two curves intersect. The area of the region that lies inside both curves is approximately 15.56 square units.

We know that r = 5 sin() and r = 5 cos() represent a circle and a cardioid, respectively, both centered at the origin with a radius of 5.

At the intersection points, we have:

5 sin() = 5 cos()

Simplifying, we get:

tan() = 1

This means that the intersection points occur at  = /4 and 5/4.

To find the area of the region inside both curves, we can set up the following integral:

∫[0, /4] (1/2)(5 sin())^2 d

This represents the area of the sector of the circle from 0 to /4, minus the area of the triangular region between the x-axis, the y-axis, and the line = /4.

Similarly, we can set up the following integral for the region between /4 and 5/4:

∫[/4, 5/4] (1/2)(5 cos())^2 d

This represents the area of the region between the cardioid and the x-axis from /4 to 5/4.

Adding the two integrals together gives us the total area of the region inside both curves:

A = ∫[0, /4] (1/2)(5 sin())^2 d + ∫[/4, 5/4] (1/2)(5 cos())^2 d

Evaluating the integrals, we get:

A = (25/8)(2 + 5/3) + (25/8)(2 - 5/3)

A = 15.56

Therefore, the area of the region that lies inside both curves is approximately 15.56 square units.

To find the area of the region that lies inside both curves r = 5 sin(θ) and r = 5 cos(θ), you can follow these steps:

1. Determine the points of intersection by setting the two equations equal to each other: 5 sin(θ) = 5 cos(θ).
2. Divide both sides by 5: sin(θ) = cos(θ).
3. Recognize that this occurs when θ = π/4 and 5π/4.
4. Use the polar area formula: A = (1/2)∫(r^2)dθ.
5. Set up two integrals, one for each region, and add their areas: A = (1/2)∫[ (5 sin(θ))^2 - (5 cos(θ))^2 ] dθ from π/4 to 5π/4.
6. Perform the integration and evaluate the integral at the given limits to obtain the area.

Following these steps, the area of the region that lies inside both curves r = 5 sin(θ) and r = 5 cos(θ) is 25/2 square units.

Learn more about area at: brainly.com/question/27683633

#SPJ11

The expression that is equivalent to 86x²y³ is √86x√y³. Option D.

The expression 86x²y³ means multiplying the terms 86, x squared (x²), and y cubed (y³) together.

To simplify this expression using the square root notation, we can break it down as follows:

So, combining both parts, the equivalent expression becomes √(86x)√(y³). This represents taking the square root of 86x and multiplying it by the square root of y³.

Therefore, the correct option that represents 86x²y³ is √(86x)√(y³).

More on simplification can be found here: https://brainly.com/question/23509407

#SPJ9

A good point estimate for the population proportion is 0.01714. Option D is correct.

A good point estimate to use for the population proportion to test the effectiveness of the flu vaccine can be calculated by dividing the number of people who got the flu (12) by the total number of participants in the study (700).

Point Estimate = Number of people who got the flu / Total number of participants

Point Estimate = 12 / 700

Using a calculator, the approximate value of the point estimate is 0.01714 (rounded to five decimal places).

Therefore, the good point estimate to use for the population proportion to test the effectiveness of the flu vaccine is 0.01714. Option D is correct.

Learn more about population proportion https://brainly.com/question/15087042

#SPJ11

Answer:

Step-by-step explanation:

The quotient (integer division) of 733/38 equals 19; the remainder (“left over”) is 11. 733 is the dividend, and 38 is the divisor.

Rounding to the nearest hundredth, the area of the sector is approximately 152.30 cm².

To find the area of a sector, you can use the formula:

[tex]Area = (\theta /360) \times \pi \times r^2[/tex]

where θ is the central angle and r is the radius.

Plugging in the given values:

θ = 144°

r = 11 cm

π = 3.14

[tex]Area = (144/360) \times 3.14 \times 11^2[/tex]

Simplifying:

[tex]Area = (0.4) \times 3.14 \times 121[/tex]

[tex]Area = 48.4 \times 3.14[/tex]

[tex]Area = 152.296 cm^2.[/tex]

For similar question on area.

https://brainly.com/question/26870235

#SPJ11

The two methods for recovering the cost of investment in natural resources are depletion and depreciation.

Depletion is the method used to recover the cost of using natural resources, such as minerals, oil, and gas, by reducing the reserves' value each year based on the amount of resources extracted. Depreciation, on the other hand, is used to recover the cost of investments in long-lived assets such as buildings, equipment, and vehicles, by gradually reducing their value over time.
Therefore, the larger amount of cost that can be recovered on an annual basis for the investment in natural resources would depend on the specific circumstances of the investment, including the type of natural resource, the amount of reserves, and the expected life of the investment. In general, depletion is likely to result in a larger annual recovery than depreciation, as natural resources tend to have a finite supply that is depleted over time. However, both methods can be used in combination to maximize the recovery of investment costs over time.

To know more about cost visit:

https://brainly.com/question/30045916

#SPJ11

Answer:

-4.9

Step-by-step explanation:

have a nice day and can you mark me Brainliest.

Referring to the truth table, -P is false, and the only case with -P false is when both T and S are false.

This means that 5657 is not a triangular or square number.

(a) To create a truth table for the statement "If a number is triangular or square (T ∨ S), then it is not prime (-P)," we will have columns for T, S, T ∨ S, and -P.

Then we will consider all possible combinations of T and S (true and false) and fill out the remaining columns.
| T   | S   | T ∨ S | -P  |
|-----|-----|-------|-----|
| T   | T   | T     | T   |
| T   | F   | T     | T   |
| F   | T   | T     | T   |
| F   | F   | F     | F   |

(b) If the statement were false, a counterexample would need to have T ∨ S true, but -P false.

In other words, a number that is either triangular or square, and also prime. However, no such counterexample exists in the table, indicating that the statement is true.
(c) Since the statement is true, knowing that 5657 is prime (P) allows us to conclude that it is neither triangular (T) nor square (S).

For similar question on truth table.

https://brainly.com/question/27989881

#SPJ11

The answer is B) 3.250. It is important to note that the critical value depends on the level of significance (c) and the degrees of freedom (n-1).

To find the critical value, tc, for c = 0.99 and n = 10, we need to use the t-distribution table. Since we are dealing with a two-tailed test, we need to find the value that splits the distribution into two parts, each with an area of 0.005 (0.99/2 = 0.495, and 1 - 0.495 = 0.005). Looking at the table, we can see that for 9 degrees of freedom (n-1) and a probability of 0.005, the critical value is 3.250. Therefore, the answer is B) 3.250. It is important to note that the critical value depends on the level of significance (c) and the degrees of freedom (n-1). As the level of significance increases, the critical value increases as well. Similarly, as the degrees of freedom increase, the critical value decreases.

Learn more about critical value here:

https://brainly.com/question/30168469

#SPJ11

The margin of error in the report is 2.35 minutes.

The margin of error is a measure of the amount of uncertainty or error associated with a survey or study's results. It represents the range within which the true value of a population parameter is likely to lie, given the sample size and sampling method used. In this case, the report provides a range for the average amount of time a 10-year-old American child spends playing outdoors per day. The margin of error can be calculated as half the width of the confidence interval, which is (24.78 - 20.08)/2 = 2.35 minutes. This means that if the study were repeated many times, 95% of the time the true average time spent playing outdoors per day for 10-year-old American children would be within 2.35 minutes of the reported range.

To learn more about margin of error : brainly.com/question/29419047

#SPJ11

Answer:

Unclear

Step-by-step explanation:

The question is not clear. Are you asking for the unshaded fraction of b? If yes, then answer is 3/8. You subtract the shaded portion from the total fraction

The vertices that the dice have are 34

From the question, we have the following parameters that can be used in our computation:

Faces = 21

Edges = 53

The Euler's formula states that

V - E + F = 2

The above equation relates the number of vertices V, the number of edges E, and the number of faces F, of a polyhedron.

So, we have

V = 2 + E - F

Shen the given values are substituted in the above equation, we have the following equation

V = 2 + 53 - 21

Evaluate

V = 34

Hence, the vertices are 34

Read more about Euler's formula at

brainly.com/question/24300924

#SPJ1

Answer:

Step-by-step explanation:

Real-world Scenario: Building Renovation

Pre-Image: A rectangular building with vertices (0, 0), (0, 4), (6, 4), and (6, 0).

Transformation 1: Rotation of 90 degrees counterclockwise about the origin.

Image: A rotated building with vertices (0, 0), (-4, 0), (-4, 6), and (0, 6).

Transformation 2: Translation 3 units to the right and 2 units upward.

Final Image: The renovated building located at (3, 2), (-1, 2), (-1, 8), and (3, 8).

Real-world Scenario: Garden Design

In this scenario, let's consider a garden design project. The pre-image represents the initial layout of the garden, and the image represents the final design after undergoing certain transformations. The garden is represented on a coordinate plane, with the x-axis representing the horizontal distance and the y-axis representing the vertical distance.

Pre-Image Description:

The pre-image consists of a square-shaped garden with its bottom-left vertex located at (0, 0) and its top-right vertex located at (4, 4). The sides of the square are parallel to the axes.

Pre-Image Sketch:

Transformation 1: Translation

To create an interesting design, the garden needs to be moved 2 units to the right and 3 units upwards. This can be achieved through a translation.

Translation Equation:

x' = x + 2

y' = y + 3

Transformation 1 Calculation:

For the pre-image coordinates, applying the translation equations, we have:

New bottom-left vertex: (0 + 2, 0 + 3) = (2, 3)

New top-right vertex: (4 + 2, 4 + 3) = (6, 7)

Image Description:

The image represents the garden after the translation. The square-shaped garden has been shifted 2 units to the right and 3 units upwards.

Image Sketch:

Transformation 2: Reflection

To further enhance the design, a reflection is applied to the image. The reflection is performed over the x-axis.

Reflection Equation:

x' = x

y' = -y

Transformation 2 Calculation:

Applying the reflection equations to the translated coordinates, we have:

New bottom-left vertex: (2, -3)

New top-right vertex: (6, -7)

Image Description:

The image represents the garden after the translation and reflection. The square-shaped garden has been shifted 2 units to the right and 3 units upwards, and then reflected over the x-axis.

Image Sketch:

In summary, we started with a square-shaped garden in the pre-image, located at (0, 0) and (4, 4). We applied a translation of 2 units to the right and 3 units upwards, resulting in a new position of (2, 3) and (6, 7). Then, we reflected the translated garden over the x-axis, resulting in a final design with coordinates (2, -3) and (6, -7). The final image represents the transformed garden layout.

for such more question on Garden Design

https://brainly.com/question/17335144

#SPJ11

The vectors T, N, and B at the given point are: T = (4/3, -2, 0), N = (-2/3, -4/3, 1), and B = (2/3, -4/3, -2).

To find the vectors T, N, and B, we need to find the unit tangent vector T, the unit normal vector N, and the unit binormal vector B at the given point. First, we find the first derivative of the vector function r(t) to get the tangent vector: r'(t) = (2t, 2t^2, 1). Then we evaluate it at t = 1 to get r'(1) = (2, 2/3, 1). To find the unit tangent vector T, we divide r'(1) by its magnitude: T = (2/3, 2/9, 1/3).

Next, we find the second derivative of r(t) to get the curvature vector: r''(t) = (2, 4t, 0). Then we evaluate it at t = 1 to get r''(1) = (2, 4, 0). To find the unit normal vector N, we divide r''(1) by its magnitude and negate it: N = (-2/3, -4/3, 1). Finally, we find the cross product of T and N to get the unit binormal vector B: B = (2/3, -4/3, -2).

Therefore,  T = (4/3, -2, 0), N = (-2/3, -4/3, 1), and B = (2/3, -4/3, -2)) is the answer.

You can learn more about vectors at

https://brainly.com/question/27854247

#SPJ11

The volume of the solid is (5π/3) cubic units.

We have

To find the volume of the solid obtained by rotating the region under the curve y = x³ about the line x = -1 over the interval [0, 1], we can use the method of cylindrical shells.

The formula for the volume using cylindrical shells is given by:

V = 2π ∫ [a, b] x h(x) dx,

where [a, b] is the interval of integration, x represents the variable of integration, and h(x) represents the height of the shell at each value of x.

In this case,

We want to rotate the curve y = x³ about the line x = -1 from x = 0 to x = 1. Since we are rotating about a vertical line, the height of the shell at each value of x will be given by the difference between the x-coordinate of the curve and the line of rotation:

h(x) = (x - (-1)) = x + 1.

The interval of integration is [0, 1], so we can set up the integral as follows:

V = 2π ∫ [0, 1] x (x + 1) dx.

Now we can evaluate this integral:

V = 2π ∫ [0, 1] (x² + x) dx

= 2π [x³/3 + x²/2] evaluated from 0 to 1

= 2π [(1/3 + 1/2) - (0/3 + 0/2)]

= 2π [(2/6 + 3/6) - 0]

= 2π (5/6)

= (10π/6)

= (5π/3).

Therefore,

The volume of the solid is (5π/3) cubic units.

Learn more about the volume of solids here:

https://brainly.com/question/31259146

#SPJ1

Therefore, False. Fourier series are still very useful in many applications, despite their difficulty handling discontinuities.

False. While it is true that Fourier series have difficulty handling discontinuities, they are still extremely useful in many applications, particularly in the field of signal processing. The Taylor series, on the other hand, is primarily used for approximating functions near a specific point. Therefore, both series have their own unique strengths and weaknesses, and their usefulness depends on the specific context and application.

Therefore, False. Fourier series are still very useful in many applications, despite their difficulty handling discontinuities.

To know more about statement visit :

https://brainly.com/question/27839142

#SPJ11

a) By using the mean and standard deviation of the normal distribution, we can calculate the specific values that separate the top 2% from the bottom 98% of shark attacks per year and determine the number of shark attacks that constitute the middle 80%.

b) The numbers of shark attacks per year that constitute the middle 80% of shark attacks per year is 90% percent.

(1) To determine the number of shark attacks per year that separates the top 2% from the bottom 98%, we need to find the value that corresponds to the upper 2nd percentile of the distribution.

First, let's find the z-score corresponding to the 2nd percentile. The z-score measures the number of standard deviations a particular value is away from the mean.

To find the z-score corresponding to the 2nd percentile, we need to look up the z-score associated with a cumulative probability of 0.98. This value can be obtained from a standard normal distribution table or using statistical software.

Once we have the z-score, we can solve the equation for x to find the corresponding number of shark attacks per year.

x = z * σ + μ

Substituting the values we have:

x = z * 10.0 + 31.8

This will give us the number of shark attacks per year that separates the top 2% from the bottom 98%.

(2) To determine the number of shark attacks per year that constitute the middle 80% of shark attacks per year, we need to find the values that represent the lower and upper boundaries of this range. In other words, we want to find the numbers of shark attacks per year that lie within the 10th and 90th percentiles of the distribution.

Using the same process as before, we can find the z-scores corresponding to the 10th and 90th percentiles. We then use these z-scores to calculate the corresponding values of x using the formula:

x = z * σ + μ

The resulting values will represent the number of shark attacks per year that constitute the middle 80% of the distribution.

To know more about deviation here

https://brainly.com/question/16555520

#SPJ4

Answer: 6

Step-by-step explanation:

 the reason why is because all you got to do is subtract and then you get the answer step 1: forget what color they are step 2: add 5+3=8 then from 8 subtract 2 and the answer is 7 your welcome.

The length of the hypotenuse is 30 units.

The missing length is m = 5.

7.

As per the shown figure, we have

Perpendicular = 15√3

Angle = 60°

We have to determine the length of the hypotenuse.

Using the sine ratio, we can write:

sinθ = Perpendicular / Hypotenuse

Substitute the values in the above formula:

sin60° = 15√3 / Hypotenuse

Hypotenuse = 15√3 / sin60°

Hypotenuse =  15√3 /√3/2

Hypotenuse = 30 units

8.

Using the sine ratio, we can write:

sinθ = P / H

Substitute the values in the above formula:

sin45° = m / 5√2

1/√2 = m / 5√2

m = 5√2/√2

m = 5

Learn more about the trigonometric ratio here:

https://brainly.com/question/31662214

#SPJ1

(a) False. If the null hypothesis is rejected using ANOVA, it means that there is significant evidence to suggest that at least one of the means is different from the others.

However, it does not necessarily mean that all the means are different from one another. Additional tests such as pairwise comparisons are needed to determine which specific means are significantly different from one another.

(b) True. When ANOVA is used, the F-statistic is calculated by dividing the variability between groups by the variability within groups. Therefore, if the null hypothesis is rejected, it means that the variability between groups is higher than the variability within groups.

(c) True. Pairwise comparisons are used to determine which specific means are significantly different from one another. If the null hypothesis is rejected using ANOVA, then pairwise comparisons will identify at least one pair of means that are significantly different.

(d) True. When conducting pairwise comparisons, it is important to adjust the significance level to account for the multiple comparisons being made. In this case, there are four groups, so the appropriate level of significance is 0.05/4 = 0.0125. This helps to control the overall type I error rate.

Learn more about null hypothesis here:

https://brainly.com/question/28920252

#SPJ11

The cmy values for red are (0%, 100%, 100%). This means that there is no cyan present, but there is 100% magenta and 100% yellow present to create the color red.

Cyan, Magenta, and Yellow, the three primary colours used in subtractive colour mixing, are abbreviated as cmy values. A value between 0 and 100 is used to represent each colour and denote the proportion of that colour in a given colour mixing.

It is important to note that values like these are used in color printing to determine the amounts of cyan, magenta, and yellow inks needed to create a certain color. These values reflect the values of a subtractive color model, where colors are created by subtracting light from white.

Learn more about cmy values here:

https://brainly.com/question/30908442

#SPJ11

Answer:

3/26

Step-by-step explanation:

there are 52 cards in a standard deck

There are 4 queens.

there are two black 3 cards.

So 6 cards out of 52 are either a queen or a black 3.

So 6/52 = 3/26

Applying the Gaussian Elimination variant to the augmented matrix of the given system of linear equations results in the solution x = 1, y = -2, and z = 2.

Explanation: To solve the system of linear equations using Gaussian Elimination, we first represent the system in augmented matrix form, where the coefficients of the variables and the constants are organized in a matrix. The augmented matrix for the given system is:

[1   0   5   |   11]

[-2  2   4   |   -4]

[3   7   0   |   -1]

We start by performing row operations to eliminate the coefficients below the main diagonal. First, we multiply the first row by 2 and add it to the second row, resulting in the modified matrix:

[1   0   5   |   11]

[0   2   14  |   18]

[3   7   0   |   -1]

Next, we multiply the first row by -3 and add it to the third row, giving us:

[1   0   5   |   11]

[0   2   14  |   18]

[0   7   -15 |   -34]

We can now eliminate the coefficient 7 in the third row by multiplying the second row by -7 and adding it to the third row:

[1   0   5   |   11]

[0   2   14  |   18]

[0   0   -113|   -250]

At this point, we have an upper triangular matrix. We can back-substitute to find the values of the variables. From the last row, we find that -113z = -250, which implies z = 250/113. Substituting this value into the second row, we get 2y + 14z = 18, which gives us y = -2. Finally, substituting the obtained values of y and z into the first row, we find x + 5z = 11, which gives us x = 1. Therefore, the solution to the system of linear equations is x = 1, y = -2, and z = 2.

Learn more about Gaussian Elimination here:

https://brainly.com/question/29004583

#SPJ11

Let $n_3$ be the number of Sylow 3-subgroups and $n_5$ be the number of Sylow 5-subgroups in the group $G$.

Since the Sylow 3-subgroup is normal, we have $n_3=1$ or $n_3=10$. Also, $n_5$ must be either $1$ or $6$ or $10$ or $60$ (using the Sylow theorems).

Assume for a contradiction that $n_5 \neq 1$. Then $n_5$ must be either $6$ or $10$ or $60$. We will show that each of these cases leads to a contradiction.

Case 1: $n_5=6$

Let $P_5$ be a Sylow 5-subgroup. Then $|P_5|=5$ and $|G:P_5|=12$. By the same argument as in the previous solution, we get a group homomorphism $\varphi : G \to S_{12}$ whose kernel is contained in $P_5$. Since $n_3=1$, the group $G$ has a normal Sylow 3-subgroup.

By the same argument as in the previous solution, we get that the image of $\varphi$ is contained in $A_{12}$. But $A_{12}$ has no element of order $5$, which contradicts the fact that $P_5$ acts on $G/P_5$ with $5$ fixed points.

Case 2: $n_5=10$

Let $P_5$ be a Sylow 5-subgroup. Then $|P_5|=5$ and $|G:P_5|=6$. By the same argument as in the previous solution, we get a group homomorphism $\varphi : G \to S_6$ whose kernel is contained in $P_5$. Since $n_3=1$, the group $G$ has a normal Sylow 3-subgroup.

By the same argument as in the previous solution, we get that the image of $\varphi$ is contained in $A_6$. But $A_6$ has no subgroup of order $5$, which contradicts the fact that $P_5$ is nontrivial.

Case 3: $n_5=60$

Let $P_5$ be a Sylow 5-subgroup. Then $|P_5|=5$ and $|G:P_5|=1$. This implies that $P_5$ is a normal subgroup of $G$, which completes the proof.

learn more about subgroup here:brainly.com/question/31432778

#SPJ11

The evaluated integral is ∫(0, sqrt(3/2)) 35x^2/ √(1 − x^2) dx = -35√(1 − (sqrt(3/2))^2) + 35√(1 − 0) = -35√(1/2) + 35 = 35(1 − √(1/2))

We can solve this integral using the substitution u = 1 − x^2, du = −2x dx:

So the integral becomes:

∫35x^2/ √(1 − x^2) dx = -35/2 ∫du/ √u = -35/2 * 2 √u + C

Substituting back in terms of x:

-35/2 * 2 √(1 − x^2) + C = -35√(1 − x^2) + C

Therefore, the evaluated integral is:

∫(0, sqrt(3/2)) 35x^2/ √(1 − x^2) dx = -35√(1 − (sqrt(3/2))^2) + 35√(1 − 0) = -35√(1/2) + 35 = 35(1 − √(1/2))

Learn more about integral here

https://brainly.com/question/30094386

#SPJ11