For ever 100 clovers that Lucy picked 22 of them had four leaves while the others only had three leave in total Lucy picked 1,000 clovers if her pattern continued, how many three - leaf and four leaf would Lucy have how many of each clover would she have if she picked 2,000 clovers

Answer:

To shift the graph of f(x) = |x| to have a domain of [-3, 6], we need to move the left endpoint from -6 to -3 and the right endpoint from 3 to 6.

A translation to the right by 3 units will move the left endpoint of the graph of f(x) to -3, but it will also shift the right endpoint to 6 + 3 = 9, which is outside the desired domain.

A translation to the left by 3 units will move the right endpoint of the graph of f(x) to 3 - 3 = 0, which is outside the desired domain.

A translation upward or downward will not change the domain of the graph, so options B and D can be eliminated.

Therefore, the correct answer is C g(x) = x - 3. This translation will move the left endpoint to -3 and the right endpoint to 6, which is exactly the desired domain.

Therefore, A, B, C, and F are the proper responses as the congruence theories or postulates based on the data.

Having three straight sides and three angles where they intersect, a triangle is a closed, two-dimensional shape. It is one of the fundamental geometric shapes and has a number of characteristics that can be used to study and resolve issues that pertain to it. The triangle inequality theory states that the sum of a triangle's interior angles is always 180 degrees, and that the longest side is always the side across from the largest angle. Triangles can be used to solve a wide range of mathematical issues in a variety of disciplines and can be categorised based on the length of their sides and the measurement of their angles.

given

We can use the following congruence theories or postulates based on the data in the diagram:

A. ASA

B. AAS

C. LL (corresponding angles hypothesis)

F. SAS

Therefore, A, B, C, and F are the proper responses as the congruence theories or postulates based on the data.

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The required probability that a household in Maryland with annual income of ,

$90,000 or more is equal to 0.3377.

$50,000 or less is equal to 0.2218.

Annual household income in Maryland follows a normal distribution ,

Median =  $75,847

Standard deviation = $33,800

Probability of household in Maryland has an annual income of $90,000 or more.

Let X be the random variable representing the annual household income in Maryland.

Then,

find P(X ≥ $90,000).

Standardize the variable X using the formula,

Z = (X - μ) / σ

where μ is the mean (or median, in this case)

And σ is the standard deviation.

Substituting the given values, we get,

Z = (90,000 - 75,847) / 33,800

⇒ Z = 0.4187

Using a standard normal distribution table

greater than 0.4187  as 0.3377.

P(X ≥ $90,000)

= P(Z ≥ 0.4187)

= 0.3377

Probability that a household in Maryland has an annual income of $90,000 or more is 0.3377(rounded to four decimal places).

Probability that a household in Maryland has an annual income of $50,000 or less.

P(X ≤ $50,000).

Standardizing X, we get,

Z = (50,000 - 75,847) / 33,800

⇒ Z = -0.7674

Using a standard normal distribution table

Probability that a standard normal variable is less than -0.7674 as 0.2218. This implies,

P(X ≤ $50,000)

= P(Z ≤ -0.7674)

= 0.2218

Probability that a household in Maryland has an annual income of $50,000 or less is 0.2218.

Therefore, the probability with annual income of $90,000 or more and  $50,000 or less is equal to 0.3377 and 0.2218 respectively.

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The function that satisfies  F Has An Inverse G That Satisfies G'(2) = 1/6 is f(x) = 2x³ (option a).

More precisely, if f(x) is a function, then its inverse function g(x) satisfies the following two conditions:

g(f(x)) = x for all x in the domain of f

f(g(x)) = x for all x in the domain of g

In other words, if we apply f(x) to an input value x, and then apply g(x) to the resulting output, we get back to the original input value.

Now, let's look at the given condition: G'(2) = 1/6. This means that the derivative of the inverse function at x=2 is 1/6. We can use this condition to eliminate some of the options.

f(x) = 2x³

If we take the derivative of f(x), we get: f'(x) = 6x²

To find the inverse function, we can solve for x in the equation y = 2x³:

x = [tex]y/2^{(1/3)}[/tex]

Now we can express the inverse function g(x) in terms of y:

g(y) = [tex]y/2^{(1/3)}[/tex]

To find the derivative of g(x), we use the chain rule:

g'(x) = f'(g(x))⁻¹

g'(2) = f'(g(2))⁻¹

g'(2) = f'([tex]1/2^{(1/3)}[/tex])⁻¹

g'(2) = 6([tex]1/2^{(1/3)}[/tex])²)⁻¹

g'(2) = 6/36 = 1/6

Since g'(2) = 1/6, option a) is the correct answer.

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Here's the completed table and the graph:

x  h(x)

-90°  1.0

-75°  -0.5

-60°  -1.0

-45° -0.0

-30° 1.0

-15° 0.5

0° 1.0

15° 0.5

30° -0.0

45° -1.0

60° -0.5

75° 1.0

90° 1.0

In mathematics, a function is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output. Functions are often represented as a set of ordered pairs, where the first element of each pair is an input and the second element is the corresponding output. Functions are a fundamental concept in many areas of mathematics and have many real-world applications, including in science, engineering, and economics.

Here,

To calculate the values of h(c) in the table, we plug in the given values of x into the function h(c) = cos 2x and evaluate. For example, to find h(c) when x = -75°:

h(c) = cos 2x

h(c) = cos 2(-75°) (substitute -75° for x)

h(c) = cos (-150°) (simplify using the double angle identity)

h(c) = -0.5 (evaluate using the unit circle or a calculator)

We repeat this process for each value of x to fill out the table.

To graph the function h(x) = cos 2x, we plot each point from the table on the given system of axes. The x-axis represents the angle x in degrees, and the y-axis represents the value of h(x) = cos 2x. We then connect the points with a smooth curve to obtain the graph.

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The expression that is not equivalent to the model shown is given as follows:

-4(3 + 2). -> Option C.

Equivalent expressions are mathematical expressions that have the same value, even though they may look different. In other words, two expressions are equivalent if they produce the same output for any input value.

The expression for this problem is given by three times the subtraction of four, plus three times the addition of 2, hence:

3(-4) + 3(2) = -12 + 6 = 3(-4 + 2) = 3(-2) = -6.

Hence the expression that is not equivalent is the expression given in option C, for which the result is given as follows:

-4(3 + 2) = -4 x 5 = -20.

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Answer:

We can use the formula for the area of a rectangle to solve this problem. Let's assume that the length of the top of the bookcase is L and the width is b. Then, we can write:

L × b = 300

Solving for b, we get:

b = 300 / L

Since we don't know the length L, we cannot find the exact value of b. However, we can use the given information to make an estimate. Let's say that the length of the bookcase is 60 inches. Then, we have:

b = 300 / 60 = 5

So, if the length of the bookcase is 60 inches, the width needs to be at least 5 inches to accommodate Bria's soap carving collection. However, if the length is different, the required width will also be different.

The integral of the function expressed as sum of partial frictions is -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C.

First, factor out 2x from the denominator to obtain:

∫[(x + 3)/(2x³ - 8x)] dx = ∫[(x + 3)/(2x)(x² - 4)] dx

Next, we use partial fractions to express the integrand as a sum of simpler fractions. To do this, we need to factor the denominator of the integrand:

2x(x² - 4) = 2x(x + 2)(x - 2)

Therefore, we can write:

(x + 3)/(2x)(x² - 4) = A/(2x) + B/(x + 2) + C/(x - 2)

Multiplying both sides by the denominator, we get:

x + 3 = A(x + 2)(x - 2) + B(2x)(x - 2) + C(2x)(x + 2)

Now, we need to find the values of A, B, and C. We can do this by equating coefficients of like terms:

x = A(x² - 4) + B(2x² - 4x) + C(2x² + 4x)

x = (A + 2B + 2C)x² + (-4A - 4B + 4C)x - 4A

Equating coefficients of x², x, and the constant term, respectively, we get:

A + 2B + 2C = 0

-4A - 4B + 4C = 1

-4A = 3

Solving for A, B, and C, we find:

A = -3/4

B = 7/16

C = -1/16

Therefore, the partial fraction decomposition is:

(x + 3)/(2x)(x² - 4) = -3/(4(2x)) + 7/(16(x + 2)) - 1/(16(x - 2))

The integral becomes:

∫[(x + 3)/(2x³ - 8x)] dx = ∫[-3/(8x) + 7/(8(x + 2)) - 1/(8(x - 2))] dx

Integrating each term separately gives:

∫[-3/(8x) + 7/(8(x + 2)) - 1/(8(x - 2))] dx

= -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C

where;

Therefore, the final answer is:

∫[(x + 3)/(2x³ - 8x)] dx = -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C

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With the help of the given Venn diagram, the answer of n(A∪B) is 44 respectively.

A Venn diagram is a visual representation that makes use of circles to highlight the connections between different objects or limited groups of objects.

Circles that overlap share certain characteristics, whereas circles that do not overlap do not.

Venn diagrams are useful for showing how two concepts are related and different visually.

When two or more objects have overlapping attributes, a Venn diagram offers a simple way to illustrate the relationships between them.

Venn diagrams are frequently used in reports and presentations because they make it simpler to visualize data.

So, we need to find:

A ∪ B

Now, calculate as follows:

The collection of all objects found in either the Blue or Green circles, or both, is known as A B. Its components number is:

8 + 7 + 14 + 6 + 1 + 8 = 44

n(A∪B) = 44

Therefore, with the help of the given Venn diagram, the answer of n(A∪B) is 44 respectively.

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Answer:

The absolute extrema is minimum at (-1, 2/9)

Step-by-step explanation:

Absolute extrema is a logical point that shows whether a the curve function is maximum or minimum.

Forexample a curve in the image attached. A, B and C are points of absolute maxima or absolute maximum. and P and Q are points of absolute minima or minimum.

Remember A, B, C, P, Q are critical points or stationary points.

How do we find absolute extrema?

The find the sign of the second derivative of the function.

From the question;

[tex]{ \sf{g(x) = \sqrt[3]{x} }} \\ \\ { \sf{g(x) = {x}^{ \frac{1}{3} } }} \\ [/tex]

Find the first derivative of g(x)

[tex]{ \sf{g {}^{l}(x) = \frac{1}{3} {x}^{ - \frac{2}{3} } }} \\ [/tex]

Find the second derivative;

[tex]{ \sf{g {}^{ll} (x) = ( \frac{1}{3} \times - \frac{2}{3}) {x}^{( - \frac{2}{3} - 1) } }} \\ \\ { \sf{g {ll}^{(x)} = - \frac{2}{9} {x}^{ - \frac{5}{3} } }}[/tex]

Then substitute for x as -1 from [-1, 1]

[tex]{ \sf{g {}^{ll}(x) = - \frac{2}{9} ( - 1) {}^{ - \frac{5}{3} } }} \\ \\ = \frac{ - 2}{9} \times - 1 \\ \\ = \frac{2}{9} [/tex]

Since the sign of the result is positive, the absolute extrema is minimum

a.(1) When both r and s are odd integers, the quadratic polynomial obtained by multiplying out (x - r)(x - s) will have a coefficient of 1 for x^2 term, and both the coefficient of x term and constant term will be odd integers.

(2) When both r and s are even integers, the polynomial obtained by multiplying out (x - r)(x - s) will also have a coefficient of 1 for x^2 term, but the coefficient of x term and constant term will be even integers.

(3) When one of r and s is even and the other is odd, the polynomial obtained by multiplying out (x - r)(x - s) will have a coefficient of 1 for x^2 term, the coefficient of x term will be an odd integer, while the constant term will be an even integer.

b. x^2 - 1253x + 255 cannot be written as a product of two polynomials with integer coefficients.

a. When both r and s are odd integers, the product (x − r)(x − s) will have a coefficient of 1 for x^2 term, and both the coefficient of x term and constant term will be odd integers. This is because the sum of two odd integers and the product of two odd integers is also an odd integer.

When both r and s are even integers, the product (x − r)(x − s) will also have a coefficient of 1 for x^2 term, but the coefficient of x term and constant term will be even integers. This is because the sum of two even integers and the product of two even integers is also an even integer.

When one of r and s is even and the other is odd, the product (x − r)(x − s) will have a coefficient of 1 for x^2 term, and the coefficient of x term will be an odd integer, while the constant term will be an even integer. This is because the sum of an odd and even integer is an odd integer, and the product of an odd and even integer is an even integer.

b. If x^2 - 1253x + 255 can be written as a product of two polynomials with integer coefficients, then we can write it as (x - r)(x - s) where r and s are integers. From part (a), we know that both r and s cannot be odd integers since the coefficient of x term would be odd, but 1253 is an odd integer. Similarly, both r and s cannot be even integers since the constant term would be even, but 255 is an odd integer. Therefore, one of r and s must be odd and the other must be even. However, the difference between an odd integer and an even integer is always odd, so the coefficient of x term in the product (x - r)(x - s) would be odd, which is not equal to the coefficient of x term in x^2 - 1253x + 255. Hence, x^2 - 1253x + 255 cannot be written as a product of two polynomials with integer coefficients.

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The sample mean is 2.1, the sample variance is 212.5% and the standard deviation is 14.57%

a. The sample mean can be computed as the average of the quarterly percent total returns:

[tex](11.2 - 20.5 + 13.2 + 12.6 + 9.5 - 5.8 - 17.7 + 14.3) / 8 = 2.1[/tex]

So the sample mean is 2.1%, which can be interpreted as the average quarterly percent total return for the stock over the sample period.

b. The sample variance can be computed using the formula:

[tex]s^2 = sum((x - mean)^2) / (n - 1)[/tex]

where x is each quarterly percent total return, mean is the sample mean, and n is the sample size. Plugging in the values, we get:

[tex]s^2 = (11.2 - 2.1)^2 + (-20.5 - 2.1)^2 + (13.2 - 2.1)^2 + (12.6 - 2.1)^2 + (9.5 - 2.1)^2 + (-5.8 - 2.1)^2 + (-17.7 - 2.1)^2 + (14.3 - 2.1)^2 / (8 - 1) = 212.15[/tex]

So the sample variance is 212.15%. The sample standard deviation can be computed as the square root of the sample variance:

[tex]s = \sqrt(s^2) = \sqrt(212.15) = 14.57[/tex]

So the sample standard deviation is 14.57%.

c. To construct a 95% confidence interval for the population variance, we can use the chi-square distribution with degrees of freedom n - 1 = 7. The upper and lower bounds of the confidence interval can be found using the chi-square distribution table or calculator, as follows:

upper bound = (n - 1) * s^2 / chi-square(0.025, n - 1) = 306.05

lower bound = (n - 1) * s^2 / chi-square(0.975, n - 1) = 91.91

So the 95% confidence interval for the population variance is (91.91, 306.05).

d. To construct a 95% confidence interval for the standard deviation (as percent), we can use the formula:

lower bound = s * √((n - 1) / chi-square(0.975, n - 1))

upper bound = s * √((n - 1) / chi-square(0.025, n - 1))

Plugging in the values, we get:

lower bound = 6.4685%

upper bound = 20.1422%

So the 95% confidence interval for the standard deviation (as percent) is (6.4685%, 20.1422%).

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Answer:

Step-by-step explanation:

1)  b      (acute is less than 90)

2)  a     (obtuse: more than 90, less than 180)

3)  c

4) c

Answer:

1. NOM, JOK, KOL

2. MOL, NOK, MOJ

3. NOJ, JOL

4. NOL, MOK

Answer: x = 5

Step-by-step explanation:

To solve for x, we can first add 13 to both sides to isolate the variable term:

5x - 13 + 13 = 12 + 13

Simplifying the left side and evaluating the right side:

5x = 25

Then, divide both sides by 5 to isolate x:

5x/5 = 25/5

Simplifying:

x = 5

Therefore, the solution to the equation 5x - 13 = 12 is x = 5.

To solve for x in the equation 5x-13=12, we want to isolate the variable x on one side of the equation. We can do this by adding 13 to both sides of the equation:

5x-13+13 = 12+13

Simplifying, we get:

5x = 25

Finally, we can solve for x by dividing both sides of the equation by 5:

5x/5 = 25/5

Simplifying, we get:

x = 5

Therefore, the solution to the equation 5x-13=12 is x = 5.

Answer:

x = 7

Step-by-step explanation:

The two angles are equal so the opposite sides are equal.

5x-2 =33

Add two to each side.

5x-2+2 = 33+2

5x=35

Divide by 5

5x/5 =35/5

x = 7

a. 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720. b. 20! = 20 x 19 x 18 x ... x 2 x 1. c. There are 16 zeros at the end of the decimal representation of (20!)2.

a. 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

b. 20! = 20 x 19 x 18 x ... x 2 x 1. To write this in factored form, we can identify the prime factors of each number and write the product using exponents. For example, 20 = 2² x 5, so we can write 20! as:

20! = (2² x 5) x 19 x (2 x 3²) x 17 x (2² x 7) x 13 x (2 x 2 x 3) x 11 x (2³) x (3) x (2) x 7 x (2) x 5 x (2) x 3 x 2 x 1

Simplifying, we get:

20! = 2¹⁸ x 3⁸ x 5⁴ x 7² x 11 x 13 x 17 x 19

c. The number of zeros at the end of (20!)² in decimal form is determined by the number of factors of 10, which is equivalent to the number of factors of 2 x 5. Since there are more factors of 2 than 5 in the prime factorization of (20!)², we only need to count the number of factors of 5. There are four factors of 5 in the prime factorization of 20!, which contribute four factors of 10 to the square. Therefore, (20!)²ends in 8 zeros when written in decimal form.

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Step-by-step explanation:

40 cans/student   X   20 students  X  15 gram/can = 12 000 gm  = 12 kg

The statement that best describes the mode of selection depicted in the figure is (b) Directional Selection, changing the average color of population over time.

The Directional selection is a type of natural selection that occurs when individuals with a certain trait or phenotype are more likely to survive and reproduce than individuals with other traits or phenotypes.

In the directional selection of evolution, the mean shifts that means average shifts to one extreme and supports one trait and leads to eventually removal of the other trait.

In this case, one end of the extreme-phenotypes which means that the dark-brown rats are being selected for. So, over the time, the average color of the rat population will change.

Therefore, the correct option is (b).

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The given question is incomplete, the complete question is

The figure below shows the change of a population over time. which statement best describes the mode of selection depicted in the figure?

(a) Disruptive Official, favoring the average individual

(b) Directional Selection, changing the average color of population over time

(c) Directional selection, favoring the average individual

(d) Stabilizing Selection, changing the average color of population over time

Answer:

19/20 , 1 1/4 , 1 4/5

Step-by-step explanation:

1.8 = 1 4/5 (fraction)

1.8 converts to 18/10. This can be simplified twice, firstly by making it 9/5 since both 18 and 10 are divisible by two, but can be simplified further to 1 4/5

1.25 = 1 1/4 (fraction)

1.25 converts to 125/100. This can be simplified to 5/4 or 1 1/4

0.95 = 19/20 (fraction)

0.95 converts to 95/100. This can be simplified to 19/20

Ascending Order (smallest to largest)

smallest - 19/20

middle - 1 1/4

largest - 1 4/5

I believe this is the right answer, but haven't done fractions in a while so may want to double check to make sure

The cardinality of set A, n(A) = 29

The cardinality of a set is the total number of elements in the set

Given the Venn diagram here shows the cardinality of each set. To find the cardinality of set A, n(A), we proceed as follows.

Since the cardinality of a set is the total number of elements in the set, then cardinality of set A , n(A) = 9 + 8 + 3 + 9

= 29

So, n(A) = 29

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From the discriminant of the give quadratic equation, the temperature of the machine will part after 50 minutes of operation.

The given equation that models the temperature of the machine is;

T = -0.005x² + 0.45x + 125

Let check if there's a value that exists for T = 135

Putting T = 135 in the given equation,

135 = -0.005x² + 0.45x + 125

We can simplify this to;

0.005x² - 0.45x + 10 = 0

From the general form of quadratic equation which is ax² + bx + c = 0, where a = 0.005, b = -0.45, and c = 10.

The discriminant of this quadratic equation is given by:

D = b² - 4ac

= (-0.45)² - 4(0.005)(10)

= 0.2025 - 0.2

= 0.0025

The discriminant of the equation is positive which indicates we have two roots. Therefore, the temperature of the machine part will cross 135°F at some point during the operation.

We can also find the roots of the quadratic equation using the formula:

[tex]x = (-b \± \sqrt(D)) / 2a[/tex]

Substituting the values of a, b, and D, we get:

[tex]x = (0.45 \± \sqrt(0.0025)) / 2(0.005)\\= (0.45 \± 0.05) / 0.01[/tex]

Taking the positive value, we get:

x = 50

Therefore, the temperature of the machine part will cross 135°F after 50 minutes of operation.

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Answer:

Show your solution ( 3. ) C + 18 = 29

Step-by-step explanation:

To solve the equation C + 18 = 29, we want to isolate the variable C on one side of the equation.

We can start by subtracting 18 from both sides of the equation:

C + 18 - 18 = 29 - 18

Simplifying the left side of the equation:

C = 29 - 18

C = 11

Therefore, the solution to the equation C + 18 = 29 is C = 11.

4.5y

subtract 2.9y from 7.4y, and you get 4.5y

Sure, let's solve this step-by-step:

First, we need to solve for x in the equation x + 1/2 = 5.

We can do this by subtracting 1/2 from both sides, giving us x = 4 1/2.

Now, we can substitute x = 4 1/2 into the equation 2*x^2 - 3x + 6 - 3/x +2/x^2.

We can simplify the equation by multiplying both sides by x^2, giving us:

2*x^2 - 3x + 6 - 3/x +2 = 10*x^2 - 3x + 6.

Now, we can combine all of the terms with x:

10*x^2 - 6x + 6 = 0.

Finally, we can solve the equation using the quadratic formula:

x = 3/5 or x = 2.

Therefore, the answer to the equation is 10*(3/5)^2 - 6(3/5) + 6 = 4.8, or 10*2^2 - 6(2) + 6 = 16.

The value of the x is 5√3 after we successfully do the application of the 30°-60°-90° Triangle theorem.

The 30°-60°-90° Triangle Theorem states that in such a triangle, the side opposite the 30° angle is half the length of the hypotenuse, and the side opposite the 60° angle is the product of the length of the hypotenuse and the square root of 3 divided by 2.

Using this theorem, we can write:

y = hypotenuse

Opposite of 30° angle = 5 = hypotenuse/2

Opposite of 60° angle = x = hypotenuse × (√(3)/2)

Solving for the hypotenuse in terms of y from the first equation, we get:

hypotenuse = 5×2 = 10

Substituting this value into the third equation, we get:

x = 10 × (√(3)/2) = 5 × √(3)

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to get the equation of any straight line, we simply need two points off of it, let's use those two in the picture below

[tex](\stackrel{x_1}{-3}~,~\stackrel{y_1}{2})\qquad (\stackrel{x_2}{3}~,~\stackrel{y_2}{-4}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-4}-\stackrel{y1}{2}}}{\underset{\textit{\large run}} {\underset{x_2}{3}-\underset{x_1}{(-3)}}} \implies \cfrac{-6}{3 +3} \implies \cfrac{ -6 }{ 6 } \implies - 1[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{2}=\stackrel{m}{- 1}(x-\stackrel{x_1}{(-3)}) \implies y -2 = - 1 ( x +3) \\\\\\ y-2=-x-3\implies {\Large \begin{array}{llll} y=-x-1 \end{array}}[/tex]

The  angles can be named with one letter  are 2.

Angles are a type of geometric shape which are formed when two straight lines intersect at a point. They are measured in degrees, usually between 0 and 360. Angles can be classified as acute, right, obtuse, straight and reflex, and the sum of any three angles in a triangle will always equal 180 degrees. Angles can be used in various ways, from providing stability in construction work to helping to identify other shapes. They can also be used to measure the size and shape of objects and to calculate distances and areas on maps. Angles are an important part of mathematics, geometry and trigonometry, and are used in a variety of applications in science and engineering.

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The function f(x,y) has only minimum value at (1,0) is -1 and maximum value does not exist.

The given function is f(x,y)=x²+y²-2x

First find the partial derivative with respect to x and y

f'(x)=2x-2

f'(y)=2y

f'(x)=0=f'(y)

2x-2=0

x=1

and y=0

Now we will cheak maxima and minima at (1,0)

f''(x,y)=2 and f"(x,y)=2 and f"(x,y)=0( derivative of first order of x with respect to y)

We know that

rt-s²≥0 and r positive then f is minimum and  r negative maximum

r=2 , t=2 and s=0

rt-s²≥0 and r is positive so f(x,y) is minimum at (1,0)

f(1,0)=1-2=-1

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According to the Venn diagram the value of [tex]n(A ^ C \cap B ^ C) = {3}[/tex] so the number of elements in that set is 1.

What is Venn diagram ?

A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. It is usually represented as a rectangle or a circle for each set and the overlapping areas between them, showing the common elements that belong to more than one set. Venn diagrams are widely used in mathematics, logic, statistics, and computer science to visualize the relationships between different sets and help solve problems related to set theory.

According to the question:
To solve this problem, we first need to understand the notation used.

n(A) denotes the set A and the numbers within the braces {} indicate the elements in set A. For example, n(A)={7,4,3,9} means that the set A contains 7, 4, 3, and 9.

n(AnB) denotes the intersection of sets A and B, i.e., the elements that are common to both A and B. For example, n(AnB)={4,3} means that the sets A and B have 4 and 3 in common.

^ denotes intersection of sets

cap denotes the intersection of sets

Now, we need to find the elements that are common to sets A and C, and sets B and C. We can do this by taking the intersection of A and C, and the intersection of B and C, and then taking the intersection of the two resulting sets.

[tex]n(A ^ C) = n(A \cap C) = {3,9}[/tex]

[tex]n(B ^ C) = n(B\cap C) = {3,5}[/tex]

Now, we take the intersection of [tex]n(A ^ C)[/tex] and [tex]n(B ^ C)[/tex]:

[tex]n(A ^ C \cap B ^ C) = {3}[/tex]

Therefore, the answer is 1.

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