which equation represent g(x)?​

(a) gcd(a, b) must equal 1.

(b) No, it is not necessarily true that gcd(a, b) = 6 just because au + bv = 6.

An integer is a whole number that can be positive, negative, or zero. It is a number without any fractional or decimal component. Integers include the counting numbers (1, 2, 3, ...) and their negatives (-1, -2, -3, ...), as well as 0. Examples of integers are -5, -3, -1, 0, 1, 2, 3, 5, and so on. Integers are used in a variety of mathematical operations, such as addition, subtraction, multiplication, and division, and they play a key role in many areas of mathematics, including number theory, algebra, and discrete mathematics.

(a) Suppose that gcd(a, b) is not equal to 1. Then, there exists a positive integer k such that k divides both a and b. Let a = ka' and b = kb', where a' and b' are relatively prime.

Since there are integers u and v satisfying au + bv = 1, it follows that ka'u + kb'v = k(a'u + b'v) = k. But this means that k divides 1, which is a contradiction. Hence, gcd(a, b) must equal 1.

(b) No, it is not necessarily true that gcd(a, b) = 6 just because au + bv = 6. Consider the counter example a = 4 and b = 2. In this case, we have 2u + 4v = 6, but gcd(a, b) = 2, not 6.

In general, if there are integers u and v satisfying au + bv = k, then gcd(a, b) divides k. However, the converse is not necessarily true; that is, just because gcd(a, b) divides k does not mean there are integers u and v satisfying au + bv = k.

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Gcd(a,b) can take on any value from 1 to the minimum of a and b (inclusive). For example, if a = 6 and b = 1, then au + bu = 6, but gcd(6,1) = 1.

GCD is an abbreviation for Greatest Common Divisor. It is the greatest positive integer that divides two or more numbers without a remainder. It is used to simplify fractions and solve linear Diophantine equations. The Euclidean algorithm or the prime factorization method can be used to compute GCD.

(a) Assume u and v are integers such that au + bv = 1. So a and b must have no common factor, because the only way to add two integers and obtain 1 is if they have no common factors. As a result, gcd(a,b) = 1.

(b) No, it is not always the case that gcd(a,b) = 6. For example, if an is 6 and b is 1, au + bu equals 6, while gcd(6,1) equals 1. In general, gcd(a,b) can have any value between 1 and the minimum of a and b. (inclusive).

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Louise the Lion says that if a preimage has parallel lines, then the image will have corresponding parallel lines after a rotation, translation, and reflection, The lion is right.

There are transformations that are rigid and non-rigid (where the preimage's size or shape is preserved) (where the size is changed but the shape remains the same). These are the basic tenets that this concept adheres to. It is an easy technique for changing 2D shapes.

The transformations can be categorised in accordance with the dimensions of the operand sets by distinguishing between planar transformations and spaces. They can also be grouped according to their traits.

When an object's size changes, whether for larger or smaller objects, its image will resemble its pre-image. The dimensions of the comparable figures are proportionately equal.

Therefore,

The lion is right.

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Since you have two sets of side lengths that are in the same ratio, you need the included angles. The included angle of two sides is the angle formed by the two sides. ∠N ≅ ∠Z

Two triangles are said to be comparable if the two sides of one are in proportion to the two sides of another triangle and the angles that the two sides inscribe in each triangle are equal.

Hence, if you align both triangles' equations, option A A. ∠N ≅ ∠Z is the answer.

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The value of a and b are 4 and 3 respectively

Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle. The ratios of sides of a right-angled triangle with respect to any of its acute angles are known as the trigonometric ratios of that particular angle.

60°, 45° ,30° are special angles with definite values

To solve for a;

sin60 = 2√3/a

√3/2 = 2√3/a

a = 2√3/√3/2

a = 2√3 ×2/√3

a = 2× 2 = 4

tan 60 = 2√3/5-b

√3 = 2√3/(5-b)

(5-b) = 2√3× 1/√3

5-b = 2

b = 3

therefore the value of a and b are 4 and 3 respectively

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Step-by-step explanation:

Customers eat 7/8. Tony says it's 14/16

[tex] \frac{7}{8} = \frac{14}{16} [/tex]

because 7 times 2 is 14 and 8 times 2 is 16, so he got to the correct conclusion

Tony says...

[tex] \frac{1}{7} + \frac{1}{14} = \frac{14}{16} [/tex]

There's no way to get this answer because neither 7 or 14 goes evenly into 16.

Tony got the correct answer, but his addition is incorrect for that said reason

Answer:

Tony's addition is correct.

Step-by-step explanation:

Tony needs to add 7 two times, since the customers ate 7 slices of out both pizzas. And, 7 + 7 is 14, and you need to also count the total slices of pizza, which is 16. So, the fraction would be 14/16, which can be simplified into 7/8, which is just 14/16 multiplied by 2.

The assumption must be false, which means that either x < 8 or y < 8 must hold true.

What is the Proof by Contradiction?

A proof by contradiction is a method of proof in which we assume the opposite of what we want to prove, and then show that this assumption leads to a contradiction. The contradiction then serves as evidence that the assumption must be false, and therefore the original statement must be true.

In this case, we want to prove that for any positive real numbers x and y, if x · y ≤ 50, then either x < 8 or y < 8.

Suppose for the sake of contradiction that x and y are positive real numbers such that x · y ≤ 50 and x ≥ 8 and y ≥ 8. Then, since x and y are positive, we have x · y = xy ≥ 64, which contradicts the assumption that x · y ≤ 50.

Therefore, we have shown that the assumption that x and y are positive real numbers such that x · y ≤ 50 and x ≥ 8 and y ≥ 8 leads to a contradiction.

Hence, the assumption must be false, which means that either x < 8 or y < 8 must hold true.

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Step-by-step explanation:

D) The volume of the pyramid is 3 times larger than the prism.

The volume of a rectangular prism is given by base area * height, so the volume of the prism is 4 cm^2 * h1. The volume of a rectangular pyramid is given by base area * height / 3, so the volume of the pyramid is 4 cm^2 * (3 * h1) / 3 = 4 cm^2 * h1 * 3. Hence, the volume of the pyramid is 3 times larger than the prism.

Answer:

Your answer is: D) The volume of the pyramid is 3 times larger than the prism

Step-by-step explanation:

The expression that is equivalent to (1 + cos(x))2Tangent (StartFraction x Over 2 EndFraction) ) is option D. (1 + cos(x))(sin (x))

The expression (1 + cos(x))2Tangent (StartFraction x Over 2 EndFraction) is equivalent to (1 + cos(x))(sin (x)) because of the double angle identity for tangent.

here, we have,

The double angle identity states that tangent of 2 times an angle is equal to 2 times the tangent of that angle divided by 1 minus the square of the tangent of that angle. In other words,

tan(2θ) = 2tan(θ)/(1 - tan2(θ))

In this expression, we have tangent of x/2,

so substituting θ = x/2 gives us:

tan(x) = 2tan(x/2)/(1 - tan2(x/2))

Since cos(x) = 1 - 2sin2(x/2),

we can simplify the expression to:

(1 + cos(x))2tan(x/2) = (1 + 1 - 2sin2(x/2))2tan(x/2)

                               = (2 - 2sin2(x/2))(2sin(x/2)/(1 - sin2(x/2)))

Expanding the product of the two factors gives us the final result:

(1 + cos(x))2tan(x/2) = (2 - 2sin2(x/2))(sin(x)) = (1 + cos(x))(sin(x))

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Answer:

  D) Yes, DEF and GEF can be proven congruent by HL.

Step-by-step explanation:

Given triangles DEF and GEF with sides DF and GF marked congruent, and a right angle at E, you want to know if congruence can be proved, and how.

The hypotenuses of the right triangles are marked congruent. The shared leg EF is congruent to itself, so enough information is provided to claim congruence by the HL theorem.

__

Additional comment

Sides DE and GE are not marked congruent, so you only can make claims about 2 of the sides. SSS congruence cannot apply.

The HL theorem only applies to right triangles, which these are.

The graph of the function f(x) = sin(x) - 2 is shifted 2 units down along the y-axis to the parent function f(x) = sin(x).

A periodic function has a range that is determined for a fixed interval and a domain that includes all real number values.

Any function that has a positive real integer p such that f (x + p) = f (x), with all x being real values, is said to be periodic.

We are familiar with the sine function f(x) = six.

Which starts At (0, 0) and completes one cycle at (0, 2π).

Now, The graph of f(x) = sin(x) - 2 will be the same as the graph of f(x) but shifted down 2 units vertically.

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Answer: The matrix A of the linear transformation T(f(t)) = f''(t) + 3f'(t) + 7f(t) from the space spanned by the functions cos(t) and sin(t) into itself with respect to the basis {cos(t), sin(t)} can be found by computing the images of the basis vectors under T and expressing those images as linear combinations of the basis vectors.

We have:

T(cos(t)) = -cos(t)'' - 3cos(t)' - 7cos(t) = -cos(t) - 3(-sin(t)) - 7cos(t) = -8cos(t) - 3sin(t)

T(sin(t)) = -sin(t)'' - 3sin(t)' - 7sin(t) = -sin(t) - 3cos(t) - 7sin(t) = -8sin(t) + 3cos(t)

So, with respect to the basis {cos(t), sin(t)}, the matrix A is:

A = [ -8, -3; 3, -8 ]

This is the matrix representation of the linear transformation T with respect to the basis {cos(t), sin(t)}.

Step-by-step explanation:

Answer:

[tex]\frac{9}{8} [W]\ and \ \frac{3}{4} [S].[/tex]

Step-by-step explanation:

1. if the weight of Whisters is 'w', then the weight of Scamp is 'w-3/8'; then

2. it is possible to make up the equation:

[tex]w+w-\frac{3}{8}=1\frac{7}{8};[/tex]

3. if to solve this equation, then w=9/8 - this is the weight of Whiskers;

the weight of Scamp is 9/8-3/8=6/8=3/4 [pounds].

x²+x-12=0 is the equation have the solution {3, -4}

A quadratic equation is a second-order polynomial equation in a single variable x , ax²+bx+c=0. with a ≠ 0 .

(x-3)(x-4)=0

x=3 and x=4

So the equation does not have the solution {3, -4}

x²+x-12=0

x²+4x-3x-12=0

x(x+4)-3(x+4)

(x-3)(x+4)=0

x=3 and x=-4

x²+x-12=0 the equation have the solution {3, -4}

(x+3)(x-4)=0

x=-3 and x=4

(x+3)(x-4)=0 does not have the solution {3, -4}

x²-x-12=0

x²-4x+3x-12=0

x(x-4)+3(x-4)=0

(x+3)(x-4)=0

x=-3 and x=-4 does not have the solution {3, -4}

Hence, x²+x-12=0 is the equation have the solution {3, -4}

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Therefore, we will need volume approximately 694.4 cubic feet of sand to fill the pit to a depth of 30 inches.

The volume of a three-dimensional item, which is measured in cubic units, describes how much room it occupies. Two instances of cubic units are cm3 and in3. To the contrary hand, a measurement of a thing's mass indicates how much material it includes. Typically, the weight of an item is measured in mass units the same as pounds or kilograms.

Here,

a.

Length: 30 feet 2 inches

Width: 9 feet 2 inches

Height: 30 inches (since the sand needs to be 30 inches deep)

To calculate the amount of wood required, we need to find the perimeter and area of the base of the frame.

Perimeter = 2(Length + Width) = 2(30 ft 2 in + 9 ft 2 in) = 2(39 ft 4 in) = 78 ft 8 in

Area of the base = Length × Width = (30 ft 2 in) × (9 ft 2 in) = 275 ft²

The amount of wood needed to surround the pit can then be calculated by finding the total surface area of the wooden frame.

Total surface area = 2(L × H) + 2(W × H) + Area of the base

= 2(30 ft 2 in × 30 in) + 2(9 ft 2 in × 30 in) + 275 ft²

≈ 240.5 square feet

Therefore, we will need approximately 240.5 square feet of wood to surround the pit.

b.

Length: 30 ft 2 in = (30 × 12) + 2 = 362 inches

Width: 9 ft 2 in = (9 × 12) + 2 = 110 inches

Depth: 30 inches

The volume of sand needed to fill the pit can be calculated using the formula:

Volume = Length × Width × Depth

Substituting the values, we get:

Volume = 362 in × 110 in × 30 in

= 1,198,800 cubic inches

To convert cubic inches to cubic feet, we divide by 12^3 (since there are 12 inches in a foot, and we need to cube this conversion factor):

Volume = 1,198,800 in³ ÷ (12³ in³/ft³)

= 694.4 ft³ (rounded to one decimal place)

Therefore, we will need approximately 694.4 cubic feet of sand to fill the pit to a depth of 30 inches.

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Step-by-step explanation:

The proceeds of the note can be calculated by multiplying the face amount by the interest rate. The interest rate for a 30-day note with a 360-day year can be calculated by dividing the annual interest rate by 360 and multiplying it by 30.

Proceeds of the note = 55,200 * (5/100/360 * 30) = 55,200 * (5/12) = 4,600

The discounted note can be calculated by subtracting the proceeds from the face amount.

Discounted note = 55,200 - 4,600 = 50,600

So, the proceeds of the note are $4,600 and the discounted note is $50,600.

A system of equations are x+y=20 and x-y=6 and the solution is (13, 7).

A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously. The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.

Two numbers x and y add up to 20.

x+y=20 --------(I)

Two numbers x and y have a difference of 6.

x-y=6 --------(II)

x=6+y

Substitute x=6+y in equation (I), we get

6+y+y=20

6+2y=20

2y=14

y=7

Substitute y=7 in equation (I), we get

x+7=20

x=13

So, the solution is (13, 7)

Therefore, a system of equations are x+y=20 and x-y=6 and the solution is (13, 7).

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Please find attached the completed equation puzzle with the number 8 representing a, created with MS Word.

An equation is an expression of equivalence of the two expressions, quantities and or numbers, which are joined by the '=' sign.

The puzzle can be completed using both variables and values that satisfies equations created in the puzzle as follows;

Let a = 8, starting from top left of the puzzle, we get;

3·a + 4 = 3 × 8 + 4 = 28

The vertical column in the top middle with 28 at the top, indicates that we get the following equation;

28 + 4·a = 28 + 4 × 8 = 60

The row that crosses the middle of the vertical row above indicates that we get;

4·a + 4·a - 3·a = 5·a

The value, 3·a, obtained above is evaluated as; 3·a = 3 × 8 = 24

The vertical column with 6·a is evaluated as follows;

4·a + x = 6·a + a = 7·a

x = 7·a - 4·a = 3·a

x = 3·a

Therefore, we get; 4·a + 3·a = 6·a + a = 7·a

The values left blank are evaluated as follows;

a + 6·a - 3·a = 4·a

2·a + 4·a = 6·a

a + 6·a = 7·a

60 - 2·a = y - 5·a

y = 60 - 2·a + 5·a = 60 + 3·a = 60 + 3 × 8 = 84

84 - 5·a = 84 - 5 × 8 = 44

2·a + 4·a = 6·a

a + 6·a = 7·a

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The required function of s and decreasing function of s are  a) h(s) = V / (s²) b) dh/ds = -2V / (s³).

Each thing in three dimensions takes up some space. The volume of this area is what is being measured. The space occupied within an object's borders in three dimensions is referred to as its volume. It is sometimes referred to as the object's capacity.

(a) We know that the volume of the box, V, is given by the product of its height, h, and its base area, which is s².

we can write the equation:

V = h * s²

Solving for h, we get:

h = V / (s²)

So the height of the box, h, as a function of s can be expressed as

h(s) = V / (s²)

(b) To determine whether h(s) is an increasing or decreasing function of s, we need to look at the sign of its derivative.

The derivative of h(s) with respect to s is given by:

dh/ds = -2V / (s³)

Since the derivative is negative for all positive values of s, h(s) is a decreasing function of s.

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Answer:The most expensive house sold, based on the stem-and-leaf plot, would have a sale price of $667,778. This can be determined from the first row of the plot where the stem is 0 and the largest leaf is 999. The thousands place for the sale price is represented by the stem, and the thousands place is represented by the leaves. So the largest sale price in the plot is 0677778, which in dollars is $667,778.

Step-by-step explanation:

The triangle is not a right triangle because the Pythagorean Theorem (c² = a² + b²) is not satisfied.

What is the Pythagorean Theorem?

Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“. The sides of this triangle have been named Perpendicular, Base, and Hypotenuse.

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side, which is opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In this case, the given values of a and b do not satisfy the equation:

c² = a² + b²

(86)² = (76.5)² + (40.1)²

7396 = 5862.25 + 1608.01

7396 = 7470.26

This means that the triangle with sides a = 76.5 yd, b = 40.1 yd, and c = 86 yd is not a right triangle.

Hence, The triangle is not a right triangle because the Pythagorean Theorem (c² = a² + b²) is not satisfied.

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Answer: The circumference of the smaller circular base of the cone can be found using the formula:

C = 2 * π * r

where "r" is the radius of the circle. The radius can be found using the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the longest side (the hypotenuse).

Since the cut plane is parallel to the base of the cone, the height of the cone and the height of the remaining section of the cone are equal. Let's call this height "h". Then, using the Pythagorean theorem, we have:

r^2 + (h/2)^2 = (h/2)^2

Simplifying, we get:

r^2 = (h/2)^2

Taking the square root of both sides:

r = h/2

Substituting "h/2" for "r" in the formula for the circumference:

C = 2 * π * (h/2) = π * h

Since we do not have a specific value for "h", we cannot find an exact value for the circumference. However, based on the given options, the closest approximate value to π * h is c) 3.36 feet.

Step-by-step explanation:

Answer:

4.2 years

Step-by-step explanation:

If a continuous random variable X is normally distributed with mean μ and variance σ², it is written as:

[tex]\boxed{X \sim\text{N}(\mu,\sigma^2)}[/tex]

Given:

Therefore, if the lifespans of an item are normally distributed:

[tex]\boxed{X \sim\text{N}(6.7,1.7^2)}[/tex]

where X is the lifespan of the item.

Converting to the Z distribution

[tex]\boxed{\textsf{If }\: X \sim\textsf{N}(\mu,\sigma^2)\:\textsf{ then }\: \dfrac{X-\mu}{\sigma}=Z, \quad \textsf{where }\: Z \sim \textsf{N}(0,1)}[/tex]

To find the number of years that 7% of the items with the shortest lifespan will last less than, we need to find the value of a for which P(X < a) = 7%:

[tex]\implies \text{P}(X < a) =0.07[/tex]

Transform X to Z:

[tex]\text{P}(X < a) = \text{P}\left(Z < \dfrac{a-6.7}{1.7}\right)=0.07[/tex]

According to the z-tables, when p = 0.07, z = -1.47579106...

[tex]\implies \dfrac{a-6.7}{1.7}= -1.47579106...[/tex]

[tex]\implies a-6.7= -2.50884480...[/tex]

[tex]\implies a=4.19115519...[/tex]

[tex]\implies a=4.2\; \sf years[/tex]

Therefore, the 7% of items with the shortest lifespan will last less than 4.2 years.

Answer:

(a) 440/264 * 3 = 5

(b) 8/3 * 264 = 704 days

For the given data, the range, variance, and standard deviation is 1.47, 0.1716, and 0.414, respectively.

To find the range, subtract the minimum value from the maximum value in the set of data. The minimum value is 0.48 and the maximum value is 1.95, so the range is: 1.95 - 0.48 = 1.47.

To find the sample variance, follow these steps.

1. Calculate the mean of the data set.

2. Subtract the mean from each value in the data set.

3. Take the summation of the squares of the result.

4. Divide the sample size minus 1.

Upon calculation, the sample variance is 0.1716.

Lastly, to find the sample standard deviation, take the square root of the sample variance: √0.1716. = 0.414.

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Answer:

x = 4/3

y = 7/3

Step-by-step explanation:

7x - y = 7

x + 2y = 6

Times the second equation by -7

7x - y = 7

-7x - 14y = -42

-15y = -35

y = 7/3

Now put 7/3 back in for y and solve for x

x + 2(7/3) = 6

x + 14/3 = 6

x + 14/3 = 18/3

x = 4/3

Let's check

4/3 + 2(7/3) = 6

4/3 + 14/3 = 6

18/3 = 6

6 = 6

So, x = 4/3 and y = 7/3 is the correct answer.

Answer: If you help me I will help you Cuz I been screaming for help all day

Step-by-step explanation:

The number that can replace * in 3x - 694 = 10* is 2

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

3x - 694 = 10*

Make 3x the subject

3x = (694 + 10*)

Divide both sides of the equation by 3

So, we have the following representation

x = (694 + 10*)/3

For x to be a whole number, 694 + 10* must be a multiple of 3

694 + 10*

The smallest value of * for this is 2

So, we have

x = (694 + 10 * 2)/3

Evalaute

x = 238

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Answer:

2

Step-by-step explanation:

a gram is 1000 milligrams (mg) so there is 2000 so then 2000 divided by 1000 =2

Answer:

13

Step-by-step explanation:

plot the 2 points on grid paper,

connect the 2 points

you'll get triangle that has 8+4 =12 units on the Y axis

and 5 units to the left on X axis

the DIRECT distance b/w the 2 points = sqrt of (5²+12²) =√169 =13

Step-by-step explanation:

You would want to use the distance formula to find this. Fill in the correct variables and solve. Let’s say (0,8) is 1 and (-5,-4) is 2. When solving use pemdas to make sure you get the correct answer. The included image shows the work on how to do the problem. Hope this helped :)

Fixed rate only have fixed payments for at least the first 6 years. Therefore, option A is the correct answer.

A fixed-rate mortgage is a home loan option with a specific interest rate for the entire term of the loan. Essentially, the interest rate on the mortgage will not change over the lifetime of the loan and the borrower's interest and principal payments will remain the same each month.

An adjustable-rate mortgage, also called an ARM, is a home loan with an interest rate that adjusts over time based on the market. ARMs typically start with a lower interest rate than fixed-rate mortgages, so an ARM is a great option if your goal is to get the lowest possible mortgage rate starting out.

A balloon mortgage begins with fixed payments for a specific period and ends with a final lump-sum payment. The one-time payment is called a balloon payment because it's much larger than the beginning payments.

Therefore, option A is the correct answer.

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"Your question is incomplete, probably the complete question/missing part is:"

A bank offers 3 mortgage.

Option1: Fixed rate mortgage at 4% for 15 years.

Option 2: Adjustable rate mortgage at 2.99% for 15 years with terms 6/1 and a cap of 2/5

Option 3: Balloon mortgage at 5% with terms 15/5.

Which mortgage (s) will have fixed payments for at least the first 6 years?

A) Fixed rate only

B) Fixed rate and the balloon only

C) Fixed rate and adjustable rate only

D) Fixed rate, adjustable rate, and balloon.