this is due tmr !!!!

To find the determinant using elementary row or column operations, we can use the following steps:

1. Rewrite the matrix in an augmented form with the identity matrix on the right:

3 -3 -2 | 1 0 0
3 1 2 | 0 1 0
-6 6 4 | 0 0 1

2. Use elementary row operations to transform the matrix into an upper triangular form:

R2 = R2 - R1
R3 = R3 + 2R1
R3 = R3 + 2R2

3 -3 -2 | 1 0 0
0 4 4 | -1 1 0
0 0 0 | -2 2 1

3. The determinant of an upper triangular matrix is the product of its diagonal elements:

det(A) = 3 x 4 x 0 = 0

Therefore, the determinant of the original matrix is 0.

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Systematic sampling is a sampling plan that selects every nth item from the population. Therefore the correct option is (d) Systematic

Systematic sampling is a statistical sampling method that involves selecting every nth item from the population to create a representative sample. This sampling method is useful when the population is large and ordered in some way, such as in a list or sequence.

To conduct a systematic sample, researchers select a starting point at random and then choose every nth item from that point forward until the desired sample size is reached. The advantage of systematic sampling is that it is simpler and more efficient than other sampling methods, such as simple random sampling, while still providing a representative sample of the population.

Therefore, the correct option is (d) systematic

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The solution to the system of equations is (x, y) = (1, 3).

We have,

We use the elimination method on the two equations:

-3x + 4y = 9

2x + 4y = 14

We can eliminate y by subtracting the second equation from the first equation:

-3x + 4y - (2x + 4y) = 9 - 14

Simplifying the left side and the right side, we get:

-5x = -5

Dividing both sides by -5, we get:

x = 1

Let's use the first equation:

-3x + 4y = 9

Substituting x = 1.

-3(1) + 4y = 9

Simplifying and solving for y.

4y = 12

y = 3

Therefore,

The solution to the system of equations is (x, y) = (1, 3).

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Derivative dy/dx by implicit differentiation is dy/dx = -2xy / (x^2 - 8y^3)

To find dy/dx using implicit differentiation, we differentiate both sides of the given equation with respect to x, using the chain rule for the terms involving y:

d/dx (x^2y - 2y^4) = d/dx (-6)

Using the product rule, we get:

2xy + x^2(dy/dx) - 8y^3(dy/dx) = 0

Now we can solve for dy/dx:

(dy/dx)(x^2 - 8y^3) = -2xy

dy/dx = -2xy / (x^2 - 8y^3)

So the derivative dy/dx can be expressed in terms of both x and y.

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If function f(x)=10(3)2x−2, then f(0) = 10/9.

Explanation:

Step 1: To evaluate f(0), we can substitute x with 0 in the given function f(x) = 10(3)^(2x-2).

f(0) = 10(3)^(2(0)-2) = 10(3)^(-2)

Step 2: Now we know that a^(-n) = 1/a^n. So, we can rewrite 3^(-2) as 1/3^2.

f(0) = 10 * (1/3^2) = 10 * (1/9)

Finally, f(0) = 10/9.

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To display both cMat(1,1) and cMat(2,2) as a row using linear indexing, we can create a logical index array that selects these elements in sequence. The linear index of cMat(1,1) is 1, and the linear index of cMat(2,2) is 4 (since there are two columns in cMat). Therefore, we can create a logical index array as follows:

logical_index = [1,4];

We can then use this logical index array to select the desired elements from cMat:

cMat(logical_index)

This will output a row vector with the values 10 and 40, which correspond to cMat(1,1) and cMat(2,2), respectively.
To display both cMat(1,1) and cMat(2,2) as a row using linear indexing, you would use the logical index array [1, 4]. In this case, cMat(1,1) corresponds to the value 10, and cMat(2,2) corresponds to the value 40. The resulting row would be [10, 40].

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Answer: Length=24ft Width=6ft

Step-by-step explanation:

Perimeter= 2L+2W= 10 W= 60 and W=6ft and L=24ft

Area= Length x Width

The correct answer is option C, which includes the sets Z⁺, Z⁻, and {0}. These sets are non-overlapping and together they cover all of the integers in Z, forming a partition. (C)

Z⁺ includes all positive integers, Z⁻ includes all negative integers, and {0} includes only the number 0. Each integer in Z belongs to exactly one of these sets.

Option A, ZZ⁻ and {0}, is not a partition because it includes 0 in both sets, violating the requirement that sets in a partition be non-overlapping.

Option B, Z and Z⁻, also does not form a partition because it does not include any positive integers. Option D, Z⁺ and Z⁻, does not include {0} and therefore does not cover all of the integers in Z.(C)

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The portobello mushroom was growing faster.

Given that, Andrew has two mushrooms, he recorded their height, enoki mushroom was 3.9 centimeters tall, after 5 days, it was 4.8 centimeters tall.

Also, the height of the portobello mushroom is given by equation,

y = 0.2x + 4.1, where y is the height of the portobello mushroom in centimeters and x is the number of days since he brought it,

So,

Considering the portobello mushroom height, after 5 days,

y = 0.2(5) + 4.1 = 5.1 cm

And the enoki mushroom was 4.8 cm tall on its 5th day,

Since, the height of portobello mushroom is more than enoki mushroom on 5th day.

Hence, the portobello mushroom was growing faster.

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Answer:

A and E

Step-by-step explanation:

They are the same shape and size if rotated properly.

Answer:

A and E

Step-by-step explanation:

when rotated they are the same shape and size

The solution is:

The proof is given below.

Here, we have,

Given a parallelogram ABCD. Diagonals AC and BD intersect at E. We have to prove that AE is congruent to CE and BE is congruent to DE i.e diagonals of parallelogram bisect each other.

In ΔACD and ΔBEC

AD=BC              (∵Opposite sides of parallelogram are equal)

∠DAC=∠BCE       (∵Alternate angles)

∠ADC=∠CBE        (∵Alternate angles)

By ASA rule, ΔACD≅ΔBEC

By CPCT(Corresponding Parts of Congruent triangles)

AE=EC and DE=EB

Hence, AE is conruent to CE and BE is congruent to DE.

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complete question:

Proving the Parallelogram Diagonal Theorem

Given ABCD is a parralelogam, Diagnals AC and BD intersect at E

Prove AE is conruent to CE and BE is congruent to DE

The value of x in the given composite figure is 11.31 units.

In accordance with the Pythagorean theorem, the square of the length of the hypotenuse (the side that faces the right angle) in a right triangle equals the sum of the squares of the lengths of the other two sides. If you know the lengths of the other two sides of a right triangle, you may apply this theorem to determine the length of the third side. By examining whether the sides of a triangle satisfy the Pythagorean equation, it can also be used to assess whether a triangle is a right triangle. Pythagoras, an ancient Greek mathematician, is credited with discovering the theorem, therefore it bears his name.

The given figure can be divided into a rectangle and a triangle.

The dimensions of the triangle are:

h = 18 - 10 = 8 units

b = 8

Now, using the Pythagoras Theorem we have:

x² = 8² + 8²

x² = 64 + 64

x² = 128

x = 11.31

Hence, the value of x in the given composite figure is 11.31 units.

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The mean of the binomial variable is 28 and the standard deviation is 2.72, given a probability of success of 0.7 with a total number of attempts of 40.

To calculate the mean (μ) and standard deviation (σ) of a binomial variable, we use the following formulas

μ = np

σ = sqrt(np × (1-p))

where n is the number of trials, and p is the probability of success for each trial.

In this case, the probability of success is 0.7, the number of trials is 40. So:

μ = 400.7 = 28

σ = sqrt(400.7 × (1-0.7)) = 2.72

Therefore, the mean of the binomial variable is 28, and the standard deviation is 2.72

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The number of  distinguishable code symbols can be formed with the letters for the words philosophical and mathematics is 24

Permutation is a mathematical technique that determines the number of possible arrangements in a set when the order of the arrangements matters.

In the word philosophical , There are 13 letters

2ps, 2is, 2Os, 2Hs and in Mathematics, there are = 11 letters

2ms, 2ts, 2As,

Therefore the number of permutations is 2!212!2! and 2!2!2!

This imples 16 + 8

Therefore, the number of distinguishable code symbols can be formed with the letters for the words philosophical and mathematics = 24

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The exponential distribution is a special case of the gamma distribution with alpha = 1, not 0.

False.

The exponential distribution is a special case of the gamma distribution when the shape parameter (alpha) is equal to 1, not 0.

The probability density function (pdf) of the gamma distribution with shape parameter alpha and rate parameter beta is:

f(x) = (1/((beta^alpha)*gamma(alpha))) * (x^(alpha-1)) * (e^(-x/beta))

When alpha = 1, this reduces to the exponential distribution with rate parameter lambda = 1/beta:

f(x) = lambda * e^(-lambda*x)

So the exponential distribution is a special case of the gamma distribution with alpha = 1, not 0.

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Okay, let's break this down step-by-step:

* f(x) = -4x3 + 15 (this is the original function)

* We want to find f(x) when x = 3

* So substitute 3 in for x:

f(3) = -4(3)3 + 15

f(3) = -81 + 15

f(3) = -66

Therefore, f(x) = -66 when x = 3.

[tex]\sf f(3)=-66.[/tex]

[tex]\sf f(3)=-4(3)^{3} +15\\ \\[/tex]

[tex]\sf f(3)=-4(3*3*3) +15\\\\\sf f(3)=-4(27) +15[/tex]

[tex]\sf f(3)=-81+15[/tex]

[tex]\sf f(3)=-66.[/tex]

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a) To find the value of β that satisfies limt→[infinity]y(t) = 0, we can first find the general solution of the differential equation. So the value(s) of β for which the solution y(t) is never equal to 0 for all t is [tex]β ∈ (-∞, -2) U (-2/3, ∞)[/tex]

The characteristic equation is [tex]r^2 + r - 2 = 0[/tex], which has roots r = 1 and r = -2.

Therefore, the general solution is[tex]y(t) = c1e^t + c2e^-2t.[/tex]

Using the initial conditions y(0) = 2 and y'(0) = β, we can solve for the constants c1 and c2:

[tex]c1 + c2 = 2[/tex]

[tex]c1 - 2c2 = β[/tex]

Solving this system of equations, we get [tex]c1 = 2 - β/3[/tex] and [tex]c2 = β/3.[/tex]

Therefore, the solution is y(t) =[tex](2 - β/3)e^t[/tex] + [tex]β/3)e^-2t[/tex]. To satisfy limt→[infinity]y(t) = 0, we need the coefficient of e^t to be 0, which gives us 2 - β/3 = 0. Solving for β, we get β = 6.

So the value of β that satisfies limt→[infinity]y(t) = 0 is β = 6.

b) To find the value(s) of β for which the solution y(t) is never equal to 0 for all t, we can use the fact that the discriminant of the characteristic equation determines the nature of the roots.

In this case, the characteristic equation is r^2 + r - 2 = 0, which has roots r = 1 and r = -2. These are distinct real roots, so the general solution is y(t) = [tex]c1e^t + c2e^-2t.[/tex]

For y(t) differential equation to never be equal to 0 for all t, we need both constants c1 and c2 to be nonzero. Using the initial condition y(0) = 2, we get c1 + c2 = 2.

Using the second initial condition y'(0) = β, we get c1 - 2c2 = β.

Solving these equations, we get [tex]c1 = (2β + 4)/5[/tex] and [tex]c2 = (6 - β)/5.[/tex]

Therefore, y(t) is never equal to 0 for all t if and only if both c1 and c2 are nonzero, which is true if and only if the coefficients satisfy the inequality (2β + 4)(6 - β) ≠ 0. Solving this inequality, we get [tex]β ∈ (-∞, -2) U (-2/3, ∞).[/tex]

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Answer:

4092

Step-by-step explanation:

We can see that this is a geometric sequence where the first term is 4 and the common ratio is 2. We can use the formula for the sum of a geometric sequence to find the sum of this series:

sum = a(1 - r^n) / (1 - r)

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

We need to find n, the number of terms. We can use the formula for the nth term of a geometric sequence:

a_n = a * r^(n-1)

We want to find the value of n when a_n = 2048:

2048 = 4 * 2^(n-1)

512 = 2^(n-1)

n-1 = log2(512) = 9

n = 10

So there are 10 terms in the series. Now we can use the formula for the sum of a geometric sequence:

sum = a(1 - r^n) / (1 - r)

sum = 4(1 - 2^10) / (1 - 2)

sum = 4(1 - 1024) / (-1)

sum = 4(1023)

sum = 4092

Rounding to the nearest hundredth, the sum is approximately 4092.00.

Answer:

Sum=8188

Step-by-step explanation:

This is a geometric series with a first term of 4 and a common ratio of 2. The formula for the sum of a geometric series is:

Sn​=1−ra(1−rn)​

where a is the first term, r is the common ratio and n is the number of terms. In this case, we have:

S11​=1−24(1−211)​

Simplifying, we get:

S11​=−14(−2047)​

S11​=8188

Therefore, the sum of the series is 8188.

The surface whose equation is given by ρ = cos ϕ is a type of conical surface in the spherical coordinate system. In this equation, ρ represents the radial distance from the origin, and ϕ denotes the polar angle.

Explanation:

The surface described by the equation ρ = cos(ϕ) is a type of conical surface in the spherical coordinate system. Let's break down the explanation step by step:

Spherical Coordinate System: The spherical coordinate system is a three-dimensional coordinate system used to represent points in space using three parameters - radial distance (ρ), polar angle (ϕ), and azimuthal angle (θ). The radial distance ρ represents the distance from the origin (0,0,0) to a point in space, ϕ represents the polar angle measured from the positive z-axis (ranging from 0 to π), and θ represents the azimuthal angle measured from the positive x-axis in the xy-plane (ranging from 0 to 2π).

Equation ρ = cos(ϕ): The equation ρ = cos(ϕ) describes a relationship between the radial distance ρ and the polar angle ϕ. It specifies that for any given value of the polar angle ϕ, the radial distance ρ should be equal to the cosine of ϕ.

Conical Surface: In the context of the spherical coordinate system, a conical surface is a surface that forms a cone shape with its apex at the origin. The equation ρ = cos(ϕ) describes a conical surface because it specifies that the radial distance ρ is determined by the cosine of the polar angle ϕ.

Shape of the Surface: As the polar angle ϕ varies, the equation ρ = cos(ϕ) determines the radial distance ρ at each point on the surface. Since the radial distance is only determined by the cosine of the polar angle, the surface will have a conical shape. Specifically, the surface will form a cone with its apex at the origin and its base expanding outward as ϕ increases from 0 to π. The radius of the base of the cone will vary with the value of ϕ, as determined by the cosine function. When ϕ = 0, the base of the cone will have its maximum radius, equal to 1 (since cos(0) = 1), and as ϕ increases towards π, the radius of the base will decrease until it reaches its minimum value of -1 (since cos(π) = -1). The surface will extend infinitely in the positive and negative z-directions.

In conclusion, the surface described by the equation ρ = cos(ϕ) in the spherical coordinate system is a type of conical surface, forming a cone with its apex at the origin and its base expanding outward as the polar angle ϕ increases, with the radius of the base varying based on the cosine of ϕ.

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The formula for y = f(x - 14) in terms of the variable x is y = (1/3)(x - 14)^2. To sketch the graph, draw a parabola and shift it 14 units to the right.

(a) To get a formula for y = f(x - 14) in terms of the variable x, substitute (x - 14) for x in the given function f(x) = (1/3)x^2:
y = f(x - 14) = (1/3)(x - 14)^2
(b) To sketch a graph of y = f(x - 14) using graph transformations, consider that the original function f(x) = (1/3)x^2 is a parabola. The transformation f(x - 14) shifts the graph 14 units to the right. Unfortunately, I cannot provide or select a graph letter from A-E, as there are no graphs provided here. However, to sketch it on paper, draw a parabola and shift it 14 units to the right.

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The probability of occurence of chocolate is 0.4 0r 40%. So the option A is the correct one.

Probability refers to the measure or quantification of the likelihood or chance of an event or outcome occurring. It is typically expressed as a numerical value ranging from 0 to 1, where 0 represents an impossible event and 1 represents a certain event.

In probability theory and statistics, a random variable is a variable whose value is determined by the outcome of a random event or process. It is often denoted by a capital letter, such as X or Y, and it can take on different values with certain probabilities associated with each value.

Based on the given condition, formulate:

8/20=2/5

0.4 or 40%

Therefore option (A) is correct.

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Since the least square curve fit method is a flexible method for approximating the best fit to a given set of data points using several mathematical models, all of these models are suitable.

The applicable model for the least square curve fit depends on the type of data being analyzed. In this case, the question mentions a sinusoidal model as one of the options. Therefore, a least square curve fit can fit data points to a sinusoidal model, which includes sine and cosine functions. However, it may not necessarily be able to fit the data points to an exponential model, polynomial model of appropriate order, or power curve.
Least square curve fit can fit the data points to the following models:

a) sinusoidal model (including sine and cosine functions)
b) exponential model
c) polynomial model of appropriate order
d) power curve ([tex]y=c1x^(c2)[/tex])

All of these models are applicable because the least square curve fit method is a versatile technique for approximating the best fit to a given set of data points using different mathematical models.

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Since the least square curve fit method is a flexible method for approximating the best fit to a given set of data points using several mathematical models, all of these models are suitable.

The applicable model for the least square curve fit depends on the type of data being analyzed. In this case, the question mentions a sinusoidal model as one of the options. Therefore, a least square curve fit can fit data points to a sinusoidal model, which includes sine and cosine functions. However, it may not necessarily be able to fit the data points to an exponential model, polynomial model of appropriate order, or power curve.
Least square curve fit can fit the data points to the following models:

a) sinusoidal model (including sine and cosine functions)
b) exponential model
c) polynomial model of appropriate order
d) power curve ([tex]y=c1x^(c2)[/tex])

All of these models are applicable because the least square curve fit method is a versatile technique for approximating the best fit to a given set of data points using different mathematical models.

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the value of the given double integral is 150

To evaluate this double integral, we first identify it as the volume of a solid. In this case, the region r represents a rectangle in the xy-plane with dimensions 6 units (from x = -3 to x = 3) and 5 units (from y = 3 to y = 8). The given integral represents the volume of a rectangular prism, where the height is given by the constant value 5.

The given double integral of 5 da represents the volume of a solid over the rectangular region r = {(x, y) | −3 ≤ x ≤ 3, 3 ≤ y ≤ 8}.

To evaluate this double integral, we integrate the given constant 5 over the given region:

∬r 5 da = ∫₃⁸ ∫₋³³ 5 dx dy

Integrating with respect to x first, we get:

∫₋³³ 5 dx = 5x ∣₋³³ = 5(3) - 5(-3) = 30

Substituting this value and integrating with respect to y, we get:

∫₃⁸ 30 dy = 30y ∣₃⁸ = 30(8) - 30(3) = 150

Therefore, the value of the given double integral is 150.

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Answer:

1. sin(A) = [tex]\frac{4}{5}[/tex]

2. cos(A) = [tex]\frac{3}{5}[/tex]

3. tan(A) = [tex]\frac{4}{3}[/tex]

4. sin(B) = [tex]\frac{3}{5}[/tex]

5. cos(B) = [tex]\frac{4}{5}[/tex]

6. tan(B) = [tex]\frac{3}{4}[/tex]

Step-by-step explanation:

Use SOHCAHTOA:

Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

1. sin(A) = opposite of A / hypotenuse of A = [tex]\frac{4}{5}[/tex]

2. cos(A) = adjacent of A / hypotenuse of A = [tex]\frac{3}{5}[/tex]

3. tan(A) = opposite of A / adjacent of A = [tex]\frac{4}{3}[/tex]

4. sin(B) = opposite of B / hypotenuse of B = [tex]\frac{3}{5}[/tex]

5. cos(B) = adjacent of B / hypotenuse of B = [tex]\frac{4}{5}[/tex]

6. tan(B) = opposite of B / adjacent of B = [tex]\frac{3}{4}[/tex]

The derivative of y with respect to x is:

[tex]y' = [(5x) / (5x^2 + 9y^2)] + [(9y) / (5x^2 + 9y^2)] * dy/dx[/tex]

or

[tex]dy/dx = [(5x) / (5x^2 + 9y^2)] + [(9y) / (5x^2 + 9y^2)] * y'[/tex]

To find y', we need to use the chain rule of differentiation because we have a composite function (i.e., the natural logarithm function is applied to a function of x and y).

Let's start by applying the chain rule:

[tex]y' = d/dx [ln(5x^2 + 9y^2)]y' = (1 / (5x^2 + 9y^2)) * d/dx [5x^2 + 9y^2][/tex]

Now, we need to apply the chain rule to find the derivative of[tex]5x^2 + 9y^2[/tex]with respect to x:

[tex]d/dx [5x^2 + 9y^2] = d/dx [5x^2] + d/dx [9y^2][/tex]

[tex]d/dx [5x^2] = 10x[/tex]

[tex]d/dx [9y^2] = 18y * dy/dx[/tex]

(Note that we used the chain rule again to find [tex]dy/dx.)[/tex]

Substituting these derivatives into the expression for y', we get:

[tex]y' = (1 / (5x^2 + 9y^2)) * (10x + 18y * dy/dx)[/tex]

Finally, we can simplify this expression by solving for dy/dx:

[tex]y' = (10x + 18y * dy/dx) / (5x^2 + 9y^2)[/tex]

Multiplying both sides by (5x^2 + 9y^2), we get:

[tex]y' * (5x^2 + 9y^2) = 10x + 18y * dy/dx[/tex]

Solving for dy/dx, we obtain:

[tex]dy/dx = (y' * (5x^2 + 9y^2) - 10x) / 18y[/tex]

Therefore, the derivative of y with respect to x is:

[tex]y' = [(5x) / (5x^2 + 9y^2)] + [(9y) / (5x^2 + 9y^2)] * dy/dx[/tex]

or

[tex]dy/dx = [(5x) / (5x^2 + 9y^2)] + [(9y) / (5x^2 + 9y^2)] * y'[/tex]

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Sine function value as 190°. θ = 350°

Cosine function value as 190°. θ = 170°.

Rotational Symmetry: A figure is said to have rotational symmetry if it looks exactly the same after rotating it some angle less than

360∘ (a full rotation).

θ angle with 0° < θ < 360° that I has the same:

sin θ is symmetric over the y-axis and cos θ is symmetric over the x-axis.

This means that if you reflect a point (cos θ, sin θ) over the y-axis, the value of sin θ will not change.

If we reflect the angle of 190° over the y-axis we get 350°

If we reflect the angle of 190° over the x-axis we get 170°

Therefore the answers are 350° and 170°.

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The only option that represents power of power exponents rule is option B: (x^a)^b = x^(ab)

The exponent of a number says how many times to use the number in a multiplication.

There are different laws of exponents such as:
Zero Exponent Law: a^0 = 1.

Identity Exponent Law: a^1 = a.

Product Law: a^m × a^n = a^(m+n)

Quotient Law: a^m/a^n = a^(m-n)

Negative Exponents Law: a^(-m) = 1/a^(m)

Power of a Power: (a^m)^n = a^(mn)

Power of a Product: (ab)^(m) = a^mb^m

Power of a Quotient: (a/b)^m = a^m/b^m

Using power of power exponents rule, we can say that only option B is correct because:

(x^a)^b = x^(ab)

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Thus, the area of new kite with its diagonal doubled is found as: 140 sq. cm.

The area a kite encloses is known as its area of flight. A quadrilateral with two sets of neighbouring sides that are equal is referred to as a kite. A kite is made up of four angles, four sides, and two diagonals.

The product of a lengths of a kite's diagonals divides its area in half.

The area of the kite ABCD = 35 cm square.

The formula for the area of kite = 1/2*(d)*(D)

d - length of small diagonal

D - length of large diagonal.

Then,

35 =  1/2*(d)*(D)

(d)*(D) = 35*2

(d)*(D) = 70 cm sq.  ..eq 1

Now, the length of diagonals of new kite are doubles that is 2d and 2D.

Area of new kite = 1/2 *(2d)*(2D)

Area of new kite = 1/2 *4*(d)*(D)

Area of new kite = 2 *(d)*(D)

Put the value of (d)*(D) from eq 1.

Area of new kite = 2*70

Area of new kite = 140 sq. cm

Thus, the area of the new kite with its diagonal doubled is found as: 140 sq. cm.

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The given function is f(x) = (x+3)^4 * (x-4)^3 * (x-6)^6. The interval on which f(x) is increasing is (4, ∞).

To determine the intervals on which f(x) is increasing or decreasing, we need to analyze the sign of f'(x), the first derivative of f(x). In this case, f'(x) can be calculated using the product and chain rules of differentiation:

f'(x) = 4(x+3)^3 * (x-4)^3 * (x-6)^6 + 3(x+3)^4 * (x-4)^2 * (x-6)^6 + 6(x+3)^4 * (x-4)^3 * (x-6)^5

Simplifying f'(x) and factoring out common terms, we get:

f'(x) = (x+3)^3 * (x-4)^2 * (x-6)^5 * [4(x-6) + 3(x+3)(x-4) + 6(x-4)]

We can now analyze the sign of f'(x) for different values of x:

Thus, the interval on which f(x) is increasing is (4, ∞).

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The given function is f(x) = (x+3)^4 * (x-4)^3 * (x-6)^6. The interval on which f(x) is increasing is (4, ∞).

To determine the intervals on which f(x) is increasing or decreasing, we need to analyze the sign of f'(x), the first derivative of f(x). In this case, f'(x) can be calculated using the product and chain rules of differentiation:

f'(x) = 4(x+3)^3 * (x-4)^3 * (x-6)^6 + 3(x+3)^4 * (x-4)^2 * (x-6)^6 + 6(x+3)^4 * (x-4)^3 * (x-6)^5

Simplifying f'(x) and factoring out common terms, we get:

f'(x) = (x+3)^3 * (x-4)^2 * (x-6)^5 * [4(x-6) + 3(x+3)(x-4) + 6(x-4)]

We can now analyze the sign of f'(x) for different values of x:

Thus, the interval on which f(x) is increasing is (4, ∞).

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To prove that δqrev in equation 6.1 is not an exact differential, we can use the criterion developed in math chapter d. The criterion states that if a differential equation is exact, then its partial derivatives must satisfy the condition ∂M/∂y = ∂N/∂x.

In equation 6.1, δqrev is defined as δqrev = TdS. If we express δqrev in terms of its partial derivatives, we get:
∂(δqrev)/∂S = T
∂(δqrev)/∂T = S

Now, let's calculate the partial derivatives of ∂(∂(δqrev)/∂S)/∂T and ∂(∂(δqrev)/∂T)/∂S:
∂(∂(δqrev)/∂S)/∂T = ∂T/∂S = 0 (since T does not depend on S)
∂(∂(δqrev)/∂T)/∂S = ∂S/∂T = 0 (since S does not depend on T)

Since these partial derivatives are equal to zero, we can conclude that δqrev is not an exact differential, as it does not satisfy the condition ∂M/∂y = ∂N/∂x.

Therefore, we have proven that δqrev in equation 6.1 is not an exact differential.

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