A plot has a concrete path within its borders on all sides having uniform width of 4m. The plot is rectangular with sides 20m and 15m. Charge of removing concrete is Rs. 6 per sq.m. How much is spent ​

The answer is neither (x + 1) nor (x - 1) is a factor of the polynomial x³ + x² - 16x - 16x + 1.

The result is a quotient of x² + 2x - 14 and a remainder of 15. Again, since the remainder is nonzero, the binomial (x - 1) is not a factor of the given polynomial. Hence, neither (x + 1) nor (x - 1) is a factor of the polynomial x³ + x² - 16x - 16x + 1.

To determine whether each binomial is a factor of the polynomial x³ + x² - 16x - 16x + 1, we can use polynomial long division or synthetic division. Let's check each binomial separately:

For the binomial (x + 1):

Performing polynomial long division or synthetic division, we divide x³ + x² - 16x - 16x + 1 by (x + 1):

(x³ + x² - 16x - 16x + 1) ÷ (x + 1)

The result is a quotient of x² - 15x - 16 and a remainder of 17. Since the remainder is nonzero, the binomial (x + 1) is not a factor of the given polynomial.

For the binomial (x - 1):

Performing polynomial long division or synthetic division, we divide x³ + x² - 16x - 16x + 1 by (x - 1):

(x³ + x² - 16x - 16x + 1) ÷ (x - 1)

The result is a quotient of x² + 2x - 14 and a remainder of 15. Again, since the remainder is nonzero, the binomial (x - 1) is not a factor of the given polynomial.

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Answer:

x ≈ 6.2

Step-by-step explanation:

using the sine ratio in the right triangle

sin37° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{AC}{AB}[/tex] = [tex]\frac{x}{10.3}[/tex] ( multiply both sides by 10.3 )

10.3 × sin37° = x , then

x ≈ 6.2 ( to the nearest tenth )

Answer:

x ≈ 6.2

Step-by-step explanation:

Apply the sine ratio rule where:

[tex]\displaystyle{\sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}}[/tex]

Opposite means a side length of a right triangle that is opposed to the measurement (37 degrees), which is "x".

Hypotenuse is a slant side, or a side length opposed to the right angle, which is 10.3 units.

Substitute θ = 37°, opposite = x and hypotenuse = 10.3, thus:

[tex]\displaystyle{\sin 37^{\circ} = \dfrac{x}{10.3}}[/tex]

Solve for x:

[tex]\displaystyle{\sin 37^{\circ} \times 10.3 = \dfrac{x}{10.3} \times 10.3}\\\\\displaystyle{10.3 \sin 37^{\circ} = x}[/tex]

Evaluate 10.3sin37° with your scientific calculator, which results in:

[tex]\displaystyle{6.19869473847... = x}[/tex]

Round to the nearest tenth, hence, the answer is:

[tex]\displaystyle{x \approx 6.2}[/tex]

If the bank pays 10 per cent interest, he is required to save each year Kshs 174,963.76.

We know that Ruto wants to have Kshs 8 million at the end of 15 years. If he saves a certain amount at the end of each year for the next fifteen years and deposits it in a bank that pays 10 per cent interest.

The formula for future value of an annuity is as follows:

FV = PMT x ((1 + r)n - 1) / r

Where,FV is the future value of an annuity

PMT is the amount deposited each yearr is the interest rate

n is the number of years

Let the amount he saves each year be x.

Therefore, the amount of deposit will be x*15.

The interest rate is 10%,

which means r=10/100

=0.10.

Using the formula of future value of an annuity,

FV = x*15 * ((1 + 0.10)^15 - 1) / 0.10FV

= x*15 * (4.046 - 1)FV

= x*15 * 3.046FV

= 45.69x

From the above, we know that the future value of the deposit after 15 years should be Kshs 8,000,000.

Therefore, we can say that:

45.69x = 8,000,000

x = 8,000,000 / 45.69x

= 174963.76 Kshs, approx.

Ruto is required to save Kshs 174,963.76 each year for the next fifteen years.

Therefore, the total amount he will save in fifteen years is Kshs 2,624,456.4, which when invested in a bank paying 10% interest, will earn him a total of Kshs 8 million in 15 years.

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Therefore, the answer to the problem is that the given set S={x∈R4:2x1​−6x2​+7x3​−8x4​=0} is indeed a subspace of R4.

To prove that S={x∈R4:2x1​−6x2​+7x3​−8x4​=0} is a subspace of R4, we must show that it satisfies the following three conditions: It contains the zero vector. The addition of vectors in S is in S. The multiplication of a scalar by a vector in S is in S. Condition 1: S contains the zero vector To show that S contains the zero vector, we must show that (0, 0, 0, 0) is in S. We can do this by substituting 0 for each x value:2(0) - 6(0) + 7(0) - 8(0) = 0Thus, the zero vector is in S. Condition 2: S is closed under addition To show that S is closed under addition, we must show that if u and v are in S, then u + v is also in S. Let u and v be arbitrary vectors in S, then: u = (u1, u2, u3, u4), where 2u1 - 6u2 + 7u3 - 8u4 = 0v = (v1, v2, v3, v4), where 2v1 - 6v2 + 7v3 - 8v4 = 0Then:u + v = (u1 + v1, u2 + v2, u3 + v3, u4 + v4)We can prove that u + v is in S by showing that 2(u1 + v1) - 6(u2 + v2) + 7(u3 + v3) - 8(u4 + v4) = 0 Expanding this out:2u1 + 2v1 - 6u2 - 6v2 + 7u3 + 7v3 - 8u4 - 8v4 = (2u1 - 6u2 + 7u3 - 8u4) + (2v1 - 6v2 + 7v3 - 8v4) = 0 + 0 = 0 Thus, u + v is in S.

Condition 3: S is closed under scalar multiplication To show that S is closed under scalar multiplication, we must show that if c is a scalar and u is in S, then cu is also in S. Let u be an arbitrary vector in S, then: u = (u1, u2, u3, u4), where 2u1 - 6u2 + 7u3 - 8u4 = 0 Then: cu = (cu1, cu2, cu3, cu4)We can prove that cu is in S by showing that 2(cu1) - 6(cu2) + 7(cu3) - 8(cu4) = 0Expanding this out: c(2u1 - 6u2 + 7u3 - 8u4) = c(0) = 0Thus, cu is in S. Because S satisfies all three conditions, we can conclude that S is a subspace of R4. Therefore, the answer to the problem is that the given set S={x∈R4:2x1​−6x2​+7x3​−8x4​=0} is indeed a subspace of R4.

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Step-by-step explanation:

here

AAA postulate can prove that the triangle BAC is congurant to triangle DEF

Answer:

A - [tex]f(x) = 1*2^x[/tex]

Step-by-step explanation:

You know that this is true, because A is the only function option that represents growth. B and D both show decay, and C stays the same.

(a) The solution of the initial value problem is x(t) = -51e^(-5t), and x(0) = 1.

(b) As t approaches infinity, the behavior of the solution x(t) is that it approaches zero. In other words, the solution decays exponentially to zero as time goes to infinity.

To find the solution of the initial value problem -51x' = x^2 - 5x, x(0) = 1, we can separate the variables and integrate.

Starting with the differential equation:

-51x' = x^2 - 5x

Dividing both sides by x^2 - 5x:

-51x' / (x^2 - 5x) = 1

Now, let's integrate both sides with respect to t:

∫ -51x' / (x^2 - 5x) dt = ∫ 1 dt

On the left side, we can perform a substitution: u = x^2 - 5x, du = (2x - 5) dx. Rearranging the terms, we get dx = du / (2x - 5).

Substituting this into the left side of the equation:

∫ -51 / u du = ∫ 1 dt

Simplifying the integral on the left side:

-51ln|u| = t + C₁

Now, substituting back u = x^2 - 5x and simplifying:

-51ln|x^2 - 5x| = t + C₁

To find the constant C₁, we can use the initial condition x(0) = 1. Substituting t = 0 and x = 1 into the equation:

-51ln|1^2 - 5(1)| = 0 + C₁

-51ln|1 - 5| = C₁

-51ln|-4| = C₁

-51ln4 = C₁

Therefore, the solution to the initial value problem is:

-51ln|x^2 - 5x| = t - 51ln4

Simplifying further:

ln|x^2 - 5x| = -t/51 + ln4

Taking the exponential of both sides:

|x^2 - 5x| = e^(-t/51) * 4

Now, we can remove the absolute value by considering two cases:

1) If x^2 - 5x > 0:

  x^2 - 5x = 4e^(-t/51)

2) If x^2 - 5x < 0:

  -(x^2 - 5x) = 4e^(-t/51)

Simplifying each case:

1) x^2 - 5x = 4e^(-t/51)

2) -x^2 + 5x = 4e^(-t/51)

These equations represent the general solution to the initial value problem, leaving it in implicit form.

As for the behavior of the solution as t approaches infinity, we can analyze each case separately:

1) For x^2 - 5x = 4e^(-t/51):

  As t approaches infinity, the exponential term e^(-t/51) approaches zero, which implies that the right side of the equation approaches zero. Therefore, the left side x^2 - 5x must also approach zero. This implies that the solution x(t) approaches the roots of the quadratic equation x^2 - 5x = 0, which are x = 0 and x = 5.

2) For -x^2 + 5x = 4e^(-t/51):

  As t approaches infinity, the exponential term e^(-t/51) approaches zero, which implies that the right side of the equation approaches zero. Therefore, the left side -x^2 + 5x must also approach zero. This implies that the solution x(t) approaches the roots of the quadratic equation -x^2 + 5x = 0, which are x = 0 and x = 5.

In both cases, as t approaches infinity, the solution x(t) approaches the values of 0 and 5.

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a) The points on E are: (0, 2), (0, 3), (1, 0), (1, 2), (1, 3), (2, 0), (2, 3), (3, 0), (3, 1), (3, 4), (4, 1), (4, 4), (infinity).

b) The sum (1, 4) + (3, 1) on the curve is (4, 3).

The given equation is E: y² = x³ + 2x² + 3 (mod 5).

To find the points on E, substitute each value of x (mod 5) into the equation y² = x³ + 2x² + 3 (mod 5) and solve for y (mod 5). The points on E are:

(0, 2), (0, 3), (1, 0), (1, 2), (1, 3), (2, 0), (2, 3), (3, 0), (3, 1), (3, 4), (4, 1), (4, 4), (infinity).

The points (0, 2), (0, 3), (2, 0), and (4, 1) all have an order of 2 as the tangent lines are vertical. So, the other non-zero points on E must have an order of 6.

b) Compute (1, 4) + (3, 1) on the curve:

The equation of the line that passes through (1, 4) and (3, 1) is given by y + 3x = 7, which can be written as y = 7 - 3x (mod 5).

Substituting this line equation into y² = x³ + 2x² + 3 (mod 5), we have:

(7 - 3x)² = x³ + 2x² + 3 (mod 5)

This simplifies to:

4x³ + 2x² + 2x + 4 = 0 (mod 5)

Solving this equation, we find that the value of x (mod 5) is 4. Substituting this value into y = 7 - 3x (mod 5), we have y = 3 (mod 5). Therefore, the sum (1, 4) + (3, 1) on the curve is (4, 3).

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We can conclude that consumer B and consumer C have the same preferences since they have the same utility levels at (91,92) of 8555 and 1044 respectively.

We can use the notion of the Indifference Curve to determine which consumers have the same preferences as given below: From the given information, we have four consumers A, B, C, and D, with utility functions:

UA (91,92) = ln(91) + 292

UB (91, 92) = 91 + (92)²

uc (91,92) = 12q₁ + 12(q2)²

Up (91,92) = 5ln(q₁) + 10q2 +3

Now, we can evaluate the utility functions of the consumers with a common set of commodities to find the utility levels that yield the same levels of satisfaction as shown below: For consumer A:

UA (91,92) = ln(91) + 292UA (91, 92) = 5.26269018917 + 292UA (91, 92) = 297.26269018917

For consumer B:

UB (91, 92) = 91 + (92)²UB (91, 92) = 91 + 8464UB (91, 92) = 8555

For consumer C:

uc (91,92) = 12q₁ + 12(q2)²uc (91,92) = 12 (91) + 12 (92)²uc (91,92) = 1044

For consumer D:

Up (91,92) = 5ln(q₁) + 10q2 +3Up (91,92) = 5ln(91) + 10(92) +3Up (91,92) = 1214.18251811136

Therefore, we can conclude that consumer B and consumer C have the same preferences since they have the same utility levels at (91,92) of 8555 and 1044 respectively.

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The correct order to solve the recurrence relation an - an-1 + 6an-2 for n ≥ 2 with the initial conditions a0 = 3 and a1 = 6 is as follows:

1. Determine the characteristic equation by assuming an = rn.

2. Solve the characteristic equation to find the roots r1 and r2.

3. Write the general solution for an in terms of r1 and r2.

4. Use the initial conditions to find the specific values of r1 and r2.

5. Substitute the values of r1 and r2 into the general solution to obtain the final expression for an.

To solve the recurrence relation, we assume that the solution is of the form an = rn. Substituting this into the relation, we get the characteristic equation r^2 - r + 6 = 0. Solving this equation gives us the roots r1 = -2 and r2 = 3.

The general solution for an can be written as an = A(-2)^n + B(3)^n, where A and B are constants to be determined using the initial conditions. Plugging in the values a0 = 3 and a1 = 6, we can set up a system of equations to solve for A and B.

By solving the system of equations, we find that A = 3/5 and B = 12/5. Therefore, the final expression for an is an = (3/5)(-2)^n + (12/5)(3)^n.

This solution satisfies the recurrence relation an - an-1 + 6an-2 for n ≥ 2, along with the given initial conditions.

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To find the groups with the most members among those with perfect 5-star ratings, you can execute the following query:

SELECT group_name

FROM groups

WHERE rating = 5

ORDER BY membership DESC

LIMIT 1;

When evaluating the quality and popularity of groups, it's important to consider both the rating and membership numbers. While a perfect 5-star rating indicates high user satisfaction, the size of the group's membership can give insight into its overall popularity and appeal.

The query above selects the group_name from the groups table, filtering only those with a rating of 5. The results are then ordered by membership in descending order, ensuring that the group with the highest membership appears at the top. Finally, the "LIMIT 1" clause ensures that only the group with the most members is returned.

By combining the criteria of a perfect rating and the highest membership, this query helps identify the group that not only maintains a stellar reputation but also attracts a significant number of members. It offers a comprehensive approach to assess a group's success and popularity based on both user satisfaction and community size.

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It is False that based only on the r and p values listed above you can come to the conclusion that age is a moderator of the bivariate relationship between the amount of texting and the number of accidents.

In the given scenario, it is not completely true that based only on the r and p values listed above, you can come to the conclusion that age is a moderator of the bivariate relationship between the amount of texting and the number of accidents.

Let's first understand what is meant by the term "moderator.

"Moderator: A moderator variable is a variable that changes the strength of a connection between two variables. If there is a statistically significant bivariate relationship between the amount of texting during driving and the number of accidents, scientists investigate whether this bivariate relationship is moderated by age.

Therefore, based on the values of r and p, it is difficult to determine if age is a moderator of the bivariate relationship between the amount of texting and the number of accidents.

As we have to analyze other factors also to determine whether the age is a moderator or not, such as the sample size, the effect size, and other aspects to draw a meaningful conclusion.

So, it is False that based only on the r and p values listed above you can come to the conclusion that age is a moderator of the bivariate relationship between the amount of texting and the number of accidents.

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Answer:

11.3%

Step-by-step explanation:

Percent error = (|theoretical value - expected value|)/(theoretical value)

= (|6.2-5.5|)/6.2

= 0.7/6.2

= 0.1129

= 11.3%

Answer :

Here trigonometric ratio will be used.

As we can see the figure where 5 is the perpendicular and we have to calculate the value of x.

x is Hypotenuse

Using trigonometric ratio:

[tex] \sf \: \dfrac{P}{H} = \sin \theta[/tex]

Where P is perpendicular and H is Hypotenuse.

Since hypotenuse is x and the value of perpendicular is 5. Therefore by substituting the values of Perpendicular and Hypotenuse in the above trigonometric ratio we will get required value of x.

Also, The value of [tex]\theta[/tex] will be 45°

[tex] \sf\dfrac{5}{x} = \sin 45\degree [/tex]

[tex] \sf\dfrac{5}{x} = \dfrac{1}{ \sqrt{2} } \: \: \: \: \: \: \: \: \: \: \: \bigg( \because \sin45 \degree = \dfrac{1}{ \sqrt{2} } \bigg)[/tex]

Further solving by cross multiplication,

[tex] \sf x = 5 \sqrt{2} [/tex]

So the value of x is [tex] \sf 5 \sqrt{2} [/tex]

a) The degree of the differential equation is first-order.

b) The P(x) term is given by [tex]\(P(x) = \frac{1}{t+1}\).[/tex]

c) The integrating factor is  [tex]\(e^{\int P(x) \, dx}\).[/tex]

a) The degree of the differential equation refers to the highest power of the highest-order derivative present in the equation.

In this case, since the highest-order derivative is [tex]\(dy/dt\)[/tex] , the degree of the differential equation is first-order.

b) The P(x) term represents the coefficient of the first-order derivative in the differential equation. In this case, the equation can be rewritten in the standard form as [tex]\(dy/dt - \frac{t+1}{t+1}y = 4t^2 + 4t\)[/tex].

Therefore, the P(x) term is given by [tex]\(P(x) = \frac{1}{t+1}\).[/tex]

c) The integrating factor is calculated by taking the exponential of the integral of the P(x) term. In this case, the integrating factor is [tex]\(e^{\int P(x) \, dt} = e^{\int \frac{1}{t+1} \, dt}\).[/tex]

d) To find the general solution for the differential equation, we multiply both sides of the equation by the integrating factor and integrate. The general solution is given by [tex]\(y(t) = \frac{1}{I(t)} \left( \int I(t) \cdot (4t^2 + 4t) \, dt + C \right)\)[/tex], where[tex]\(I(t)\)[/tex]represents the integrating factor.

e) To find the particular solution for the differential equation given the boundary condition[tex]\(t = 1\) and \(y = 5\),[/tex] we substitute these values into the general solution and solve for the constant [tex]\(C\).[/tex]

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The maximum possible percentage error in computing the volume is 1.5625%.

To approximate the maximum possible error in computing the volume of a circular cone, we can use the concept of total differential.

The volume V of a circular cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height.

Let's denote the radius as r = 3 inches and the height as h = 12 inches. The possible measurement error is given as Δr = Δh = 1/16 inch.

To find the maximum possible error in the volume, we can use the total differential:

dV = (∂V/∂r)Δr + (∂V/∂h)Δh

First, let's find the partial derivatives of V with respect to r and h:

∂V/∂r = (2/3)πrh

∂V/∂h = (1/3)πr^2

Substituting the values of r and h, we have:

∂V/∂r = (2/3)π(3)(12) = 24π

∂V/∂h = (1/3)π(3)^2 = 3π

Now, we can calculate the maximum possible error in the volume:

dV = (24π)(1/16) + (3π)(1/16)

= (3/4)π + (3/16)π

= (9/16)π

Therefore, the maximum possible error in the volume is (9/16)π cubic inches.

To calculate the percentage error, we divide the absolute error by the actual volume and multiply by 100:

Percentage Error = [(9/16)π / (1/3)π(3^2)(12)] * 100

= (9/16) * (1/36) * 100

= 1/64 * 100

= 1.5625%

Therefore, the maximum possible percentage error in computing the volume is 1.5625%.

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A conjecture about a quadrilateral with a pair of opposite sides that are both congruent and parallel is that it is a parallelogram.

A parallelogram is a quadrilateral with two pairs of opposite sides that are both parallel and congruent. If we have a quadrilateral with just one pair of opposite sides that are congruent and parallel, we can make a conjecture that the other pair of opposite sides is also parallel and congruent, thus forming a parallelogram.

To understand why this conjecture holds, we can consider the properties of congruent and parallel sides. If two sides of a quadrilateral are congruent, it means they have the same length. Additionally, if they are parallel, it means they will never intersect.

By having one pair of opposite sides that are congruent and parallel, it implies that the other pair of opposite sides must also have the same length and be parallel to each other to maintain the symmetry of the quadrilateral.

Therefore, based on these properties, we can confidently conjecture that a quadrilateral with a pair of opposite sides that are both congruent and parallel is a parallelogram.

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The solution to the system of linear equations is x = 3 and y = 5.

To solve the system of linear equations using elimination, we manipulate the equations to eliminate one variable. Let's consider the given system:

Equation 1: 5x + 3y = 30

Equation 2: 10x + 3y = 45

We can eliminate the variable y by multiplying Equation 1 by -2 and adding it to Equation 2:

-10x - 6y = -60

10x + 3y = 45

The x-term cancels out, and we are left with -3y = -15. Solving for y, we find y = 5. Substituting this value back into Equation 1 or Equation 2, we can solve for x:

5x + 3(5) = 30

5x + 15 = 30

5x = 15

x = 3

Therefore, the solution to the system of linear equations is x = 3 and y = 5.

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In this scenario, Rachel and Simon have been running a restaurant business together for 15 years. Rachel is responsible for managing the front-of-house operations and staffing, while Simon is a trained chef who takes care of the kitchen. However, they have differing opinions on how to allocate the profits.

Rachel wants to save most of the profits, but also believes it's important to spend a small portion on advertising to promote the restaurant. On the other hand, Simon wants to use a large portion of the profits to redecorate the restaurant. Conflicts like these regarding money are quite common in business partnerships.
To address this issue, Rachel and Simon need to communicate and find a middle ground that satisfies both of their interests. They can start by discussing their individual perspectives and concerns openly. For example, Rachel can explain the importance of advertising in attracting more customers and increasing revenue, while Simon can explain how the redecoration can enhance the overall dining experience and potentially attract new customers as well.
Once they understand each other's viewpoints, they can brainstorm potential solutions together. One option could be allocating a portion of the profits to both advertising and redecoration, finding a balance that satisfies both parties. They can also explore other possibilities, such as seeking funding for the redecoration project through external sources, or gradually saving for it over a longer period of time.
It's crucial for Rachel and Simon to have open and respectful communication throughout this process. They should listen to each other's concerns, be willing to compromise, and ultimately make decisions that benefit the long-term success of their restaurant business. By finding a solution that considers both their needs and goals, they can navigate this conflict and continue running their restaurant successfully.

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The solution to the quadratic equation x² −6x+6=0 by completing the square is 3+√3 , 3-√3

To complete the square, we first move the constant term to the right-hand side of the equation:

x² − 6x = -6

We then take half of the coefficient of our x term, square it, and add it to both sides of the equation:

x² − 6x + (-6/2)² = -6 + (-6/2)²

x² − 6x + 9 = -6 + 9

(x - 3)² = 3

Taking the square root of both sides of the equation, we get:

x - 3 = ±√3

x = 3 ± √3

Therefore, the solutions to the quadratic equation x² − 6x+6=0 are:

x = 3 + √3

x = 3 - √3

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Answer: 378π in³

Step-by-step explanation:

The given statement is false. A square matrix A (m × n) has an eigenvalue λ if there is a nonzero vector x in Rn such that Ax = λx.

If the only eigenvalue of A is zero, it is called a zero matrix, and all its entries are zero. The matrix A is a scalar matrix with an eigenvalue λ if it is diagonal, and each diagonal entry is equal to λ.The matrix A will not necessarily be zero if 0 is its only eigenvalue. To prove the statement is false, we will provide an example; Let A be the following 3 x 3 matrix:

{0, 1, 0} {0, 0, 1} {0, 0, 0}A=0

is the only eigenvalue of A, but A is not equal to 0. The statement "If 0 is the only eigenvalue of A (matrix M3x3 (C)), then A = 0" is false. A matrix A (m × n) has an eigenvalue λ if there is a nonzero vector x in Rn such that

Ax = λx

If the only eigenvalue of A is zero, it is called a zero matrix, and all its entries are zero.The matrix A will not necessarily be zero if 0 is its only eigenvalue. To prove the statement is false, we can take an example of a matrix A with 0 as the only eigenvalue. For instance,

{0, 1, 0} {0, 0, 1} {0, 0, 0}A=0

is the only eigenvalue of A, but A is not equal to 0.

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(a) (i) To prove that the rule holds true in every group G, we need to show that if x° = 1, then x = 1 for all elements x in the group. This rule is indeed true in every group because the identity element, denoted by 1, satisfies this property.

(b)

(i) In array notation, a = [4, 7, 9, 5, 6, 1, 8, 2, 3].

(c) Given that o' = 1, we want to prove that o is even. In permutations, the identity element is considered an even permutation.

For any element x in the group, if x° (the identity element operation) results in the identity element 1, then x must be equal to 1.

(ii) To prove or disprove this rule, we need to find a counterexample where xy = 1 but yx ≠ 1. Consider the group of non-zero real numbers under multiplication. Let x = 2 and y = 1/2. We have xy = 2 * (1/2) = 1, but yx = (1/2) * 2 = 1, which is not equal to 1. Therefore, this rule is false in some groups.

(iii) To prove or disprove this rule, we need to find a counterexample where (xy)2 ≠ x²y2. Consider the group of non-zero real numbers under multiplication. Let x = 2 and y = 3. We have (xy)2 = (2 * 3)2 = 36, whereas x²y2 = (2²) * (3²) = 36. Thus, (xy)2 = x²y2, and this rule holds true in every group.

(iv) To prove or disprove this rule, we need to find a counterexample where xyx-ly-1 = 1 but xy ≠ yx. Consider the group of permutations of three elements. Let x be the permutation that swaps elements 1 and 2, and let y be the permutation that swaps elements 2 and 3. We have xyx-ly-1 = (2 1 3) = 1, but xy = (2 3) ≠ (3 2) = yx. Thus, this rule is false in some groups.

(b)

(i) In array notation, a = [4, 7, 9, 5, 6, 1, 8, 2, 3].

(ii) In cyclic notation, a = (4 5 6 1)(7 8 2)(9 3).

(iii) The sign of a permutation can be determined by counting the number of inversions. An inversion occurs whenever a number appears before another number in the permutation and is larger than it. In this case, a has 6 inversions: (4, 1), (4, 2), (7, 2), (9, 3), (9, 5), and (9, 6). Since there are an even number of inversions, the sign of a is positive or +1. The order of a can be determined by finding the least common multiple of the lengths of the disjoint cycles, which in this case is lcm(4, 3, 2) = 12. Therefore, the sign of a is +1 and the order of a is 12.

(iv) To compute a2022, we can simplify it by taking the remainder of 2022 divided by the order of a, which is 12. The remainder is 2, so a2022 = a2. Computing a2, we get:

a2 = (4 5 6 1)(7 8 2)(9 3) * (4 5 6 1)(7 8 2)(9 3)

= (4 5 6 1)(7 8 2)(9 3) * (4 5 6 1)(7 8 2)(9 3)

= (4 5 6 1)(7 8 2)(9 3)(4 5 6 1)(7 8 2)(9 3)

= (4 1)(5 6)(7 2)(8)(9 3)

= (4 1)(5 6)(7 2)(9 3)

Therefore, a2022 = (4 1)(5 6)(7 2)(9 3).

(c) Given that o' = 1, we want to prove that o is even. In permutations, the identity element is considered an even permutation. If o' = 1, it means that the number of inversions in o is even. An even permutation can be represented as a product of an even number of transpositions. Since the identity permutation can be represented as a product of zero transpositions (an even number), o must also be even.

Regarding o^-1 (the inverse of o), the inverse of an even permutation is also even, and the inverse of an odd permutation is odd. Therefore, if o is even, its inverse o^-1 will also be even.

In summary, if o' = 1, o is even, and o^-1 is also even.

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Answer:

ST = 108km

Step-by-step explanation:

In ΔPQR and ΔTSR,

∠PRQ = ∠TRS (vertically opposite)

∠PQR = ∠TSR (alternate interior)

∠QPR = ∠ STR (alternate interior)

Since all the angles are equal,

ΔPQR and ΔTSR are similar

Therefore, their corresponding sides have the same ratio

[tex]\implies \frac{ST}{PQ} = \frac{RT}{PR}\\ \\\implies \frac{ST}{48} = \frac{81}{36}\\\\\implies ST = \frac{81*48}{36}[/tex]

⇒ ST = 108km

Answer:

Hexagon

Step-by-step explanation:

the hexagon is the only one that can tessellate without leaving gaps. A tessellation is a tiling of a plane with shapes, such that there are no gaps or overlaps. Hexagons have the unique property that they can fit together perfectly without leaving any spaces between them. This is why hexagonal shapes, such as honeycombs, are often found in nature, as they provide an efficient use of space. The octagon, pentagon, and circle cannot tessellate without leaving gaps because their shapes do not fit together seamlessly like the hexagons.

Answer:Equilateral triangles, squares and regular hexagons

Step-by-step explanation:

Answer:

See below

Step-by-step explanation:

An easy example of an inequality where you need to flip the sign to be true is something like [tex]-2x > 4[/tex]. By dividing both sides by -2 to isolate x and get [tex]x < -2[/tex], you would need to also flip the sign to make the inequality true.

One that wouldn't need to be reversed is [tex]2x > 4[/tex]. You can just divide both sides by 2 to get [tex]x > 2[/tex] and there's no flipping the sign since you are not multiplying or dividing by a negative.

a) The probability that both Ekene and Amina will be alive in 5 years time is 3/10.

b) The probability that only Ekene will be alive in 5 years time is 9/20.

a) Probability that both Ekene and Amina will be alive:

To find the probability that both Ekene and Amina will be alive in 5 years time, we use the principle of multiplication. Since Ekene's probability of being alive is 3/4 and Amina's probability is 2/5, we multiply these probabilities together to get the joint probability.

The probability of Ekene being alive is 3/4, which means there is a 3 out of 4 chance that he will be alive. Similarly, the probability of Amina being alive is 2/5, indicating a 2 out of 5 chance of her being alive. When we multiply these probabilities, we get:

P(Both alive) = (3/4) * (2/5) = 6/20 = 3/10

Therefore, the probability that both Ekene and Amina will be alive in 5 years time is 3/10.

b) Probability that only Ekene will be alive:

To find the probability that only Ekene will be alive in 5 years time, we need to subtract the probability of both Ekene and Amina being alive from the probability of Amina being alive. This gives us the probability that only Ekene will be alive.

P(Only Ekene alive) = P(Ekene alive) - P(Both alive)

We already know that the probability of Ekene being alive is 3/4. And from part (a), we found that the probability of both Ekene and Amina being alive is 3/10. By subtracting these two probabilities, we get:

P(Only Ekene alive) = (3/4) - (3/10) = 30/40 - 12/40 = 18/40 = 9/20

Therefore, the probability that only Ekene will be alive in 5 years time is 9/20.

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Answer:

3

Step-by-step explanation:

First find all the factors of 48:

1, 2, 3, 4, 6, 8, 12, 16, 24, 48

These are the only values that x can be.  Try them all and see which results in a whole number:

√48/1 = 6.93  not whole

√48/2 = 4.9  not whole

√48/3 = 4  WHOLE

√48/4 = 3.46  not whole

√48/6 = 2.83  not whole

√48/8 = 2.45  not whole

√48/12 = 2  WHOLE

√48/16 = 1.73  not whole

√48/24 = 1.41  not whole

√48/48 = 1  WHOLE

Therefore, there are 3 values of x for which √48/x = whole number.  The numbers are x = 3, 12, 48