Prove that the roots of x² + (1-k)x+k-3=0 are real for all real values of k.​

Step-by-step explanation:

Out of the given options, only √196 is a rational number.

A rational number is a number that can be expressed as the ratio of two integers. In other words, it can be written in the form of p/q where p and q are integers and q is not equal to zero.

√56 cannot be simplified further and has no integer factors that can be canceled out to express it as a ratio of two integers. Therefore, it is an irrational number.

Similarly, √63 and √240 cannot be simplified further and do not have any integer factors that can be canceled out to express them as a ratio of two integers. Therefore, they are also irrational numbers.

On the other hand, √196 simplifies to 14 which is a ratio of two integers (14/1). Hence, it is a rational number.

In summary, only √196 is a rational number out of the given options.

The domain of the function is x ≥ 1 and the range of the function is f(x) ≤ 0.

The given function is:

f(x) = -√(x-6+5)

Simplifying inside the square root gives:

f(x) = -√(x-1)

For this function, the value inside the square root must be non-negative, so we have:

x - 1 ≥ 0

x ≥ 1

So, the domain of the function is all real numbers greater than or equal to 1.

To find the range of the function,

We can note that the square root is always non-positive since it has a negative coefficient.

So, the range of the function is all real numbers less than or equal to 0.

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The company should produce 5 units of Metakeb tractors, 8 units of Mewesson tractors, and 16 units of Metekem tractors in a month to use up all the available resources.

What is equations ?

An equation is a mathematical statement that shows that two expressions are equal. It consists of two expressions separated by an equals sign. For example, 3x + 5 = 14 is an equation that states that the sum of 3 times x and 5 is equal to 14.

Let x, y, and z be the number of units of Metakeb, Mewesson, and Metekem tractors produced, respectively.

Then we have the following system of equations based on the time required in each department:

[tex]1200x + 1800y + 3000z &\leq 63000 \\&&\text{(machining department)}\2400x + 1200y + 1200z &\leq 69600 \\&&\text{(assembly department)}\600x + 3000y &\leq 49800 \\&&\text{(inspection department)}[/tex]

Converting the hours to minutes, we get:

Machining department: 1050 hours = 63,000 minutes

Assembly department: 1160 hours = 69,600 minutes

Inspection department: 830 hours = 49,800 minutes

We can rewrite the system of equations as follows:

[tex]20x + 30y + 50z &\leq 1050 &&\text{(machining department)}\\ \\10x + 5y + 5z &\leq 580 &&\text{(assembly department)}\\ \\x + 5y &\leq 83 &&\text{(inspection department)}\\ \\[/tex]

To use Gaussian elimination to solve the system of equations, we first write the augmented matrix:

[tex]\begin{bmatrix}20 & 30 & 50 & | & 1050 \\10 & 5 & 5 & | & 580 \\1 & 5 & 0 & | & 83\end{bmatrix}[/tex]

Performing row operations to obtain row echelon form:

[tex]\begin{bmatrix}20 & 30 & 50 & | & 1050 \\0 & -55 & -95 & | & -790 \\0 & 1 & -5/4 & | & -1.35\end{bmatrix}[/tex]

Next, perform additional row operations to get the matrix in reduced row echelon form:

[tex]\begin{bmatrix}20 & 30 & 50 & | & 1050 \\0 & 1 & 19/55 & | & -14.36 \\0 & 0 & 367/55 & | & 105.36\end{bmatrix}[/tex]

Finally, solve for the variables:

[tex]z &= \frac{105.36}{367/55} = 16 \\ \\y + \frac{19}{55}z &= 14.36 \\ \\y + \frac{19}{55}(16) &= 14.36 \\ \\y &= 8 \\ \\20x + 30y + 50z &= 1050 \\ \\20x + 30(8) + 50(16) &= 1050 \\ \\20x &= 90 \\ \\x &= 4.5[/tex]

Therefore, the company should produce 4.5 units of Metakeb tractors, 8 units of Mewesson tractors, and 16 units of Metekem tractors in a month to use up all the available resources. Since we cannot produce fractional units, we can round up the values of x and y to obtain a feasible production plan of 5 units of Metakeb tractors and 8 units of Mewesson tractors.

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Lambert will have R22 500 left over after paying tax.

How to solve and what is tax?

If Lambert earns R30 000 every month and 1/4 of that amount is paid as tax, then the amount of tax he pays every month can be calculated as:

Tax = (1/4) x R30 000

= R7 500

Therefore, after paying tax, Lambert will have the remaining amount left over, which can be calculated as:

Remaining amount = R30 000 - R7 500

= R22 500

So, Lambert will have R22 500 left over after paying tax.

Tax is a compulsory financial charge or levy imposed by a government or other authority on income, goods, services, or transactions. Taxes are typically levied as a percentage of income or the value of goods and services, and they are collected to fund public services and infrastructure, such as schools, hospitals, roads, and public utilities.

The tax system is an important source of revenue for governments around the world, and it is used to finance various public goods and services that are essential for the functioning of society.

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Answer:

8,415

Step-by-step explanation:

2+2+2+2+20+41+20+8252+74 = 8,415

Answer:

8,415

Step-by-step explanation:

Copy and pasted into a calc

Answer:

y = -1/16 (x+9)^2 - 4

Step-by-step explanation:

y = -1/16 (x^2 + 18x + 81) - 4

y = -1/16 x^2 - 18/16 x - 81/16 - 4

y = -1/16 x^2 - 18/16 x - 85/16

The correct answer is 5.88 × 1013 miles. This is because a light-year is the distance that light can travel in one year, or approximately 5.88 × 1012 miles.

A light year is a unit of distance used to measure astronomical distances. It is the distance that light travels in a vacuum in one year, which is equal to 9.46 trillion kilometres. This means that light travelling at a speed of 300,000 kilometres per second would take one year to travel nine trillion kilometres. T

To find the distance of a star from Earth, one must multiply the number of light-years by 5.88 × 1012. In this case, the star is 9.8 × 101 light-years away, meaning that it is 9.8 × 101 times 5.88 × 1012 miles, or 5.88 × 1013 miles, away from Earth.

Light-years are a unit of distance used by astronomers to measure the great distances between stars and galaxies. Light travels at a speed of approximately 186,000 miles per second, and one light-year is the distance that light can travel in one year. For example, a star that is 9.8 × 101 light-years away from Earth is approximately 5.88 × 1013 miles away from Earth. To convert the number of light-years to the number of miles, one must multiply the number of light-years by 5.88 × 1012.

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The present value of the security is $2142.86, rounded to the nearest cent.

Value is a subjective concept that is based on personal beliefs, experiences, and backgrounds. It is the appreciation of something, whether tangible or intangible, that is based on a person's perception of its worth. Value is often used to describe the worth or importance of something or someone, and can be expressed in monetary terms or in terms of the usefulness of something. Ultimately, value is the measure of how much something is worth to an individual.

The present value of this security is $2142.86. This is determined by using the formula:

Present Value = Dividend / (1 + R)t

Where Dividend is the annual dividend of $150, R is the rate of return which is 7%, and T is the time in years which is 1.

Therefore, the present value of the security is:

150 / (1 + 0.07)¹ = $2142.86

Thus, the present value of the security is $2142.86, rounded to the nearest cent.

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The required function is f(m) = -53.33m + 360, where f(m) represents the number of frogs in the lake after m months.

What is function ?

In mathematics, a function is a rule that assigns a unique output or "dependent variable" to each input or "independent variable" in a set. It 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.

Based on the given graph, it appears that the number of frogs in the lake is decreasing linearly over time. We can use the slope-intercept form of a linear equation to represent this relationship:

y = mx + b

where y is the number of frogs, x is the time in months, m is the slope (or rate of decrease) of the line, and b is the initial number of frogs at time zero.

To find the slope of the line, we can use the two points (0, 360) and (3, 200) from the graph:

m = (change in y) / (change in x) = (200 - 360) / (3 - 0) = -53.33

So the slope of the line is -53.33, which means that the number of frogs is decreasing by an average of 53.33 per month.

To find the initial number of frogs, we can use the y-intercept of the line, which appears to be around 360 on the graph. So we have:

b = 360

Putting it all together, the function that represents the number of frogs in the lake after m months is:

f(m) = -53.33m + 360

Therefore, the required function is f(m) = -53.33m + 360, where f(m) represents the number of frogs in the lake after m months.

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The formula for the linear function in the graph is f(x) = -2x + 4.

The graph is a data structure that is used to represent relationships between objects. It is composed of nodes (or vertices) that are connected by edges (or lines). Graphs can be used to represent many types of data, such as networks of people, roads, and even abstract concepts like language. Graphs are used in a variety of fields, such as computer science, mathematics, and sociology. They allow for efficient storage and retrieval of data, as well as efficient traversal of the data.

The graph shown below is a linear graph with a slope of -2 and a y-intercept of 4.

To find the formula for the linear function, we can use the slope-intercept form, which is y = mx + b. In this equation, m is the slope of the line and b is the y-intercept.

Therefore, the formula for the linear function in the graph is f(x) = -2x + 4.

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complete question:

Find a formula for the linear function depicted in the following graph.

f(x)=

Answer:

2502

Step-by-step explanation:

8,794-2,670-3,622=

The linear functions of the graph is y = 2x + 5

The complete question is added as attachment

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

(0, 5) and (-2.5, 0)

From the question, we understand that the function is a linear function

A linear function is represented as

y = mx + c

Using the above as a guide, we have the following equations

0 * m + c = 5

-2.5 * m + c = 0

When evaluated, we have

c = 5

So, we have

-2.5 * m + 5 = 0

This gives

-2.5m = -5

Evaluate

m = 2

So, the equation is y = 2x + 5

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Answer:

90(3^10) is divisible by 90.

Step-by-step explanation:

9^7+3^12=(3^2)^7+3^12

=(3^7)^2+3^12

=(3^14)+(3^12)

=(3^12)(3^2)+3^12

=(3^12)(3^2+1)

=(3^12)(4)

=90(3^10)

An angle whose vertex lies on a circle and whose sides intercept the circle (the sides contain chords of the circle) is called an inscribed angle. The measure of an inscribed angle is half the measure of the arc it intercepts

Answer:

One possible equation that represents the data in the table is: y = 4x^2.

Answer:

429

Step-by-step explanation:

Due to the ladder's 13-foot height, its angle with the ground is approximately 55.1 degrees.

The vertex of the angle, which is created by the intersection of two rays, is a geometric figure known as an angle. The sides or legs of the angle are often used terms to describe the rays. Angles can be acute (less than 90 degrees), right (exactly 90 degrees), obtuse (more than 90 degrees but less than 180 degrees), straight (exactly 180 degrees), or reflex. Angles are measured in units of degrees or radians (greater than 180 degrees but less than 360 degrees).

given

In order to solve this puzzle, we must determine the angle that the ladder creates with the ground using the neighbouring side and hypotenuse of a right triangle. The trigonometric function cosine can be used to calculate the angle.

adjacent/hypotenuse = cos

cos θ = 7/13

θ = cos⁻¹(7/13) (using the inverse cosine of both sides) (taking the inverse cosine of both sides)

Calculating the answer, we obtain:

θ ≈ 55.1°

Due to the ladder's 13-foot height, its angle with the ground is approximately 55.1 degrees.

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The complete question is:-   1) A ladder that is 13 ft. tall has fallen against a house. If the top of the ladder touches the house 7 ft. above the ground, what is the angle that the ladder makes with the ground? Round your degrees to one decimal place.

Label the triangle and show all of your work! Circle your final answer.

Rounding to the nearest whole number, we get that approximately 340 students in the random sample of 431 IRSC students are under the age of 22.

Percentages are commonly used to express ratios, proportions, or rates in many fields, such as mathematics, science, economics, and everyday life. They can be used to compare quantities, indicate changes, or make predictions based on data. For instance, we can use percentages to calculate discounts, taxes, interest rates, or probabilities, or to analyze trends, surveys, or experiments. Percentages are also often used in visual representations, such as pie charts, bar graphs, or maps, to show the distribution or magnitude of data.

Here,

To determine how many IRSC students are under the age of 22, we need to find 67% of the total number of students. Using the percentage formula, we get:

67% × 19,026 = 12,739.42

Rounding to the nearest whole number, we get that approximately 12,739 IRSC students are under the age of 22.

To determine how many students in the random sample of 431 students are under the age of 22, we need to find 79% of the sample size. Using the percentage formula, we get:

79% × 431 = 340.49

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Answer:

Step-by-step explanation:

[tex]i=2: \frac{3}{2} \times\frac{1}{3^2}=\frac{1}{6}[/tex]

[tex]i=3: \frac{3}{2} \times\frac{1}{3^3}=\frac{1}{18}[/tex]

[tex]i=4: \frac{3}{2} \times\frac{1}{3^4}=\frac{1}{54}[/tex]

[tex]i=5: \frac{3}{2} \times\frac{1}{3^5}=\frac{1}{162}[/tex]

[tex]i=6: \frac{3}{2} \times\frac{1}{3^6}=\frac{1}{486}[/tex]

[tex]sum=\frac{121}{486}[/tex]

The corresponding angle of ∠3 in the parallel line is ∠8.

When parallel lines are cut by a transversal line, angle relationships are formed such as corresponding angles, vertically opposite angles, alternate interior angles, alternate exterior angles, linear angles etc.

Corresponding angles are the angles which are formed in matching corners or corresponding corners with the transversal when two parallel lines are intersected by any other line.

Corresponding angles are congruent.

Hence, the corresponding angle of ∠3 is ∠8.

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The value of x is given as follows:

x = 4.

The exponential expression for this problem is defined as follows:

8^x = 4096.

Both 8 and 4096 are powers of 2, and can be represented as follows:

Hence the expression can be written as follows:

(2³)^x = 2^12.

Applying the power of power rule, we have that:

2^(3x) = 2^(12).

Applying the one-to-one property of the family of functions powers of x, the value of x is obtained as follows:

3x = 12

x = 4.

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Option C, The two packages have the same volume because their bases are congruent and they have the same height.

Volume is a measure of the amount of space occupied by a three-dimensional object. It is typically measured in cubic units, such as cubic meters or cubic centimeters.

The volume of cylinder A = πr²h

= 6πr²

The volume of cylinder B= πr²h

= 6πr²

As both the cylinder have same values, Option C, The two packages have the same volume because their bases are congruent and they have the same height.

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The seismic waves of the Chernobyl disaster were approximately 400 times stronger than the standard seismic waves.

The Chernobyl disaster was a catastrophic nuclear accident that occurred on April 26, 1986 at the Chernobyl Nuclear Power Plant in Ukraine. The accident occurred due to a sudden power increase that caused an unexpected power surge in one of the reactors, resulting in a series of explosions and a fire that released large quantities of radioactive particles into the atmosphere.

This is because the Chernobyl disaster was a nuclear explosion, which released a vast amount of energy, much greater than that released by a standard earthquake.

The nuclear reactor exploded at the Chernobyl site, releasing a large amount of energy into the atmosphere. This energy was then converted into seismic waves, which spread out in all directions. These seismic waves were much stronger than the waves produced by a standard earthquake, and they were felt around the world.

The seismic waves generated by the Chernobyl disaster had an estimated magnitude of 5.5, while standard seismic waves usually have a magnitude of 2.0 or less. This means that the Chernobyl disaster seismic waves were around 400 times more powerful than the standard seismic waves.

The Chernobyl disaster is one of the most devastating nuclear accidents in history. The seismic waves that were generated were far stronger than those of a standard earthquake, and the energy released was devastating for the region and the world. The effects of the disaster are still felt today, and it serves as a reminder of the dangers of nuclear energy.

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The graph of y=2ˣ-4 is as shown.

To graph the function y=2ˣ-4 and plot some points with integer coordinates, we can use a graphing calculator or an online graphing tool. Here are the steps:

Open a graphing tool, such as Desmos or GeoGebra.

Enter the function y=2ˣ-4 in the equation editor or function input box.

Adjust the x-axis and y-axis to fit the desired range. For example, we can set the x-axis from -5 to 5 and the y-axis from -5 to 10.

Plot some points with integer coordinates that fit the axes. Here are four possible points:

When x=0, y=2⁰-4= -2, so the point (0,-2) is on the graph.

When x=1, y=2¹-4= -2, so the point (1,-2) is on the graph.

When x=-1, y=2⁻¹-4= -4.5, so the point (-1,-4) is on the graph.

When x=2, y=2²-4= 0, so the point (2,0) is on the graph.

Plot these points on the graph by clicking or tapping on the corresponding locations. The graphing tool will connect the points and draw the graph of the function.

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Answer:

We'll start with the left-hand side (LHS) of the identity, which is:

sin x (1 + cot^2 x)

We can rewrite cot^2 x as (cos x / sin x)^2, since cotangent is the reciprocal of tangent. Substituting this into the LHS, we get:

sin x (1 + (cos x / sin x)^2)

Now we can simplify the expression in the parentheses by using the identity:

tan^2 x + 1 = sec^2 x

Rearranging this identity, we get:

tan^2 x = sec^2 x - 1

Substituting this into our expression, we get:

sin x (1 + (cos x / sin x)^2) = sin x (1 + (cos^2 x / sin^2 x))

= sin x (sin^2 x / sin^2 x + cos^2 x / sin^2 x)

= sin x ((sin^2 x + cos^2 x) / sin^2 x)

= sin x (1 / sin^2 x)

= sin x / sin^2 x

= 1 / sin x

Now we'll simplify the right-hand side (RHS) of the identity, which is:

csc x

We know that csc x is the reciprocal of sin x, so we can rewrite the RHS as:

1 / sin x

This is the same as the expression we obtained for the LHS, so we have shown that:

sin x (1 + cot^2 x) = csc x

And this proves the identity!

Remember, when you're proving trigonometric identities, it's important to be familiar with the fundamental trigonometric identities and the basic algebraic rules of manipulating equations

good luck with your exam

The total cost for d drinks and p popcorns can be calculated by multiplying the number of drinks by the cost per drink ($4) and adding it to the number of popcorns multiplied by the cost per popcorn ($6), like this:

Total cost = (cost per drink x number of drinks) + (cost per popcorn x number of popcorns)

Total cost = ($4 x d) + ($6 x p)

So the answer would be:

Total cost = $4d + $6p

The total charge for d drinks is 4d, and the total charge for p popcorns is 6p. Therefore, the total charge for d drinks and p popcorns is:

4d + 6p