4.3.1 Investigate whether the following statements are true for all x € N. i. x² - x is divisible by 2. ii. Try to generalise each of the above statements. Prove your generalisation. iii.​

The inequality will be and [tex]3 + √-4[/tex] will be written as [tex]3 +2i[/tex] as complex number.

What are complex numbers?

In mathematics, a complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1). Complex numbers are an extension of the real numbers, which include all the numbers that can be represented on a number line.

The real part of a complex number a + bi is the real number a, and the imaginary part is the real number bi.

A. Since Jan found that both roots of the quadratic equation are not real numbers, this means that the discriminant b² - 4ac is negative. Therefore, the inequality that must always be true for Jan's quadratic equation is:

[tex]b² - 4ac < 0[/tex]

B. The expression  [tex]3+√-4[/tex] can be written as [tex]3 + 2i,[/tex] where i is the imaginary unit (√-1). This is because √-4 is equal to 2i, so [tex]3+√-4[/tex] can be written as [tex]3 + 2i.[/tex] Therefore, [tex]3+√-4[/tex] written as a complex number is 3 + 2i.

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Answer:

[tex]112cm^2[/tex]

Step-by-step explanation:

Volume of a rectangluar prism = Length x Width x Height

Volume, Length, and Width is given now we can solve for Height, c

[tex]V = l * w * h[/tex]

[tex]80m^3 = 4m * 4m * h[/tex]

[tex]h = 80m^3 / 16m^2[/tex]

[tex]h = 5m[/tex]

Surface area of a rectangular prism is A = 2(wl +hl +hw)

[tex]A = 2(4m * 4m + 5m * 4m + 5m * 4m)[/tex]

[tex]= 2(16m^2 + 20m^2 + 20m^2)[/tex]

[tex]= 2(56cm^2)\\A= 112cm^2[/tex]

Hope this helps!
Brainliest is much appreciated!

supongo q es es 8 cm pero no creo q sería 2 o l

The Japanese island that became the center of power in Japan is Honshu.

Honshu is the largest and most populous island in Japan and has historically been the center of political, economic, and cultural power in Japan. It is home to Tokyo, the capital city of Japan, as well as many other major cities such as Osaka, Kyoto, and Yokohama. Honshu's central location and natural resources have made it a strategic location for political power throughout Japan's history, including during the feudal period when various powerful clans vied for control. Honshu's position as the economic and cultural hub of Japan has also helped solidify its status as the center of power, as it has been a center for education, trade, and innovation for centuries.

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The perimeter of a rectangular outdoor patio is 106 ft. the length is 9 ft greater than the width.  Hence, the dimensions of the patio are 38 ft and 47 ft.

Given that the perimeter of a rectangular outdoor patio is 106 ft. The length is 9 ft greater than the width. Now we need to find the dimensions of the patio.

Step 1: Let's consider the width of the patio be x feet.

The length of the patio is given as 9 ft greater than the width.

So the length of the patio will be (x + 9) feet.

Step 2: The perimeter of a rectangle is given by P = 2(l + w).

So the perimeter of the patio is given as 106 ft.

Thus,2(l + w) = 1062(x + x + 9)

                    = 1062(2x + 9)

                    = 1062x + 1821

                     = 106x = 106 - 182x

                     = -76 (not possible)

Therefore, x = 38 ftSo the width of the patio is 38 ft The length of the patio is 38 + 9 = 47 ft

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Using matrix inversion method and perimeter of rectangle, the dimension of the rectangle are 16cm and 7cm

Let the length of the rectangle be l and the width be w. We know that l - w = 3.5 and 2l + 2w = 67. We can rewrite these equations as a system of linear equations in matrix form:

[tex]$$\begin{bmatrix}1 & -1 \ 2 & 2\end{bmatrix} \begin{bmatrix}l \ w\end{bmatrix} = \begin{bmatrix}3.5 \ 67/2\end{bmatrix}$$[/tex]

We can solve for [tex]$\begin{bmatrix}l \ w\end{bmatrix}$[/tex] by multiplying both sides of the equation by the inverse of the coefficient matrix:

[tex]$$\begin{bmatrix}l \ w\end{bmatrix} = \begin{bmatrix}1 & -1 \ 2 & 2\end{bmatrix}^{-1} \begin{bmatrix}3.5 \ 67/2\end{bmatrix}$$[/tex]

To find the inverse of the coefficient matrix, we can use the following formula:

[tex]$$\begin{bmatrix}a & b \ c & d\end{bmatrix}^{-1} = \frac{1}{ad-bc} \begin{bmatrix}d & -b \ -c & a\end{bmatrix}$$[/tex]

Plugging in the values for our matrix, we get:

[tex]$$\begin{bmatrix}1 & -1 \ 2 & 2\end{bmatrix}^{-1} = \frac{1}{1 \cdot 2 - (-1) \cdot 2} \begin{bmatrix}2 & 1 \ -2 & 1\end{bmatrix} = \begin{bmatrix}1/2 & 1/4 \ -1/2 & 1/4\end{bmatrix}$$[/tex]

Now we can substitute this matrix and the vector on the right-hand side of the equation into our formula to obtain the solution:

[tex]$$\begin{bmatrix}l \ w\end{bmatrix} = \begin{bmatrix}1/2 & 1/4 \ -1/2 & 1/4\end{bmatrix} \begin{bmatrix}3.5 \ 67/2\end{bmatrix} = \begin{bmatrix}16 \ 7\end{bmatrix}$$[/tex]

Therefore, the dimensions of the rectangle are 16 cm by 7 cm.

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Answer:

Line A has a slope of -4

Line A and line B have the same slope

Step-by-step explanation:

Line A is in the slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.  As we can see, -4 is m in line A so -4 is the slope of line A.

If we have two points on a line, we can find the slope using the slope formula

[tex]m = \frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex], where x1 and y1 are one of the points and x2 and y2 come from the other point.

Thus, if we allow (-2, 1) to be x1 and y1 and (1, -11) to be x2 and y2,

[tex]m=\frac{-11-1}{1-(-2)}\\ m=-4[/tex]

Since the slope of Line B is -4, it has the same slope as line A

Answer:

[tex]\boxed{\mathtt{Area \approx 104.7m^{2}}}[/tex]

Step-by-step explanation:

[tex]\textsf{We are asked to find the area of 1/3 of a circle.}[/tex]

[tex]\textsf{Let's review the formula needed to find the area of a \underline{whole} circle.}[/tex]

[tex]\large\underline{\textsf{Formula:}}[/tex]

[tex]\mathtt{Area = \pi (radius)^{2}.}[/tex]

[tex]\textsf{We should know that a circle is 360}^{\circ}. \ \textsf{We are given 120}^{\circ} \textsf{of a circle.}[/tex]

[tex]\textsf{The area of \underline{1/3} of a Circle is the area of a whole circle \underline{divided by 3.}}[/tex]

[tex]\textsf{Let's begin solving for the area.}[/tex]

[tex]\large\underline{\textsf{Substitute:}}[/tex]

[tex]\mathtt{Area = \pi (10)^{2}}[/tex]

[tex]\large\underline{\textsf{Evaluate:}}[/tex]

[tex]\mathtt{Area = 100\pi }[/tex]

[tex]\large\underline{\textsf{Divide by 3:}}[/tex]

[tex]\mathtt{\frac{Area}{3} = \frac{100\pi }{3}} [/tex]

[tex]\boxed{\mathtt{Area \approx 104.7m^{2}}}[/tex]

The investigation of a possible health concern in an office building due to an environmental toxin involves following the health status of all employees for a period of time, identifying potential sources of toxins, and taking steps to remove the toxins and protect the health of the employees.

The possible health concern in an office building that may be due to an environmental toxin can be investigated by following all 300 of their office employees for 30 days and observing their health status. On day 10, 35 employees report respiratory distress and on day 20, 70 employees report respiratory distress.

The remaining employees do not report any respiratory distress over the 30-day period. This suggests that there may be an environmental toxin in the office building that is causing respiratory distress in the employees.

It is important to further investigate the source of the toxin and take steps to remove it in order to protect the health of the employees. This may include conducting air quality tests, examining the building for potential sources of toxins, and implementing measures to improve ventilation and air quality.

It is also important to continue monitoring the health of the employees and providing them with appropriate medical care if needed.

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The correct statement regarding the quadratic equation is given as follows:

C. There are two complex solutions.

A quadratic equation is modeled by the general equation presented as follows:

y = ax² + bx + c

The discriminant of the quadratic function is given by the equation as follows:

Δ = b² - 4ac.

The numeric value of the coefficient and the number of solutions of the quadratic equation are related as follows:

The coefficients of the function for this problem are given as follows:

a = -2, b = 9, c = -12.

Hence the discriminant is given as follows:

Δ = 9² - 4(-2)(-12)

Δ = -15.

Negative discriminant, hence there are two complex solutions.

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The quadratic equation has two real solutions

Here we have the quadratic equation:

y = -2x^2 + 9x - 12

We want to study the solutions of the equations, so we need to look at the discriminant:

Generally for a*x^2 + b*x + c = 0 the discriminant is b^2 - 4ac

Here it is:

D = 9^2 - 4*-12*-2 = 33

A positive determinant means that there are 2 real solutions so the correct option is A.

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The volume is increasing at a rate of 16755.21 mm³/s when the diameter is 100 mm.

We are given that the radius of a sphere is increasing at a rate of 4 mm/s.

Let's first write the formula of the volume of a sphere;

[tex]V=43πr3V  =\frac{4}{3}\pi r^3V  =34πr3[/tex]

Now we can differentiate it by using the chain rule;

[tex]V'=43π3r2drdt+4πr3(4)orV'   =4πr2drdt[/tex]

The problem asks for how fast the volume is increasing, so we need to find V' when the diameter is 100 mm, or r=50 mm.

Thus;

[tex]V'=4π(50²)(4)   =40,000πV'   =125663.71 mm³/s[/tex]

Therefore, the volume is increasing at a rate of 125663.71 mm³/s when the diameter is 100 mm. Now we round to two decimal places to get;V'=16755.21 mm³/s.

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[tex] \tt \: c)77[/tex]

_________

[tex] \tt{7x} \\ \: \: \: \: \: = \tt 7(11) \\ \tt= 77[/tex]

[tex] \: [/tex]

━━━━━━━━━━━━━━━━━━━━━━━

hope it helps☘️

Therefore, the values of a that make the expression true are approximately 0.378 and 14.121.

In mathematics, an expression is a combination of numbers, variables, and operations that represents a quantity or a mathematical relationship. Expressions can be simple or complex, and can involve arithmetic, algebraic, trigonometric, or other mathematical operations. Expressions can also include constants, which are fixed values that do not change. Expressions can be evaluated by substituting numerical values for the variables and performing the indicated operations.

Here,

To solve for the values of a that make the given expressions true, we need to isolate the variable a on one side of the equation. We can start by simplifying the left side of the equation:

1/2a - 5 1/2 = 5 - a

Multiplying both sides by 2a gives:

1 - 10a* 1/2 = 10a - 2a²

Simplifying the left side by multiplying both terms by 2:

2 - 20a = 10a - 2a²

Rearranging the terms gives a quadratic equation:

2a² - 30a + 2 = 0

We can solve for a by using the quadratic formula:

a = (-b ± √(b² - 4ac)) / 2a

where a = 2, b = -30, and c = 2.

Plugging in these values gives:

a = (30 ± √((-30)² - 4(2)(2))) / 4

a = (30 ± √(892)) / 4

a ≈ 0.378 or a ≈ 14.121

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Answer:

The length of the shortest side is 3 units.

Step-by-step explanation:

Let's assume that the shortest side has a length of 3x, where x is a constant.

Then, the other sides would have lengths of 10x and 7x, respectively.

To check if this is a valid triangle, we need to make sure that the sum of the lengths of any two sides is greater than the length of the third side:

3x + 10x > 7x (sum of shortest and middle side is greater than longest side)

3x + 7x > 10x (sum of shortest and longest side is greater than middle side)

10x + 7x > 3x (sum of middle and longest side is greater than shortest side)

Simplifying these inequalities, we get:

13x > 7x

10x > 3x

17x > 0

All of these inequalities are true if x > 0, so our assumption is valid.

Therefore, the length of the shortest side is 3x. To find the value of x, we can set up the following equation based on the given ratio:

3x : 10x : 7x = 3 : 10 : 7

Simplifying this equation by dividing all sides by x, we get:

3 : 10 : 7 = 3 : 10 : 7

This is true, so any value of x will satisfy the given ratio.

However, to find the length of the shortest side, we can simply substitute x = 1 into our assumption:

Shortest side = 3x = 3(1) = 3

Therefore, the length of the shortest side is 3 units.

Hope this helps! Sorry if this is wrong. :]

Answer:

The statement "The function is decreasing on the interval (0, ∞)" is true about h(x).

To see why, let's take the derivative of h(x) and examine its sign:

h(x) = -2√x

h'(x) = -2/(2√x) = -1/√x

Since √x is always positive, h'(x) is negative for all x > 0. This means that h(x) is decreasing on the interval (0, ∞).

Therefore, the correct answer is: The function is decreasing on the interval (0, ∞).

Answer:

statistical experiments performed on a well defined sample space

Step-by-step explanation:

experiments can be performed either by probability, or other repetitive actions on sample space for possible outcomes.

The use of consonance in "Thundered the battery's double brass" best conveys the violent sounds of the battle.

Consonance is a literary device where consonant sounds are repeated in close proximity, often at the ends of words, to create a musical or rhythmic effect in a text.

The passage uses consonance, which is the repetition of consonant sounds within or at the end of words, to convey the violent sounds of the battle.

In particular, the repeated "t" and "d" sounds in "Thundered the battery's double brass" and "Difficult music for men to face" create a harsh, jarring effect that emphasizes the chaos and danger of the fighting.

Similarly, the repeated "v" and "l" sounds in "Undulate like the living waves" create a sense of movement and fluidity, conveying the unrelenting and overwhelming force of the rebel advance.


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The trend line equation y = 0.05x + 0.25 can be used to accurately predict the cost of gas per gallon for any year since 1970.

The equation of the trend line in the scatter plot is given by y = 0.05x + 0.25, where x is the number of years since 1970 and y is the price of gas per gallon in dollars. This equation can be used to calculate the price of gas in a given year since 1970. For example, if we want to calculate the price of gas in 2017, we can plug in x = 47 (the number of years since 1970) into the equation to get y = 0.05(47) + 0.25 = 2.85. This means that in 2017, the price of gas per gallon was approximately 2.85 dollars. This equation can be used to accurately predict the cost of gas for any given year since 1970.

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The probability that a random driver reports 0 injuries is 0.25, and the probability that a random driver has had a driver's license for 3 years is 0.25.

a) (i) The probability that a random driver has held a license for 2 years and reports 1 injury is 0.125.
(ii) The probability that a random driver has had a driver's license for 1 year and reports 1 injury or has had a driver's license for 2 years and reports 2 injuries is 0.3125.

b) The marginal probability distributions of ???? and ???? are given in the table below:

   ????
   ???? = 0
   ???? = 1
   ???? = 2
   ???? = 3
 
   ???? = 0
   0.25
   0.125
   0.0625
   0.03125
 
   ???? = 1
   0.25
   0.25
   0.125
   0.0625
 
 
   ???? = 2
   0.25
   0.25
   0.25
   0.125
 
   ???? = 3
   0.25
   0.25
   0.25
   0.25
 The probability that a random driver reports 0 injuries is 0.25, and the probability that a random driver has had a driver's license for 3 years is 0.25.

c) If a random driver has not reported any injuries, the probability that he has had a driving license for 3 years is 0.25.

d) ????(????) =  1, ????(????) = 1.5, ????????????(????) = 0.5, ????????????(????) = 0.9.

e) The covariance between ???? and ???? is 0.225, and the correlation coefficient is 0.45. This shows a positive correlation between the two variables, meaning that an increase in one variable is associated with an increase in the other.

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Answer: 5cm

Step-by-step explanation:

You divide the area of green (93.75) by pink's length (7.5)

that gives you 12.5.

12.5 is the length of green.

12.5 - 7.5 = 5

You subtract the 7.5 because to get yellow's length because Yellow's length plus pink's length is equal to green's total length.

Answer:

1300

Step-by-step explanation:

1864-564= 1300 students

(total - no. graduated)

Hope this helps! :)

The length of the opposite side (y) is equal to 3.2 centimeters.

In order to determine the length of the opposite side (y), we would apply the sine trigonometry ratio because the given side lengths represent the opposite side and hypotenuse of a right-angled triangle.

cos(θ) = Adj/Hyp

Where:

Based on the information provided in the image, we can logically deduce the following parameters:

By substituting the parameters into the sine trigonometry ratio formula, we have the following;

sin(θ) = Opp/Hyp

sin(39) = y/5

y = 5sin(39)

y = 3.2 centimeters.

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

Answer:

392.5g

Step-by-step explanation:

Density =mass/volume

Mass = Density x volume

Mass = 7.85g/cm³ x 50 cm³

Mass = 392.5g

Answer:

[tex]\textsf{B)} \quad a_n=-22+7(n-1)[/tex]

Step-by-step explanation:

An explicit formula for a sequence allows you to find the nth term of the sequence.

A recursive formula for a sequence allows you to find the nth term of the sequence provided you know the value of the previous term in the sequence.

Given recursive rule:

[tex]\begin{cases}a_{n+1}=a_n+7\\a_1=-22\end{cases}[/tex]

From the given recursive rule, we can see that each term is 7 more than the previous term.  Therefore, the common difference between terms is d = 7.

[tex]a_n = a_1 + (n - 1)d[/tex]

Input the values of a₁ and d into the explicit formula:

[tex]\implies a_n=-22+(n-1) \cdot 7[/tex]

[tex]\implies a_n=-22+7(n-1)[/tex]

Therefore, the explicit formula for the given sequence is:

[tex]\boxed{a_n=-22+7(n-1)}[/tex]

If the rate of the current was 3 kilometers per hour, Christine's speed in still water is 5 kilometers per hour.

Let's start by assuming that Christine's speed in still water is x kilometers per hour. We know that the current is flowing at a rate of 3 kilometers per hour.

When Christine is swimming against the current, her effective speed is reduced by the speed of the current, so her speed is (x-3) kilometers per hour. If it takes her the same amount of time to swim 2 kilometers against the current as it does to swim 8 kilometers with the current, we can set up an equation:

time to swim 2 km against current = time to swim 8 km with current

We can use the formula distance = speed x time to express the distance that Christine covers in each scenario. The distance she covers against the current is 2 kilometers, and the distance she covers with the current is 8 kilometers. So our equation becomes:

2 / (x - 3) = 8 / (x + 3)

To solve for x, we can cross-multiply and simplify:

2(x + 3) = 8(x - 3)

2x + 6 = 8x - 24

6x = 30

x = 5

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The values would be

[tex]15. = \frac{1}{1000y^3}\\\\16. =-\dfrac{1024}{n^5}\\\\17. =\dfrac{1}{32k^{10}}\\\\18. =$\dfrac{49}{36c^2}$\\\\\\19. = $64^t$\\\\20. = $5 \cdot 5^t$\\\\21. = $81 \cdot 3^{4t}$\\\\22. = $\dfrac{2^{2t}}{2}$ or $2^{2t-1}$[/tex]

What is an expression?

In mathematics, an expression is a combination of symbols and/or numbers that represents a value or a relationship between values. Expressions can be as simple as a single number or variable, or they can be complex, involving multiple operations and functions. Expressions can be used to describe relationships between variables, to represent mathematical formulas or equations, and to perform calculations.

[tex]15. = \frac{1}{1000y^3}\\\\16. =-\dfrac{1024}{n^5}\\\\17. =\dfrac{1}{32k^{10}}\\\\18. =$\dfrac{49}{36c^2}$\\\\\\19. = $64^t$\\\\20. = $5 \cdot 5^t$\\\\21. = $81 \cdot 3^{4t}$\\\\22. = $\dfrac{2^{2t}}{2}$ or $2^{2t-1}$[/tex]

Hence, the values would be

[tex]15. = \frac{1}{1000y^3}\\\\16. =-\dfrac{1024}{n^5}\\\\17. =\dfrac{1}{32k^{10}}\\\\18. =$\dfrac{49}{36c^2}$\\\\\\19. = $64^t$\\\\20. = $5 \cdot 5^t$\\\\21. = $81 \cdot 3^{4t}$\\\\22. = $\dfrac{2^{2t}}{2}$ or $2^{2t-1}$[/tex]

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Answer:

36

Step-by-step explanation:

see image for explanation

The required answers are a) 17.1 km b) 7.1 km, c) 18.6 km away from town A on a bearing of 293°.

To solve this problem, we can use vector addition to find the displacement vector from town A to the cyclist's destination.

a) To find how far east the cyclist's destination is from town A, we need to find the east component of the displacement vector. We can break down the displacement vector into its north and east components using trigonometry:

[tex]$$\text{East displacement} = 5\text{ km} + 10\text{ km}\cos(45^\circ) = 5\text{ km} + 10\text{ km}\frac{\sqrt{2}}{2} = 10\text{ km} + 5\sqrt{2}\text{ km} \approx 17.1\text{ km}$$[/tex]

So the cyclist's destination is approximately 17.1 km east of town A.

b) Similarly, to find how far north the cyclist's destination is from town A, we need to find the north component of the displacement vector:

[tex]$$\text{North displacement} = 10\text{ km}\sin(45^\circ) = 10\text{ km}\frac{\sqrt{2}}{2} = 5\sqrt{2}\text{ km} \approx 7.1\text{ km}$$[/tex]

So the cyclist's destination is approximately 7.1 km north of town A.

c) To find the distance and bearing of the cyclist's destination from town A, we can use the Pythagorean theorem and trigonometry. The displacement vector is the hypotenuse of a right triangle with legs of length 17.1 km and 7.1 km, so its length is:

[tex]$$\text{Displacement} = \sqrt{(17.1\text{ km})^2 + (7.1\text{ km})^2} \approx 18.6\text{ km}$$[/tex]

To find the bearing of the displacement vector, we can use the inverse tangent function:

[tex]$$\text{Bearing} = \tan^{-1}\left(\frac{\text{East displacement}}{\text{North displacement}}\right) \approx 67^\circ$$[/tex]

However, this angle is measured clockwise from north, so we need to subtract it from 360° to get the bearing measured counterclockwise from north:

[tex]$$\text{Bearing} = 360^\circ - 67^\circ = 293^\circ$$[/tex]

So the cyclist's destination is approximately 18.6 km away from town A on a bearing of 293°.

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