Answer:
a.)
Statement - If x and y are a pair of consecutive integers, then x and y have opposite parity.
Contra-positive statement - If x and y have same parity, then x and y can not be a pair of consecutive integers.
Assumed - x and y have same parity
To prove - x and y can not be a pair of consecutive integers.
Contradiction statement - If x and y are a pair of consecutive integers and they have same parity then then it is not possible.
Assumed - x and y are a pair of consecutive integers and they have same parity
To prove - It is not possible
Proof - Let x and y are even numbers
⇒x = 2a , y = 2b
then , y - x = 2b - 2a = 2(b - a) is alteast 0 if x ≠y
Contradiction.
b.)
Statement - For all integers n, if n² is odd, then n is also odd.
Contrapositive statement - If n is even , then n² is even
assumed - n is even
To prove - n² is even
Contradiction - If n² is odd and n is even then it is not possible
Assumed - n² is odd and n is even
To prove - Our assumption is not true
Proof - As n is even
⇒n = 2a
⇒n² = (2a)² = 4a²
⇒n² is even
Contradiction.
You spend 100 minutes in 2 classes. How many minute will you spend in 3 classes?
Answer:150
Step-by-step explanation:
Answer:
150
Step-by-step explanation:
since half of 100 is 50, 2 classes will have 50 minutes. If you add 100+50 you get 150. So therefore ur answer is 150!
Find the number of ways of arranging the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 in a row so that the product of any two adjacent numbers is even or a multiple of 3
Answer:
3,10,8,5,4,1,7,2,9,6
Step-by-step explanation:
Which is the rate of change for the interval between –6 and –3 on the x-axis?
The rate of change for the interval between –6 and –3 on the x-axis is; 2
How to find the rate of change of the graph?To find the rate of change for the interval between –6 and –3 on the x-axis, we will first identify the ordered pairs at those two points. The ordered pairs from the graph would be (3, -2) and (6, 4).
Using the slope formula with those two points, we can find the rate of change as;
slope; m = (y₂ - y₁)/(x₂ - x₁)
m = (4 + 2)/(6 - 3)
m = 6/3
m = 2
This value of m represents the rate of change
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-2 is the answer i cant exaplin it bc i jsut guessed and got it right
Here is a completed box challenge puzzle. If you look at it closely, you'll see that the umber at the top is the product of the two numbers on the left and right, and the number at the bottom is the sum of the two numbers on the left and right. *** 18 15 3x15=45 3+15= 18
The product of the two numbers 45, is the highest number, and the sum of the two numbers 18, is the lowest number follow symmetry .
what is symmetry ?When an object is split into two identical, mirrored components, it is said to have symmetry in mathematics. What does "axis of symmetry" refer to? An "axis of symmetry" is a fictitious line that has the ability to fold or split a thing into two identical mirror parts. Any object is symmetrical if two parts fit together. To determine the shape's symmetry, draw a mirror line across the center, making sure that all parts are identical. Another way of putting it is that symmetry is when two objects are facing one another or when two items have mated components that circle around an axis.
given
The two numbers on the left and right are multiplied together to create the number at the top.
The two digits on the left and right add up to the number at the bottom.
middle number + top number = 3 * 15 = 45 ( top number )
3+15 Equals 18 ( lowest number ) ( bottom number )
The product of the two numbers on the left and right, 45, is the highest number, and the sum of the two numbers on the left and right, 18, is the lowest number.
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The complete question is :-
Here is a completed box challenge puzzle. If you look at it closely, you'll see that the number at the top is the product of the two numbers on the left and right, and the number at the bottom is the sum of the two numbers on the left and right.
Select the correct answer. The given equation has been solved in the table. Step Statement 1 2 3 x − 9 = − 13 2 2 3 x − 9 + 9 = − 13 + 9 3 2 3 x = − 4 4 3 2 ⋅ 2 3 x = 3 2 ⋅ ( − 4 ) 5 x = − 6 In which step was the addition property of equality applied? A. step 2 B. step 3 C. step 4 D. The addition property of equality was not applied to solve this equation.
In step 2 the addition property of equality was applied. Thus, the correct option is A.
What is addition property of equality?When the same amount is added to both sides of an equation, according to the addition property of equality, the equation remains the same. An equation that reads as A = B still holds true if a number x is added to both sides. The equation is A + x = B + x in mathematics.
When the same amount is added to both sides of an equation, the equation remains true, which is known as the addition property of equality. One of the basic characteristics of equality in mathematics, it is also very significant. The division, multiplication, and addition properties of equality are the additional characteristics.
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Full question:-
Select the correct answer. The given equation has been solved in the table. (see attachment)
Use the number line below, where RS=4y+4, ST=3y+8, and RT=54.
a. What is the value of y?
b. Find RS and ST.
The value of y is 6 and the length of RS and ST are 28, 26 respectively.
What is a line ?
line is a straight one dimensional figure having no thickness is called as the line.
Given , number lines
RS = 4y+4
ST = 3y+8
RT = 54
From the given figure ,
RS+ST = RT
4y+4+3y+8 = 54
7y+12 = 54
7y = 54-12
7y = 42
y =42/7
y = 6
The RS = 4y+4 = 4*6+4 = 28
ST = 3y+8 = 3*6+8 = 26
Hence, The value of y is 6 and the length of RS and ST are 28, 26 respectively.
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Can you solve this?
A savings account is opened with an initial deposit of $425.00. Determine the account balance after 9 years if the account earns 2.7% simple interest annually.
$3,928.28
$1,457.75
$528.28
$103.28
PLEASE HELPP
Which is an irrational number?
A
317, because it cannot be rewritten as an integer.
B
−6, because it is less than 0.
C
13−−√, because it cannot be expressed as either a terminating or a repeating decimal.
D
0.2971¯¯¯¯, because it is a repeating decimal.
The maximum resistance for a new spark plug wire is 840 ohms per inch. A 25-in. wire had a resistance of 20,250 ohms. Does this fall within the acceptable limit?
Answer:
Yes
Step-by-step explanation:
Let's calculate the resistance of one inch for our wire and compare it with the maximum:
20250 / 25 = 810 Ohm
810 < 840 ? Yes!
Yes, this is within the acceptable limit. Wire can be used.
HOW FAR FROM HOME PLATE
TO SECOND BASE?
The catcher at home plate often needs
to make a quick throw to second base.
According to the measurements shown on
the baseball diamond, the exact distance
is √16,200 feet. How far is that in decimal
form? (Round to the nearest hundredth.)(please hurry )
The exact distance from home plate to second base on a baseball diamond is √16,200 feet. To convert this distance to decimal form, we must first understand the concept of a square root and how it relates to the distance formula.
A square root is a mathematical operation that represents the value whose square is equal to a given value. In this case, the square root of 16,200 is equal to the distance from home plate to second base.
To find the decimal form of the square root of 16,200, we can use a calculator or we can use the long division method. With a calculator, we simply input √16,200, which will give us the result of 126.84955592153876
To round to the nearest hundredth we look at the hundredths digit (8) if it's 5 or above we round up otherwise we round down so 126.84955592153876 rounded to the nearest hundredth is 126.85
The distance from home plate to second base on a baseball diamond is 126.85 feet in decimal form.
It's worth noting that the distance from home plate to second base is a critical aspect of the game and it's really important for the catcher to make a quick and accurate throw to second base.
1. What is the measure of 21? (1 point)
047°
037⁰
074⁰
053⁰
2. What is the measure of 22? (1 point)
047⁰
074°
037⁰
053⁰
Answer:
The measure of 21 and 22 doesn't make sense as they are not specific angles and they are not defined as to what angle you are refering to.
Without proper context and information, I am unable to provide an accurate answer to your question. Can you please provide more context, or explain what exactly you are trying to ask ?
Step-by-step explanation:
Find the x intercepts of the graph y= 2/pie squared
The x-intercept is the point where the graph of a line intersects the x-axis. The y-intercept is the point where the graph of a line intersects the y-axis.
What is the formula for x-intercept form?Take into account a line with the slope intercept form y = mx + c, an intercept of (c, 0), and a slope of m. To find the x-intercept, enter y = 0 in the equation.Yb = m (xa), where m is the line's slope and (a,b) is a point on the line, is the point-slope form of a straight line. You may find the line's x-intercept by entering y = 0 and x-intercept = (amb)/m.Where the line crosses the x-axis is known as the x-intercept. Because the point won't be on the x-axis if y doesn't equal 0, you must always use a value of 0 for y when calculating the x-intercept.Given data :
y = ( 2 / π )²
y = 2² / π²
y = 4 / π²
Here i attached the graph of x interception
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ANSWER THESE QUESTIONS PLEASE. (picture provided)
Answer: oh easy its 1220703250
Step-by-step explanation:
1) Health officials routinely check sanitary conditions of restaurants. Assume you visit a popular tourist spot and read in the newspaper that in 3 of every 7 restaurants checked, there were unsanitary health conditions found. Assuming you are planning to eat out 10
times while you are there on vacation, answer the following questions:
a) How likely is it that you will eat at three restaurants with unsanitary conditions?
b) How likely is it that you will eat at 4 or 5 restaurants with unsanitary conditions?
c) Explain how you would compute the probability of eating in at least one restaurant with unsanitary conditions. Could you use the complement to solve this problem?
d) What is the most likely number to occur in this experiment?
e) How variable will the data be around the most likely number?
f) Is this is a binomial distribution?
g) If it is a binomial distribution, does that mean that the likelihood of success is always
50% since there are only two possible outcomes?
a) The probability of eating at a restaurant with unsanitary conditions is 3/7. If you plan to eat at 10 different restaurants, the probability of eating at exactly 3 restaurants with unsanitary conditions is given by the binomial probability formula:
P(X = 3) = (10 choose 3) * (3/7)^3 * (4/7)^7 = 120 * (0.42857)^3 * (0.57143)^7 = 0.1437
So, the likelihood of eating at three restaurants with unsanitary conditions is 14.37%.
b) The probability of eating at 4 or 5 restaurants with unsanitary conditions can be found by summing the binomial probabilities for each case:
P(X = 4) = (10 choose 4) * (3/7)^4 * (4/7)^6 = 210 * (0.42857)^4 * (0.57143)^6 = 0.1735
P(X = 5) = (10 choose 5) * (3/7)^5 * (4/7)^5 = 252 * (0.42857)^5 * (0.57143)^5 = 0.1428
P(X = 4 or X = 5) = P(X = 4) + P(X = 5) = 0.1735 + 0.1428 = 0.3163
So, the likelihood of eating at 4 or 5 restaurants with unsanitary conditions is 31.63%.
c) The probability of eating in at least one restaurant with unsanitary conditions can be found by subtracting the probability of eating in no restaurants with unsanitary conditions from 1. The probability of eating in no restaurants with unsanitary conditions is given by the binomial probability formula:
P(X = 0) = (10 choose 0) * (3/7)^0 * (4/7)^10 = 1 * (1) * (0.57143)^10 = 0.0139
P(X >= 1) = 1 - P(X = 0) = 1 - 0.0139 = 0.9861
So, the probability of eating in at least one restaurant with unsanitary conditions is 98.61%.
d) The most likely number is 3, as it has the highest probability of occuring.
e) The data is not very variable around the most likely number, as the probability of the other numbers is relatively low.
f) Yes, this is a binomial distribution as the experiment has only two possible outcomes, "success" (eating at a restaurant with unsanitary conditions) or "failure" (eating at a restaurant with sanitary conditions) and the number of trials is fixed (10)
g) No, the likelihood of success is not always 50% in a binomial distribution. The probability of success is given by the proportion of successful trials in the population, which in this case is 3/7.
Use a graph to find the value of f(x) when x = -2 for each function
f(x) = -x + 7
Answer:
9
Step-by-step explanation:
X= -2, substitute -2 for X in the function
-(-2)+7, = 2+7
=9
hello! can someone please help me? i'll give brainly. Thanks a lot! :)
Answer:
Brand z
Step-by-step explanation:
Answer:
X
Step-by-step explanation:
for W its 20/12 which is 5/3 or roughly 1.66. for X its 1500/1125 which is 4/3 or around 1.33. for Y its 100/65 which is 20/13 or around 1.54. and for Z its 50/25 or just 2.
Solve equation 7x+8=43
x = 5
7x+8=43
First, we want to separate the variable, so we subtract 8 from both sides:
7x=35
Then we divide both sides by 7 to get x all by itself:
x=5
This is because 35/7=5
sing the variables from the problem statement together with the acceleration due to gravity, g , write an equation for the magnitude of the acceleration of the system. hint: based upon the numeric choices, block 1 will accelerate up the incline while the suspended block, block 2, accelerates vertically downward.
The equation for the magnitude of the acceleration of the system is equals to the
a = g (m₂ - m₁μ꜀ cosθ - m₁sinθ )/(m₁ + m₂ ).
We have, two blocks with masses say m₁ = 3.8 kg and m₂ = 8.6 kg inclined on the plane at angle θ=39° above the horizontal. Also, cofficient of friction, μ=0.39. According to problem, In case of block 1 , accelerate up the inclined plane, Then forces along y-axis,
mass m₁gcosθ , where g is gravity constant moves vertically downward normal force Fₙ moves upwardNet force during vertical motion = 0
=> Fₙ - m₁gcosθ = 0 --(1)
On horizontal direction,
Tension T acts upward acceleration, "a" upward motionforce of friction, f꜀ acts downward m₁gsinθ moves downwardAgain net force = 0
=> T - f꜀ - m₁gsinθ = m₁a --(2) ( using Newton's second law)
For block 2, Vertical motion
mass m₂g moving downward T acting upward Acceleration, " a"acting downwardSo, net vertical force = 0
=> - T + m₂g = m₂a --(3)
Now, f꜀ = μ꜀ Fₙ= μ꜀ m₁g cosθ
so, T - μ꜀ m₁g cosθ - m₁gsinθ = m₁ a --(4)
Using the equation (4) and (3), we have to determine the equation for acceleration, a.
Adding (3) from (4),
m₁ a + m₂a = -μ꜀ m₁g cosθ - m₁gsinθ + m₂g + T - T
=> (m₁ a + m₂a) = g ( -m₁cosθ - m₁ sinθ + m₂)
=> a ( m₁ + m₂ ) = g (-m₁μ꜀ cosθ - m₁ sinθ + m₂)
=> a = g (- m₁μ꜀ cosθ - m₁sinθ + m₂)/(m₁ + m₂ )
by plugging all the above values we can calculate the magnitude of acceleration.
Hence, required equation is a = g (m₂ - m₁μ꜀cosθ - m₁sinθ)/(m₁ + m₂) .
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Complete question:
Referring to the image to the above figure , the mass of block 1 is m1=3.8 kg while the mass of block 2 is m2=8.6 kg. The coefficient of friction between m1 and the inclined surface is μ=0.39. The inclined surface is at an angle θ=39∘ above the horizontal. Using the variables from the problem statement together with the acceleration due to gravity, g , write an equation for the magnitude of the acceleration of the system. hint: based upon the numeric choices, block 1 will accelerate up the incline while the suspended block, block 2, accelerates vertically downward.
A sample proportion of 0.62 is found. To determine the margin of error for this statistic, a simulation of 200 trials is run, each with a sample size of 100 and a point estimate of 0.62.
The minimum sample proportion from the simulation is 0.73, and the maximum sample proportion from the simulation is 0.97.
What is the margin of error of the population proportion using an estimate of the standard deviation?
A simulation of 200 trials is run, each with a sample size of 100 and a point estimate of 0.62=0.08 to establish the margin of error for this statistic.
What is margin of error?Your results will vary from the actual population value by how many percentage points, as indicated by the margin of error.
A 95% confidence interval with a 4% margin of error, for instance, indicates that your statistic will, 95% of the time, be within 4% of the true population figure.
However, the official definition goes a little bit beyond. The range of values in a confidence interval that lie below and above the sample statistic is known as the margin of error.
The confidence interval is a tool for displaying the level of uncertainty surrounding a certain statistic (i.e. from a poll or survey).
According to our question-
Because the sample proportions' range is 0.97-0.73 = 0.24
Calculate the standard deviation:
Range = 6* (s-bar)
= 0.24
s-bar = 0.04
S/Sqrt (100) then equals 0.04
and s = 0.4
S.D = 0.04 + 0.04
= 0.08
Hence, A simulation of 200 trials is run, each with a sample size of 100 and a point estimate of 0.62=0.08 to establish the margin of error for this statistic.
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Recall the equation for a circle with center (h, k) and radius r. At what point in the first quadrant does
the line with equation y = x + 1 intersect the circle with radius 4 and center (0, 1)?
The point of intersection of the line [tex]y = x + 1[/tex] and the circle with radius 4 and center [tex](0,1)[/tex] is [tex](x, y) = (-1, 0)[/tex] which is in the first quadrant.
How to find the point in the first quadrant?The equation of a circle with center (h, k) and radius r is:
[tex](x - h)^2 + (y - k)^2 = r^2.[/tex]
The equation of the line is given as [tex]y = x + 1[/tex].
Given the center of the circle is (0,1) and the radius is 4, the equation of the circle is [tex](x-0)^2 + (y-1)^2 = 4^2[/tex]
This can be simplified as [tex]x^2 + y^2 - 2y + 1 = 0[/tex]
Now we can substitute [tex]y = x + 1[/tex] in the circle equation:
[tex]x^2 + (x+1)^2 - 2(x+1) + 1 = 0\\x^2 + x^2 + 2x + 1 + 2x + 1 = 0\\2x^2 + 4x = -2\\x^2 + 2x = -1\\(x+1)^2 = 0\\x = -1[/tex]
Now that we have the value of x, we can substitute it in the equation [tex]y = x + 1[/tex] to find the corresponding y value:
[tex]y = -1 + 1\\y = 0[/tex]
So, the point of intersection of the line [tex]y = x + 1[/tex] and the circle with radius 4 and center (0,1) is [tex](x, y) = (-1, 0)[/tex] which is in the first quadrant.
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Write an equation of the line that passes through (−6, 0)
and (0, −24)
.
The equation of the line will be y = -4x - 24.
What is an equation of the line?An equation of the line is defined as a linear equation having a degree of one. The equation of the line contains two variables x and y. And the third parameter is the slope of the line which represents the elevation of the line.
The general form of the equation of the line:-
y = mx + c
m = slope
c = y-intercept
Slope = ( y₂ - y₁ ) / ( x₂ - x₁ )
Given that the line is passing through the point (−6, 0) and (0, −24).
Ther slope of the point is calculated as,
Slope = ( y₂ - y₁ ) / ( x₂ - x₁ )
Slope = ( -24 - 0 ) / (0 - (-6)
Slope = -4
The y-intercept will be calculated as,
y = mx + c
y = -4x + c
0 =-4 x - 6 + c
c = -24
The equation of the line will be,
y = mx + c
y = -4x - 24
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which of the following tests analyze the difference between the means of two independent samples?
a. Correlated T Test
B. T test for independent groups
c. sign test
d. all the above
T-test for independent groups analyzes the difference between the means of two independent samples. Hence, option B is the correct answer.
B. T-test for independent groups is the test that analyzes the difference between the means of two independent samples. This test assumes that the two samples are drawn from populations that have normal distributions and that the variances of the two populations are equal. The T-test for independent groups compares the means of the two samples and determines if there is a statistically significant difference between them. This test is also known as a two-sample T-test or independent T-test.
The other options A and C are not the correct ones.
A. Correlated T-Test is used when comparing means of dependent samples, it's also known as a dependent t-test
C. Sign test is a non-parametric test that is used to compare two dependent samples.
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Subtract.
(5y − 3) - (-2y - 7) =
Which of the following sets of numbers could represent the three sides of a triangle? { 12 , 27 , 39 } {12,27,39} { 12 , 18 , 29 } {12,18,29} { 5 , 9 , 16 } {5,9,16} { 9 , 24 , 33 } {9,24,33}
The sets of numbers that could represent three sides of a triangle is given as follows:
{12, 18, 29}.
What is the condition for 3 lengths to represent a triangle?In a triangle, the sum of the lengths of the two smaller sides has to be greater than the length of the greater side.
Hence the set in this problem is given as follows:
{12, 18, 29}.
The larger measure is of:
29.
The sum of the two smaller measures is given as follows:
12 + 18 = 30.
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Greatest common factor 10x2+28x-6
Answer:
The equation 10x2+28x-6 can be simplified using a few steps. Firstly, the greatest common factor (GCF) needs to be identified. The GCF of this equation is 2, which can be identified by factoring out the coefficient of the first term, 10. After factoring out 2, the equation becomes -3 + -14x + 5x2.
The next step is to factor the remaining polynomial. This can be done by multiplying the coefficient of the first term by the constant and then finding two factors whose sum equals the coefficient of the middle term. In this case, the multiplication of the coefficient of the first term (5) and the constant (-3) gives -15. Then, two factors of -15 whose sum adds up to -14 must be found. These factors are -15 and 1. Therefore, the equation can be rewritten as 5x2 - 15x + 1x - 3.
Afterwards, the four terms in the equation can be added up in pairs, pulling out like factors. This results in (5x+1)(x-3), which is the desired factorization. Finally, the expression can be written as 2(x-3)(5x+1). Thus, the GCF of 10x2+28x-6 is 2.
Step-by-step explanation:
Answer:
2 is the answer .
if u r not sure u can try doing it yourself
the answer will be the same as I told
The higher your credit score, the
A) lower your savings interest rate
B) higher your car loan rate
C) lower your mortgage interest rate
D) higher risk you are to a creditor
Answer: c lower your mortgage interest rate
Step-by-step explanation
The higher your credit score, the less risk you represent for a lender.
My dad taught me this.
Brenna is considering which brand of cereal to buy. She knows some brands contain a lot of sodium, or salt, which she is trying to avoid. Crispee Flakes has 0.23 grams of sodium per cup. Corn Puffers has 0.28 grams of sodium per cup. Energy Doodles has 0.26 grams of sodium per cup.
How many of the brands have less than 1\4
of a gram of sodium per cup?
Using mathematical operations, we can say that only 1 brand has less than 1/4 grams of sodium per cup which is Crispy Flakes.
What are mathematical operations?An operation is a function in mathematics that transforms zero or more input values into a clearly defined output value.
The operation's arity is determined by the number of operands.
Basic arithmetic operations or calculations, such as addition, subtraction, multiplication, and division, are covered in Fundamentals of Mathematics and are concepts we learn in primary school.
Students will eventually study fundamental ideas like algebra, geometry, factors, ratios, etc. in upper classes.
So, we have that:
Crispy Flakes have 0.23 grams of sodium per cup.
Corn Puffers have 0.28 grams of sodium per cup.
Energy Doodles have 0.26 grams of sodium per cup.
Now, brands that have less than 1/4 gram of sodium per cup:
100 * 1/4 = 25
Then, 1/4 of 1.00 = 0.25
0.23 > 0.25
Therefore, using mathematical operations, we can say that only 1 brand has less than 1/4 grams of sodium per cup which is Crispy Flakes.
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Determine the dimensions of Nul A and Col A for the matrix shown below. A = [1 0 0 0 4 0 0 0 -5 1 0 0 2 -4 1 0 -4 -3 -3 0 6 -1 -4 0 -1 -6 6 0] The dimension of Nul A is and the dimension of col A is
The dimension of Nul A = 0 and the dimension of Col A = 4. means that the matrix A has no solutions for the homogeneous equation A × x = 0, which means that its null space is trivial, and it has 4 linearly independent columns, which means that its column space is 4-dimensional.
In other words, it is the number of linearly independent vectors x that satisfy the equation Ax = 0.
The dimension of Col A, also known as the column space of the matrix A, is the dimension of the subspace spanned by the columns of the matrix A. In other words, it is the number of linearly independent columns of matrix A.
In general, it is not possible to determine the dimensions of Nul A and Col A of a matrix just by looking at its elements. In order to find the dimensions, we need to perform the row echelon form or reduced row echelon form of the matrix to find the rank and the nullity which is the difference in the number of columns and rank.
For the specific matrix A provided, the rank of the matrix is not provided, thus we can not determine the dimension of the Nul A and Col A.
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A nurse sets an intravenous
fluid drip rate at 1000 mL
every 8 hr. How much fluid would the patient receive in
3 hr? Step by step (applications of rational equations)
Answer:
375 mL
Step-by-step explanation:
Let x = Amount of fluid patient receives in 3 hours
Fluid drip rate: 1000 mL --------- 8hr
x --------- 3 hr
Cross multiplication is applied:
(x).(8 hr) = (1000 mL).(3hr)
x = [(1000mL).(3 hr)]/ (8 hr)
x = 375 mL
how many linear factors does a polynomial function f of degree n have, where n > 0?
There will be n linear factors of a polynomial function f of degree n
We derive the above answer based on the following explanation,
The Linear Factorization Theorem states that . . . if f(x) is a polynomial of degree n, where n > 0, then f has precisely n linear factors f(x) = an(x – c1)( x – c2) . . . (x – cn) where c1, c2, . . . , cn are complex numbers.
Let us assume f to be a polynomial function with real coefficients and suppose a+bi, b≠0, is a zero of f(x). Then, by the Factor Theorem, x−(a+bi) is a factor of f(x). For f to have real coefficients, x-(a−bi) must also be a factor of f(x). This is true because any factor other than x−(a−bi) when multiplied by x−(a+bi), will leave imaginary components in the product. Only conjugate pair multiplication will get rid of the imaginary components and produce real coefficients.
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