The false statement among the following statements is: |ez| = |e³|. Here, e is the Euler number, and z is a complex number. Therefore, the correct answer is option (i) |ez| = |e³|.
We know that e^(ix) = cos(x) + i sin(x)
It is also known as Euler's formula, where e is the Euler number, i is the imaginary unit, x is the angle in radians. This formula connects the trigonometric functions with the exponential function. In this question, e is the Euler number, and z is a complex number. So, ez = |ez| × e^(iθ), where θ is the angle of the complex number z from the positive real axis. In the same way, e³ = |e³| × e^(i3θ)Here, the modulus of ez is |ez|, and the modulus of e³ is |e³|. It is not necessary that both will be equal because the value of θ may differ. Hence, the false statement is |ez| = |e³|.
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1/5 x 11 simplified if can
Answer:
2.2
Step-by-step explanation:
Answer:
11/5
Step-by-step explanation:
a
& b
5. Find the following limits. (a) lim40 12 (b) limz+1+1 +22-22+2 i 2-iz-1-1
The limits are,
(a) lim(x→0) 4x/(x² + 1) = 0
(b) lim(z→-1) (1 + √(2 - 2z + z²))/(2 - iz - 1) = ((1 + √(5))(3 - i))/10
(a) To find the limit of lim(x→0) 4x/(x² + 1), we can directly substitute 0 for x in the expression:
lim(x→0) 4x/(x² + 1) = (4 × 0)/(0² + 1) = 0/1 = 0
Therefore, the limit is 0.
(b) To find the limit of lim(z→-1) (1 + √(2 - 2z + z²))/(2 - iz - 1), we can again substitute -1 for z in the expression:
lim(z→-1) (1 + √(2 - 2z + z²))/(2 - iz - 1) = (1 + sqrt(2 - 2(-1) + (-1)^2))/(2 - i(-1) - 1)
= (1 + √(2 + 2 + 1))/(2 + i + 1)
= (1 + √(5))/(3 + i)
To simplify this expression further, we need to rationalize the denominator. We can multiply the numerator and denominator by the conjugate of the denominator, which is (3 - i):
lim(z→-1) (1 + √(5))/(3 + i) × (3 - i)/(3 - i)
= ((1 + √(5))(3 - i))/(9 - i²)
= ((1 + √(5))(3 - i))/(9 + 1)
= ((1 + √(5))(3 - i))/10
Therefore, the limit is ((1 + √(5))(3 - i))/10.
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The question is -
Find the following limits:
(a) lim(x->0) 4x/(x^2 + 1)
(b) lim(z->-1) (1 + sqrt(2 - 2z + z^2))/(2 - iz - 1)
The diameter of a circle is 63 centimetres find its circumference use pie = 3.14
Answer:
197.9
Step-by-step explanation:
The formula for circumference is 2(pi)r and r is the radius
The diameter is two times the size of the radius, so by dividing the diameter by two, you can get the radius
So, r=63/2
r= 31.5
That means that 2(pi)(31.5) is the circumference
2(pi)(31.5) = 197.9 (rounded to the nearest tenth)
You drop a ball vertically from a height of 1 m. It returns to a height of 0.6 m. What is the coefficient of restitution between the ball and the ground?
The coefficient of restitution between the ball and the ground is 0, indicating a completely inelastic collision.
The coefficient of restitution (e) is a measure of the elasticity or bounciness of a collision between two objects. It is defined as the ratio of the relative velocity of separation after the collision to the relative velocity of approach before the collision.
In this case, the ball is dropped vertically from a height of 1 m and returns to a height of 0.6 m. We can assume that the collision with the ground is approximately elastic, meaning that kinetic energy is conserved.
When the ball hits the ground, its initial velocity is zero, and the final velocity after the collision is also zero since it momentarily comes to rest before bouncing back up. Therefore, the relative velocity of separation is zero.
The relative velocity of the approach is the velocity just before the collision. Since the ball is dropped vertically, its velocity just before hitting the ground is given by the equation:
[tex]v = \sqrt(2gh)[/tex]
where v is the velocity, g is the acceleration due to gravity (approximately[tex]9.8 m/s^2[/tex]), and h is the initial height (1 m).
Plugging in the values:
[tex]v = \sqrt(2 * 9.8 * 1)[/tex]
[tex]= \sqrt(19.6)[/tex]
≈ 4.427 m/s
Therefore, the relative velocity of the approach is approximately 4.427 m/s.
Since the relative velocity of separation is zero, we can calculate the coefficient of restitution (e) as:
e = 0 / 4.427
= 0
Therefore, the coefficient of restitution between the ball and the ground, in this case, is 0, indicating a completely inelastic collision where the ball comes to a stop upon hitting the ground and does not bounce back.
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The ____ sequence begins with two ones, and then each new term is formed by adding the two terms before it: 1, 1, 2, 3, 5, 8, 13, 21,...
Answer:
Fibonacci
Step-by-step explanation:
the Fibonacci sequence
Suppose () = 1/8 for 0 ≤ ≤ 4 for x being a continuous random variable Is () a probability density function? Prove or disprove.
Answer:
The expected value of x ; E(x) = 1
Step-by-step explanation:
F(x) = 1/8 for 0 ≤ x ≤ 4
To prove that it is a probability density function we will find E(x )
attached below is the required prove
It is proven that F(x) = 1/8 for 0 ≤ x ≤ 4 is probability density function
The expected value of X = 1
In a poll, students were asked to choose which of six colors was their favorite. The circle graph shows how the students answered. If students participated in the poll, how many chose Orange?
Answer:
1666.70
Step-by-step explanation: 10,000/6=1666.70
11. A bag contains 2 blue marbles and 2 green marbles. What is the probability of drawing a blue marble followed by a green marble, without replacing the first marble before drawing the second marble?
Please show work ty
Answer: your answer should be 50%
Step-by-step explanation: This is because there are only four marbles in the bag total and only 2 are blue and only 2 are green so your chances of pulling out either is 50%
Answer:
33%
Step-by-step explanation:
2 blue marbles + 2 green marbles = 4 marbles
1st draw for blue: 2/4 (2 blue marbles out of 4 marbles)
2nd draw for green: 2/3 (1 less marble from 4, marble not put back in)
2/4 x 2/3 = 4/12 = 1/3 = 0.33 or 33%
I need help imm struggling
Answer:
180in3 (180 inch cubed)
Step-by-step explanation:
12 x 5 x 3
Answer: I would assume the answer would be 180
Step-by-step explanation: The formula for volume is Length x Width x Height. So multiply all the number above and the answer will be 180
Mrs. Smith washed 2 5 of her laundry. Her son washed 1 3 of it. Who washed most of the laundry? How much of the laundry still needs to be washed?
Answer:
a) The person who washed the most of the laundry is Mrs Smith
b) 4/15 of the laundry is left to wash
Step-by-step explanation:
Mrs. Smith washed 2/5 of her laundry. Her son washed 1/3 of it.
a) Who washed most of the laundry?
We convert the fraction of laundry each person washed to decimal
Mrs Smith = 2/5 = 0.4
Her son = 1/3 = 0.333
Therefore, the person who washed the most of the laundry is Mrs Smith
b) How much of the laundry still needs to be washed?
Let us total laundry = 1
=1 - ( 2/5 + 1/3)
Lowest Common Denominator is 15
=1- (3 × 2 + 5 × 1/15)
= 1 - (6 +5/15)
=1 - 11/15
= 4/15
Evaluate the following integral over the Region R. (Answer accurate to 2 decimal places).
10(x +y))dA
R = (1, y) 16 < x² + y2 < 25, x < 0
∫ ∫R 10(x+y) dA R={(x,y)∣16≤x2+y2≤25,x≤0} Hint: The integral and Region is defined in rectangular coordinates.
The value of the integral is 15.87.
The given integral is:∫∫R 10(x+y) dAwhere R={(x,y)∣16≤x²+y²≤25,x≤0} in rectangular coordinates.In rectangular coordinates, the equation of circle is x²+y² = r², where r is the radius of the circle and the equation of the circle is given as: 16 ≤ x² + y² ≤ 25 ⇒ 4 ≤ r ≤ 5We need to evaluate the integral over the region R using rectangular coordinates and integrate first with respect to x and then with respect to y.∫∫R 10(x + y) dA = 10∫ from 4 to 5 ∫ from -√(25-y²) to -√(16-y²) (x+y) dx dy...[since x < 0]
Now, integrating ∫(x+y) dx we get ∫(x+y) dx = (x²/2 + xy)Therefore, 10∫ from 4 to 5 ∫ from -√(25-y²) to -√(16-y²) (x+y) dx dy = 10∫ from 4 to 5 ∫ from -√(25-y²) to -√(16-y²) [ (x²/2 + xy) ] dy dx= 10∫ from 4 to 5 [∫ from -√(25-y²) to -√(16-y²) (x²/2 + xy) dy] dxNow integrating with respect to y we get∫(x²/2 + xy) dy = (xy/2 + y²/2)
Putting the limits and integrating we get10∫ from 4 to 5 [∫ from -√(25-y²) to -√(16-y²) (x²/2 + xy) dy] dx = 10∫ from 4 to 5 [(∫ from -√(25-y²) to -√(16-y²) (x²/2 + xy) dy)] dx = 10∫ from 4 to 5 [(x²/2)[y]^(-√(16-x²) )_(^(-√(25-x²))] + [(xy/2)[y]^(-√(16-x²) )_(^(-√(25-x²)))] dx = 10∫ from 4 to 5 [-(x²/2)√(16-x²) + (x²/2)√(25-x²) - (x/2)(16-x²)^(3/2) + (x/2)(25-x²)^(3/2) ] dxNow integrating with respect to x, we get10∫ from 4 to 5 [-(x²/2)√(16-x²) + (x²/2)√(25-x²) - (x/2)(16-x²)^(3/2) + (x/2)(25-x²)^(3/2) ] dx = [ (10/3) [(25/3)^(3/2) - (16/3)^(3/2)] - 5√3 - (5/3)[(25/3)^(3/2) - (16/3)^(3/2) ] ]Ans: The value of the integral is 15.87.
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if 121 ml of a 1.0 m glucose solution is diluted to 550.0 ml , what is the molarity of the diluted solution?
The molarity of the diluted solution is approximately 0.220 M.
The concentration of a solute in a solution is measured by its molarity. The amount of solute that dissolves in one liter (L) of solution is the number of moles. One of the most used units of concentration is t, represented by the symbol M. Number of moles of solute contained in 1 liter of solution is how it is defined.
To calculate the molarity of a solution, you need to use the formula:
M₁V₁ = M₂V₂
Substituting these values into the formula:
(1.0 M)(121 ml) = M₂(550.0 ml)
Rearranging the equation to solve for M₂:
M₂ = (1.0 M)(121 ml) / (550.0 ml)
M₂ = 121 / 550 ≈ 0.220 M
Therefore, the molarity of the diluted solution is approximately 0.220 M.
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A cheese shop has some bulk cheese in blocks measuring 30 cm x 20 cm x 8 cm. How much paper is needed to cover the block of cheese
Answer:
232cm
Step-by-step explanation:
A cheese usually has the shape of a cuboid.
Characteristics of a Cuboid
A cuboid is a convex polyhedronIt has 12 edgesIt has 8 facesIts base shape is a rectangleIt has 6 facesThe amount of paper that would be needed to cover the block of cheese can be determined by calculating the perimeter of the block of cheese which is shaped as a cuboid
Perimeter of a cuboid = 4 x (length + breadth + height)
4 x (30 + 20 + 8) = 232cm
Kelly received two gift cards to her favorite store. One card was worth $25 and the other was
worth $40. She went shopping and used the cards to buy 3 shirts for $9 each and 2 skirts for
$17 each. How much gift card money did she have left?
Two particles, Alpha and Beta, race from the y-axis to the vertical line x = 6*pi. For t >= 0, Alpha's position is given by the parametric equations xalpha = 3t - 4sin(t) and yalpha = 3 - 3cos(t) while Beta's position is given by xbeta = 3t - 4sin(t) and ybeta = 3 - 4sin(t). Which sentence best describes the race and its outcome?
(A) Beta starts out in the wrong direction and loses.
(B) Alpha takes a shorter path and wins.
(C) Alpha moves slower and loses.
(D) Beta moves faster but loses.
(E) Alpha and Beta tie
The outcome of the race between Alpha and Beta, as described by their parametric equations, is that Beta moves faster but loses. Although Beta has a higher speed, Alpha consistently maintains a higher vertical position, leading to Alpha winning the race.
To determine the outcome of the race between Alpha and Beta, let's compare their positions using the given parametric equations:
Alpha's position:
[tex]x_{alpha} = 3t - 4sin(t)\\y_{alpha}= 3 - 3cos(t)[/tex]
Beta's position:
[tex]x_{beta} = 3t - 4sin(t)\\y_{beta} = 3 - 4sin(t)[/tex]
From the equations, we can see that the x-coordinate of both Alpha and Beta is the same, given by 3t - 4sin(t). Therefore, their horizontal positions are identical throughout the race.
To determine the vertical positions, we compare their y-coordinates. Alpha's y-coordinate is given by 3 - 3cos(t), while Beta's y-coordinate is given by 3 - 4sin(t).
Since cos(t) ranges from -1 to 1, and sin(t) ranges from -1 to 1, we can observe the following:
For Alpha, the y-coordinate (3 - 3cos(t)) ranges from 0 to 6, inclusive.
For Beta, the y-coordinate (3 - 4sin(t)) ranges from 2 to 4, inclusive.
Based on the range of their y-coordinates, we can conclude that Beta remains at a higher position throughout the race. Therefore, the correct answer is:
(D) Beta moves faster but loses.
Despite Beta moving faster, it loses the race because Alpha consistently maintains a higher vertical position.
Therefore, the outcome of the race between Alpha and Beta, as described by their parametric equations, is that Beta moves faster but loses. Although Beta has a higher speed, Alpha consistently maintains a higher vertical position, leading to Alpha winning the race.
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Find the volume of this square pyramid
Answer:
216
Step-by-step explanation:
Answer:
72yd
Step-by-step explanation:
Hope that helps
hsobsnsjns
Solve the logarithmic equation with Properties of Logs
Answer:
Simplifying
logx + -4 = 0
Reorder the terms:
-4 + glox = 0
Solving
-4 + glox = 0
Solving for variable 'g'.
Move all terms containing g to the left, all other terms to the right.
Add '4' to each side of the equation.
-4 + 4 + glox = 0 + 4
Combine like terms: -4 + 4 = 0
0 + glox = 0 + 4
glox = 0 + 4
Combine like terms: 0 + 4 = 4
glox = 4
Divide each side by 'lox'.
g = 4l-1o-1x-1
Simplifying
g = 4l-1o-1x-1
Answer:
[tex]x=40000[/tex]
Step-by-step explanation:
[tex]log(x)-log(4)=4\\log(\frac{x}{4})=4\\\frac{x}{4} = 10^4 \\x=4*10^4=40000[/tex]
A rectangular park, 90 meters by 60 meters, is to be built on a city block having an area of 9000 m^2. A uniform strip borders all four sides of the park for parking. How wide is the strip? Use quadratic formula and show your work.
Answer:
x = 10.52 m
Step-by-step explanation:
Given that,
Length of a park = 90 m
Width of a park = 60 m
Area, A = 9000 m²
A uniform strip borders all four sides of the park for parking. We need to find the width of the strip. Let it is x. Now the area becomes,
(90+2x)(60+2x) = 9000
[tex]4x^2 +120x +180x =5400 = 9000\\\\x=10.52\ m[/tex]
So, the width of the strip is equal to 10.52 m.
What is 3.72 of 0.6?
Answer:
Since I do not know the context of the question I will list answers I think it could be based on what you asked:
1. 3.72 x 0.6 = 2.232
2. 3.72 ÷ 0.6 = 6.2
3. 3.72% of 0.6 = 0.02232
The answer is probably the first one. I can't give a definite solution without knowing the exact question being asked, sorry!
What is the biggest difference between exponential functions and other functions you have learned about up to this point?
Answer:
No no don't click the link
Answer:
The biggest difference between exponential and linear functions is that linear functions change at a constant rate, while exponential functions change at a rate proportional to it's value, or exponent.
Basically, that's also what separates exponential functions from all others. It's the only function that changes at a rate proportional to its exponent.
Step-by-step explanation:
Find m so that x + 4 is a factor of 5x3 + 18x2 + mx + 16
The value of 'm' for which (x + 4) is a factor of the polynomial [tex]5x^3 + 18x^2 + mx + 16[/tex] is -4.
To find the value of 'm' for which the expression (x + 4) is a factor of the polynomial[tex]5x^3 + 18x^2 + mx + 16[/tex], we can apply the factor theorem. According to the factor theorem, if (x + 4) is a factor of the polynomial, then the polynomial evaluated at (-4) should be equal to zero.
Substituting (-4) into the polynomial, we get:
[tex]5(-4)^3 + 18(-4)^2 + m(-4) + 16 = 0[/tex]
-320 + 288 + (-4m) + 16 = 0
-16 + (-4m) = 0
Simplifying the equation, we have:
-4m - 16 = 0
-4m = 16
m = 16 / -4
m = -4
Therefore, the value of 'm' for which (x + 4) is a factor of the polynomial [tex]5x^3 + 18x^2 + mx + 16[/tex] is -4.
By substituting -4 for 'm' in the given polynomial, we obtain:
[tex]5x^3 + 18x^2 - 4x + 16[/tex]
When this polynomial is divided by (x + 4), the remainder will be zero, confirming that (x + 4) is indeed a factor.
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Solve for x.
-3(x+5)=-9
Enter your answer in the box. X=__
Answer:
x=-2
Step-by-step explanation:
Divide both sides by the numeric factor on the left side, then solve.
A rubber ball is dropped from a height of 26 feet, and on each bounce it rebounds up 62% of its previous height. Step 2 of 2: Assuming the ball bounces indefinitely, find the total vertical distance traveled. Round your answer to two decimal places.
The total vertical distance traveled by the rubber ball, assuming it bounces indefinitely, is approximately 85.71 feet.
To find the total vertical distance traveled, we need to sum up the heights achieved by the ball during each bounce. The ball initially drops from a height of 26 feet, so we start with this value. On each bounce, the ball rebounds up 62% of its previous height. This means that after the first bounce, the ball reaches a height of 26 feet * 0.62 = 16.12 feet.
For subsequent bounces, we continue to multiply the previous height by 0.62 to find the new height. Therefore, after the second bounce, the height becomes 16.12 feet * 0.62 = 9.99 feet.
We can see that the heights achieved during each bounce form a geometric sequence with a common ratio of 0.62. The sum of an infinite geometric sequence can be calculated using the formula,
Sum = a / (1 - r), first term is a and 'r' is the common ratio is r.
In this case, 'a' is the initial height of 26 feet and 'r' is 0.62. Plugging these values into the formula, we get,
Sum = 26 / (1 - 0.62) = 26 / 0.38 ≈ 68.42 feet.
Therefore, adding all the distances,
Distance = 68.42 + 9.99 + 16.12
Distance = 85.71 feet, total vertical distance traveled by the rubber ball, rounded to two decimal places, is approximately 85.71 feet.
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PLEASE HELP!! I only have 5 mins
Please help. No files allowed or you will be reported
plsssssd help me find the anwser
One angle of an isosceles triangle measures 46°. Which other angles could be in that isosceles triangle?
Answer:
67 degrees for both of the other angles or 46 degrees and 88
Step-by-step explanation:
An isosceles triangle has two angle that are the same size so it could only be these.
Verify that f_xy = f_yx, for the function f(x,y) = 3x^7 + 4y^7 + 12.
For the function f(x,y) = 3x^7 + 4y^7 + 12, f_xy = f_yx since fx = ______ and fy = ____
Therefore, fxy= _______ and fyx = _______
Given the function: f(x,y) = 3x^7 + 4y^7 + 12To verify that f_xy = f_yx, we need to find the partial derivatives of the given function with respect to x and y. We can find them as follows: ∂f/∂x = 21x^6 ∂f/∂y = 28y^6
Now, to verify that f_xy = f_yx, we need to find f_xy and f_yx. We can find them as follows: f_xy = ∂^2f/∂y∂x = ∂/∂y(∂f/∂x) = ∂/∂y(21x^6) = 0 (since we have no y terms in the derivative of ∂f/∂x) f_yx = ∂^2f/∂x∂y = ∂/∂x(∂f/∂y) = ∂/∂x(28y^6) = 0 (since we have no x terms in the derivative of ∂f/∂y)Since f_xy = f_yx = 0, we can say that f_xy = f_yx.
Therefore, the value of fx is 21x^6 and the value of fy is 28y^6. Hence, the value of fxy is 0 and fyx is also 0.
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factorise:x^2-y^2-x-y
9514 1404 393
Answer:
(x -y -1)(x +y)
Step-by-step explanation:
The expression can be factored by grouping.
x^2 -y^2 -x -y = (x^2 -y^2) -(x +y)
= (x -y)(x +y) -1(x +y)
= (x -y -1)(x +y)
_____
It is useful to know that a difference of squares is factored as ...
a^2 -b^2 = (a -b)(a +b)
NEED HELP WHAT ARE THSES TWOO!!