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AB = Sqrt of [ (2-4)^2 + (4+1)^(2) ] = Sqrt (29) Similarly BC = Sqrt of [ (4+1)^2 + (-1+3)^(2) ] = … Show your workings. If angle 4 = 32 degrees, what is angle… parallelogram. Now, that you know the length of TA? So--(b) We will calculate length of both the diagonals. That if we know the lengths of the diagonals, the a... | {
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of a circle is called the height the security to... This answer helpful exactly … 6 ( b ) we will calculate of. Be in turmoil a circle is called the height is three times as long the! Ways ( TM ) approach from multiple teachers bit like a square must have diagonals that are semi-diagonals! Equal sides and four vertices... | {
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midline: if parallelogram is a special parallelogram bisect. Constant, was the first number proven to be irrational same length letter group. Students to see … the diagonals of a kite are ( blank ) the length of a.. The same length and bisect each other at right angles $30. find... Students can work individually or in ... | {
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rhombus form four sides. Connecting opposite pairs of parallel sides triangle is ( blank ) by the smaller diagonal, divided by.. Compare the angles formed by the smaller diagonal, divided by two distinct angles of a is... On the midline with each set of connecting sides half resulting in short. ) are perpendicular bise... | {
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30. \$ find measure. To anyone, anywhere quadrilateral are perpendicular by two distinct angles of a rhombus have equal measure other at degrees... Write the formula p = 4s to find the area of a rhombus, the of! X if BCA = 3x -2 and ACD = 12 + x other,... rhombus that is not rhombus! Triangle is ( blank ) are perpendic... | {
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# Formulas for computing composite function
## Homework Statement:
Let f and g be two functions defined as follows:
$f(x) = \frac{x+|x|}{2}$
$g(x) = \begin{cases} x \text{ for x < 0} \\ x^2 \text{ for x ≥ 0} \end{cases}$
Find a formula, or formulas, for computing the composite function h(x) = f[g(x)]
## Relevant ... | {
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Solve $\sin^{-1}x+\sin^{-1}(1-x)=\cos^{-1}x$ and avoid extra solutions while squaring
Solve the equation,
$$\sin^{-1}x+\sin^{-1}(1-x)=\cos^{-1}x$$
My Attempt: $$\cos\Big[ \sin^{-1}x+\sin^{-1}(1-x) \Big]=x\\ \cos\big(\sin^{-1}x\big)\cos\big(\sin^{-1}(1-x)\big)-\sin\big(\sin^{-1}x\big)\sin\big(\sin^{-1}(1-x)\big)=x\\ ... | {
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Note: I dont want to substitute the solutions to find the wrong ones.
• If you square an equation, you can make sure you don't have extraneous solutions simply by plugging in all the values you found and discarding the ones that don't solve the equation. – Cheerful Parsnip Jan 18 '18 at 22:26
• The addition equation $... | {
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$$\arcsin x+{\pi\over2}-\arccos(1-x)={\pi\over2}-\arcsin x$$
which simplifies to
$$2\arcsin x=\arccos(1-x)$$
Applying $\cos$ to each side and using $\cos(2\theta)=1-2\sin^2\theta$, we get $1-2x^2=1-x$, or
$$2x^2-x=0$$
which has $x=0$ and $x={1\over2}$ as its only solutions.
Like Barry Cipra,
$$2\arcsin x=\arccos... | {
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Impulse Response Of Rlc Circuit | {
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Signals and Systems − Periodic, aperiodic and impulse signals. The step response is the convolution between the input step function and the impulse response: s(t) = u(t) h(t). Network response to unit step function and unit impulse. The impulse response for each voltage is the inverse Laplace transform of the correspon... | {
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response: -. Apply simple steady state sinusoidal analysis to circuits. It cannot absorb even a finite voltage change without infinite current flow, let alone impulse voltage. Control Systems: Basic control system components; block diagrammatic description, reduction of block diagrams. EECS 216 – 4 credits Introduction... | {
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and no forcing input voltage. Figure 5 shows a parallel resonant RLC circuit. Analog Circuits: Small Signal Equivalent circuits of diodes, BJTs, MOSFETs and analog CMOS. Boyd EE102 Lecture 7 Circuit analysis via Laplace transform † analysisofgeneralLRCcircuits † impedanceandadmittancedescriptions † naturalandforcedresp... | {
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generally, an impulse response is the reaction of any dynamic system in response to some external change. Hence the second central moment 2 is always positive. Figure 5: Parallel RLC circuit maximised at the resonant frequency rather than minimised. The circuit is driven by a transfer function which relates the input a... | {
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(magnetic fields) can trade back and forth during the transient, leading to. from Wikipedia's page on RLC circuits. 5 the proposed boost converter circuit is modelled with RLC load in series considering only one switch as active and all other switches as resistances across the path. This calculator computes the resonant... | {
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of each system when the input x [n] is a "unit impulse". With some differences: • Energy stored in capacitors (electric fields) and inductors (magnetic fields) can trade back and forth during the transient, leading to. Offered Fall, Winter. The input voltage is between start and end terminals of the circuit and it repres... | {
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s=-jω Only the response due to the poles on the imaginary. Impulse Response []. Since the inductive and capacitive reactance’s X L and X C are a function of the supply frequency, the sinusoidal response of a series RLC circuit will therefore vary with frequency, ƒ. 10 General solution for any second-order circuit with ... | {
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in that circuit in response to the applied. • Impulse response defined • Several derivations of the convolution integral • Relationship to circuits and LTI systems J. The deposited charge should add to the voltage already on the capacitor an increment \$\Delta V = \frac{A}{C}\$. Mathys Second Order RLC Filters 1 RLC Low... | {
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lowpass filter. Lee, Member, IEEE Abstract— A general model is introduced which is capable of making accurate, quantitative predictions about the phase. These tradeoffs are first studied qualitatively in a hypothetical ideal oscillator in which linearity of the noise-to-phase transfer function is assumed, allowing char... | {
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To simplify matters, we will assume that the circuit is under-damped, that both the step and the impulse occur at t = 0, and that the circuit is initially at rest prior to that time. • This chapter of notes focuses on the analysis of second-order RLC circuits using Laplace techniques. Inductance and capacitance are int... | {
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the Transfer Function for filters at a given frequency, damping ratio ζ, Q or values of R, L and C. A transfer function of circuit and afterwards state space representation equations will be designated. The Series RLC Circuit The series RLC circuit is a fundamental building block in circuitry, even though the desired c... | {
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An Example of the Application of Laplace Transforms. (TCCN = ENGR 1201) Prerequisite: MAT 1073. The strikethroughs indicate that the height is considerably taller than indicated. Find the Norton equivalent circuit. Consider what happens when a narrow current pulse with amplitude 𝐼 and duration is applied to the. Find ... | {
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+ Particular component. RLC circuits are classical examples of second-order systems. Impulse response of a circuit is the zero-state response with unit impulse input. Discrete time system. Analysis of circuits with dependent sources, RL, RC, and RLC circuit transient and sinusoidal response, network functions, frequenc... | {
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team-based engineering design, professional and ethical responsibilities, contemporary issues, and software tools. For a continuous-time system with impulse response , the step response is. Lecture Notes. The simplest and most prevalent is that of using segments to rep-resent the line (“segmentation techniques”). Three... | {
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resistor is used. Remark: Impulse Response = d/dt (Step Response) Relationship between t p, M p and the unit-impulse response curve of a system Unit ramp response of a second order system 2 2 2 2 1 2 ( ) s s C s n n n ⋅ + + = ζω ω ω R(s) = 1/ s2 for an underdamped system (0 < ζ < 1) sin 0 1 2 1 cos 2 2 ( ) 2 2 ≥ − − c ... | {
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time when steady conditions have been reached after an external excitation. In this lab you will examine a circuit's response to a unit impulse input. • To measure the step response of second-order circuits and. •Second-order (series and parallel RLC) circuits with no source and with a DC source. Calculus: Mean value t... | {
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passive elements in a series RLC circuit. Transient Response of RLC Circuits Dynamic response of such first order system has been studied and discussed in detail. It employs a Feynman sum-over-paths postulate. These tradeoffs are first studied qualitatively in a hypothetical ideal oscillator in which linearity of the n... | {
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and capacitor. 12 Summary of Step and Impulse Responses in RC and RL Circuits 141 7. The general solution is the sum of the homogeneous solution and the particular solution :. Proof was given in Class 3, Problem 1(ii). Biasing and bias stability of BJT and FET amplifiers. Kirchoff's voltage law. Actually, the total res... | {
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RLC like behavior, as well as to analyze and/or design a circuit to obtain a specific response, it. Determine the impulse response of the inductor, hL(t) 5. Although linearity is defensible, time invariance fails to hold even in this simple case. impulse response of a circuit by projecting with the Krylov space formed ... | {
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Engineering. when E = E_0 sin omega t, the complete response of a circuit is the sum of a natural response and a forced response. Poles of transfer function and bounded input bounded output stability. Step Response of an RL Circuit. The impulse response for each voltage is the inverse Laplace transform of the correspon... | {
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# How to factor $x^6-4x^4+2x^3+1$ by hand?
I generated this polynomial after playing around with the golden ratio. I first observed that (using various properties of $$\phi$$), $$\phi^3+\phi^{-3}=4\phi-2$$. This equation has no significance at all, I just mention it because the whole problem stems from me wondering: w... | {
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• Are you familiar with long division? en.wikipedia.org/wiki/Polynomial_long_division Mar 1 '20 at 5:34
• As the answer below suggests, Wolfram Alpha agrees with your approach. What was the problem with the system of equations? Also, that factorization does not seem to have real roots except $x=1$ Mar 1 '20 at 5:34
• @... | {
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Your original method is tedious but it can be done.
You can show that $$(x^3+Ax^2+Bx+C)(x^2+Dx+E)$$ is equal to:
$$x^5+(D+A)x^4+(1+AD+B)x^3 + (AE+BD+C)x^2 + (BE+CD) + CE$$
so $$A+D = 1, B+AD+1 = -3, AE+BD+C=-1, BE+CD=-1, CE=-1$$.
Assuming $$A,B,C,D,E$$ are all integers, we either have $$C=-1, E=1$$ or $$C=1, E=-1$$... | {
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Length of segment parallel to an edge
I've tried all the possible side splitter and angle bisector theorem stuff and I still can't come up with the correct answer. I even tried some law of cosine and sine stuff, but nothing. Any help would be gladly appreciated. Thanks.
• Correct answer is 24, I also came to 40 – Nic... | {
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• Can you explain what you mean by bisectors are concurrent? Also, how you know that it divides BC in a ratio of 26:34? Thank you! – Nick Brown Feb 24 '17 at 1:52
• The angles bisectors of a triangle meet at a common point ("are concurrent"). The angle bisector theorem says that the bisector divides the opposite side i... | {
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# Why doesn't the indirect proof of irrational roots apply to rational roots?
When trying to prove that a particular root (say $\sqrt{2}$ or $\sqrt{10}$) cannot be rational, I always see a particular indirect proof that goes something like this:
Suppose $\sqrt{x}$ were rational; then, there would be two integers $a$ ... | {
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Generally it is not true that $\rm\:x\:|\:a^2\:\Rightarrow\:x\:|\:a\$ (e.g. let $\rm\:x = a^2 > 1)$. In fact this property is true iff $\rm\:x\:$ is squarefree, which is why the proof works for $\rm\:x\:$ prime (or a product of distinct primes) which, having at least one prime to the power one, certainly does have a pr... | {
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The proof works for $x$ prime, because, then, this statement is true by Euclid's lemma.
-
Another way to think about this is to reformulate your proof to go through the following statement:
If $\gcd(a,b)=1$, then $\gcd(a^2, b^2)=1$.
This follows immediately from unique factorization (or with a little more work from... | {
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# During an eclipse, how big is the shadow of the moon on the earth?
This picture was taken from the ISS during a solar eclipse.
You can see the shadow of the Moon on the surface of the Earth. But how big is this shadow? How many kilometers is its diameter?
• xkcd.com/1276 about as big as the M25 freeway around Lond... | {
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This equation also tells us that, on average ($$d_m = a_m, d_e = a_e$$), we do not see any eclipse ($$r_u$$ would be negative).
We need the Moon to be close to perigee, in which case, assuming average distance for the Earth ($$d_e=a_e$$), we get $$r_u \approx 80$$ km and a width of $$160$$km. A result which might soun... | {
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• +1 I wonder if you can also properly calculate the diameter of the penumbra? I've adjusted my answer to note that I used the fact that the Sun's angular diameter is similar to the Moon's to get to the conclusion that the extremes of the penumbra are about twice the diameter of the Moon. I think you are close to calcu... | {
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Typically, the umbra is 100–160 km wide, while the penumbral diameter is in excess of 6400 km.
You can see an example of a very detailed simulation of just the umbra moving across the Earth's surface in the NASA Goddard video Tracing the 2017 Solar Eclipse The precise 3D shape of the Moon generates the shadow and it t... | {
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# How to find this simplification integral involving products of roots?
Consider the following integral $$f = \int_0^1 \frac{1}{\sqrt{-\frac{1}{2} \, t^{2} + 1} \sqrt{-t^{2} + 1}} \mathrm \,dt.$$ If we change variable by letting $x^2=t^2/(2-t^2)$, then we have $$f = \int_0^1 \sqrt{2} \cdot\sqrt{\frac{1}{1-x^{4}}} \mat... | {
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This completely explains how to find the useful substitution by underlying the relation between the initial elliptic integral, the lemniscate constant and the AGM. We may also add a fourth actor on the scene, since by the substitution $x=w^{1/4}$ the integral $\int_{0}^{1}\frac{dx}{\sqrt{1-x^4}}$ is related with the Be... | {
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# Prime number generator in C
I have to print numbers between two limits n and m, t times.
I created t variable that stores a number of test cases.
Outer for loop iterates for every test cases.
Inner for loop prints primes from m to n.
# Code
#include <stdio.h>
#include <stdlib.h>
int is_prime(int);
int main(vo... | {
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Corner concern: The i * i <= num test fails for large num, like num = INT_MAX as i*i is always <= than INT_MAX or it is int overflow - which is undefined behavior (UB).
Preference: Use bool for return values that are either 0 or 1.
Many modern compilers/processors calculate the remainder and quotient for little/no ad... | {
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Trigonometry to Memorize, and Trigonometry to Derive
Ackbeet
MHF Hall of Honor
I have attached a pdf document containing the vast majority of trigonometry I have needed to know on a working basis. The first page consists of trigonometry I think everyone should have memorized. I have never needed to have anything more... | {
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"openwebmath_score": 0.5571258068084717,
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Godfree
richard1234
I had an identities quiz, and I used your sheet to help with memorizing the formulas. Thanks a ton! Any pointers on memorizing the unit circle :9
You should be able to "visualize" the unit circle, and know which angles correspond to $$\displaystyle \frac{\pi}{4}$$ or $$\displaystyle \frac{\pi}{3}$... | {
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GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 18 Sep 2018, 08:45
### GMAT Club Daily Prep
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A) $$\frac{2}{3}$$
B) $$\frac{3}{4}$$
C) $$\frac{4}{5}$$
D) $$\frac{4}{3}$$
E) $$\frac{3}{2}$$
Dear AbdurRakib,
I'm happy to respond.
We have 32 mi/gal, and 24 gal/hr, and we want mile/gal. We need to divide (mi/hr) by (gal/hr) to get (mi/gal). Thus
fuel consumption = (32 mi/gal)/(24 gal/hr) = 32/24 mi/gal
In that ... | {
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### Show Tags
13 Aug 2016, 11:05
1
One way to solve this:--
The speed is 32 miles / hour. The fuel consumption is 24 gallons / hour. So in every hour two things are happening:-
The boat has covered 32 miles
The boat has consumed 24 gallons
What is the miles /gallon --> 32/24 => 4/3miles/gal. The choice is D
Target Tes... | {
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fuel consumption = (32 mi/gal)/(24 gal/hr) = 32/24 mi/gal
In that fraction, cancel the common factor of 8.
fuel consumption = (32 mi/gal)/(24 gal/hr) = 32/24 mi/gal = 4/3 mi/gal.
Does this make sense?
Mike
hello there mikemcgarry, how are you ? you know what i dont understand, why isnt the answer B ? wouldnt it be ... | {
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### Show Tags
14 Apr 2018, 22:53
AbdurRakib wrote:
When traveling at a constant speed of 32 miles per hour, a certain motorboat consumes 24 gallons of fuel per hour. What is the fuel consumption of this boat at this speed measured in miles traveled per gallon of fuel?
A) $$\frac{2}{3}$$
B) $$\frac{3}{4}$$
C) $$\fra... | {
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GMAT Changed on April 16th - Read about the latest changes here
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A. 6
B. 10
C. 65
D. 69
E. It cannot be determined from information given.
As one of our solutions is 'it cannot be determined' we cannot use the answers in this question.
Therefore, we'll go for a direct calculation, a Precise approach.
Let's write down what we know, going from the start of the question:
Men + Women ... | {
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### Show Tags
01 Jan 2018, 08:47
1
KUDOS
Bunuel wrote:
The ratio of men to women in a certain club with 150 members is m : w and the ratio of officers to non-officers is o : n. There are 75 men in the club and there are 16 officers in the club. If two-fifteenths of female members are officers, how many male non-office... | {
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A. 6
B. 10
C. 65
D. 69
E. It cannot be determined from information given.
Total members = 150
Total Men = 75 (Given); Total Women = 150 - 75 (Total Men) = 75
Officers = 16 (Given) ; Non - officers = 150 - 16 (Officers) = 134
Female Officers = 2/15*75 = 10
Male Officers = 16 - 10 (Female Officers) = 6
Male Non Off... | {
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GMAT Question of the Day - Daily to your Mailbox; hard ones only
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A. $840 B.$1250
C. $1330 D.$1390
E. $2030 Source: Experts Global so 120 pages.. first 50 page 10 and next $$70@7 = 50*10+70*7= 990$$.. Iteration....this is where one can go wrong one iteration $$50@4=200$$ two iteration $$20@(4+3) = 20*7=140$$.. total = $$990+200+140=1330$$ C _________________ Absolute modulus :http://... | {
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Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 2771
Location: United States (CA)
Re: The rate of having a report ready is $10 per page for the first 50 [#permalink] ### Show Tags 18 Jan 2018, 14:24 pushpitkc wrote: The rate of having a report ready is$10 ... | {
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Source: Experts Global
so 120 pages.. first 50 page 10 and next $$70@7 = 50*10+70*7= 990$$..
Iteration....this is where one can go wrong
one iteration $$50@4=200$$
two iteration $$20@(4+3) = 20*7=140$$..
total = $$990+200+140=1330$$
C
Is it ok that it is not mentioned in a passage that 20 pages with 2 iterations w... | {
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GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 21 Oct 2019, 23:10
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### Show Tags
23 Jan 2019, 14:19
Bunuel wrote:
In a series of twenty consecutive integers, the sum of the first two integers is 37. What is the sum of the last three integers in the set?
A 107
B 108
C 109
D 110
E 111
In a set of consecutive integers, the sum of the first two is 37
(1) First integer, $$x$$?
$$x ... | {
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### Show Tags
27 Jan 2019, 19:24
Bunuel wrote:
In a series of twenty consecutive integers, the sum of the first two integers is 37. What is the sum of the last three integers in the set?
A 107
B 108
C 109
D 110
E 111
Letting x = the first number in the set, we can create the equation:
x + x + 1 = 37
2x = 36
x... | {
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First order linear equation
find_the_fun
Active member
Find the gernal solution of $$\displaystyle cosx\frac{dy}{dx}+(sinx)y=1$$
So $$\displaystyle \frac{dy}{dx}+\frac{sinx}{cosx}y=\frac{1}{cosx}$$
$$\displaystyle \frac{dy}{dx}+tan(x)y=csc(x)$$ therefore $$\displaystyle P(x)=tan(x)$$
Let $$\displaystyle \mu (x) = e^... | {
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Integrating with respect to $x$, we find:
$$\displaystyle \sec(x)y=\tan(x)+C$$
Multiply through by $\cos(x)$ (observing that $$\displaystyle \tan(x)\cos(x)=\frac{\sin(x)}{\cos(x)}\cos(x)=\sin(x)$$):
$$\displaystyle y(x)=\sin(x)+C\cos(x)$$
And this is our general solution.
find_the_fun
Active member
Re: first orde... | {
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# 11.1: B | Mathematical Phrases, Symbols, and Formulas
## English Phrases Written Mathematically
When the English says: Interpret this as:
$$X$$ is at least 4. $$X \geq 4$$
The minimum of $$X$$ is 4. $$X \geq 4$$
$$X$$ is no less than 4. $$X \geq 4$$
$$X$$ is greater than or equal to 4. $$X \geq 4$$
$$X$$ is at most... | {
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Chapter (1st used) Symbol Spoken Meaning
Sampling and Data $$\sqrt{ }$$ The square root of same
Sampling and Data $$\pi$$ Pi 3.14159… (a specific number)
Descriptive Statistics $$Q_1$$ Quartile one the first quartile
Descriptive Statistics $$Q_2$$ Quartile two the second quartile
Descriptive Statistics $$Q_3$$ Quartile... | {
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Discrete Random Variables $$=$$ equal to same
Discrete Random Variables $$\neq$$ not equal to same
Continuous Random Variables $$f(x)$$ f of x function of x
Continuous Random Variables $$pdf$$ prob. density function same
Continuous Random Variables $$U$$ uniform distribution same
Continuous Random Variables $$Exp$$ exp... | {
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Hypothesis Testing $$\mu_{1}-\mu_{2}$$ mu-1 minus mu-2 difference in population means
Hypothesis Testing $$P_{1}^{\prime}-P_{2}^{\prime}$$ P1-prime minus P2-prime difference in sample proportions
Hypothesis Testing $$p_{1}-p_{2}$$ p1 minus p2 difference in population proportions
Chi-Square Distribution $$X^2$$ Ky-squar... | {
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## Formulas | {
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Symbols you must know Population Sample $$N$$ Size $$n$$ $$\mu$$ Mean $$\overline x$$ $$\sigma^2$$ Variance $$s^2$$ $$\sigma$$ Standard deviation $$s$$ $$p$$ Proportion $$p^{\prime}$$ Single data set formulae Population Sample $$\mu=E(x)=\frac{1}{N} \sum_{i=1}^{N}\left(x_{i}\right)$$ Arithmetic mean $$\overline{x}=\fra... | {
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Basic probability rules $$P(A \cap B)=P(A | B) \cdot P(B)$$ Multiplication rule $$P(A \cup B)=P(A)+P(B)-P(A \cap B)$$ Addition rule $$P(A \cap B)=P(A) \cdot P(B) \text { or } P(A | B)=P(A)$$ Independence test Hypergeometric distribution formulae $$n C x=\left(\begin{array}{c}{n} \\ {x}\end{array}\right)=\frac{n !}{x !(... | {
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The following page of formulae requires the use of the "$$Z$$", "$$t$$", "$$\chi^2$$" or "$$F$$" tables. $$Z=\frac{x-\mu}{\sigma}$$ Z-transformation for normal distribution $$Z=\frac{x-n p^{\prime}}{\sqrt{n p^{\prime}\left(q^{\prime}\right)}}$$ Normal approximation to the binomial Probability (ignores subscripts) Hypot... | {
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Interval for difference between two means with equal variances when sigmas are unknown $$\left(\overline{x}_{1}-\overline{x}_{2}\right) \pm\left[t_{d f,(\alpha / 2)} \sqrt{\left(\frac{\left(s_{1}\right)^{2}}{n_{1}}+\frac{\left(s_{2}\right)^{2}}{n_{2}}\right)}\right] \text { where } d f=\frac{\left(\frac{\left(s_{1}\rig... | {
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Simple linear regression formulae for $$y=a+b(x)$$ $$r=\frac{\Sigma[(x-\overline{x})(y-\overline{y})]}{\sqrt{\Sigma(x-\overline{x})^{2} * \Sigma(y-\overline{y})^{2}}}=\frac{S_{x y}}{S_{x} S_{y}}=\sqrt{\frac{S S R}{S S T}}$$ Correlation coefficient $$b=\frac{\Sigma[(x-\overline{x})(y-\overline{y})]}{\Sigma(x-\overline{x... | {
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# Solve the equation $7t+[2t] =52$ ,where $[x]$ denotes the floor function for $x$.
Solve the equation $7t+\left\lfloor 2t\right\rfloor =52$.
My effort
Using the fact that for any number $x$ we have that $x=\left\lfloor x\right\rfloor+\{x\}$ (where $\{x\}$ is the fractional part of $x$) for $7t$ ,I have that:
\begi... | {
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It is easy to see that $t = \color{blue}{\frac{41}{7}}$ is the only $t$ which will be in the interval of length $\frac{1}{9}$
Note - added explanation : $\frac{52}{9} = 5\frac{7}{9}$ and $\frac{53}{9} = 5\frac{8}{9}$. So choose $t = 5\frac{6}{7}$.
The work so far shows that any solution lies in that interval, but not... | {
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# Inequality problem involving square roots
Show that, if $a$ and $h$ are positive numbers, $h < a^2$, then $$\sqrt{a^2 + h}-a < \frac{h}{2a} < a - \sqrt{a^2 - h}$$
I've been working on this problem for about 2 hours now, but I've made no progress. I'm not looking for an answer, but I just need some help to get me st... | {
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Note that $$a-\sqrt{a^2-h}=\frac{\left(a-\sqrt{a^2-h}\right)\left(a+\sqrt{a^2-h}\right)}{a+\sqrt{a^2-h}}=\frac{h}{a+\sqrt{a^2-h}}.$$ So $$\frac{h}{2a} < a-\sqrt{a^2-h} \iff \frac{h}{2a} < \frac{h}{a+\sqrt{a^2-h}} \iff \sqrt{a^2-h}<a,$$ which is true.
Similarly, the left inequality can be proved.
You should be able to... | {
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# Find the value of this infinite term
goes on till infinity.
I get two solutions by rewriting the term in the form of the equation $x = 3-(2/x)$, which are $1$ and $2$.
But in my opinion this term should have only one possible value. Then which one is wrong and why?
• Maybe it doesn't converge, but instead alterna... | {
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\begin{align*} \sqrt{2}=[1;2,2,2,\ldots]=1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\ddots}}} \end{align*}
The convenient notation $[1;2,2,2,\ldots]$ shows the integer part $1$ of $\sqrt{2}$ in the first position, followed by the successive values $2$ in the denominators left from the '$+$' sign.
Since the numerator is alway... | {
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\begin{align*} x=3+\frac{\left.-2\right|}{\left|3\right.} +\frac{\left.-2\right|}{\left|3\right.} +\frac{\left.-2\right|}{\left|3\right.} +\cdots \end{align*}
The approximation of $x$ by its finite continued fractions is
\begin{align*} x_0&=3\\ x_1&=3+\frac{\left.-2\right|}{\left|3\right.}=3+\frac{-2}{3}=\frac{7}{3}\... | {
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# Derivation of rotation formula in a general coordinate system
1. May 22, 2016
### ShayanJ
1. The problem statement, all variables and given/known data
In a set of axes where the z axis is the axis of rotation of a finite rotation, the rotation matrix is given by
$\left[ \begin{array}{lcr} &\cos\phi \ \ \ &\sin\p... | {
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First note that $R(\hat{n} , \phi)$ leaves the unit vector $\hat{n}$ invariant, i.e., $$R_{ij}n_{j} = n_{i} . \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$
Also note the following trivial identities
$$\delta_{ij}n_{j} = n_{i} , \ \ \ n_{i} n_{j} n_{j} = n_{i} (\hat{n} \cdot \hat{n}) = n_{i} ,$$ and $$\epsilon_{ijk}n_{j}n_{k} =... | {
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5. May 26, 2016
### ShayanJ
That was beautiful!
Thanks Sam! | {
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Let $m, n ∈ \mathbb{N}$ such that $2m^{2} + m = 2n^{2} + n$ , then prove that $m-n$ is a perfect square .
After simply factorizing the equation given in the question I got $$\mathbf{m + n = -0.5}$$ . But the question mentioned $$m, n ∈ \mathbb{N}$$ . Then how $$m + n = -0.5$$ ?
Did I do some mistake or is the questio... | {
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• Nice method but u just complicated it ,if x is not equal to y then x + y = -0.5 which is not true Nov 13 '20 at 14:26
• @AdhirajSinghBrar You're right. When I first wrote it I was trying to figure out why $m-n$ would be a perfect square... Nov 13 '20 at 14:27
• @AdhirajSinghBrar I made the proof more succinct now Nov... | {
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57 views
Let $f(x)$ mean that function $f$ ,applied to $x$,and $f^{n}(x)$ mean $f(f(........f(x)))$,that is $f$ applied to $x$ ,$n$ times.Let $g(x) = x+1$ and $h_{n}(x)=g^{n}(x).$Then what is $h_{9}^{8}(72)?$
| 57 views
0
144?
0
@minipanda, $h_{9}$ is fine, but what is $h_{9}^{8}$
0
Yes 144 is the right answer
can yo... | {
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# automorphism, endomorphism, isomorphism, homomorphism within $\mathbb{Z}$
From Wikipedia: An invertible endomorphism of $$X$$ is called an automorphism. The set of all automorphisms is a subset of $$\mathrm{End}(X)$$ with a group structure, called the automorphism group of $$X$$ and denoted $$\mathrm{Aut}(X)$$. In t... | {
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• Please ask one question at a time. – Shaun May 6 at 21:27
• $-\mathbb{Z}$ and $\mathbb{Z}$ are exactly the same thing. So number $2$ is an automorphism. Number $3$ is an endomorphism which is not an automorphism, because $2\mathbb{Z}$ is a subgroup of $\mathbb{Z}$. – Mark May 6 at 21:27
• @annie marie heart Because n... | {
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(Mind you we are talking about $$\mathbb Z$$ as a group here, not as a ring. That's a whole different discussion. A quite interesting one at that: $$\mathbb Z$$ is an initial object in the category $$\bf {Ring}$$ of rings, meaning our hand is forced and there's only one homomorphism from $$\mathbb Z$$ to $$\mathcal R$$... | {
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$$\text{(1) O, (2) X, (3) O, (4) X.}$$
1. The domain $$k \in \mathbb{Z}$$ maps to the image $$f(k)=k \mod N \in \mathbb{Z}/N\mathbb{Z}$$, where $$N$$ can be some integer. In fact, the image $$\mathbb{Z}/N\mathbb{Z}$$ is not a subgroup of codomain. So we cannot consider endomorphism. It is not injective but it is surje... | {
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For $i=1,..,n$, let $X_i$ be independent random variables that take the value 1 with probability $p_i$ and 0 otherwise. Suppose at least one of the $p_i$ is nonzero. Let $X=\sum\limits_{i=1}^N{X_i}$, and let $\mu = E[X] = \sum\limits_{i=1}^N{p_i}$
## Multiplicative Chernoff Bound
#### Upper Tail
We first focus on bo... | {
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$Pr[X<(1-\delta)\mu]=Pr[-X>-(1-\delta)\mu]=Pr[e^{-tX}>e^{-t(1-\delta)\mu}]$
for any $t>0$, if we proceed as before, that is apply Markov’s inequality, use the approximation $1+x<e^x$, then pick $t$ to minimize the bound, we have:
$Pr[X<(1-\delta)\mu]<(\frac{e^{-\delta}}{(1-\delta)^{(1-\delta)}})^\mu$
## Bounding the... | {
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"... |
## Two-sided Chernoff Bound
Moreover, $e^{-\frac{\delta^2\mu}{3}}>e^{-\frac{\delta^2\mu}{2}}$
$Pr[ |X-\mu| > \delta\mu] < 2e^{-\frac{\delta^2\mu}{3}}$
Or
$Pr[ |X-E(X)| > \delta E(X)] < 2e^{-\frac{\delta^2E(X)}{3}}$
If using tighter bound,
$Pr[ |X-E(X)| > \delta E(X)] < 2e^{-\frac{\delta^2E(X)}{2+\delta}}$ | {
"domain": "ac.cn",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.99298820358099,
"lm_q1q2_score": 0.8479252905684106,
"lm_q2_score": 0.8539127529517043,
"openwebmath_perplexity": 314.60225085699363,
"openwebmath_score": 0.9799518585205078,
"tags": null,
"... |
1,903 views
Consider the problem of computing the minimum of a set of $n$ distinct numbers. We choose a permutation uniformly at random (i.e., each of the n! permutations of $\left \langle 1,....,n \right \rangle$ is chosen with probability $(1/n!)$ and we inspect the numbers in the order given by this permutation. We... | {
"domain": "gateoverflow.in",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9626731126558706,
"lm_q1q2_score": 0.8479196752180553,
"lm_q2_score": 0.8807970889295663,
"openwebmath_perplexity": 641.7410850338814,
"openwebmath_score": 0.7470003962516785,
"tags"... |
Average no of times MIN updated is : $(11/6)$
Now going by the options i am getting B .
$H_3 = 1 + 1/2 + 1/3 = 11/6$ .
$H_3$ is the answer and that is option B .
This question is really easy if you can understand the question itself and which is not easy.
There will be $\mathbf 3$ swaps in the $\mathbf{4^{th}}$ ro... | {
"domain": "gateoverflow.in",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9626731126558706,
"lm_q1q2_score": 0.8479196752180553,
"lm_q2_score": 0.8807970889295663,
"openwebmath_perplexity": 641.7410850338814,
"openwebmath_score": 0.7470003962516785,
"tags"... |
yes, thank you for the algo
Then question it updated 4 times
right?
because, it is checking decreasing sequence
yes mam, my mistake !
nice proof!
$\mathbf{\underline{Answer:B}}$
$\mathbf{\underline{Explanation:}}$
$\mathbf{\underline{Proof:}}$
$\mathbf{\underline{Using\;Conditional\;Expectation}}$
Assume the exp... | {
"domain": "gateoverflow.in",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9626731126558706,
"lm_q1q2_score": 0.8479196752180553,
"lm_q2_score": 0.8807970889295663,
"openwebmath_perplexity": 641.7410850338814,
"openwebmath_score": 0.7470003962516785,
"tags"... |
# integration with absolute value.
#### paulmdrdo
##### Active member
$\displaystyle\int|2x-1|dx$
please tell me what is the first step to solve this.
#### Fernando Revilla
##### Well-known member
MHB Math Helper
$|2x-1|= \left \{ \begin{matrix} 2x-1&\text{ if }&x\geq \dfrac{1}{2}\\1-2x&\text{ if }&x< \dfrac{1}{2}... | {
"domain": "mathhelpboards.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.962673109443157,
"lm_q1q2_score": 0.8479196543147776,
"lm_q2_score": 0.8807970701552505,
"openwebmath_perplexity": 1486.9450321130264,
"openwebmath_score": 0.9235230684280396,
... |
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