text stringlengths 100 932k | input_ids listlengths 26 131k | length int64 26 131k |
|---|---|---|
Commutative Property Of Addition 2. If A is an n×m matrix and O is a m×k zero-matrix, then we have: AO = O Note that AO is the n×k zero-matrix. Matrix Matrix Multiplication 11:09. We have 1. To understand the properties of transpose matrix, we will take two matrices A and B which have equal order. The identity matrix i... | [
1092,
6984,
1388,
8655,
4940,
78646,
220,
17,
13,
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362,
374,
458,
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76,
6172,
323,
506,
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264,
296,
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74,
7168,
1448,
2555,
11,
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582,
614,
25,
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284,
506,
7036,
429,
65269,
374,
279,
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17568,
74,
... | 4,479 |
# Comparing the magnitudes of expressions of surds
I recently tackled some questions on maths-challenge / maths-aptitude papers where the task was to order various expressions made up of surds (without a calculator, obviously).
I found myself wondering whether I was relying too much on knowing the numerical value of ... | [
2,
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279,
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315,
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315,
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271,
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12,
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3906,
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1380,
279,
3383,
572,
311,
1973,
5257,
23393,
1865,
705,
315,
1729,
5356,
320,
... | 1,385 |
# In Calculus, how can a function have several different, yet equal, derivatives?
I've been pondering this question all night as I work through some problems, and after a very thorough search, I haven't found anything completely related to my question. I guess i'm also curious how some derivatives are simplified as we... | [
2,
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355,
11,
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729,
614,
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287,
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975,
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11,
323,
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264,
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17423,
2711,
11,
358,
8990,
... | 1,546 |
# Math Help - working backwards - cubics
1. ## working backwards - cubics
Write an equation that has the following roots: 2, -1, 5
Answer key: x^3 - 6x^2 + 3x + 10 = 0
For quadratic equations, I use the sum and product of roots, this is a cubic equation, how do I solve this?
Thanks.
2. Originally Posted by shento... | [
2,
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11,
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16,
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20,
271,
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1376,
25,
856,
61,
18,
481... | 1,704 |
# Work and time, when work is split into parts
I'm stuck on a particular type of work and time problems.
For example,
1) A,B,C can complete a work separately in 24,36 and 48 days. They started working together but C left after 4 days of start and A left 3 days before completion of the work. In how many days will the... | [
2,
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11,
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975,
374,
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5479,
271,
40,
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975,
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16,
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362,
8161,
11289,
646,
4583,
264,
975,
25156,
304,
220,
17,
19,
11,
18,
2... | 2,315 |
# Algebraic Manipulation
## Definition
Algebraic manipulation involves rearranging variables to make an algebraic expression better suit your needs. During this rearrangement, the value of the expression does not change.
## Technique
Algebraic expressions aren't always given in their most convenient forms. This is ... | [
2,
76443,
292,
60911,
2914,
271,
565,
19709,
271,
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292,
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17601,
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311,
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292,
7493,
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7781,
697,
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13,
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419,
55327,
56633,
11,
279,
897,
315,
279,
7493,
1558,
537,
2297,
... | 1,243 |
Inequality involving sums with binomial coefficient
I am trying to show upper- and lower-bounds on
$$\frac{1}{2^n}\sum_{i=0}^n\binom{n}{i}\min(i, n-i)$$
(where $$n\geq 1$$) in order to show that it basically grows as $$\Theta(n)$$.
The upper-bound is easy to get since $$\min(i, n-i)\leq i$$ for $$i\in\{0, \dots n\}... | [
641,
81638,
15860,
36398,
448,
9544,
20855,
35606,
271,
40,
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4460,
311,
1473,
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17,
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11035,
1242,
15159,
72,
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92,
61,
77,
59,
6863,
316,
913... | 3,385 |
The main condition of matrix multiplication is that the number of columns of the 1st matrix must equal to the number of rows of the 2nd one. The dimensions of $B$ are $3\times 2$ and the dimensions of $A$ are $2\times 3$. The dot product involves multiplying the corresponding elements in the row of the first matrix, by... | [
785,
1887,
2971,
315,
6172,
46444,
374,
429,
279,
1372,
315,
8147,
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220,
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576,
15336,
315,
400,
33,
3,
525,
400,
18,
4955,
1733,
220,
... | 5,723 |
# Definite Integral: $\int_0^1\frac{\ln^4(x)}{x^2+1}\,dx$
I'm trying to derive a closed-form expression for
$$I=\int_0^1\frac{\ln^4(x)}{x^2+1}\,dx$$
Letting $u=-\ln(x), x=e^{-u}, dx=-e^{-u}\,du$ yields
$$I=\int_0^{\infty}\frac{u^4e^{-u}}{e^{-2u}+1}\,du$$
Setting $u\to-u$ and manipulating the integrands yield
$$I=... | [
2,
3892,
15856,
91660,
25,
57960,
396,
62,
15,
61,
16,
59,
37018,
35702,
2261,
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90,
87,
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17,
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11,
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66426,
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4460,
311,
42430,
264,
7877,
8460,
7493,
369,
271,
14085,
40,
34433,
396... | 6,336 |
# Divisibility Rule for 9
I'm working through an elementary number theory course right now and I think I've come up with a proof here but wanted some feedback on my logic.
Question: If the sum of the digits in base 10 is divisible by 9, then the number itself is divisible by 9.
Proof: Suppose that $9|d_1+d_2+...+d_n... | [
2,
8765,
285,
3147,
18100,
369,
220,
24,
271,
40,
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3238,
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458,
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323,
358,
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358,
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705,
448,
264,
11064,
1588,
714,
4829,
1045,
11055,
389,
847,
12218,
382,
14582,
25,
1416,... | 1,385 |
# What is the remainder when a_{2013} is divided by 7?
#### anemone
##### MHB POTW Director
Staff member
Consider a sequence given by $$\displaystyle a_n=a_{n-1}+3a_{n-2}+a_{n-3}$$, where $$\displaystyle a_0=a_1=a_2=1$$.
What is the remainder of $$\displaystyle a_{2013}$$ divided by $$\displaystyle 7$$?
#### chisig... | [
2,
3555,
374,
279,
26313,
979,
264,
15159,
17,
15,
16,
18,
92,
374,
17779,
553,
220,
22,
1939,
820,
458,
336,
603,
271,
67331,
386,
30725,
61502,
54,
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198,
33211,
4462,
198,
37175,
264,
8500,
2661,
553,
26107,
59,
5493,
3528,
... | 1,028 |
The radius of convergence is half the length of the interval of convergence. We noticed that, at least in the case of the geometric series, there was an interval in which it converged, but it didn’t converge at the endpoints. Show that the following alternating harmonic series converges: Series of Both Positive and Neg... | [
785,
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315,
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374,
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279,
3084,
315,
279,
9873,
315,
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13,
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304,
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315,
279,
52484,
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11,
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572,
458,
9873,
304,
892,
432,
96775,
11,
714,
432,
3207,
1405,
79767,
5... | 7,878 |
# Math Help - Trigonometry three dimensional question
1. ## Trigonometry three dimensional question
I need help to solve:
A cylinder with radius 4 cm and perpendicular height 15 cm is tilted so that it will just fit inside a 12 cm high box. At what angle must it be tilted?
Answer given at the back of the text book i... | [
2,
4149,
11479,
481,
1163,
73881,
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2326,
55887,
3405,
271,
16,
13,
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510,
32,
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448,
10578,
220,
19,
9961,
323,
76833,
2608,
220,
16,
20,
9961,
374,
... | 1,189 |
# Golden Ratio and Fibonacci Numbers
Golden Ratio is considered to be one of the greatest beauties in mathematics. Two numbers $$a$$ and $$b$$ are said to be in Golden Ratio if $a>b>0,\quad and\quad \frac { a }{ b } =\frac { a+b }{ a }$ If we consider this ratio to be equal to some $$\varphi$$ then we have $\varphi =\... | [
2,
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323,
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34713,
271,
59790,
50748,
374,
6509,
311,
387,
825,
315,
279,
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550,
304,
37596,
13,
9043,
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26107,
64,
14085,
323,
26107,
65,
14085,
525,
1053,
311,
387,
304,
17809,
50748,
421,
400,
64,
51... | 1,828 |
# Evaluating Triple integral
1. May 7, 2005
### MattL
I'm having trouble with evaluating
[Triple Integral] |xyz| dx dy dz
over the region (x/a)^2 + (y/b)^2 + (z/c)^2 <= 1
Do I need to use some sort of parametrisation for the region, and is there some way of dealing with the absolute value function without integra... | [
2,
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1095,
36649,
25098,
271,
16,
13,
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220,
22,
11,
220,
17,
15,
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20,
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12264,
448,
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271,
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461,
694,
91660,
60,
760,
28854,
91,
13822,
13955,
25718,
271,
1975,
279,
... | 1,120 |
# Proving or disproving basic facts about sequences in Real Analysis
I am self-learning real analysis from Stephen Abott's Understanding Analysis. In Exercise 2.3.7, the author asks to prove or disprove basic results on convergence. I'd like to verify my solution, to ensure, I've understood the concepts, and if proof ... | [
2,
1298,
4405,
476,
41503,
4405,
6770,
13064,
911,
23700,
304,
8800,
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271,
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656,
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1931,
6358,
504,
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13,
758,
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220,
17,
13,
18,
13,
22,
11,
279,
3150,
17064,
311,
12118,
... | 3,046 |
Is it true that the only matrix that is similar to a scalar matrix is itself Hot Network Questions Was the title "Prince of Wales" originally claimed for the English crown prince via a trick? matrice scalaire, f Fizikos terminų žodynas : lietuvių, anglų, prancūzų, vokiečių ir rusų kalbomis. An example of a diagonal mat... | [
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429,
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429,
374,
4428,
311,
264,
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6172,
374,
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8141,
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14804,
279,
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330,
67932,
315,
22931,
1,
13214,
11660,
369,
279,
6364,
26533,
41726,
4566,
264,
14068,
30,
5517,
23120,
15771... | 3,567 |
posted by Dina
Simplify the expression.
3 [ (15 - 3)^2 / 4]
a. 36
b. 108
c. 18
d. 9
1. TutorCat
http://www.jiskha.com/display.cgi?id=1285004701
2. Dina
???108
3. PsyDAG
3 (12^2/4) = 3 (144/4) = 3 * 36 = 108
Yes!
## Similar Questions
1. ### Math(radicals or square roots)
i need a explanation for this equatio... | [
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422,
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271,
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279,
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220,
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21,
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65,
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220,
16,
15,
23,
198,
66,
13,
220,
16,
23,
198,
67,
13,
220,
24,
27... | 750 |
# Is “ln” (natural log) and “log” the same thing if used in this answer?
Find $$x$$ for $$4^{x-4} = 7$$.
Answer I got, using log, was $${\log(7)\over 2\log(2)} + 4$$
but the actual answer was $${\ln(7)\over2\ln(2)} + 4$$
I plugged both in my calculator and turns out both are the equivalent value.
Anyways, is using... | [
2,
2160,
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320,
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8,
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839,
854,
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421,
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12,
19,
92,
284,
220,
22,
14085,
382,
16141,
358,
2684,
11,
1... | 2,547 |
# $\{a_n\}$ be a sequence such that $a_{n+1}^2-2a_na_{n+1}-a_n=0$, then $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in…
Let $$\{a_n\}$$ be a sequence of positive real numbers such that
$$a_1 =1,\ \ a_{n+1}^2-2a_na_{n+1}-a_n=0, \ \ \forall n\geq 1$$.
Then the sum of the series $$\sum_1^{\infty}\frac{a_n}{3^n}$$ lies in...
... | [
2,
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90,
64,
1089,
59,
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387,
264,
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429,
400,
64,
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15,
54876,
1221,
57960,
1242,
62,
16,
61,
35702,
258,
36958,
11035,... | 2,811 |
For which primes p is $p^2 + 2$ also prime?
Origin — Elementary Number Theory — Jones — p35 — Exercise 2.17 —
Only for $p = 3$. If $p \neq 3$ then $p = 3q ± 1$ for some integer $q$, so $p^2 + 2 = 9q^2 ± 6q + 3$ is divisible by $3$, and is therefore composite.
(1) The key here looks like writing $p = 3q ± 1$. Where d... | [
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16,
22,
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7308,
369,
400,
79,
284,
220,
18,
12947,
14... | 1,591 |
1. ## Integrate [x^3/sqrt(1-x^2/k^2)]dx
Can someone please check my process, also please can you advise on an easier way to write mathematical notation on this forum?
Integrate ( x^3/(1-x^2/k^2))dx
where k is a constant.
Let u = x^2, dv = x/sqrt(1-x^2/k^2)
Integration by parts.
Integral fx = uv - integral vdu
v = -... | [
16,
13,
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508,
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61,
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7,
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271,
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847,
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11,
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646,
498,
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389,
458,
8661,
1616,
311,
3270,
35972,
44197,
389,
... | 2,422 |
It is currently 20 Apr 2018, 13:06
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based o... | [
2132,
374,
5023,
220,
17,
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5076,
220,
17,
15,
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11,
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498,
369,
1667,
279,
9021,
481,
419,
10847,
5392,
646,
16045,
697,
5068,
323,
4190,... | 1,540 |
# A Most Curious Algebraic Identity
I recently found a very interesting Algebraic Identity: $xyz+(x+y)(y+z)(z+x)=(x+y+z)(xy+yz+zx)$
What's so special about it? Note that going from one side of the equality to the other, all products are switched with sums, and all sums are switched with products!
This may be seen a ... | [
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271,
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73173,
87,
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92952,
2376,
4130,
10,
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10,
66700,
15087,... | 1,462 |
1 Definition; 2 Examples; 3 Symmetric part of a tensor; 4 Symmetric product; 5 Decomposition; 6 See also; 7 Notes; 8 References; 9 External links; Definition. Decomposition of tensor power of symmetric square. (1.5) Usually the conditions for µ (in Eq. Decomposition of Tensors T ij = TS ij + TA ij symmetric and anti-sy... | [
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220,
22,
18068,
26,
220,
23,
44078,
26,
220,
24,
30936,
7746,
... | 3,441 |
# Yet another proof for $\zeta (2) = \frac { { \pi }^{ 2 } }{ 6 }$
Let us consider the integral
$\displaystyle i\int _{ 0 }^{ \pi }{ ln(1-{ e }^{ i\theta } } )d\theta$
Now using
$\displaystyle i\int _{ 0 }^{ \pi }{ ln(1-{ e }^{ i\theta } } )d\theta \\ \\ 1-{ e }^{ i\theta } = 1-cos\theta -isin\theta = 2sin(\frac { ... | [
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2441,
11064,
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320,
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8,
284,
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220,
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601,
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279,
25098,
271,
58736,
5493,
3528,
600,
59,
396,
716,
90,
... | 1,938 |
# Thread: Solving Exponential Equations (logs involved)
1. ## Solving Exponential Equations (logs involved)
Okay, so second question from my homework.... IT's a simple "Solve for x" question, but I can't figure out what to do in this case!
6^(3x)=4^(2x-3)
6^(3x)=2^(4x-6)
We can use the log laws .... And it's likel... | [
2,
8752,
25,
11473,
4405,
1374,
59825,
11702,
804,
320,
22081,
6398,
692,
16,
13,
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11473,
4405,
1374,
59825,
11702,
804,
320,
22081,
6398,
692,
32313,
11,
773,
2086,
3405,
504,
847,
28459,
1934,
8700,
594,
264,
4285,
330,
50,
394... | 1,523 |
# difference between scalar matrix and diagonal matrix
{ For example, $$A =\begin{bmatrix} 0\\ â3\\-1 \\1/2 \end{bmatrix}$$ is a column matrix of order 4 Ã 1. In a scalar matrix, all off-diagonal elements are equal to zero and all on-diagonal elements happen to be equal. When the order is clear from the context, we... | [
2,
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65,
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92,
140... | 5,346 |
# Bipolar coordinate system
I want to extend some work presented in a paper "Analysis of TM and TE Modes in Eccentric Coaxial Lines Based on Bipolar Coordinate System" using SageMath. Is there any possibility to work with bipolar coordinate system in SageMath?
Thanks
edit retag close merge delete
Sort by » oldest n... | [
2,
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In how many ways can we pair ourselves?
Say we have an even number of elements and we sort them into pairs in a way that every element belongs to a pair and no element belongs in two pairs.
Given $2n$ elements how many different arrangements of this sort can be made?
For example given elements named $1$, $2$, $3$ an... | [
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304,
1378,
13530,
382,
... | 1,368 |
Does the "field" over which a vector space is defined have to be a Field?
I was reviewing the definition of a vector space recently, and it occurred to me that one could allow for only scalar multiplication by the integers and still satisfy all of the requirements of a vector space.
Take for example the set of all or... | [
21468,
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10017,
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825,
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369,
1172,
17274,
46444,
553... | 1,067 |
# If $g\circ f$ is injective and $f$ is surjective then $g$ is injective
Let $f:A\rightarrow B$ and $g:B\rightarrow C$ be functions, prove that if $g\circ f$ is injective and $f$ is surjective then $g$ is injective.
I need advise or correction if something is incorrect with my proof. Thank you beforehand.
We must sh... | [
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3,
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11,
... | 2,308 |
It is currently 22 Nov 2017, 04:20
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based o... | [
2132,
374,
5023,
220,
17,
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17,
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279,
9021,
481,
419,
10847,
5392,
646,
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5068,
323,
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# Necklace combinations with three group of beads
I have a hard question about a way how many different necklaces can be made.
Suppose that we have the following restrictions:
1. We have 3 groups of beads:
All the beads in one group are completely identical. This means that if you put two triangle beads next to eac... | [
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220,
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... | 3,146 |
# Is $V$ isomorphic to direct sum of subspace $U$ and $V/U$?
Given a vector space $$V$$ and a subspace $$U$$ of $$V$$. $$V \cong U \oplus(V/U)$$ Does the above equation always hold? Where $$\oplus$$ is external direct sum. For finite dimensional vector space $$V$$, here is my attemp of prove:
Let dimension of $$U$$ b... | [
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53,
3,
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53,
11... | 2,433 |
# Query on a Solution to the Problem: $\gcd(5a+2,7a+3)=1$ for all integer $a$.
I wish to show that the numbers $$5a+2$$ and $$7a+3$$ are relatively prime for all positive integer $$a$$.
Here are my solutions.
Solution 1. I proceed with Euclidean Algorithm. Note that, for all $$a$$, $$|5a+2|<|7a+3|$$. By Euclidean Al... | [
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# For any arrangment of numbers 1 to 10 in a circle, there will always exist a pair of 3 adjacent numbers in the circle that sum up to 17 or more
I set out to solve the following question using the pigeonhole principle
Regardless of how one arranges numbers $1$ to $10$ in a circle, there will always exist a pair of t... | [
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700,
31... | 1,339 |
# To find number of real roots
Consider the equation $$x^5-5x=c$$ where c is a real number.
Determine all c such that this equation has exactly 3 real roots.
I know that between consecutive real roots of $$f$$ there is a real root of $$f'$$. Now $$f'$$ in this case is $$5x^4-5$$ which always has two real roots. So t... | [
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... | 1,273 |
# Difference between continuity and uniform continuity
Tags:
1. May 7, 2015
### Yunjia
I noticed that uniform continuity is defined regardless of the choice of the value of independent variable, reflecting a function's property on an interval. However, if on a continuous interval, the function is continuous on every... | [
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315,
9489,
... | 2,124 |
## Linear Algebra And Learning From Data Github
Each point correspondence generates one constraint on F. Complete Linear Algebra for Data Science & Machine Learning 4. My research interests lie in network science, statistical inference, causal inference, information theory, machine learning, data mining, and signal. 0... | [
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It is also a matrix and also an array; all scalars are also vectors, and all scalars are also matrix, and all scalars are also array You can put this solution on YOUR website! See the picture below. Equal Matrices: Two matrices are said to be equal if they are of the same order and if their corresponding elements are e... | [
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1334,
1446,
646,
2182,
419,
6291,
389,
20922,
3910,
0,
3496,
... | 2,841 |
# Binomial theorem
Series binomial theorem series contents page contents binomial theorem notation n as a nonnegative integer proof of the binomial theorem proof when n and k are. The binomial theorem date_____ period____ find each coefficient described 1) coefficient of x2 in expansion of (2 + x)5 80 2) coefficient o... | [
2,
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11064,
979,
308,
323,
595,
525,
13,
576,
9544,
20855,
57817,... | 1,009 |
# Smallest set such that all arithmetic progression will always contain at least a number in a set
Let $$S= \left\{ 1,2,3,...,100 \right\}$$ be a set of positive integers from $$1$$ to $$100$$. Let $$P$$ be a subset of $$S$$ such that any arithmetic progression of length 10 consisting of numbers in $$S$$ will contain ... | [
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59,
92,
14085,
3... | 2,469 |
# Mutlivariable Calculus: Surface Area
This was a question a students had asked me earlier today regarding surface area.
Find the surface area of the hemisphere $x^2+y^2+z^2 = 4$ bounded below by $z=1$.
I decided to approach this problem using spherical coordinates and got the following
\begin{eqnarray} \int_{0}^{2... | [
2,
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277,
2156,
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12030,
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61,
17,
92952,
61,
1... | 921 |
Tag Archives: factoring
Infinite Ways to an Infinite Geometric Sum
One of my students, K, and I were reviewing Taylor Series last Friday when she asked for a reminder why an infinite geometric series summed to $\displaystyle \frac{g}{1-r}$ for first term g and common ratio r when $\left| r \right| < 1$. I was glad s... | [
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1537,
6602,
979,
1340,
4588,
369,
264,
26528,
3170,
458,
23809,
... | 6,440 |
# Distinguishable objects into distinguishable boxes
How many ways are there to distribute $15$ distinguishable objects into $5$ distinguishable boxes so that the boxes have one, two, three, four, and five objects in them respectively?
$\begin{gather} &\_\_\_ &\_\_\_ &\_\_\_ &\_\_\_ &\_\_\_ \\ &1 &2 &3 &4 &5 \end{gat... | [
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773,
429,
279,
14697,
614,
825,
11,
1378,
11,
2326,
11,
3040,
... | 809 |
GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 17 Aug 2018, 08:09
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customize... | [
20637,
828,
15846,
315,
279,
6059,
481,
13385,
311,
697,
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2011,
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220,
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11,
220,
15,
23,
25,
15,
24,
271,
14374,
19172,
828,
10140,
13385,
48069,
27... | 1,717 |
Evaluate: $\frac{1}{(1+1)!} + \frac{2}{(2+1)!}+...+\frac{n}{(n+1)!}$ using combinatorics.
Evaluate $\frac{1}{(1+1)!} + \frac{2}{(2+1)!}+...+\frac{n}{(n+1)!}$. This is from a combinatorics textbook so I'd like a combinatorial proof. I find doing this kind of problem difficult especially when you have to sum - I don't k... | [
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7,
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10,
16,
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31716,
1667,
3614,
17272,
1211,
382,
82345,
5... | 2,409 |
Since a right kite can be divided into two right triangles, the following metric formulas easily follow from well known properties of right triangles. Okay, so that sounds kind of complicated. The triangle ABD is isosceles. It looks like the kites you see flying up in the sky. A second identifying property of the diago... | [
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13,
576,
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9548,
374,
374,
... | 3,771 |
Fibonacci equality, proving it someway
$F_{2n} = F_n(F_n+2F_{n-1})$
$F_n$ is a nth Fibonacci number. I tried by induction but i didn't get anywhere
-
An answer just by induction on $n$ of the equality $F_{2n}=F_n(F_{n-1}+F_{n+1})$ is as follows:
For $n=2$ we have $3=F_4=(1+3)\cdot 1=(F_1+F_3)F_2$.
To go from $n$ ... | [
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21777,
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352,
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3,
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264,
55129,
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13,
358,
6679,
553,
37056,
71... | 2,219 |
# Solutions to Congruence Modulo 50
1. Jan 12, 2013
### knowLittle
1. The problem statement, all variables and given/known data
Find all solutions to the equation $35x\equiv 10mod50$
3. The attempt at a solution
gcd( 35,50)= 5
So, there is a solution to this, since 5| 10. Also, there is a theorem that guarantees t... | [
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11,
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323,
2661,
14,
5278,
821,
198,
9885,
678,
... | 1,402 |
# Math Help - Complex numbers-finding real number pairs
1. ## Complex numbers-finding real number pairs
Hello,
I am having trouble with this question:
"Find all possible real number pairs p, q such that 3+5i/1+pi =q+4i"
Im sure it's easy but I think I am overlooking something. I multiplied both sides by the conjuga... | [
2,
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481,
22096,
5109,
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1931,
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419,
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9885,
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3204,
1931,
1372,
13530,
281,
1... | 1,545 |
# What is the probability that Jan and Jon are chosen?
Jan , Jon and $10$ other children are in a classroom. The principal of the school walks in and choose $3$ children at random. What is the probability that Jan and Jon are chosen?
My approach:
Including Jon and Jan, number of ways of selection is $1\cdot1\cdot{10... | [
2,
3555,
374,
279,
18927,
429,
4350,
323,
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2906,
22479,
304,
323,
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400,
18,
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2841,
518,
4194,
13,
... | 760 |
# Number of ways to place $k$ non-attacking rooks on a $100\times 100$ chess board
I need to show that the number of ways to place $k$ non-attacking rooks (no two share the same column or row) on a $100\times 100$ chessboard is $k!{100 \choose k}^2$.
When I try to formulate this equation I end up getting ${100 \choos... | [
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400,
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311,
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429,
279,
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315,
5510,
311,
1992,
400,
74,
3,
2... | 1,392 |
# Permutation representation argument validity
#### kalish
##### Member
Hello,
I would like to check if the work I have done for this problem is valid and accurate. Any input would be appreciated. Thank you.
**Problem statement:** Let $G$ be a group of order 150. Let $H$ be a subgroup of $G$ of order 25. Consider th... | [
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... | 2,289 |
# Spring Gun
1. Mar 7, 2014
### Steven60
I have a question about a spring gun. Suppose the barrel of a spring gun is placed horizontally at the edge of a horizontal table. You put say a marble in the barrel and compress the spring x cm and after releasing the marble it travels a horizontal distance of y cm before hi... | [
2,
12252,
21707,
271,
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16,
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279,
20311,
315,
264,
10464,
6038,
374,
9099,
58888,
518,
279,
6821,
315,
... | 792 |
# Is this induction procedure correct? ($2^n<n!$)
I am rather new to mathematical induction. Specially inequalities, as seen here How to use mathematical induction with inequalities?. Thanks to that question, I've been able to solve some of the form $1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \leq \frac{n}{2... | [
2,
2160,
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37056,
10324,
4396,
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17,
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0,
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311,
990,
35972,
37056,
448,
92234,
4607,
11114,
311,
429,
3405,
11,
3... | 1,267 |
### Bfs Time Complexity
Optimality : It is optimal if BFS is used for search and paths have uniform cost. For example, if the heuristic evaluation function is an exact estimator, then A* search algorithm runs in linear time, expanding only those nodes on an optimal solution path. In this tutorial, We are going to lear... | [
14374,
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271,
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318,
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11,
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66609,
16460,
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374,
458,
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11,
1221,
362,
9,
271... | 13,884 |
The Chain Rule is used when we want to differentiate a function that may be regarded as a composition of one or more simpler functions. J'ai constaté que la version homologue française « règle de dérivation en chaîne » ou « règle de la chaîne » est quasiment inconnue des étudiants. While its mechanics appears relatively... | [
785,
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76728,
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1... | 4,304 |
# Kernel of successive powers of a matrix
For any $n \times n$ matrix $A$, is it true that $\ker(A^{n+1}) = \ker(A^{n+2}) = \ker(A^{n+3}) = \dots$ ? If yes, what is the proof and is there a name to this theorem? If not, for what matrices will it be true? How can I find a counterexample in the latter case?
I know that... | [
2,
36603,
315,
48924,
13430,
315,
264,
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271,
2461,
894,
400,
77,
1124,
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10,
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284,
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4346,
47822,
77,
10,
17,
5410,
284,
... | 1,093 |
Relative to the air, the paper is moving downwards, and so there will be an upward resistive force on the paper. https://study.com/academy/lesson/air-resistance-and-free-fall.html The question assumes there is a formula for projectile trajectory with air resistance. Q.1: A plane moving with a velocity of $$50 ms^{-1}$$... | [
28442,
311,
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374,
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905,
14,
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4524,
14,
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14,
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11588,
3924,
9777,
12577,... | 6,672 |
# How does the middle term of a quadratic $ax^2 + bx + c$ influence the graph of $y = x^2$?
Every parabola represented by the equation $$y = ax^2 + bx + c$$ can be obtained by stretching and translating the graph of $$y = x^2$$.
Therefore:
The sign of the leading coefficient, $$-a$$ or $$a$$, determines if the parab... | [
2,
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279,
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7924,
15251,
553,
279,
23606,
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284,
3859,
61,
17,
... | 1,232 |
# Cone shaped related rates of change question
A container is in the shape of a cone of semi-vertical angle $30^\circ$, with it's vertex downwards.
Liquid flows into the container at ${{\sqrt {3\pi } } \over 4}{\rm{ }}c{m^{^3}}/s$
At the instant when the radius of the circular surface of the liquid is 5 cm, find the... | [
2,
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315,
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11936,
91270,
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518,
400,
2... | 1,203 |
1. watchmath
The term of the absolute series is $$|\arctan(1/n)|/n^2<\frac{\pi/2}{n^2}$$. But the series $$\sum 1/n^2$$ is convergent (since it is a p-series with p=2), so the absolute series converges. Then the series is absolutely convergent.
2. anonymous
how did you come up with pi/2/x^2
3. watchmath
the range ... | [
16,
13,
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10374,
271,
785,
4647,
315,
279,
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374,
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15170,
77,
61,
17,
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12947,
1988,
279,
4013,
26107,
59,
124... | 692 |
# Power set of a set with an empty set
When a set has an empty set as an element, e.g.$\{\emptyset, a, b \}$. What is the powerset?
Is it: $$\{ \emptyset, \{ \emptyset \}, \{a\}, \{b\}, \{\emptyset, a\} \{\emptyset, b\}, \{a, b\}, \{\emptyset, a, b\}\}$$
Or
$$\{ \emptyset, \{a\}, \{b\}, \{\emptyset, a\} \{\emptyset... | [
2,
7420,
738,
315,
264,
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448,
458,
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11,
293,
1124,
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12947,
3555,
374,
279,
13430,
295,
1939,
3872,
432,... | 781 |
GMAT Changed on April 16th - Read about the latest changes here
It is currently 26 May 2018, 12:52
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized... | [
20637,
828,
46294,
389,
5813,
220,
16,
21,
339,
481,
4457,
911,
279,
5535,
4344,
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16,
17,
25,
20,
17,
271,
14374,
19172,
828,
10140,
13385,
48069,
271,
... | 4,890 |
# Integration of secant
\begin{align} \int \sec x \, dx &= \int \cos x \left( \frac{1}{\cos^2x} \right) \, dx \\ &= \int \cos x \left( \frac{1}{1-\sin^2x} \right) \, dx \\ & = \int\cos x\cdot\frac{1}{1-\frac{1-\cos2x}{2}} \, dx \\ &= \int \cos x \cdot\frac{2}{1+\cos2x} \, dx \end{align}
I am stuck in here. Any help t... | [
2,
40069,
315,
5701,
517,
271,
59,
7265,
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6612,
92,
1124,
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1124,
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61,
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1124,
11,
... | 1,657 |
But you might be surprised because nature seems to favor a particular numbers like 1, 2, 3, 5, 8, 13, 21 and 34. Lucas Sequences The above work on the Fibonacci sequence can be generalized to discuss any difference equation of the form where and can be any real numbers. The Fibonacci sequence is a series where the next... | [
3983,
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2578,
387,
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311,
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264,
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5109,
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17,
16,
323,
220,
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24811,
2380,
576,
340... | 8,415 |
# What is the dimension of two subspaces?
Let $V$ be the vector space with a basis $x_1, \ldots, x_9$ and $$V_1 = \{(a,a,a,b,b,b,c,c,c): a,b,c \in \mathbb{C}\}, \\ V_2=\{(a,b,c,a,b,c,a,b,c): a,b,c \in \mathbb{C}\}.$$ Then $V_1,V_2$ are subspaces of $V$. What is the dimension of $V_1 \cap V_2$?
I first try to find the... | [
2,
3555,
374,
279,
12871,
315,
1378,
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44285,
1939,
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400,
53,
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26107,
53,
62,
16,
284,
1124,
96065,
64,
15012,
1501... | 1,008 |
# Evaluating a double integral in polar coordinates
#### skatenerd
##### Active member
I've done this problem and I have a feeling it's incorrect. I've never done a problem like this so I am kind of confused on how else to go about doing it. The goal is to change the cartesian integral
$$\int_{-a}^{a}\int_{-\sqrt{a^2... | [
2,
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264,
1990,
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304,
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13934,
271,
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432,
594,
15114,
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1075,
419,
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# What kind of combinatorial problem is this?
Is there a theory from which the following problem comes? Does this type of problem have a name?
Find the largest possible number of $k$-element sets consisting of points from some finite set and have pairwise singleton or empty intersections.
I hope that was clear. If n... | [
2,
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315,
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9885,
279,
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3204,
1372,
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400,
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6101,
7289,
... | 1,558 |
# Math Help - 1+2+3+...+n
1. ## 1+2+3+...+n
Hi, can anyone tell me why
1+2+3+...+n=n(n+1)/2
I can see that it works when I choose a number for n, but I don't really see how I could have come up with it myself.
2. ## Re: 1+2+3+...+n
I think I can explain it like that:
1, 2, 3, 4, 5, 6, ... n
This is an arithmeti... | [
2,
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220,
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21747,
14... | 1,835 |
Question
# Assertion :If $$bc+qr=ca+rp=ab+pq=-1$$, then $$\begin{vmatrix} ap & a & p \\ bq & b & q \\ cr & c & r \end{vmatrix}=0\quad (abc,pqr\neq 0)$$ Reason: If system of equations $${ a }_{ 1 }x+{ b }_{ 1 }y+{ c }_{ 1 }=0,\quad { a }_{ 2 }x+{ b }_{ 2 }y+{ c }_{ 2 }=0,{ \quad a }_{ 3 }x+{ b }_{ 3 }y+{ c }_{ 3 }=0$$ ... | [
14582,
271,
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46730,
549,
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10,
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924,
10,
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92,
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293,
609,
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24984,
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Binary search works on sorted arrays. Finding the Predecessor and Successor Node of a Binary Search Tree All implementation of finding sucessor or predecessor takes O(1) constant space and run O(N) time (when BST is just a degraded linked list) - however, on average, the complexity is O(LogN) where the binary … Repeate... | [
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323,
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506,
8204,
8,
882,
... | 4,477 |
Question
# PQ and RS are two parallel chords of a circle whose centre is O and radius is $$10$$ cm. If PQ $$= 16$$ cm and RS $$= 12$$ cm. Then the distance between PQ and RS, if they lie.(i) on the same side of the centre O and(ii) on the opposite of the centre O are respectively
A
8 cm & 14 cm
B
4 cm & 14 cm
C
2 c... | [
14582,
271,
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323,
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525,
1378,
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55659,
315,
264,
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6693,
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220,
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9961,
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23229,
26107,
28,
220,
16,
17,
... | 758 |
# If matrix product $AB$ is a square, then is $BA$ a square matrix?
## Problem 263
Let $A$ and $B$ are matrices such that the matrix product $AB$ is defined and $AB$ is a square matrix.
Is it true that the matrix product $BA$ is also defined and $BA$ is a square matrix? If it is true, then prove it. If not, find a co... | [
2,
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400,
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374,
264,
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11,
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374,
400,
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400,
33,
3,
525,
35195,
1741,
429,
279,
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1985,
400,
1867,
3,
... | 1,307 |
# Math Help - somebody please teach me to complete the square
1. ## somebody please teach me to complete the square
hello
could you please be that nice to teach me how to complete the square, step by step, i think i understand most of it, except when it comes to factorize, i'm being taught about ellipses and hyperbo... | [
2,
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481,
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752,
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271,
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387,
429,
6419,
311,
4538,
752,
1246,
311,
4583,
279,
9334,
11,
3... | 1,377 |
# Proof by mathematical induction in Z
Is it possible to proof the following by mathematical induction? If yes, how?
$a\in \mathbb{Z} \Rightarrow 3$ | $(a^3-a)$
I should say no, because in my schoolcarrier they always said that mathematical induction is only possible in $\mathbb{N}$. But I never asked some questions... | [
2,
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553,
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304,
1863,
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57,
92,
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220,
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4930,
64,
61,
18,
... | 3,324 |
# Conics Section : Properties
This Note is for Those who love to use Properties and some innovative Techniques while Solving Question of Conics Section ! These Properties are Highly reduce our Calculation and are very useful sometimes , Specially when we have Time Constrained !
So Please Share Properties and Techniqu... | [
2,
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374,
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11831,
525,
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1039,
74216,
323,
525,
1602,
... | 1,626 |
# Having sign problem with algebraic fractions
• Jan 26th 2013, 05:48 PM
KevinShaughnessy
Having sign problem with algebraic fractions
Hi,
I'm having a problem with an algebraic fractions equation. It goes:
5/3(v-1) + (3v-1)/(1-v)(1+v) + 1/2(v+1)
The first thing I do is factor out the negative in the first fraction... | [
2,
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3491,
448,
46876,
292,
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271,
6667,
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7443,
2090,
88,
198,
28032,
1841,
3491,
448,
46876,
292,
64895,
198,
13048... | 1,508 |
# 'If…then…' and '…if…' and '…only if…' and 'If… only then…' statements?
Suppose you have two statements A and B and "If A then B". I am trying to think of what this implies and alternative ways of writing this.
I think
"If A then B"
= A$\rightarrow$B
= "A is sufficient but not necessary for B. B is neither necess... | [
2,
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6,
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498,
614,
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323,
425,
323,
330,
2679,
362,
1... | 1,351 |
Confirm definite integral equals zero $\frac{\sin(x)}{(1-a\cos(x))^{2}}$
Is this statement about the definite integral of a particular function $F$ true? $$\int_0^{2\pi}F(x)\, \mathrm{d}x = \int_0^{2\pi}\frac{\sin(x)}{(1-a\cos(x))^2}\, \mathrm{d}x = 0 \ \text{ for }\ 0<a<1$$
I have evaluated this expression (in Wolfr... | [
16728,
43770,
25098,
16819,
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57960,
37018,
35702,
15940,
2075,
9139,
96065,
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43770,
25098,
315,
264,
3953,
729,
400,
37,
3,
830,
30,
26107,
59,
396,
... | 1,627 |
# knapsack problem optimization
i {\displaystyle v_{i}} 2 The bounded knapsack problem (BKP) removes the restriction that there is only one of each item, but restricts the number In this variation, the weight of knapsack item items numbered from 1 up to , [ ∪ The knapsack problem or rucksack problem is a problem in co... | [
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473,
3491,
25262,
271,
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5493,
3528,
348,
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473,
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320,
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8,
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279,
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429,
1052,
374,
1172,
825,
315,
1817,
1509,
11,
714,
8891,
82,
279,... | 2,103 |
# Is my proof, by strong induction, of for all $n\in\mathbb{N}$, $G_n=3^n-2^n$ correct?
Let the sequence $G_0, G _1, G_2, ...$ be defined recursively as follows:
$G_0=0, G_1=1,$ and $G_n=5G_{n-1}-6G_{n-2}$ for every $n\in\mathbb{N}, n\ge2$.
Prove that for all $n\in\mathbb{N}$, $G_n=3^n-2^n$.
Proof. By strong induct... | [
2,
2160,
847,
11064,
11,
553,
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37056,
11,
315,
369,
678,
400,
77,
59,
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59,
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90,
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400,
38,
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18,
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17,
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3,
4396,
1939,
10061,
279,
8500,
400,
38,
62,
15,
11,
479,
716,
1... | 2,054 |
# applications using rational equations distance rate time answers
Translate the sentence to get the equation. Solve work-rate applications. Work problems often ask us to calculate how long it will take different people working at different speeds to finish a task. While traveling on a river at top speed, he went 10 m... | [
2,
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6010,
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882,
11253,
271,
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279,
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311,
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975,
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13,
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5322,
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601,
311,
11047,
1246,
1293,
432,
686,
1896,
2155,
1251,
3238,
518,
2155,
24722,... | 808 |
# Rate of water level for cone shaped water tank
A water tank in the form of an inverted cone is being emptied at the rate of $6$ ft$^3$/min. The altitude of the cone is $24$ ft, and the base radius is $12$ ft. Find how fast the water level is lowering when the water is $10$ ft deep.
I am not how to do this problem, ... | [
2,
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315,
3015,
2188,
369,
22161,
26271,
3015,
12896,
271,
32,
3015,
12896,
304,
279,
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315,
458,
46701,
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374,
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518,
279,
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315,
400,
21,
3,
10482,
3,
61,
18,
17034,
1065,
13,
576,
35858,
315,
279,
221... | 928 |
GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 25 Sep 2018, 02:33
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You... | [
20637,
828,
15846,
315,
279,
6059,
481,
13385,
311,
697,
14874,
2011,
26,
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271,
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220,
17,
20,
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220,
17,
15,
16,
23,
11,
220,
15,
17,
25,
18,
18,
271,
20637,
828,
10140,
13385,
48069,
271,
130... | 1,591 |
# rank of product of matrices
is a linear combination of the rows of dimension of the linear space spanned by its columns (or rows). Proposition All Rights Reserved. be a the dimension of the space generated by its rows. If , The number of non zero rows is 2 ∴ Rank of A is 2. ρ (A) = 2. such C. Canadian0469. Find a Ba... | [
2,
7077,
315,
1985,
315,
35195,
271,
285,
264,
13482,
10601,
315,
279,
6978,
315,
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568,
86755,
2009,
10512,
15163,
13,
387,
264,
279,
12871,
315,
279,
3550,
79... | 4,250 |
Doing the math to determine the determinant of the matrix, we get, (8) (3)- … If the generated inverse matrix is correct, the output of the below line will be True. You don’t need to use Jupyter to follow along. Yes! Subtract 1.0 * row 1 of A_M from row 3 of A_M, and Subtract 1.0 * row 1 of I_M from row 3 of I_M, 5... | [
83046,
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311,
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279,
86960,
315,
279,
6172,
11,
582,
633,
11,
320,
23,
8,
320,
18,
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279,
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374,
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11,
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315,
279,
3685,
1555,
686,
387,
3007,
13,
1446,
1513,
1405,
1184,
... | 3,480 |
# How do you calculate Arcsin?
Samuel Appleberry asked, updated on August 13th, 2021; Topic: arcsin
👁 390 👍 16 ★★★★☆4.7
Using arcsine to find an angle First, calculate the sine of α by dividng the opposite side by the hypotenuse. This results in sin(α) = a / c = 52 / 60 = 0.8666. Use the inverse function with this ... | [
2,
2585,
653,
498,
11047,
19689,
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1939,
23950,
4000,
8162,
15357,
4588,
11,
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25,
66750,
258,
198,
145633,
220,
18,
24,
15,
61804,
235,
220,
16,
21,
37234,... | 1,248 |
# x^2 = 4 versus x = 4 ^ (1/2)
#### avr5iron
##### New member
Can someone explain why the solution for x in x^2 = 4 is x = 2, -2
while the solution for x in x = 4 ^ (1/2) is 2
#### MarkFL
Staff member
1.) $\displaystyle x^2=4$
Now, using the square root property, we find:
$\displaystyle x=\pm\sqrt{4}=\pm2$
2.) $... | [
2,
856,
61,
17,
284,
220,
19,
19041,
856,
284,
220,
19,
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320,
16,
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81,
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271,
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4462,
198,
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10339,
3170,
279,
6291,
369,
856,
304,
856,
61,
17,
284,
220,
19,
374,
856,
... | 1,410 |
If you're seeing this message, it means we're having trouble loading external resources on our website.
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# 𝘶-substitution warmup
AP.CALC:
FUN‑6 (EU)
,
FUN‑6.D (LO)
,
FUN‑6.D.1 (EK)
Before diving into ou... | [
2679,
498,
2299,
9120,
419,
1943,
11,
432,
3363,
582,
2299,
3432,
12264,
8277,
9250,
4963,
389,
1039,
3910,
382,
2679,
498,
2299,
4815,
264,
3482,
4051,
11,
4486,
1281,
2704,
429,
279,
30476,
31530,
74,
559,
774,
2659,
323,
31530,
628... | 1,316 |
# Every function $f$ can be factored into $f = i \circ s$, with $i$ injective and $s$ surjective
As in the title, if $$X$$ and $$Y$$ are two arbitrary sets and $$f:X \to Y$$, my proof was by taking $$x_1 \sim x_2 \iff f(x_1) = f(x_2),$$ $$s: X \to X/\sim$$ to be the canonical surjection of $$X$$ into the quotient set ... | [
2,
7209,
729,
400,
69,
3,
646,
387,
2097,
3018,
1119,
400,
69,
284,
600,
1124,
43298,
274,
54876,
448,
400,
72,
3,
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533,
323,
400,
82,
3,
1729,
50491,
271,
2121,
304,
279,
2265,
11,
421,
26107,
55,
14085,
323,
26107,
56,
1... | 1,944 |
# Prove $\gcd(nn!, n!+1)=1$
For any $n \in \mathbb{N}$, find $\gcd(n!+1,(n+1)!+1)$. First come up with a conjecture, then prove it.
By testing some values, it seems like $\gcd(n!+1,(n+1)!+1) = 1$
I can simplify what's given to me to $\gcd(nn!, n!+1)=1$ but I can't find out how to get it into the form I want it. Can ... | [
2,
1298,
586,
57960,
91289,
33478,
17142,
308,
0,
10,
16,
11730,
16,
66426,
2461,
894,
400,
77,
1124,
258,
1124,
10374,
6066,
90,
45,
31716,
11,
1477,
57960,
91289,
1445,
0,
10,
16,
12950,
77,
10,
16,
41295,
10,
16,
8,
12947,
5512... | 1,890 |
Is it possible to accomplish calculations of complex numbers specially in polar form with scientific calculators? This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. A calculator to calculate the equivalent impedance of a resistor, a capacitor and and induct... | [
3872,
432,
3204,
311,
22054,
28117,
315,
6351,
5109,
34326,
304,
24660,
1352,
448,
12344,
5812,
2973,
30,
1096,
29952,
1558,
6770,
34784,
389,
6351,
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66249,
23393,
304,
279,
738,
315,
6351,
5109,
13,
362,
29952,
311,
11047,
27... | 3,442 |
# Probability chord of bigger circle intersects smaller circle
You are given two concentric circles $$C_1$$ and $$C_2$$ of radius $$r$$ and $$r/2$$ respectively. What is the probability that a randomly chosen chord of $$C_1$$ will intersect $$C_2$$?
Answer: $$1/2, 1/3$$ or $$1/4$$
The first method I used (gives 1/4)... | [
2,
86639,
43221,
315,
11243,
12671,
88184,
9155,
12671,
271,
2610,
525,
2661,
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26107,
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323,
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62,
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315,
10578,
26107,
81,
14085,
323,
26107,
81,
14,
17,
14085,
15576,
13,
3... | 975 |
# What is the expected number of coin tosses needed to obtain a head?
Due to my recent misunderstandings regarding the 'expected value' concept I decided to post this question. Although I have easily found the answer on the internet I haven't managed to fully understand it.
I understood that the formula for the expec... | [
2,
3555,
374,
279,
3601,
1372,
315,
16254,
25187,
288,
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311,
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264,
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358,
6635,
311,
1736,
419,
3405,
13,
10328,
358,
614,
6707,
1730,
279,
4... | 1,752 |
# Prove that A Real Root Exists in [-1, 1]
#### anemone
##### MHB POTW Director
Staff member
Given $$\displaystyle f(x)=5tx^4+sx^3+3rx^2+qx+p$$ for $f(x)\in R$. If $r+t=-p$, prove that there is a real root for $f(x)=0$ in $[-1,1]$.
#### Ackbach
##### Indicium Physicus
Staff member
Some ideas:
We rewrite the polyno... | [
2,
1298,
586,
429,
362,
8800,
18854,
72426,
304,
10055,
16,
11,
220,
16,
2533,
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458,
336,
603,
271,
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386,
30725,
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54,
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198,
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4462,
198,
22043,
26107,
59,
5493,
3528,
282,
2075,
11730,
20,
3998,
61,
19,
10... | 760 |
# find arc BN=?
#### Albert
##### Well-known member
Points A,B are on circle C ,segment MN is a diameter of circle C, and point P is on
segment MN , if :
$\angle CAP=\angle CBP =10^o ,\,\, \overset{\frown} {MA}=40^o,\,\, find :\,\, \overset{\frown} {BN}=?$
Last edited:
#### HallsofIvy
##### Well-known member
MHB... | [
2,
1477,
15580,
45316,
28,
1939,
820,
17513,
271,
67331,
8325,
21309,
4462,
198,
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362,
8161,
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389,
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356,
1154,
23169,
34995,
374,
264,
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315,
12671,
356,
11,
323,
1459,
393,
374,
389,
271,
23169,
34995,
1154,
421,
... | 773 |
# Factoring a quadratic polynomial (absolute beginner level), are both answers correct?
I'm following video tutorials on factoring quadratic polynomials. So I'm given the polynomial:
$$x^2 + 3x - 10$$
And I'm given the task of finding the values of $a$ and $b$ in:
$$(x + a) (x + b)$$
Obviously the answer is: $$(x ... | [
2,
36712,
5503,
264,
79151,
47311,
320,
17182,
48948,
2188,
701,
525,
2176,
11253,
4396,
1939,
40,
2776,
2701,
2766,
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389,
2097,
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79151,
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76961,
13,
2055,
358,
2776,
2661,
279,
47311,
1447,
14085,
87,
61,
17,
488,
220,
... | 1,209 |