url stringlengths 13 1.98k | text stringlengths 100 932k | date timestamp[s] | meta dict | prompt_college stringlengths 821 1.72k | prompt_grade_school stringlengths 748 1.65k |
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http://shop.track-it.nz/6bowhhs1/properties-of-matrix-addition-346a66 | 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... | 2021-09-28T02:16:51 | {
"domain": "track-it.nz",
"url": "http://shop.track-it.nz/6bowhhs1/properties-of-matrix-addition-346a66",
"openwebmath_score": 0.7671064734458923,
"openwebmath_perplexity": 360.7558924577382,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_q1_score": 0.9914225156000334,
"lm_q2_score":... | Write an educational piece in Arabic suited for college students related to the following text snippet:
"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 proper... | Here's an extract from a webpage:
"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... |
https://math.stackexchange.com/questions/2922644/comparing-the-magnitudes-of-expressions-of-surds | # 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 ... | 2021-10-19T03:38:04 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2922644/comparing-the-magnitudes-of-expressions-of-surds",
"openwebmath_score": 0.7826164364814758,
"openwebmath_perplexity": 198.0380721381055,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9905... | Write an educational piece in Arabic suited for college students related to the following text snippet:
"# 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... | Here's an extract from a webpage:
"# 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... |
https://math.stackexchange.com/questions/1693964/in-calculus-how-can-a-function-have-several-different-yet-equal-derivatives | # 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... | 2019-04-24T06:18:11 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1693964/in-calculus-how-can-a-function-have-several-different-yet-equal-derivatives",
"openwebmath_score": 0.8208966851234436,
"openwebmath_perplexity": 249.28708375250756,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. Y... | Write an educational piece in Arabic suited for college students related to the following text snippet:
"# 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 anyth... | Here's an extract from a webpage:
"# 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 so... |
http://mathhelpforum.com/algebra/8958-working-backwards-cubics.html | # 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... | 2014-09-19T23:58:39 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/algebra/8958-working-backwards-cubics.html",
"openwebmath_score": 0.8881317973136902,
"openwebmath_perplexity": 861.8028331615614,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9914225133191064,
"lm_q2_sco... | Write an educational piece in Arabic suited for college students related to the following text snippet:
"# 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 ... | Here's an extract from a webpage:
"# 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?
Than... |
https://math.stackexchange.com/questions/202052/work-and-time-when-work-is-split-into-parts | # 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... | 2019-04-26T15:57:27 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/202052/work-and-time-when-work-is-split-into-parts",
"openwebmath_score": 0.6277446150779724,
"openwebmath_perplexity": 439.837859763646,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. Yes\n2. Yes",
"lm_q1_score": 0.99058741263... | Write an educational piece in Arabic suited for college students related to the following text snippet:
"# 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 bu... | Here's an extract from a webpage:
"# 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 ... |
https://brilliant.org/discussions/thread/algebraic-manipulation/ | # 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 ... | 2018-04-22T10:22:09 | {
"domain": "brilliant.org",
"url": "https://brilliant.org/discussions/thread/algebraic-manipulation/",
"openwebmath_score": 1.0000100135803223,
"openwebmath_perplexity": 2798.1191500231757,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_q1_score": 0.9899864287859482,
"lm_q2_score": 0... | Write an educational piece in Arabic suited for college students related to the following text snippet:
"# 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 ... | Here's an extract from a webpage:
"# 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 th... |
https://math.stackexchange.com/questions/4454637/inequality-involving-sums-with-binomial-coefficient | 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 $$i\in\{0, \dots n\}... | 2022-06-26T11:34:07 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/4454637/inequality-involving-sums-with-binomial-coefficient",
"openwebmath_score": 0.9992073178291321,
"openwebmath_perplexity": 304.1227486557242,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9... | Write an educational piece in Arabic suited for college students related to the following text snippet:
"Inequality involving sums with binomial coefficient
I am trying to show upper- and lower-bounds on
$$\frac{1}{2^n}\sum_{i=0}^n\binom{n}{i}\min(i, n-i)$$
(where $$n\geq 1$$) in order to show that it basically grow... | Here's an extract from a webpage:
"Inequality involving sums with binomial coefficient
I am trying to show upper- and lower-bounds on
$$\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-... |
http://fileppi.com/mjau20u/product-of-matrix-5feeab | 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... | 2022-07-02T14:42:21 | {
"domain": "fileppi.com",
"url": "http://fileppi.com/mjau20u/product-of-matrix-5feeab",
"openwebmath_score": 0.8542640209197998,
"openwebmath_perplexity": 590.7593543781824,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. Yes\n2. Yes",
"lm_q1_score": 0.990731985524492,
"lm_q2_score": 0.9372107878954105,
... | Write an educational piece in Arabic suited for college students related to the following text snippet:
"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\tim... | Here's an extract from a webpage:
"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... |
https://math.stackexchange.com/questions/2893117/definite-integral-int-01-frac-ln4xx21-dx | # 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
$$I=... | 2019-08-24T09:30:21 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2893117/definite-integral-int-01-frac-ln4xx21-dx",
"openwebmath_score": 0.9958059787750244,
"openwebmath_perplexity": 907.547974804488,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9888419667789... | Write an educational piece in Arabic suited for college students related to the following text snippet:
"# Definite Integral: $\int_0^1\frac{\ln^4(x)}{x^2+1}\,dx$
I'm trying to derive a closed-form expression for
$$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_... | Here's an extract from a webpage:
"# 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 mani... |
https://math.stackexchange.com/questions/1217594/divisibility-rule-for-9 | # 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... | 2020-01-26T03:01:52 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1217594/divisibility-rule-for-9",
"openwebmath_score": 0.834643542766571,
"openwebmath_perplexity": 168.80065897347535,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9875683469514965,
"lm_q2_sc... | Write an educational piece in Arabic suited for college students related to the following text snippet:
"# 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 bas... | Here's an extract from a webpage:
"# 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.
Pro... |
https://mathhelpboards.com/threads/what-is-the-remainder-when-a_-2013-is-divided-by-7.5195/ | # 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... | 2021-06-13T11:35:22 | {
"domain": "mathhelpboards.com",
"url": "https://mathhelpboards.com/threads/what-is-the-remainder-when-a_-2013-is-divided-by-7.5195/",
"openwebmath_score": 0.9229724407196045,
"openwebmath_perplexity": 998.2764884749569,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.987568346... | Write an educational piece in Arabic suited for college students related to the following text snippet:
"# 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 $$\displaystyle a_0=a_1=a... | Here's an extract from a webpage:
"# What is the remainder when a_{2013} is divided by 7?
#### anemone
##### MHB POTW Director
Staff member
Consider a sequence given by $$\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 b... |
http://nizm.bwni.pw/interval-of-convergence-alternating-series.html | 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... | 2019-12-08T10:57:03 | {
"domain": "bwni.pw",
"url": "http://nizm.bwni.pw/interval-of-convergence-alternating-series.html",
"openwebmath_score": 0.9055711030960083,
"openwebmath_perplexity": 325.46529907420165,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. Yes\n2. Yes",
"lm_q1_score": 0.9911526449170697,
"lm_q2_score": 0.931462... | Write an educational piece in Arabic suited for college students related to the following text snippet:
"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 ... | Here's an extract from a webpage:
"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 converg... |
http://mathhelpforum.com/trigonometry/36050-trigonometry-three-dimensional-question.html | # 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... | 2014-09-03T02:41:40 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/trigonometry/36050-trigonometry-three-dimensional-question.html",
"openwebmath_score": 0.8148919343948364,
"openwebmath_perplexity": 1087.1382273861045,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.987758722546... | Write an educational piece in Arabic suited for college students related to the following text snippet:
"# 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 f... | Here's an extract from a webpage:
"# 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 g... |
https://brilliant.org/discussions/thread/golden-ratio-and-fibonacci-numbers/ | # 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$$ then we have $\varphi =\... | 2018-11-18T00:53:39 | {
"domain": "brilliant.org",
"url": "https://brilliant.org/discussions/thread/golden-ratio-and-fibonacci-numbers/",
"openwebmath_score": 0.9912343621253967,
"openwebmath_perplexity": 1404.0567632852903,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_q1_score": 0.9928785708328176,
"lm_... | Write an educational piece in Arabic suited for college students related to the following text snippet:
"# Golden Ratio and Fibonacci Numbers
Golden Ratio is considered to be one of the greatest beauties in mathematics. Two numbers $$a$$ and $$b$$ are said to be in Golden Ratio if $a>b>0,\quad and\quad \frac { a }{ b ... | Here's an extract from a webpage:
"# Golden Ratio and Fibonacci Numbers
Golden Ratio is considered to be one of the greatest beauties in mathematics. Two numbers $$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 $... |
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