text stringlengths 172 11.8k | corrupted_text stringlengths 179 11.8k | input_ids sequencelengths 2.05k 2.05k | attention_mask sequencelengths 2.05k 2.05k | labels sequencelengths 2.05k 2.05k |
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[CLS]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 mat... | [CLS]Commutative Property Of Addition 2.” If A is ω n× mm > and O is gave malgebrak More-mat, then weak have: AO &=& O changes that AO is the n×k zero-matrix. Matrix Matrix Multiplication 11:09. We hour 1. two understand the properties of transpose matrix,. needed will take twice matrices A and B which have equal order... | [
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[CLS]# 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 valu... | [CLS]# Comparing the magnitudes of expressions of surds
I recently tackled some space on math!-challenge / maths- Ititude papers where the task was to order various expressions made up find startds (without a calculator,uous).
I found myself wondering Herm I was relying too much on knowing There numerical value of Sl... | [
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[CLS]# 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 ... | [CLS]# In Calculus, how can a function have several defined, yet equal, derivatives)*(
I've been ponderIntegr this question triangle night as I work Thus some problems, and after away very thorough search, Is haven't found anything outcomes related to my question. identity guess i'm also curious (. some derivatives ar... | [
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[CLS]# 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 s... | [CLS]!\ Math Help - working backwards (* cubics
1. ## working backwards - cubics
Write an equation to has typ following roots: 2, -1, 5
Answer key:nx^3 - 6 Excel^2 + 3 quant + 10 = 0
For quadratic equations, I use the sum and product of got, this is a C expressions, how do Its seven this\,\
Thanks.cccoc2bys True ... | [
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[CLS]# 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 wil... | [CLS]# Work Rad time, when work is split into parts
I'm stuck on a particular type of work and time problems.centercccFor ex,
1) A,B,C can complete a work separately in 24These36 and 48 days. They states working together but C polyg after 4 days of start and A left 3 days before completion of test work. In how many |... | [
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[CLS]# 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. Thi... | [CLS]# Algebraic Manipulation
## Definition
Algebraic manipulation involves rearranging variation to make an algebraic expression Att suit your needs` During this rearrangement, the value of the expression does not change.
{\ Technique
Algebra Contin expressions aren't always given in their most convenient forms. T... | [
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"[CLS]Inequality involving sums with binomial coefficient\n\nI am trying to show upper- and lower-bo(...TRUNCATED) | "[CLS]I NOTquality involving sums with binomial coefficient\n\nI am trying too show upper- any lower(...TRUNCATED) | [50281,42,5803,15177,7668,22661,342,47585,10235,187,187,42,717,2820,1512,921,5170,14,667,2406,14,358(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1(...TRUNCATED) |
"[CLS]The main condition of matrix multiplication is that the number of columns of the 1st matrix mu(...TRUNCATED) | "[CLS]The main condition of matrix multiplication is that the number of columns of tell *)st May mus(...TRUNCATED) | [50281,510,2022,1617,273,4315,25219,310,326,253,1180,273,9930,273,2028,9657,296,2552,1364,4503,281,8(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0(...TRUNCATED) |
"[CLS]# Definite Integral: $\\int_0^1\\frac{\\ln^4(x)}{x^2+1}\\,dx$\n\nI'm trying to derive a closed(...TRUNCATED) | "[CLS] ## Definite Integral: $\\int[{0^1\\frac{\\ln^4(x)}{x^2+1}\\,dx$\n\nI'm trying to derive ''re-(...TRUNCATED) | [50281,20263,3366,8234,17712,267,27,669,565,39487,17,63,18,61,1124,464,6677,63,21,9,89,9783,89,63,19(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0(...TRUNCATED) |
"[CLS]# Divisibility Rule for 9\n\nI'm working through an elementary number theory course right now (...TRUNCATED) | "[CLS]# Divisibility Rule for 9 Circ\nI'm working through an elementary number theory course right n(...TRUNCATED) | [50281,4,6852,261,2322,7804,323,898,5819,187,42,1353,2444,949,271,18307,1180,3762,2282,987,1024,285,(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0(...TRUNCATED) |
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