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Maths Class 11 Straight Lines Notes are available in free pdf format that you can easily download and access it anywhere and anytime. These Straight Lines Class 11 notes are prepared by subject experts at Vedantu with close to 20 years of experience in teaching Mathematics after reviewing the last year question papers. This helps us to offer short, precise, and productive Class 11 Maths Notes of Straight Lines for the Class 11 students. Download free Maths Class 11Straight Lines Notes pdf with just a single click on the pdf link given below.
In Class 11 Maths Chapter 10 Straight Lines, the basic concepts of lines such as slope, the angle between two lines, different forms of lines, and the distance between lines are described in detail. Class 11 Revision Notes Straight Lines include important topics along with the formula for the students so that they can learn and prepare for the exams accordingly.
These revision notes will surely help students to score well in exams. Read the article below to get further information about Class 11 Chapter A straight line is defined as a line drawn up by the points traveling in a constant direction with zero curvature.
In other words, we can say that the straight line is the shortest distance between two points. The relation between variables such as x and y agrees with all points on the curve.
The general form of the equation of a straight line is given as:. Where, A, B, and C are constants and x, y are variables. The slope of the line is the line whose inclination is not equal to 90 degrees. It is seen that the slope of the x- axis is 0, if the slope of the y-axis is not defined.
The straight - line equation in slope-intercept form is given as:. Where m represents the slope of the line and C is the y-intercept. In general, it follows from these bounds that n d always equals either. The formula.
Conversely, if n p ; d denotes the number of random integers drawn from [1, d ] to obtain a probability p that at least two numbers are the same, then. This is exploited by birthday attacks on cryptographic hash functions and is the reason why a small number of collisions in a hash table are, for all practical purposes, inevitable. The theory behind the birthday problem was used by Zoe Schnabel [14] under the name of capture-recapture statistics to estimate the size of fish population in lakes.
The basic problem considers all trials to be of one "type". The birthday problem Class 10 Maths Ch 12 All Formulas Pdf has been generalized to consider an arbitrary number of types. Shared birthdays between two men or two women do not count. The probability of no shared birthdays here is. A related question is, as people enter a room one at a time, which one is most likely to be the first to have the same birthday as someone already in the room? The answer is 20�if there is a prize for first match, the best position in line is 20th.
In the birthday problem, neither of the two people is chosen in advance. By contrast, the probability q n that someone in a room of n other people has the same birthday as a particular person for example, you is given by. Another generalization is to ask for the probability of finding at least one pair in a group of n people with birthdays within k calendar days of each other, if there are d equally likely birthdays.
Thus in a group of just seven random people, it is more likely than not that two of them will have a birthday within a week of each other.
The expected total number of times a selection will repeat a previous selection as n such integers are chosen equals [17]. In an alternative formulation of the birthday problem, one asks the average number of people required to find a pair with the same birthday.
If we consider the probability function Pr[ n people have at least one shared birthday], this average is determining the mean of the distribution, as opposed to the customary formulation, which asks for the median. The problem is relevant to several hashing algorithms analyzed by Donald Knuth in his book The Art of Computer Programming. An analysis using indicator random variables can provide a simpler but approximate analysis of this problem.
An informal demonstration of the problem can be made from the list of Prime Ministers of Australia , of which there have been 29 as of [update] , in which Paul Keating , the 24th prime minister, and Edmund Barton , the first prime minister, share the same birthday, 18 January.
An analysis of the official squad lists suggested that 16 squads had pairs of players sharing birthdays, and of these 5 squads had two pairs: Argentina, France, Iran, South Korea and Switzerland each had two pairs, and Australia, Bosnia and Herzegovina, Brazil, Cameroon, Colombia, Honduras, Netherlands, Nigeria, Russia, Spain and USA each with one pair. Voracek, Tran and Formann showed that the majority of people markedly overestimate the number of people that is necessary to achieve a given probability of people having the same birthday, and markedly underestimate the probability of people having the same birthday when a specific sample size is given.
The reverse problem is to find, for a fixed probability p , the greatest n for which the probability p n is smaller than the given p , or the smallest n for which the probability p n is greater than the given p. Some values falling outside the bounds have been colored to show that the approximation is not always exact. A related problem is the partition problem , a variant of the knapsack problem from operations research.
Some weights are put on a balance scale ; each weight is an integer number of grams randomly chosen between one gram and one million grams one tonne. The question is whether one can usually that is, with probability close to 1 transfer the weights between the left and right arms to balance the scale. In case the sum of all the weights is an odd number of grams, a discrepancy of one gram is allowed.
If there are only two or three weights, the answer is very clearly no; although there are some combinations which work, the majority of randomly selected combinations of three weights do not.
If there are very many weights, the answer is clearly yes. The question is, how many are just sufficient? That is, what is the number of weights such that it is equally likely for it to be possible to balance them as it is to be impossible? Often, people's intuition is that the answer is above Most people's intuition is that it is in the thousands or tens of thousands, while others feel it should at least be in the hundreds.
The correct answer is The reason is that the correct comparison is to the number of partitions of the weights into left and right. Arthur C. Clarke 's novel A Fall of Moondust , published in , contains a section where the main characters, trapped underground for an indefinite amount of time, are celebrating a birthday and find themselves discussing the validity of the birthday problem. As stated by a physicist passenger: "If you have a group of more than twenty-four people, the odds are better than even that two of them have the same birthday.
These factors tend to increase the chance of identical birth dates, since a denser subset has more possible pairs in the extreme case when everyone was born on three days, there would obviously be many identical birthdays. The problem of a non-uniform number of births occurring during each day of the year was first understood by Murray Klamkin in 10th Class Maths All Formulas Pdf Free Download Unity The reasoning is based on important tools that all students Ch 8 Maths Class 10 All Formulas Queens of mathematics should have ready access to.
The birthday problem used to be a splendid illustration of the advantages of pure thought over mechanical manipulation; the inequalities can be obtained in a minute or two, whereas the multiplications would take much longer, and be much more subject to error, whether the instrument is a pencil or an old-fashioned desk computer.
What calculators do not yield is understanding, or mathematical facility, or a solid basis for more advanced, generalized theories. From Wikipedia, the free encyclopedia. Mathematical problem.
For yearly variation in mortality rates, see birthday effect. For the mathematical brain teaser that was asked in the Math Olympiad, see Cheryl's Birthday. Main article: Birthday attack. Royal Statistical Society. Rouse Ball and H. Selected Papers of Richard von Mises. Providence, Rhode Island: Class 10 Maths All Ch Formulas Data Amer. Michael Cambridge: Cambridge University Press. ISBN June SIAM Review. ISSN JSTOR OCLC The On-line Encyclopedia of Integer Sequences. Retrieved 17 February In Rhee M.
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