- Angle and Its Measurement
- Trigonometric Functions
- Trigonometric Functions of Compound Angles
- Factorization Formulae
- Straight Line
- Circle and Conics (Ellipse, Hyperbola and Parabola)
- Linear Inequations
It is a branch of mathematics. It involves collection, analysis, interpretation, presentation, and organization of data. (Dictionary Definition).
Descriptive Statistics summarizes the data with the help of few indices, such as mean (the central tendency) and standard deviation (the dispersion). Inferential Statistics draws conclusions from data that are subject to random variation. Inferential statistics uses the probability theory.
If the data values are arranged in ascending or descending order, the minimum and maximum values are revealed. The quantity represented by the numerical data is termed as a variate or a variable. Often, data values repeatedly occur in the dataset. The number of times a value occurs in the dataset is termed as the frequency. The representation using frequencies is the frequency distribution.
If the range of the data is wide, instead of mentioning individual values, they are grouped into class intervals. In general, in a class interval of all values and are included.
Sometimes, a cumulative frequency distribution table is prepared.
Histogram, Frequency Polygon and Ogive are commonly used to represent the data.
As mentioned earlier, in descriptive statistics, the central tendency and the dispersion are studied. The indices of central tendency are as follows :
Arithmetic Mean :
If the data are in frequency distribution format,
If the data are in grouped frequency distribution format,
where is the assumed mean, generally the middle value in the dataset. is given by . is the width of class interval and is the mid-value of the class interval.
Joint Mean of 2 distributions with and values is given by
Geometric Mean :
The geometric mean is given by
Harmonic Mean :
Harmonic mean is the reciprocal of arithmetic mean of the reciprocals of the given values.
Median/ Positional Average :
Median is that value of variate, which divides the dataset into 2 equal parts. Thus, equal number of values exist on either side of the median. If the number of values is odd, the arithmetic mean of the two middle terms is the median.
In a cumulative frequency distribution, the median is that value of variate, whose cumulative frequency is equal to or just greater than $N/2$, where $N$ is the total number of values.
It is the most frequently occurring value in the data-set.
These indices measure the spread of the values. So, 2 datasets can have same mean, but the values may be spread over a wider range for one of them.
Mean Deviation :
It is given by
Standard Deviation :
The square of the standard deviation is known as the variance.
Root Mean Square Deviation :
where is any arbitrary number.
and are related by the following relation :
In grouped distributions, the standard deviation is calculated as follows :
Here, is the width of class interval, . is the assumed mean, is the mid-value of the interval. is the sum of all frequencies.
th moment of a distribution about the mean is denoted by and is given by
This is similar to moment of a force about a point, where we define the moment as force the perpendicular distance.
th moment of a distribution about any arbitrary number is denoted by and is given by
It can be shown that and are related by the following equation :
It tells how skew the frequency distribution curve is from the symmetry.
I) Frequency curve stretches towards right : Mean to the right of mode – Right/Positively skewed
II) Frequency curve stretches towards left : Mean to the left of mode – Left/Negatively skewed
Skewness is measured by
where is the median.
The coefficient of skewness is given by
In Greek, means . The coefficient of kurtosis, is given by
It measures how peaked the frequency distribution curve is.
The curve, which is neither flat nor peaked is . If $\beta_2 > 3$, the curve is or peaked. If , the curve is .
When the dataset contains 2 variables, each data point is an ordered pair, say . Such distributions are known as bivariate distributions. We may wish to test the relationship between the 2 variables, if any.
Examples : Amount of time spent by students on social media vs. marks obtained, rainfall in a year and the crop production in the following year.
We may be tempted to come up with a conclusion without having a look at the actual values. The variables in the first example may seem to be correlated, but we cannot be very much sure unless we have the dataset.
In the second example, we can provide a good reasoning for the correlation.
When a change in one variable leads to a corresponding change in the other, we say that the variables are correlated. If one increases and the other decreases, it is a negative correlation. If both increase, it is a positive correlation.
If the ratio of values of variables in each pair is constant, the correlation is said to be linear. According to Karl Pearson, the strength of linear relationship between the variables is given by the correlation coefficient :
is the covariance. When the covariance is divided by the product of the standard deviations, we get the correlation coefficient.
Note that the value of always lies between and .
If the variables are known to be correlated, value of one variable can be obtained, if the value of the other variable is known. This is known as regression. In order to do that, a relationship between the variables, say and is developed, in the form of an equation. Further, if it is known that the relationship is linear, the equation will be of the form
I) Line of Regression of on
II) Line of Regression of on
I) of on ,
I) of on ,
Almost all MCQs are formula-based. So, make sure that you know all formulas with the terms forming them.
The set theory, developed by George Cantor, forms a basis for advanced mathematical topics, such as the calculus, real and complex analysis etc. It defines functions, which are an extremely important entity in mathematics.
A set is a collection of well defined objects. For example, the set of all subjects studied by you in class 10. This includes English, Marathi, Hindi, French, Mathematics, Science and Social Science.
The collection of all intelligent people in your locality does not qualify as a set. The intelligence cannot be quantied and therefore, it is not a well-defined object. The set of all prime numbers less than 15 is another example. This set includes 2,3,5,7,11 and 13.
The objects which belong to a set are called elements of that set. The repetition of elements is NOT allowed in set theory. So, if there are 2 elements, which are identical, they will be treated as a single element.
A set is represented by capital Roman alphabets, such as A, P etc. It is customary to write down the elements of set within curly brackets. So, the set of all prime numbers less than 15 will be written as (let us call the set as P)
This representation of a set, where we list down all the elements, is known as the Roster form. The same set can also be written in Set-builder form as follows:
The set of natural numbers, also known as counting numbers, is denoted by
The set of whole numbers is denoted by
Thus, is made up of the number zero as well as .
The set of integers is denoted by
The next set of numbers is the set of rational numbers. These numbers can be represented in the form , where and are any integers and . It is denoted by .
There are numbers, such as , which cannot be represented in the form . Such numbers are known as irrational numbers. The set of irrational numbers is denoted by .
When the set of rational numbers and the set of irrational numbers are combined together, the set of real numbers, is obtained.
When all the elements of set and set are identical, we say that and set are equal. So,
In roster form, equal sets are identical; but their set-builder forms may differ. Note that they are one and the same.
An empty set (or a null set) is the one, which contains no element. It is denoted by
For example, .
A set with only 1 element in it is known as singleton set. For example, . This contains only 1 element, which is .
Set is said to be a subset of set , when all elements in set are contained in set . Set is then called the superset of . We denote this by,
For example, . Notice that contains all elements which are there in .
A non-empty set is said to be a proper subset of a set , such that there exists at least one element in , which does not belong to . i.e.
A set, which has a finite number of elements is called a finite set. e.g.
A set, where the process of counting elements does not end is known as an infinite set. e.g. the set of natural numbers, .
A universal set is the one, which in a discussion, serves as a parent set. In other words, all other sets are subsets of this set. Note that, for all kinds of numbers we discussed earlier, the set of real numbers, is the universal set. Universal set is generally denoted by .
Union of 2 sets and is the set of all elements, which are present either in or in or in both and . It is denoted by
Intersection of 2 sets and is the set of all elements, which are common to both A and B. It is denoted by
Let be any set. The complement of set is the set of all elements, which do not belong to ; but are contained in the universal set . It is denoted by or .So,
The power set of a set is the set of ALL subsets of , including the null set and itself. So, if , the power set will be
Note that if a set has elements, its power set will have elements. So, power set of A will have elements.
A Venn diagram of a set is the pictorial representation of the set. Generally, Venn diagrams are geometrical figures, such as circle, triangle, ellipse etc. The operations on set can be represented readily using these diagrams.
We use the words sequence and series interchangeably and we generally mean a continuation of numbers/events by them. In mathematics, however, these words have a particular meaning.
A sequence (also known as progression) is an arrangement of numbers in a specific order, in such a way that a definite relation exists between the numbers and their positions.
A series is what one gets, when all the terms of a sequence are added.
The numbers which make a sequence are known as the terms. The th term is generally denoted by .
Summing up the terms of the sequence , we get a series .
It is easy to prove that .
1) The investment on compound interest grows annually forming a geometric sequence.
2) Population models of bacteria (2 from 1, 4 from 2 etc.) follow a geometric sequence.
3) Difficulty level of problems increases in such a way that each problem takes 5 minutes more than the previous one to be solved. This follows an arithmetic sequence.
4) The harmonic sequence diverges to , and is a fundamental result in mathematics. It has got applications in estimating traffic and in destroying noise!
Terms of a sequence form a function defined from , the set of natural numbers to , the set containing the terms in the sequence; e.g. consider the sequence . Clearly, .
Arithmetic Progression is a sequence for which is constant . The constant difference , , is known as the common difference and generally, the first term is denoted by .
So, th term will be and sum of first terms will be
Let be a non-zero number. Let be an A.P. Then,
I) is also an A.P. with first term and common difference
II) is also an A.P. with first term and common difference .
Geometric Progression is a sequence for which is constant . The constant ratio , , is known as the common ratio.
If is the first term of a G.P., then the th term is and the sum of first terms is given by
Let be a non-zero number. Let be a G.P. Then,
I) is also a G.P. with first term and common ratio
II) is also a G.P. with first term and common ratio .
Having studied limits, we can show that if , then sum of all terms of a G.P. (up to infinity) is given by
In other words,
Harmonic Progression is a sequence such that reciprocals of its terms are in arithmetic progression. Thus, if is a harmonic progression, then
is an A.P.
Arithmetic mean, , of and is given by . It can be shown that
are in A.P.
Geometric mean,, of and is given by . It can be shown that
are in G.P.
The harmonic mean, , of and is given by . Thus,
are in A.P.
It can be shown that
An arithmetico-geometric progression is a combination of an A.P. and a G.P. It is of the form
Sum of first terms of such a progression is given by
Let there be a sequence . The sum of these $m$ terms is briefly written as
I) Sum of first natural numbers is given by
II) Sum of squares of first natural numbers is given by
III) Sum of cubes of first natural numbers is given by
I) The exponential series is an infinite series given by
II) For ,
NOTE: We are going to use the section formula to derive the equation of hyperbola, as we did for ellipse. Since there are 2 variants of the formula (external and internal division), we get 2 foci and 2 directrices.
A hyperbola is a conic section, whose eccentricity is greater than 1. In other words, if is a focus and is the directrix, then for any point ,
The standard equation of hyperbola is
I) The black lines on a basketball or the red lines on the baseball
II) Orbits of comets around the Sun (or any star)
III) Interference patterns by 2 circular waves
IV) Potato chips (:P)
V) Cooling towers in an industrial plant
NOTE: We are going to use the section formula to derive the equation of ellipse. Since there are 2 variants of the formula (external and internal division), we get 2 foci and 2 directrices.
An ellipse is a conic section, whose eccentricity is less than 1. In other words, if is a focus and is the directrix, then for any point ,
The standard equation of ellipse is
I) Earth’s orbit around Sun, with Sun as one of the foci of the ellipse
II) The rugby ball
III) Batman logo (boundary)
IV) Eggs, lemons
V) Whispering galleries
The word parabola originated as a combination of 2 Greek words, para, which means besides and bole, which means throw. When any heavy object is thrown in air, its trajectory is a parabola. Hence the name.
The eccentricity of parabola, is . Let be the focus and be the directrix. According to the focus-directrix property,
where is any point on the parabola. Using this condition for locus of a point, we get the standard equation of parabola,
I) Shape of satellite dish
II) Automobile Headlights (The dim-dip feature)
III) McDonald’s Arches
IV) Mirror Furnace (capable of producing temperatures up to from rays of the Sun)