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Set plays a very crucial role in understanding the fundamentals of mathematics. To be precise, a set is an assembling or collection of items. With the concept of set, we can analyse many real life situations and other branches like biology, computer science, engineering etc. in a better way. It consists of set representation, finite and non-finite sets, equal sets, universal set, subset, venn diagrams, Union and intersection, complement of set, etc. It forms the basis for relation and functions. Venn diagrams are pictorial representations of sets as closed curves.
With the knowledge of relations and functions, you can associare different pairs of objects from two sets which are represented as closed - curves. Basically they derive a relation between two objects. Functions are nothing but a special type of relation. This chapter consists of ordered pairs, cartesian product of sets, finding the number of elements, domain, co-domain , Range of functions. Real valued functions like polynomial, signum, etc. and their graphs. Idea of function is very much needed for association of one object to a particular type of object.
In previous class, we have discussed basic identities, trigonometric ratios and their applications in finding the distances between two real life objects. Now we extend the concept of trigonometric ratios to trigonometric functions. In this chapter, we will discuss about measurement of angles, conversion of angles from radian to degree and vice-versa, Definition of trigonometric functions with the help of unit circle, signs of functions in quadrants, domain, range and graph of functions, sum & product formulas, multiple angles formulas. General Solution of trigonometric equations.
This chapter is all about proving the given statement is true or not by the process of induction. We deal with natural numbers because it is the least inductive subset of real numbers. Least inductive means it has the least fixed point for an operation definable by a positive formula for some natural number n. In this chapter, we will discuss the principle of mathematical induction and its simple applications.
Upto now, we are very much familiar with real numbers. But in solving quadratic equations, real numbers have no answer about -1 i.e, square root of negative numbers. Set of real numbers is a subset of complex numbers. Complex numbers give answers to these questions. This chapter consists of algebraic properties of complex numbers, argand plane , polar representation of complex numbers , fundamental theorem of algebra , Solution of quadratic equations in complex number systems. Square root of a complex number.
We are familiar with equations, which we get when a polynomial equates to some constant. But when this equals sign replaced with greater than or less than sign, we get inequalities. Not only in maths, it has some real life applications like how many products should be produced for maximisation of profit and comparison of heights of two persons, etc. This chapter consists of linear inequalities, algebraic solution of linear inequalities in one variable and their representation on number line, Graphical solution of linear inequalities and system of linear inequalities in two variables.
When it comes to counting or arranging the things, we can do it manually but for that data should not be very large. When the data is large then we have to adopt some methods. These methods will be discussed in this chapter. Permutation is a form of arrangement of 'n' items taking 'y' at a time. Combination is the number of ways for choosing 'r' items out of 'n' items. This chapter consists of fundamental principle of counting, factorial (n!), permutation and combination and derivation of their formulae and their connections, also their simple applications.
An algebraic expression containing two terms and connected by (+) & (-) operation is called binomial. When small positive powers are raised to a binomial it can be solved manually. For higher powers it becomes very difficult to solve. But the binomial theorem helps to solve expressions which have large powers. It has many applications like in permutation and combination, probability, etc. The topics which are included in this chapter - proof of binomial theorem for positive integral indices, Pascal's triangle, general and middle term in binomial expansion and its applications.
Sequence is a collection of objects taken one by one in which repetitions may present but order matters. It can have any number of terms. When these terms are added it is called a series. We get some general expressions to solve sequences and series. It solves very tedious calculations which are very difficult to solve manually. This chapter consists of arithmetic progression and mean, geometric progression and mean, general terms, sum of n terms, arithmetic and geometric series, infinite G.P, relation between A.M and G.M, some special series.
In earlier classes, you have read about lines in the Euclid geometry section. That is just an introduction but now we go in depth of lines and its various aspects. Concepts of lines are very essential to know about conics and 2d-3d geometry, which we will discuss later. This chapter consists of basics of 2d geometry, shifting of origin, slope of line, angle between two lines, various forms of equation of line point - slope form slope intercept form, two points form, intercept form, normal form, etc., general equation of line, distance of a point from a line, equation of family of lines.
We are familiar with various concepts of lines, now we have to know about conic section. For various sections of cones like circle, ellipse, parabola, hyperbola etc. we need to go through the details of this chapter. The knowledge which we gain through it has great importance. These figures are related to our real life also like the shape of an egg is elliptical. It will be very interesting to know about them. This chapter has topics such as circle, ellipse, parabola, hyperbola, a point, a straight line, a pair of intersecting lines as a degenerate case of a conic section, standard equations of sections of cone.
After the knowledge of 2d geometry, it is time to add a new dimension to the geometry after that it is called 3d geometry. This is just an introduction but details of this chapter is in the next class. This chapter is for basics. This chapter consists of coordinate axis and coordinate planes in three dimensions, coordinates of a point, distance between two points and section formula.
This chapter is an introduction for a branch called calculus. Calculus is that branch of mathematics which is associated with the study of change in the value of functions as the points of domain change. This chapter will help us to understand differentiation and integration which we study in next class. Topics of this chapter are limit of function introduced as rate of change of distance function and its geometric meaning, definition of derivative, derivatives of sum, difference product and quotient of functions, derivatives of polynomial and trigonometric functions.
If we consider mathematics, there are mainly two types of reasoning - 1) Inductive reasoning, 2.) Deductive reasoning. Inductive reasoning is studied in principle of mathematical induction. Deductive reasoning will be discussed in this chapter. Examples or situations in this chapter mostly related to real life and some with mathematics. This chapter consists of mathematically acceptable statements, connecting words/phrases - "it and only if", "implies”, "and/or", "implied by", "there exists” and their use through variety of examples related to real life and mathematics, difference between contradiction, converse and contrapositive.
This branch of Mathematics deals with a large number of data. When data is large it is very difficult to handle and we can not reach the exact result if we do it manually. Some methods are necessary for this and these methods will be provided in this chapter. Some topics are studied in earlier classes such as 8,9,10. Now, we extend our periphery. This chapter consists of measures of dispersion; mean deviation, variance and standard deviation of ungrouped/grouped data, analysis of frequency distributions with equal means but different variances.
Before happening at any event, we can only think about the possibilities of happening but we cannot be sure about it until it happens. This chapter has always been an interesting topic in Mathematics. We are familiar with the basics of probability in earlier classes. Now some advanced topics will be discussed. These topics are random experiments; outcomes, sample spaces, occurrence of events, 'not', 'and' & 'or’ events, exhaustive events, mutually exclusive events. Probability of an event, probability of 'not', 'and' & ‘or’ events, axiomatic probability.