CLASS 11 MATHS FORMULA, Cheat Sheet of Mathematics

Download CLASS 11 MATHS FORMULA and more Cheat Sheet Mathematics in PDF only on Docsity! MATHEMATICS FORMULA FOR IIT JEE & ADVANCED PART — 1 CLASS—11th CHAPTERS COVERED: NAME PAGE NUMBER SETS 3 to 6 RELATION & FUNCTION 7 to 13 TRIGNOMETRY 14 to 18 PRINCIPLE OF MATHEMATICAL INDUCTION(PMI) 19 to 23 COMPLEX NUMBERS 24 to 28 LINEAR INEQUALITIES 29 TO 31 PERMUTATION & COMBINATION 32 TO 34 BINOMIAL THEOREM 35 & 36 Chapter 1- SETS The concept of set serves as a fundamental part of the present day mathematics. Today this concept is being used in almost every branch of mathematics. Sets are used to define the concepts of relations and functions. The study of geometry, sequences, probability, etc. requires the knowledge of sets. The theory of sets was developed by German mathematician Georg Cantor (1845-1918). He first encountered sets while working on “problems on trigonometric series”. In this Chapter, we discuss some basic definitions and operations involving sets. We give below a few examples of sets used particularly in mathematics, N : the set of all natural numbers Z : the set of all integers Q : the set of all rational numbers R : the set of real numbers Z+ : the set of positive integers Q+ : the set of positive rational numbers, and R+ : the set of positive real numbers. Formulas of Sets These are the basic set of formulas from the set theory. If there are two sets P and Q, 1. n(P U Q) represents the number of elements present in one of the sets P or Q. 2. n(P ⋂ Q) represents the number of elements present in both the sets P & Q. 3. n(P U Q) = n(P) + (n(Q) – n (P ⋂ Q) • The union of two sets A and B are said to be contained elements that are either in set A and set B. The union of A and B is denoted as, A∪B. • The intersection of two sets A and B are said to be contained elements that are common in both sets. The intersection of A and B is denoted as, A∩B.  • The complement of a set A is the set of all elements given in the universal set U that are not contained in A. The complement of A is denoted as, A’. • For any two sets A and B, the following holds true: • (A∪B)ʹ=Aʹ∩Bʹ • (A∩B)ʹ=Aʹ∪Bʹ • If the finite sets A and B are given such that, (A∩B)=ϕ, then: n(A∪B)=n(A)+n(B) • If (A∪B)=ϕ, then: n(A∪B)=n(A)+n(B)−n(A∩B) • • Some other important formulas of Sets for any three sets A, B, and C are as follows: 1. A – A = Ø 2. B – A = B⋂ A’ 3. B – A = B – (A⋂B) • Algebra of functions: If the function f : X → R and g : X → R; we have: • (f + g)(x) = f(x) + g(x) ; x ϵ X • (f – g)(x) = f(x) – g(x) • (f . g)(x) = f(x).g(x) • (kf)(x) = k(f(x)) where k is a real number • {f/g}(x) = f(x)/g(x), g(x)≠0 Types of Relation There are 8 main types of relations which include: 1. Empty Relation- There is no relation between any elements of a set. 2. Universal Relation- Every element of the set is related to each other. 3. Identity Relation- In an identity relation, every element of a set is related to itself only. 4. Inverse Relation- Inverse relation is seen when a set has elements that are inverse pairs of another set. 5. Reflexive Relation- In a reflexive relation, every element maps to itself. 6. Symmetric Relation- In asymmetric relation, if a=b is true then b=a is also true. 7. Transitive Relation- For transitive relation, if (x, y) ∈ R, (y, z) ∈ R, then (x, z) ∈ R. 8. Equivalence Relation- A relation that is symmetric, transitive, and reflexive at the same time. Types of Function:  • Identity function: The function defined by y = f (x) = x for each x ∈ R. • Constant function: The function defined by y = f(x) = C, x ∈ R. • Polynomial function: f(x) = anxn   + an-1xn-1  + ….. + a0 • Rational function: These are function of the form p(x)/q(x). Graphs of the Functions All the functions can be plotted on a graph, with input values on the x-axis and their outputs on the y-axis. For example:    Let’s say f(x) = x, where x can be any real number. The graph for this function will look similar to the graph of y = x.  From the condition that was described above, it’s concluded that there cannot be two values of the function at a single value of x.  So, let’s see some examples of functions and non- function. This is a graph of f(x) = √4x In the above graph, the dashed vertical line represents a single value of x and two values of y. That means two values when given a single value of x. This violates one of the properties of the function stated above. So, this is not a function.  Let’s take another example,  In this graph above, there is no value of ‘x’ which gives two different outputs. A vertical dashed line cannot cut the graph at two places. Although a horizontal dashed line cuts the graph at two places, this indicates two inputs mapping to the same output.  • Trigonometric Ratios of Complementary Angles: • sin (90° – θ) = cos θ • cos (90° – θ) = sin θ • tan (90° – θ) = cot θ • cot (90° – θ) = tan θ • sec (90° – θ) = cosec θ • cosec (90° – θ) = sec θ • Periodic Trigonometric Ratios • sin(π/2-θ) = cos θ • cos(π/2-θ) = sin θ • sin(π-θ) = sin θ • cos(π-θ) = -cos θ • sin(π+θ)=-sin θ • cos(π+θ)=-cos θ • sin(2π-θ) = -sin θ • cos(2π-θ) = cos θ • Trigonometric Identities • sin2 θ + cos2 θ = 1 ⇒ sin2 θ = 1 – cos2 θ ⇒ cos2 θ = 1 – sin2 θ • cosec2 θ – cot2 θ = 1 ⇒ cosec2 θ = 1 + cot2 θ ⇒ cot2 θ = cosec2 θ – 1 • sec2 θ – tan2 θ = 1 ⇒ sec2 θ = 1 + tan2 θ ⇒ tan2 θ = sec2 θ – 1 • Product to Sum Formulas • sin x sin y = 1/2 [cos(x–y) − cos(x+y)] • cos x cos y = 1/2[cos(x–y) + cos(x+y)] • sin x cos y = 1/2[sin(x+y) + sin(x−y)] • cos x sin y = 1/2[sin(x+y) – sin(x−y)] • Sum to Product Formulas • sin x + sin y = 2 sin [(x+y)/2] cos [(x-y)/2] • sin x – sin y = 2 cos [(x+y)/2] sin [(x-y)/2] • cos x + cos y = 2 cos [(x+y)/2] cos [(x-y)/2] • cos x – cos y = -2 sin [(x+y)/2] sin [(x-y)/2] • General Trigonometric Formulas: • sin (x+y) = sin x × cos y + cos x × sin y • cos(x+y)=cosx×cosy−sinx×siny • cos(x–y)=cosx×cosy+sinx×siny sin(x–y)=sinx×cosy−cosx×siny • If there are no angles x, y and (x ± y) is an odd multiple of (π / 2); then: • tan (x+y) = tan x + tan y / 1 − tan x tan y • tan (x−y) = tan x − tan y / 1 + tan x tan y • If there are no angles x, y and (x ± y) is an odd multiple of π; then: • cot (x+y) = cot x cot y−1 / cot y + cot x • cot (x−y) = cot x cot y+1 / cot y − cot x • Formulas for twice of the angles: • sin2θ = 2sinθ cosθ = [2tan θ /(1+tan2θ)] • cos2θ = cos2θ–sin2θ = 1–2sin2θ = 2cos2θ– 1= [(1-tan2θ)/(1+tan2θ)] • tan 2θ = (2 tan θ)/(1-tan2θ) • Formulas for thrice of the angles: • sin 3θ = 3sin θ – 4sin 3θ • cos 3θ = 4cos 3θ – 3cos θ • tan 3θ = [3tan θ–tan 3θ]/[1−3tan 2θ] CHAPTER 4- PRINCIPLE OF MATHEMATICAL INDUCTION As the name suggests, the chapter explains the concept of the Principle of Mathematical Induction. The topics discussed are the process to prove the induction and motivating the application taking natural numbers as the least inductive subset of real numbers. One key basis for mathematical thinking is deductive reasoning. In contrast to deduction, inductive reasoning depends on working with different cases and developing a conjecture by observing incidences till we have observed each and every case. Thus, in simple language we can say the word ‘induction’ means the generalisation from particular cases or facts. Below mentioned is the list of some important terms and steps used in the chapter mentioned above: • Statement: A sentence is called a statement if it is either true or false. • Motivation: Motivation is tending to initiate an action. Here Basis step motivate us for mathematical induction. • Principle of Mathematical Induction: The principle of mathematical induction is one such tool that can be used to prove a wide variety of mathematical statements. Each such statement is CHAPTER 5- COMPLEX NUMBERS As the name of the chapter suggests, therefore, this chapter explains the concept of complex numbers and quadratic equations and their properties. The topics discussed are the square root, algebraic properties, argand plane and polar representation of complex numbers, solutions of quadratic equations in the complex number system. A few important points related to the Complex Numbers and Quadratic Equations are as follows: • Complex Numbers: A number that can be expressed in the form a + b is known as the complex number; where a and b are the real numbers and i is the imaginary part of the complex number. • Imaginary Numbers: The square root of a negative real number is called an imaginary number, e.g. √-2, √-5 etc. The quantity √-1 is an imaginary unit and it is denoted by ‘i’ called iota. i = √-1, i2 = -1, i3 = -i, i4 = 1 • Equality of Complex Number: Two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are equal, iff x1 = x2 and y1 = y2 i.e. Re(z1) = Re(z2) and Im(z1) = Im(z2) Algebra of Complex Numbers • Addition: Consider z1 = x1 + iy1 and z2 = x2 + iy2 are any two complex numbers, then their sum is defined as z1 + z2 = (x1 + iy1) + (x2 + iy2) = (x1 + x2) + i (y1 + y2) • Subtraction: Consider z1 = (x1 + iy1) and z2 = (x2 + iy2) are any two complex numbers, then their difference is defined as z1 – z2 = (x1 + iy1) – (x2 + iy2) = (x1 – x2) + i(y1 – y2) • Multiplication: Consider z1 = (x1 + iy1) and z2 = (x2 + iy2) be any two complex numbers, then their multiplication is defined as z1z2 = (x1 + iy1) (x2 + iy2) = (x1x2 – y1y2) + i (x1y2 + x2y1) • Division: Consider z1 = x1 + iy1 and z2 = x2 + iy2 be any two complex numbers, then their division is defined as Conjugate of Complex Number Consider z = x + iy, if ‘i’ is replaced by (-i), then it is called to be conjugate of the complex number z and it is denoted by z¯, i.e. Modulus of a Complex Number Consider z = x + y be a complex number. So, the positive square root of the sum of square of real part and square of imaginary part is called modulus (absolute values) of z and it is denoted by |z| i.e. |z| = √x2+y2 Argand Plane Any complex number z = x + y can be represented geometrically by a point (x, y) in a plane, called argand plane or gaussian plane.  • A pure number x, i.e. (x + 0i) is represented by the point (x, 0) on X-axis. Therefore, X-axis is called real axis. • A purely imaginary number y i.e. (0 + y) is represented by the point (0, y) on the y-axis. Therefore, the y-axis is called the imaginary axis. Argument of a complex Number The angle made by line joining point z to the origin, with the positive direction of X-axis in an anti-clockwise sense is called argument or amplitude of complex number. It is denoted by the symbol arg(z) or amp(z). arg(z) = θ = tan-1(y/x) • Principal Value of Argument • When x > 0 and y > 0 ⇒ arg(z) = θ • When x < 0 and y > 0 ⇒ arg(z) = π – θ • When x < 0 and y < 0 ⇒ arg(z) = -(π – θ) • When x > 0 and y < 0 ⇒ arg(z) = -θ Polar Form of a Complex Number When z = x + iy is a complex number, so z can be written as,  • z = |z| (cosθ + isinθ), where θ = arg(z). • The symbol ≠ means the quantities on the left and right sides are not equal to. Algebraic Solutions for Linear Inequalities in One Variable and its Graphical Representation Using the trial-and-error method, the solution to the linear inequality can be determined. However, this method isn’t always possible, and computing the solution takes longer. So, using a numerical approach, the linear inequality can be solved. When solving linear inequalities, remember to follow these rules: Rule 1: Don’t change the sign of an inequality by adding or subtracting the same integer on both sides of an equation. Rule 2: Add or subtract the same positive integer from both sides of an inequality equation. CHAPTER 7- PERMUTATION & COMBINATION The present chapter explains the concepts of permutation (an arrangement of a number of objects in a definite order) and combination (a collection of the objects irrespective of the order). The topics discussed are the fundamental principle of counting, factorial, permutations, combinations, and their applications along with the concept of restricted permutation. If a certain event occurs in ‘m’ different ways followed by an event that occurs in ‘n’ different ways, then the total number of occurrences of the events can be given in m × n order. Find the important Maths formulas for class 11 Permutations and Combinations are as under: • Factorial: The continued product of first n natural number is called factorial ‘n’. It is denoted by n! which is given by, n! = n(n – 1)(n – 2)… 3 × 2 × 1 and 0! = 1! = 1 • Permutations: Permutation refers to the various arrangements that can be constructed by taking some or all of a set of things. The number of an arrangement of n objects taken r at a time, where 0 < r ≤ n, denoted by nPr is given by nPr = n! / (n−r)! • The number of permutation of n objects of which p1 are of one kind, p2 are of second kind,… pk are of kth kind such that p1 + p2 + p3 + … + pk = n is n! / p1! p2! p3! ….. pk! • Combinations: Combinations are any of the various selections formed by taking some or all of a number of objects, regardless of their arrangement. The number of r objects chosen from a set of n objects is indicated by nCr, and it is given by nCr = n! / r!(n−r)! • Relation Between Permutation and combination: The relationship between the two concepts is given by two theorems as, • nPr = nCr r! when 0 < r ≤ n. • nCr + nCr-1 = n+1Cr