- Logarithms
- Integrations
- Derivatives
- Limits
- Probability
#Credit – NCERT
My Notes:
ICSE XI- Mathematics
Chapters: 16
Chapter1: Sets
Chapter2: Relations & functions
Chapter3: Trignometric functions
Chapter4: Principle of Mathematical Induction
Chapter5: Complex numbers and quadratic equations
Chapter6: Linear inequalities
Chapter7: Permutations & Combinations
Chapter8: Binomial theorem
Chapter9: Sequences & Series
Chapter10: Straight Lines
Chapter11: Conic sections
Chapter12: Introduction to the Three dimensional geometry
Chapter13: Limits & Derivatives
Chapter14: Mathematical reasoning
Chapter15: Statistics
Chapter16: Probability
Chapter17: Infinite Series
Chapter18: Mathematical modeling
Chapter1: Sets
. Theory of sets developed by German mathematician George Cantor(1845-1918) while working on ‘problems on trigonometric series’.
. A set is a well defined collection of objects.
. Sets are denoted by capital letter eg. A, Z, N etc
. Elements of set are denoted by small letter eg. a,b,c
. If a is an element of set A, then
. a belongs to A
. a ϵ A (epsilon – belongs to)
. If a is not an element of set A, then
. a does not belongs to A
. a ∉ A (epsilon not – not belongs to)
. Methods of representation
. Roster / tabular form
. set builder form
. Roster – all even number less than 7 => {2, 4, 6}
. Here order of elements are immaterial
. Same elements are not repeated
. Set builder – V = {x: x is even number and 0 < x < 7}
Q1) Write the solution set of the equation x^2 + x – 2 = 0 in roster form
A1) (x – 1)(x + 2) = 0 i.e. x = 1, – 2
therefore X= {1, -2}
Q2) Write the set {x: x is a positive integer and x^2 < 40} in the roaster form
A2) X = {1, 2, 3, 4, 5,6}
Q3) Write the set A = {1, 4, 9,16,25, …} in builder form
A3) {x: x is a square of natural numbers}
{x: x = n^2, where x ϵ N}
Q4) Write the set {1/2, 2/3, 3/4, 4/5, 5/6, 6/7} in the set builder form.
A4) X = {x: x = n / (n + 1) and n ϵ N and 1 <= n <= 6}
Q5) {P,R,I,N,C,A,L} = { x: x is letter of the word PRINCIPAL}
{0} = {x: x is an integer and x + 1 = 1}
{1,2,3,6,9,18} = {x : x is positive integer and is divisor of 18 }
{3, -3} = {x: x is an integer and x^2 + 9 = 0}
#2 – The empty set
. A set which does not contains any element is called the empty set or the null set or the void set.
. Empty set denoted by Ø or { }
. eg. A = {x: 1 < x < 2, x is natural number}
C = {x: x is an even prime number greater than 2}
#3 – Finite and infinite sets
. Finite set – A = {1,2,3,4} or W = {x: x is day of week}
. Infinite set – B = {men living presently in different part of world}
G = {x: x is point on line}
. A set which is empty or consists of a definite number of elements is called finite otherwise, the set is called infinite.
. Infinite set is written as ending with … eg {1,2,3,…}
. All infinite sets can not be described in the roaster form. (means few can be)
. eg set of real numbers can not be described in roaster form. (set which starts with … and ends …, can not be described in roaster form).
A6) {x: x ϵ N and (x-1)(x-2) = 0} => finite => {1,2}
{x: x ϵ N and x^2 = 4} => finite => {2}
{x: x ϵ N and 2x – 1 = 0} => finite => { }
{x: x ϵ N and x is prime} => Infinite => {1,2,3,5, …}
{x: x ϵ N and x is odd} => Infinite => {1,3,5,7, …}
#4 – Equal sets
. Two sets A and B are said to be equal if they have exactly the same elements and we write A = B. Otherwise, the sets are said to be unequal and we write A ≠ B.
. Equal : A = {1,2,3,4} B = {4,2,3,1} A = B
. Unequal : A = {-1,2,3,4} B = {4,2,3,1} A ≠ B
. A set does not change if an element is repeated
A = {1,2,3} and A={1,2,1,3} are same
A7) A={0}, B={x: x>15 and x<5} = { }, C={x: x-5 = 0 } = {5}
D={x: x^2 = 25} = {-5, 5} E={x: x is an integral positive root of the equation x^2 -2x -15 = 0} = {5}
: C=E
A8) X = {A,L,O,Y}, B = {L,O,Y,A} => X=B
A9)
योगेश कलवार 