Principal of mathematical induction

Q1. n(n+1)(n+5) is a multiple of 3.

Proof:-

 We will prove it by using the formula of mathematical induction for all n ϵ N

Let P(n)=n(n+1)(n+5)=3d where d ϵ N

For n=1

P(1)=1(2)(6)=12 which is divisible by 3

Let P(k) is true 

P(k)=k(k+1)(k+5)=3m where m ϵ N

Since P(k+1) is true whenever P(k) is true.

So, by the principle of induction, P(n) is divisible by 3 for all n ϵ N

1.1 PROGRESSIONS :

 

Those sequence whose terms follow certain patterns are called progression. Generally there are three types of progression.

(i) Arithmetic Progression (A.P.)

(ii) Geometric Progression (G.P.) 

(iii) Harmonic Progression (H.P.)

 1.2 ARTHMETIC PROGRESSION :

A sequence is called an A.P., if the difference of a term and the previous term is always same. i.e. 

d = tn+1 – tn

= Constant for all

nN

. The constant difference, generally denoted by ‘d’ is called the common difference.

Ex.1 Find the common difference of the following A.P. : 1,4,7,10,13,16 ......

Sol. 4 - 1 = 7 - 4 = 10 - 7 = 13 - 10 = 16 - 13 = 3 (constant).

 Common difference (d) = 3.

6.3 GENERAL FORM OF AN A.P. :

If we denote the starting number i.e. the 1st number by ‘a’ and a fixed number to the added is ‘d’ then a, a +

d, a + 2d, a + 3d, a + 4d, ...... forms an A.P.

Ex.2 Find the A.P. whose 1st term is 10 & common difference is 5.

Sol. Given : First term (a) = 10 & Common difference (d) = 5.

 A.P. is 10, 15, 20, 25, 30, .....

6.4 nth TERM OF AN A.P. :

Let A.P. be a, a + d, a + 2d, a + 3d, .....

Then, First term (a1

) = a + 0.d

Second term (a2

) = a + 1.d

Third term (a3

) = a + 2.d

. .

. .

. .

nth term (an) = a + (n - 1) d

 an = a + (n - 1) d is called the nth term.

Ex.3 Determine the A.P. whose their term is 16 and the difference of 5th term from 7th term is 12.

Sol. Given : a3

= a + (3 - 1) d = a + 2d = 16 .....(i)

a7

- a5

= 12 ....(ii)

(a + 6d) - (a + 4d) = 12

a + 6d - a - 4d = 12

2d = 12

d = 6

Put d = 6 in equation (i)

a = 16 - 12

a = 4

 A.P. is 4, 10, 16, 22, 28, ......

problems on probability

 OBJECTIVE 

1. If there coins are tossed simultaneously, then the probability of getting at least two heads, is

(A) 1/4 (B) 3/8 (C) 1/2 (D) 1/4

2. A bag contains three green marbles four blue marbles, and two orange marbles. If marble is picked at random, then the probability that it is not a orange marble is

(A) 1/4  (B) 1/3 (C) 4/9  (D) 7/9

3. A number is selected from number 1 to 27. The probability that it is prime is

(A) 2/3  (B) 1/6 (C) 1/3 (D) 2/9

4. IF (P(E) = 0.05, then P (not E) =

(A) -0.05 (B) 0.5 (C) 0.9 (D) 0.95

5. A bulb is taken out at random from a box of 600 electric bulbs that contains 12 defective bulbs. Then the probability of a non-defective bulb is

(A) 0.02 (B) 0.98 (C) 0.50 (D) None

SUBJECTIVE 

1. To dice are thrown simultaneously. Find the probability of getting :

(i) An even number of the sum

(ii) The sum as a prime number

(iii) A total of at least 10

(iv) A multiple of 2 on one dice and a multiple of 3 on the other.

2. Find the probability that a leap year selected at random will contain 53 Tuesdays.

3. A bag contains 12 balls out of which x are white.(i) If one ball is drawn at random, what is the probability it will be a white ball ? (ii) If 6 more white balls are put in the box. The probability of drawing a white ball will be double than that is (i). Find x.

4. In a class, there are 18 girls and 16 boys. The class teacher wants to choose one pupil for class monitor. What she does, she writes the name of each pupil a card and puts them into a basket and mixes thoroughly. A child is asked to pick one card from the basket. What is the probability that the name written on the card is: (i) The name of a girl (ii) The name of boy ?

5. The probability of selecting a green marble at random from a jar that contains only green, white and yellow marbles is 1/4. The probability of selecting a white marble from the same jar is 1/3. If this jar contains 10 yellow marbles. What is the total number of marbles in the jar ?

6. A card is drawn at random from a well suffled desk of playing cards. Find the probability that the card drawn is (i) A card of spade or an ace (ii) A red king (iii) Neither a king nor a queen (iv) Either a king or a queen

7. There are 30 cards of same size in a bag on which number 1 to 30 are written. One card is taken out of the bag at random. Find the probability that the number of the selected card is not divisible by 3.

8. In figure points A,B,C and D are the centers of four circles that each have a radius of length on unit. If a point is selected at random from the interior of square ABCD. What is the probability that the point will be chosen from the shaded region ?

9. A bag contains 5 white balls, 6 red balls, 6 black balls and 8 green balls. One ball is drawn at random from the bag. Find the probability that the ball drawn is (i) White (ii) Red or black (iii) Note green (iv) Neither white nor black  [CBSE - 2006]

10. A bag contains 4 red and 6 black balls. A ball is taken out of the bag at random. Find the probability of getting a black ball. [CBSE - 2008]

11. Cards. marked with number 5 to 50, are placed in a box and mixed thoroughly. A card is drawn from the box at random. Find the probability that the number on the taken out card is (i) a prime number less than 10. (ii) a number which is a perfect square.

#jee #math #probability 

Probability

1.1 EXPERIMENT : The word experiment means an operation, which can produce well defined outcomes. 

The are two types of experiment : 

                   (i) Deterministic experiment 

                    (ii) Probabilistic or Random experiment 

(i) Deterministic Experiment : Those experiment which when repeated under identical conditions, produced the same results or outcome are known as deterministic experiment. For example, Physics or Chemistry experiments performed under identical conditions.

 (ii) Probabilistic or Random Experiment :- In an experiment, when repeated under identical conditions donot produce the same outcomes every time. For example, in tossing a coin, one is not sure that if a head or tail will be obtained. So it is a random experiment. 

Sample space : The set of all possible out comes of a random experiment is called a sample space associated with it and is generally denoted by S. 

For example, When a dice is tossed then S = {1, 2, 3, 4, 5, 6}.

 Event : A subset of sample space associated with a random experiment is called an event. For example, In tossing a dive getting an even no is an event. 

Favorable Event : Let S be a sample space associated with a random experiment and A be event associated with the random experiment. The elementary events belonging to A are know as favorable events to the event A. 

For example, in throwing a pair of dive, A is defined by “Getting 8 as the sum”. Then following elementary events are as out comes : (2, 6), (3, 5), (4, 4) (5, 3) (6, 2). So, there are 5 elementary events favorable to event A. 

1.2 PROBABILITY : If there are n elementary events associated with a random experiment and m of them are favorable to an event A, then the probability of happening or occurrence of event A is denoted by P(A)

 Thus,

    

P(A)     =   (Total number of favourable outcomes)/(Total number of possible outcomes)

                   =  m/n 

And 0 £ P(A) £ 1

 If, P(A) = 0, then A is called impossible event

If, P(A) = 1, then A is called sure event

                                P(A) + P (A") = 1

Where P(A) = probability of occurrence of A. 

            P (A') = probability of non - occurrence of A.

 

Ex.1 A box contains 5 red balls, 4 green balls and 7 white balls. A ball is drawn at random from the box. Find the probability that the ball drawn is (i) white (ii) neither red nor white

Sol. Total number of balls in the bag = 5 + 4 + 7 = 16

\ Total number of elementary events =16

(i)                There are 7 white balls in the bag.

\ Favorable number of elementary events = 7

P(Getting a white ball ) = (Total No. of elementaryevents) /(Total No. favourable elementaryevents)

                    = 7 /16

 (ii) There are 4 balls that are neither red nor white 

 Favorable number of elementary events = 4 

Hence, P (Getting neither red not white ball) = 4/16 =1/4

 Ex.2 All the three face cards of spades are removed from a well-shuffled pack of 52 cards. A card is then drawn at random from the remaining pack. Find the probability of getting  [CBSE - 2007] (i) black face card                     (ii) a queen             (iii) a black card.

 Sol. After removing three face cards of spades (king, queen, jack) from a deck of 52 playing cards, there are 49 cards left in the pack. 

Out of these 49 cards one card can be chosen in 49 ways. 

Therefore, Total number of elementary events = 49 

(i) There are 6 black face cards out of which 3 face cards of spades are already removed. 

So, out of remaining 3 black face cards one black face card ban be chosen in 3 ways. 

Therefore, Favorable number of elementary events = 3 

Hence, P (Getting a black face card ) = 3/49

 (ii) There are 3 queens in the remaining 49 cards.

 So, out of these three queens, on queen can be chosen in 3 ways 

Therefore, Favorable number of elementary events = 3

 Hence P (Getting a queen) = 3/49

 (iii) There are 23 black cards in the remaining 49 cards, 

So, out to these 23 black card, one black card can be chosen in 23 ways 

Therefore, Favorable number of elementary events = 23 

Hence, P (Getting a black card) =  23/49

 Ex.3 A die is thrown, Find the probability of (i) prime number (ii) multiple of 2 or 3 (iii) a number greater than 3.

Sol.

In a single throw of die any one of six numbers 1,2,3,4,5,6 can be obtained. 

Therefore, the tome number of elementary events associated with the random experiment of throwing a die is 6.

 (i) Let A denote the event “Getting a prime no”. 

Clearly, event A occurs if any one of 2,3,5 comes as out come. 

Therefore, Favorable number of elementary events = 3 

Hence, P (Getting a prime no.) = 3/6=1/2

(ii) An multiple of 2 or 3 is obtained if we obtain one of the numbers 2,3,4,6 as out comes.

Therefore, Favorable number of elementary events = 4 

Hence, P (Getting multiple of 2 or 3) = 4/6=2/3

 (iii) The event “Getting a number greater than 3” will occur, if we obtain one of number 4,5,6 as an out come. 

Therefore, Favorable number of out comes = 3

 Hence, required probability = 3/6=1/2

 Ex.4 Two unbiased coins are tossed simultaneously. Find the probability of getting (i) two heads (ii) at least one head (iii) at most one head. 

Sol. 

If two unbiased coins are tossed simultaneously, 

we obtain any one of the following as an out come : HH, HT, TH, TT 

Therefore, Total number of elementary events = 4

 (i) Two heads are obtained if elementary event HH occurs. 

Therefore, Favorable number of events = 1 Hence, P (Two heads) =  1/4

 (ii) At least one head is obtained if any one of the following elementary events happen : HH, HT, TH 

Therefore, favorable number of events = 3 

Hence P (At least one head) = 3/4

 (iii) If one of the elementary events HT, TH, TT occurs, than at most one head is obtained 

Therefore, favorable number of events = 3 

Hence, P (At most one head) = 3/3

Ex.5 A box contains 20 balls bearing numbers, 1,2,3,4…...20. A ball is drawn at random from the box. What is the probability that the number of the ball is (i) an odd number (ii) divisible by 2 or 3 (iii) prime number 

Sol. Here, total numbers are 20. 

 Total number of elementary events = 20 

(i) The number selected will be odd number, if it is elected from 1,3,5,7,9,11,13,15,17,19 

 Therefore, Favorable number of elementary events = 10 

Hence, P (An odd number ) = 10/20=1/2

 (ii) Number divisible by 2 or 3 are 2,3,4,6,8,9,10,12,14,15,16,18,20 

Therefore, Favorable number of elementary events = 13

 P (Number divisible by 2 or 3) =  13/20

 (iii) There are 8 prime number from 1 to 20 i.e., 2,3,5,7,11,13,17,19

Therefore, Favorable number of elementary events = 8 

P (prime number ) = 8/20=2/5

 Ex.6 A die is drop at random on the rectangular region as shown in figure. What is the probability that it will land inside the circle with diameter 1m ? 

comment answer of this question

New syllabus of class 11 session 2021-2022 MathematicsCOURSE STRUCTURE MATHEMATICS (class XI) (Code No. 041) Session – 2021-22

TERM 1

Unit-I: Sets and Functions

1. Sets

Sets and their representations. Empty set. Finite and Infinite sets. Equal sets. Subsets. Subsets of a set

of real numbers especially intervals (with notations). Power set. Universal set. Venn diagrams. Union and

Intersection of sets.

2. Relations & Functions

Ordered pairs. Cartesian product of sets. Number of elements in the Cartesian product of two finite sets.

Cartesian product of the set of reals with itself ( R x R only).Definition of relation, pictorial diagrams, domain,

co-domain and range of a relation. Function as a special type of relation. Pictorial representation of a

function, domain, co-domain and range of a function. Real valued functions, domain and range of these

functions, constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic and greatest

integer functions, with their graphs.

Unit-II: Algebra

1. Complex Numbers and Quadratic Equations

Need for complex numbers, especially√−1, to be motivated by inability to solve some of the quardratic

equations. Algebraic properties of complex numbers. Argand plane. Statement of Fundamental Theorem

of Algebra, solution of quadratic equations (with real coefficients) in the complex number system.

2. Sequence and Series

Sequence and Series. Arithmetic Progression (A. P.). Arithmetic Mean (A.M.) Geometric Progression

(G.P.), general term of a G.P., sum of n terms of a G.P., infinite G.P. and its sum, geometric mean (G.M.),

relation between A.M. and G.M.

Unit-III: Coordinate Geometry

1. Straight Lines

Brief recall of two dimensional geometry from earlier classes. Slope of a line and angle between two lines.

Various forms of equations of a line: parallel to axis, point -slope form, slope-intercept form, two-point form,

intercept form and normal form. General equation of a line. Distance of a point from a line.

Unit-IV: Calculus

1. Limits

Intuitive idea of limit. Limits of polynomials and rational functions trigonometric, exponential and

logarithmic functions

Unit-V: Statistics and Probability

1. Statistics

Measures of Dispersion: Range, mean deviation, variance and standard deviation of ungrouped/grouped

data.

TERM - II

Unit-I: Sets and Functions

1. Trigonometric Functions

Positive and negative angles. Measuring angles in radians and in degrees and conversion from one

measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the identity

sin2x + cos2x = 1, for all x. Signs of trigonometric functions. Domain and range of trigonometric functions

and their graphs. Expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple

applications. Deducing identities like the following:

Identities related to sin2x, cos2x, tan2 x, sin3x, cos3x and tan3x.

Unit-II: Algebra

1. Linear Inequalities

Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation on the

number line. Graphical solution of linear inequalities in two variables. Graphical method of finding a solution

of system of linear inequalities in two variables.

2. Permutations and Combinations

Fundamental principle of counting. Factorial n. (n!) Permutations and combinations, formula

for nPr and nCr, simple applications.

Unit-III: Coordinate Geometry

1. Conic Sections

Sections of a cone: circles, ellipse, parabola, hyperbola. Standard equations and simple properties of

parabola, ellipse and hyperbola. Standard equation of a circle.

2. Introduction to Three-dimensional Geometry

Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two

points and section formula.

Unit-IV: Calculus

1. Derivatives

Derivative introduced as rate of change both as that of distance function and geometrically. Definition of

Derivative, relate it to scope of tangent of the curve, derivative of sum, difference, product and quotient of

functions. Derivatives of polynomial and trigonometric functions.

Unit-V: Statistics and Probability

1. Probability

Random experiments; outcomes, sample spaces (set representation). Events; occurrence of events, ‘not’,

‘and’ and ‘or’ events, exhaustive events, mutually exclusive events, Probability of an event, probability of

‘not’, ‘and’ and ‘or’ events

Sunday math class notes 25.07.2021

Click here to download class notes oF AP

Class 10 new syllabus session 2021-2022

             COURSE STRUCTURE

               CLASS –X (2021-22)

       FIRST TERM

 One Paper

 90 Minute

UNIT-NUMBER SYSTEMS

1. REAL NUMBER

Fundamental Theorem of Arithmetic - statements after reviewing work done earlier and 

after illustrating and motivating through examples. Decimal representation of rational 

numbers in terms of terminating/non-terminating recurring decimals.

UNIT-ALGEBRA

2. POLYNOMIALS 

Zeroes of a polynomial. Relationship between zeroes and coefficients of quadratic 

polynomials only.

3. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 

Pair of linear equations in two variables and graphical method of their solution, 

consistency/inconsistency. Algebraic conditions for number of solutions. Solution of a 

pair of linear equations in two variables algebraically - by substitution and by elimination. 

Simple situational problems. Simple problems on equations reducible to linear 

equations.

UNIT-COORDINATE GEOMETRY

4. COORDINATE GEOMETRY

 LINES (In two-dimensions) 

 Review: Concepts of coordinate geometry, graphs of linear equations. Distance formula. 

Section formula (internal division)

 UNIT-GEOMETRY

 5. TRIANGLES 

Definitions, examples, counter examples of similar triangles. 

1. (Prove) If a line is drawn parallel to one side of a triangle to intersect the other two sides 

in distinct points, the other two sides are divided in the same ratio. 

2. (Motivate) If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side.

3. (Motivate) If in two triangles, the corresponding angles are equal, their corresponding

sides are proportional and the triangles are similar.

4. (Motivate) If the corresponding sides of two triangles are proportional, their corresponding

angles are equal and the two triangles are similar.

5. (Motivate) If one angle of a triangle is equal to one angle of another triangle and the sides

including these angles are proportional, the two triangles are similar.

6. (Motivate) If a perpendicular is drawn from the vertex of the right angle of a right triangle

to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole

triangle and to each other.

7. (Motivate) The ratio of the areas of two similar triangles is equal to the ratio of the

squares of their corresponding sides.

8. (Prove) In a right triangle, the square on the hypotenuse is equal to the sum of the

squares on the other two sides.

9. (Motivate) In a triangle, if the square on one side is equal to sum of the squares on the

other two sides, the angle opposite to the first side is a right angle.

UNIT- TRIGONOMETRY

6. INTRODUCTION TO TRIGONOMETRY

Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well

defined). Values of the trigonometric ratios of 30

, 45 and 60

. Relationships between the

ratios.

TRIGONOMETRIC IDENTITIES

Proof and applications of the identity . Only simple identities to be given

UNIT-MENSURATION

7. AREAS RELATED TO CIRCLES

Motivate the area of a circle; area of sectors and segments of a circle. Problems based on

areas and perimeter / circumference of the above said plane figures. (In calculating area of

segment of a circle, problems should be restricted to central angle of 60° and 90° only.

Plane figures involving triangles, simple quadrilaterals and circle should be taken.)

UNIT- STATISTICS & PROBABILITY

 8. PROBABILITY

Classical definition of probability. Simple problems on finding the probability of an event

UNIT-ALGEBRA

1. QUADRATIC EQUATIONS (10 Periods)

Standard form of a quadratic equation ax2 + bx + c = 0, (a ≠ 0). Solutions of quadratic

equations (only real roots) by factorization, and by using quadratic formula. Relationship

between discriminant and nature of roots. Situational problems based on quadratic

equations related to day to day activities (problems on equations reducible to quadratic

equations are excluded)

2. ARITHMETIC PROGRESSIONS

Motivation for studying Arithmetic Progression Derivation of the nth term and sum of

the first n terms of A.P. and their application in solving daily life problems.

(Applications based on sum to n terms of an A.P. are excluded)

UNIT- GEOMETRY

3. CIRCLES

Tangent to a circle at, point of contact

1. (Prove) The tangent at any point of a circle is perpendicular to the radius through the

point of contact.

2. (Prove) The lengths of tangents drawn from an external point to a circle are equal.

4. CONSTRUCTIONS

1. Division of a line segment in a given ratio (internally).

2. Tangents to a circle from a point outside it.

UNIT-TRIGONOMETRY

5. SOME APPLICATIONS OF TRIGONOMETRY

 HEIGHTS AND DISTANCES-Angle of elevation, Angle of Depression.

Simple problems on heights and distances. Problems should not involve more than two

right triangles. Angles of elevation / depression should be only 30°, 45°, 60°.

UNIT-MENSURATION

6. SURFACE AREAS AND VOLUMES

1. Surface areas and volumes of combinations of any two of the following: cubes,

cuboids, spheres, hemispheres and right circular cylinders/cones.

2. Problems involving converting one type of metallic solid into another and other mixed

problems. (Problems with combination of not more than two different solids be taken).

UNIT-STATISTICS & PROBABILITY

 7. STATISTICS

Mean, median and mode of grouped data (bimodal situation to be avoided). Mean by

Direct Method and Assumed Mean Method only

Relation and function NCERT CHAPTER 2

Relation Let A and B be two non – empty sets. Then, a relation R from A to B is a subset of (A×B) .

Thus , R is a relation from A to B iff R⊆(A×B).

If (a,b)∈R then we say that’a is related to b’ and we write , a R b.

If (a,b)∉R then ‘a is not related to b’ 

DOMAIN OF A RELATION

The set of all first coordinates of elements of R is called the domain of R, written as dom (R) .

RANGE  OF A RELATION

The set of all second coordinates of element of R is called the range of R. dented by range (R).

CO – DOMAIN OF A RELATION

The set B is called the co- domain of R.

Thus, dom (R) = { a : (a,b)⊆R} and 

range (R) = {b : (a,b)⊆R}.

ARROW DIAGRAM 

Let R be a relation from A to B.

Then we draw two bounded figures to denote A and B.

We put dots to represent the elements of A and B.

For each (a,b)∈R, we draw an arrow from a to b.

Set theory class 11 mathematics

Set : - A well defined collection of objects is called a set. The objects of the set are called its members or elements. 

 Example : - i ) The collection of vowels in english alphabet.

ii) The collection of all rivers of India.

Representation of a set

 There are two ways to represent a given set.

1) Roster or Tabular Form

 In this form , we list all the members of the set withing curly brackets and separate these by commas.  

For Example: -

i) The set A of all even numbers less than 15

A={2,4,6,8,10,12,14}

ii) The set B of vowels in English alphabet

B = { a, e, i,o,u}

2) Set builder form 

In this form , we write a variable (say x ) representing any member of the set which is followed by a colon ‘:’ and there after we write the property satisfied by each member of the set and then enclose the whole description within braces.  

For Example

i) The set A of all even natural numbers less than 15

A = {x : x is an even natural number less than 15}

ii) The set B = { 1, 4, 9, 16, 25 ,…….} in the builder form can be written as

B = { x : x is the square of a natural number }

Type of Sets: -

1. Empty Set - A set which does not contain any element is called the empty set or the null set or the void set. It is denoted by Φ or { }.

For Example : -

i) The collection of natural numbers less than 1

2. Singleton Set - A set that contains only one element is called a singleton set.

For example

i) {O}

ii) { x : 3x – 1 = 8}

3 Finite Set - A set that contains a limited number of different elements is called a finite set.

For Example

i) A = {a,e,i,o,u}

ii) B = { x : x is a month of year}

4) Infinite Set - A set that contains an unlimited number of different elements is called an infinite set.

For Example: -

i)The set of even natural numbers i.e. {2, 4, 6….}

ii) The set of all points on a line segment.

Cardinal Number of a Finite Set : -

The number of different elements in a finite set A is called the cardinal number of A and it is denoted by n(A) or O(A).

For Example.

i) A = {a,e,i,o,u} , then n(A) = 5

ii) A = {2, 3, 5, } then n (A) = 3

Note - The cardinal number of the empty set is zero and the cardinal number of a singleton set is one. The cardinal number of an infinite set is never defined. 

Some Standard sets of number: -

1. Natural numbers - N = { 1,2,3,…………}

2. Whole numbers - W = { 0, 1, 2, 3, ……….}

3. Integers - I or Z = {………, -2, -1, 0, 1,2,……….}

4. Rational numbers - Any number which can be expressed in the form q/p where  p, q ∈ I and  q ≠0 is called a rational number. Thus 8/6 , -6/7 etc. are rational numbers . The set of rational number is denoted by Q.  

5. Real numbers: - All rational as well as irrational numbers are real numbers. Thus – 3, 0, 5/3,√2 etc.are all real numbers. The set of real numbers is denotes by R.  

6. Irrational numbers:- T ={x:x∈R and x∉Q} So, √2, √5, √11, √21 are irrational numbers .  

7. Positive rational numbers: - The set of positive rational numbers is denoted by Q+

8. Positive real numbers : - The set of positive real numbers is denoted by R+

5. Equivalent sets: - Two (finite) sets A and B are called equivalent if they have same number of elements : -

For example: - A = {a,b,c,d,e} and B = {2,3,5,7,9} then n(A) = 5 = n(B)

6. Equal sets : - Two sets A and B are said to be equal if they have exactly the same elements, we write it as A = B.

For example 

 A = {1,2} and B = {2, 1,1,2,1} , then A = B

Subset : - Let A, B be any two sets , then a is called a subset of B if every member of A is also a member of B. We write it as  A⊂ B (read as “A is a subset of B’).  

If A ⊂B, then B is a supper set of A. write it as  B⊃ A. 

Propers Subset : - Let A be any set and B be a non – empty set, then A is called a proper subset of B if every member of A is also a member of B and ther exist atleast one element in B which is not a member of A.  

If A is a proper subset of B, then A⊂B, A≠B. 

Note: -

1. A⊂ A i.e. every set is a subset of itself , but not a proper subset. A subset which is not a proper subset is called an improper subset.  

2. Every set has only improper subset.

3. Empty set is a proper subset of every set except itself.

4. If A is a set with n(A) = m, then the number of subset of A = (2)^m  and the number of proper subset of A =  (2)^m- 1.

Operations on sets: -

1 Union of two sets : - the union of two sets A and B written as A⋃B  (read as ‘A union B’)  is the set consisting of all the elements which belongs to A or B or both.

 Thus A⋃B={x:x€Aor x€B} 

For Example

If A = { 1, 2, 3, 4, 5} and B = {1,3,5, 7, 9} , then

A⋃B={1,2,3,4,5,7,9}

2 Intersection of two sets: - The intersection of two sets A and B, written as A∩B (read as ‘A intersection B’) is the set consisting of all the elements which belong to both A and B. Thus  

A∩B={x:x€Aand x€B}

For Example

If A = { 1, 2, 3, 4, 5} and B = { 1,3,5,7,9} , then

A∩B={1,3,5}

3 Difference of two Sets: - Let A,B be two sets, then A – B is the set consisting of all the elements

which belongs to A but do not belong to B. Thus.

A-B={x:x∈A or x∉ B}

Similarly,

B-A={x:x∈Bor x∉A}

For Example

Let A = {1,2,3,4,5} and B = { 1, 3, 5, 7, 9}

 then A –B = { 2,4} and b- A = { 7, 9}

Remark: - The sets A – B , A∩B and  B-A are mutually disjoint i.e. the intersection of any two of these set is the empty  

4 Complement of a set : - Let U be the universal set and A be any set then the complement of A,

denoted by A’, is the set consisting of all the elements of U which do not belong to A.  

Thus.

A' =( x:x∈U and x∉A)= U - A 

For Example , Let U = {1,2,3, …..,10} 

and A ={ 1,2,3,5,7,9} , then A’ = {4,6,8,10}

Remark: -

i) If A is a sub set of the universal set U, then A’ is also a subset of U.

ii) (A’)’ = A

Venn diagram: - Most of the ideas about sets and the various relationship between them can be visualized by means of geometric figure known as ‘Venn diagrams’. Usually the universal set U is denoted by a  rectangle and its subsets by circles within the rectangle.

Venn Diagrams in different Situations

1. SUBSET : -

2. UNION OF SETS : -

3. INTERSECTION OF SETS:

4. DIFFERENCE OF SETS:

5. COMPLEMENTS SET :

CBSE class 11 Mathematics syllabus

          Topics Name.                         Marks

I          Sets and Functions                     29

II        Algebra                                          37

III       Co-ordinate Geometry.               13

IV.      Calculus                                          6

V        Mathematical Reasoning.           3

VI       Statistics and Probability.         12

                                                     Total 100 marks

Course Syllabus

Unit-I: Sets and Functions

Chapter 1: Sets

*Sets and their representations

*Empty set

*Finite and Infinite sets

*Equal sets. Subsets

*Subsets of a set of real numbers especially intervals (with notations)

*Power set

*Universal set

*Venn diagrams

*Union and Intersection of sets

*Difference of sets

*Complement of a set

*Properties of Complement Sets

*Practical Problems based on sets

Chapter 2: Relations & Functions

*Ordered pairs

*Cartesian product of sets

*Number of elements in the cartesian product of two finite sets

*Cartesian product of the sets of real (up to R × R)

*Definition of Relation

*Pictorial diagrams

*Domain

*Co-domain

*Range of a relation

*Function as a special kind of relation from one set to another

*Pictorial representation of a function, domain, co-domain and range of a function

*Real valued functions, domain and range of these functions −Constant,Identity,Polynomial,Rational,Modulus,Signum,Exponential,Logarithmic,Greatest integer functions (with their graphs)

*Sum, difference, product and quotients of functions.

Chapter 3: Trigonometric Functions

*Positive and negative angles

*Measuring angles in radians and in degrees and conversion of one into other

*Definition of trigonometric functions with the help of unit circle

*Truth of the sin2x + cos2x = 1, for all x

*Signs of trigonometric functions

*Domain and range of trigonometric functions and their graphs

*Expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple application

*Identities related to sin 2x, cos2x, tan 2x, sin3x, cos3x and tan3x

*General solution of trigonometric equations of the type

 sin y = sin a, 

cos y = cos a and

tan y = tan a.

Unit-II: Algebra

Chapter 1: Principle of Mathematical Induction

*Process of the proof by induction −

*Motivating the application of the method by looking at natural numbers as the least inductive subset of real numbers

*The principle of mathematical induction and simple applications

Chapter 2: Complex Numbers and Quadratic Equations

*Need for complex numbers, especially √-1, to be motivated by inability to solve some of the quadratic equations

*Algebraic properties of complex numbers

*Argand plane and polar representation of complex numbers

*Statement of Fundamental Theorem of Algebra

*Solution of quadratic equations in the complex number system

*Square root of a complex number

Chapter 3: Linear Inequalities

*Linear inequalities

*Algebraic solutions of linear inequalities in one variable and their representation on the number line

*Graphical solution of linear inequalities in two variables

*Graphical solution of system of linear inequalities in two variables

Chapter 4: Permutations and Combinations

*Fundamental principle of counting Factorial n (n!) 

*Permutations and combinations

*Derivation of formulae and their connections *Simple applications.

Chapter 5: Binomial Theorem

*History

*Statement and proof of the binomial theorem for positive integral indices

*Pascal's triangle

*General and middle term in binomial expansion

*Simple applications

Chapter 6: Sequence and Series

*Sequence and Series

*Arithmetic Progression (A.P.)

*Arithmetic Mean (A.M.)

*Geometric Progression (G.P.)

*General term of a G.P.

*Sum of n terms of a G.P.

*Arithmetic and Geometric series infinite G.P. and its sum

*Geometric mean (G.M.)

*Relation between A.M. and G.M.

Unit-III: Coordinate Geometry

Chapter 1: Straight Lines

*Brief recall of two dimensional geometries from earlier classes

*Shifting of origin

*Slope of a line and angle between two lines

*Various forms of equations of a line -Parallel to axis

*Point-slope form

*Slope-intercept form

*Two-point form

*Intercept form

*Normal form

*General equation of a line

*Equation of family of lines passing through the point of intersection of two lines

*Distance of a point from a line

Chapter 2: Conic Sections

*Sections of a cone −

Circles-Standard equation of a circle,Ellipse,Parabola,Hyperbola − a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section.

*Standard equations and simple properties of −ParabolaEllipse,Hyperbola

Chapter 3. Introduction to Three–dimensional Geometry

*Coordinate axes and coordinate planes in three dimensions

*Coordinates of a point

*Distance between two points and section formula

Unit-IV: Calculus

Chapter 1: Limits and Derivatives

*Derivative introduced as rate of change both as that of distance function and geometrically

*Intuitive idea of limit

*Limits of −Polynomials and rational functions

Trigonometric, exponential and logarithmic functions

*Definition of derivative, relate it to slope of tangent of a curve, derivative of sum, difference, product and quotient of functions

*The derivative of polynomial and trigonometric functions

Unit-V: Mathematical Reasoning

Chapter 1: Mathematical Reasoning

*Mathematically acceptable statements

Connecting words/ phrases - consolidating the understanding of "if and only if (necessary and sufficient) condition", "implies", "and/or", "implied by", "and", "or", "there exists" and their use through variety of examples related to real life and Mathematics

*Validating the statements involving the connecting words difference between contradiction, converse and contrapositive

Unit-VI: Statistics and Probability

Chapter 1: Statistics

*Measures of dispersion −Range,Mean deviation,Variance

*Standard deviation of ungrouped/grouped data

*Analysis of frequency distributions with equal means but different variances.

Chapter 2: Probability

*Random experiments −Outcomes

Sample spaces (set representation)

Events − Occurrence of events, 'not', 'and' and 'or' events

*Exhaustive events

*Mutually exclusive events

*Axiomatic (set theoretic) probability

Connections with the theories of earlier classes

*Probability of −An event

*probability of 'not', 'and' and 'or' events

SET THEORY CLASS 11 NCERT EXERCISE QUESTION AND SOLUTION MATH

1: Write the following sets in the roster form.

(i) A = {x | x is a positive integer less than 10 and 2x – 1 is an odd number}

(ii) C = {x : x2 + 7x – 8 = 0, x ∈ R}

(i) 2x – 1 is always an odd number for all positive integral values of x since 2x is an even number.

In particular, 2x – 1 is an odd number for x = 1, 2

Q. 2: Write the following sets in roster form:

(i) A = {x : x is an integer and –3 ≤ x < 7}

(ii) B = {x : x is a natural number less than 6}

Q. 3: Given that N = {1, 2, 3, …, 100}, then

(i) Write the subset A of N, whose elements are odd numbers.

(ii) Write the subset B of N, whose elements are represented by x + 2, where x ∈ N.

Q. 4: Let X = {1, 2, 3, 4, 5, 6}. If n represent any member of X, express the following as sets:

(i) n ∈ X but 2n ∉ X

(ii) n + 5 = 8

(iii) n is greater than 4

Q. 5: Let U = {1, 2, 3, 4, 5, 6}, A = {2, 3} and B = {3, 4, 5}. Find A′, B′, A′ ∩ B′, A ∪ B and hence show that ( A ∪ B )′ = A′∩ B′.

Q. 6: Use the properties of sets to prove that for all the sets A and B, A – (A ∩ B) = A – B

Q. 7: Let U = {1, 2, 3, 4, 5, 6, 7}, A = {2, 4, 6}, B = {3, 5} and C = {1, 2, 4, 7}, find

(i) A′ ∪ (B ∩ C′) (ii) (B – A) ∪ (A – C)

Q. 8: In a class of 60 students, 23 play hockey, 15 play basketball,20 play cricket and 7 play hockey and basketball, 5 play cricket and basketball, 4 play hockey and cricket, 15 do not play any of the three games. Find

(i) How many play hockey, basketball and cricket

(ii) How many play hockey but not cricket

(iii) How many play hockey and cricket but not basketbal

Q. 9: Let U = {x : x ∈ N, x ≤ 9}; A = {x : x is an even number, 0 < x < 10}; B = {2, 3, 5, 7}. Write the set (A U B)’.

Q. 10: In a survey of 600 students in a school, 150 students were found to be drinking Tea and 225 drinking Coffee, 100 were drinking both Tea and Coffee. Find how many students were drinking neither Tea nor Coffee.