Showing posts with label 12-Maths. Show all posts
Showing posts with label 12-Maths. Show all posts

Thursday, September 16, 2021

Relations and Functions free pdf and video

 Relations and functions

NCERT Solutions  for Maths - Relations and Functions  by Sonika Anand Academy

Multiple Choice Questions

Video Lectures

Types of functions

Domain and Range of a function

How to find cartesian product of two sets A×B

Representation of two sets using arrow diagram

Find a relation between two sets where one set is cube root of other

How to find number of relations from one site to another

How to find number of elements in relation from a to b


Topic covered in given lecture are

Composition of a function 

How to find whether a  given function is function or not
How to calculate the value of function if given function is in algebraic form or trignometric form

How to add two functions
How to subtract two functions
How to multiply and divide two functions

How we can calculate value of fof and gof


Topic explained 
Find value of f(2),
Or f(x)etc 




Topics
Addition of two functions
Substraction of two functions
How to multiply two functions
Division of two functions




Functions are given in ordered form then how will you add subtract multiply divide the two functions and what is the range and domain of the resulting functions as they are given in the order form so must watch the video attached below




Cartesian product 

What is difference between function and Relation
How we find ordered pairs from two sets using cartesian product 
How ordered pair of Relations can be obtained from Cartesian product of two sets
If one set  A has m elements and another set B  has n elements then how will you find the number of functions obtained from two sets A and B = m×n


Content of Lecture

How to find domain and range and codomain of any function

How we will distinguish between one one and many one function

It is also explained as what is the other name of one one function that is one one function is also named as injective function whereas many one function is also known as surjective function

A  function as both one one and onto then it is also known as bijective function

 what are the different conditions which are satisfied when any function is one one function and when it is many one function

How meni define whether a function is strictly increasing or decreasing using graph method

How to find the number of relations in between two given sets that is what is the formula to find the number of relations between two sets

What is the formula to find the number of mappings from one set to another



Inverse of a function 

What is an invertible function and how to find inverse of any function os explained in given lecture 





Binary Composition
This concept is  started in class 12 that is what are the commutative and associative properties of the binary functions

a*b=b*a. Commutative property
a*(b*c)=(a*b)*c associative property

How to check whether a function is infimum function or superinfimum function
How will we obtain a binary composition table for any given function



Domain and  Range of a Function 
Domain and range of a function is very important topic and generally asked in examinations but question is always asked there in IIT for internal exams of class 12th level so how to calculate to main and range of a function should be there in a there are one or more videos which I have attached below I must watch them properly and I think the concept of domain and range of a function header with the help of a diagram for with the help of order you can easily calculate how to find the domain and range


DOMAIN AND RANGE


Domain of Trignometric functions
In order to calculate the domain of trigonometric functions the concept is little bit different
So I made a separate video and audit explain how to find the domain and range for the trigonometric functions



Domain of modulus function
How to find domain of any modulus function ie |x|
Que discussed in attached lecture are --

Que-- Find domain and range for  f(x)= |x-3|
Que -- f(x) = (|x|-x)/2x
Que f(x) =( |x-4|)/x-4


Equivalence Relation
A function is an equivalence relation if a function  satisfies the three conditions
1 the function must be reflexive
2 a function must be symmetric
3 function  is transitive
Now what is reflexive and what are symmetric and what is transitive function and how will be show
All these concepts are explained in the lecture given below along with the questions

How to find a relation is equivalence or not 
For a equivalence relation equation is 
a) reflexive 
{(a,a)(b,b)}
b) symmetric 
{(a,b),(b,a)}
c) transitive 
{(a,b)(b,c),(a,c)}

Problem discussed 
If X= { 1,2,4...7}
R={(x,y)|(x-y) is divisible by 3} prove it is equivalence relation or not 

How to find Digraph of a Relation 


INJECTIVE or Surjective or Bijective 
How to find INVERSE of a function.              
If a function is oneone(injective) then
f(x)=f(x')
x=x' 
If a function is onto (surjective) then
 range = codomain 
Is a function is both one one and onto then that function is termed as bijective function
Problem 
Que --If f:N---N given by f(x) = 5x then find function is injective surjective or bijective

 Que --If f:A---B f(x) = (x-2)/(x-3) then show weather it is bijective or not Also find inverse 
Que -- f:R--[4, ♾️) f(x) = x²+4 show f is invertible find also its inverse




Questions discussed in next lecture --

How to find 

subset ,proper subset

 cardinal number of a set 

Subset =2^n

Proper subset = 2^n-1

Find number of subset 

If A={x:x = 3x+1,2<=x<=5}

Set is x= {2,3,4,5}

A={7,10,13,16}

So subset =2^n = 2⁴= 2×2×2×2= 16

How to find number of Relations from A to B containing m and n elements 

(2)^m×n

If two finite  sets A and B  have m and n  elements total number of relations is 64 find the value of m and n

If two sets have m and n elements and total number of subsets of first set is 56 more than subset of B  find the value of m and n



Summary - if A={1,2,3...m}. B ={1,2,3,....n}
Subset of set A - 2^m
Subset of B -2^n
Proper subset of A 2^m - 1



Set Theory

Problems discussed in given lecture--
Relations class 11 ex 2.1 

1)  If (x/3+1,y-2/3)= (5/3,1/3) find value of x and y 
2) If A set has 3 elements and the set
 B ={3,4,5} find the number of elements in A×B 

3) If G ={7,8} and H= {5,4,2}
Find G×H and H×G 

4) if A = {-1,1} find A×A×A

5) If A×B = {(a,x),(a,y),(b,x),(b,y)}
Find A and B 

6) If A ={1,2} B={ 3,4} find A×B and how many subsets A×B have 






How to prove DeMorgon's Law --

1) AUB = BUA 
U means union U means or 
2) AπB=BπA 
3) A-(BUC)= (A-B) π(A-C)
4) (AUB)'=A'πB'
5) A-B= AπB'


Problems  discussed in given lecture are --

1) --If X and Y are two sets that x has 40 elements XUY has 60 elements and X intersection  Y has 10 elements how many elements Y has?

2)--A college awarded 38 medals in football 15 in basketball 20 in cricket there medal  went to total of 58 men and only three men got medal in all three source how many received medals in exactly two or three sports



Multiple choice questions


1) The relation is defined on the set

A= {1,2,3,4,5} by R={(a,b):|a^2-b^2|<16} is given by

a) {(1,1),(2,1),(3,1),(4,1),(2,3)}

b) {(2,2),(3,2),(4,2),(2,4)}

c){(3,3),(4,3),(5,4),(3,4)}

d) none of these

Ans 

d )none of these


2) The smallest equivalence relation on the set A= {1,2,3}is

a) {(1,1),(2,2),(3,3)}

b){(1,1)}

c){(1,1),(2,1),(3,1),(2,2),(2,3),(3,2)}

d) none of these

Ans 

a) {(1,1),(2,2),(3,3)}


3) If A = {1,2,3}and B ={1,4,6,9} and R is a relation from Ato  B defined  X is greater than y the range of R is 

a) {1,4,6,9}

b){4,6,9}

c) {1}

d) none of these

Ans

c) {1}


4) A relation R is define from { 2,3,4,5 }  {3,6,7 ,10} by x R y = x is relatively prime to y then domain of R is 

a) {2,3,5}

b.){3,5}

c ) {2,3,4}

d) {2,3,4,5}

Ans 

a) {2,3,5}


5) R is a realation from {11,12,13} to {8,10,12} defined by y=x-3 Then inverse of R is

a) {(8,11),(10,13)}

b.) {(11,8),(13,10)}

c) {(10,13),(8,11),(8,10)}

d) none of these


Ans

a) {(8,11),(10,13)}


6) Let R be a relation on the set N of natural numbers defined by n R m iff n divides m Then R is 

a) Reflexive and symmetric 

b) Transitive and symmetric 

c) Equivalence

 d) Reflexive,transitive,but not symmetric

Ans 

d)

7) Maximum number of equivalence relations on the set A={1,2,3} is

a) 1

b). 2

c) 3

d) 5

Ans 

d ) 5


8) If the set A contains 7 elements and set B  contains 10 elements then number of one one functions from A to B is


Ans  b)

9) If A = {1,2,3,.....n} and B = {a,b}.Then number of subjections from A to B is 



Ans   (b)

10) If f(x)= px/x+1,xis not equal to -1 then for what values of p f(f(x))= x

a) √2

b) -√2

c) 1

d) -1

Ans 

d ) -1

Wednesday, September 1, 2021

Vector

Vector


Vector is a quantity which have both magnitude and direction it is represented by Bold capital letters A


Types of Vector

Unit vector  vector having magnitude one


Negative vector The vector having same magnitude but its direction is opposite


 Equal vector The two vectors are equal if they have same magnitude and equal directions


 Addition of  two vectors

A= i+j-2k.      B= 2i-j+k 

A+B =( i+j-2k ) +( 2i+j+k)

       = 3i+2j-k


Substraction of two vectors 

A-B = ( i+j-2k) -(2i-j+k)

        =i-2i+j-j-2k-k

         =-i-3k





Find unit vector of a vector A= 3i+ 4j

|A|= ✓(3)²+(4)² = ✓ 9+16 = ✓ 25 =5 
unit vector is 
3i/5+4j /5


Find unit vector of A = 3i- 5j + 7k 






Projection of a vector 



Scalar and cross product of two vectors





How to find angle between two vectors






How to find coplanar vector 



How to find a vector perpendicular to a vector



Extra questions on Vector





Ncert Solutions for Maths -Three Dimensional Geometry

 Ncert Solutions -Maths -Three Dimensional Geometry

Cartesian system of rectangular coordinates

Magnitude or Length of a vector 

If vector 

A=xi+yj+zk then magnitude is ✓(x²+y²+z²)

To find position vector --

b-a

Section formula---

For internal division

OR=mb+na/m+n

For external division 

OR = mb-na/m-n



Practice question

1) write position vector of mid point of vector joining points P(2,3,4) and Q(4,1,-2)

2) Find magnitude of vector a= 3i-2j +6k 

How to find unit vector and magnitude of vector 

Formula to find unit vector 

a=a/|a|


Practice questions

1) find a unit vector in the direction of the vector

a= 6i-2j+ 3k

2) find the magnitude of the vector

a= 2i-6j-3k

3) write a vector of magnitude 9 units in the direction of the vector -2i+j+2k


Direction  Cosine and Direction Ratio's

Direction cosine of a vector == ai+bj+ck is 

a/(✓a²+b²+c²),. b/(✓a²+b²+c²). ,c/(✓a²+b²+c²) 

Direction Ratio of Vector 
if P(x,y,z) and Q(a,b,c)  then direction ratio are 

(x-a),(y-b),(z-c) 

Watch the attach lecture ---



Practice question


1)  If  P (1,5,4) and Q(4,1,-2) find direction ratio of PQ

2) if a line has direction ratio (3,4,5) find direction cosines


Exercise 11.1 

Que 1

If a line makes angles of 90°,135°, 45° with x axis  y axis and z Axis respectively find its direction cosines

Solution--


Que 2

Find the direction cosines of a line which makes equal angle with coordinate  axes 

Solution--


Que 4

Show that the points (2,3,4) ,(-1,-2,1) and (5,8,7) are collinear

Solution




Que 4

If a line has direction ratios   (-18,12,-4) find its direction cosines

Solution






Ex 11.2 




Cartesian and vector equation of a line 


Lines can be expressed in two forms vector  form and cartesian form

Vector form r=a+μ(b-a) 

Where a and b are the position vectors of the points through which the line is passing


Watch the lecture  to understand the concept---



Practice Questions---

Write the vector equation of the line given by 

(x-5)/3=(y+4)/7=(z-6)/2

Write cartesian equation of given line given in form of vector 

r=(i+2j-4k)+μ(2i+3j+6k)

Shortest distance between two lines

Vector form 

r=a+Πb and 

r=A+ΠB then 

Shortest distance 

d= |(b×B).(A-a)|

        -------------------

         |B×b|


The formula to find the shortest between two lines in cartesian form is very long so if you want to understand that what you are dash lecture given below and which I explain how will you find the shortest distance between two lines when they are given in vector form and when they are given in cartesian form



Practice question

1) find shortest distance between lines

r= (i+2j +k) +μ(i-2j+2k)

r= (-4i-k) + ¥(3i-2j-2k)

2). Find shortest distance between the lines

L1--- (x-1)/1. = (y-2)/-1. = (z-1)/1

L2 -_- (x-2)/2. = (y+1)/ 1= (z+1)/2

Equation of a plane in Cartesian and vector form

Find the vector and cartesian equation of a plane containing the two lines

r= (2i+j-3k) +μ(i+2j+5k)

Solution



Equation of plane in cartesian form and vector form when three points are given 

Find the vector equation of a plane passing through the points A(2,2,-1) B(3,4,2) ,C(7,0,6) and also find the cartesian equation of the plane

Solution




Distance of plane from a point and origin

Distance from a point (a,b,c) to the plane Ax+By+Cz+D=0  is

(Aa+Bb+Cc+D)/✓A²+B²+C²




Practice questions

Find the distance of the plane  3x-4y+12z=3 from the origin

Find the distance of the plane 2x-y+2z+1=0  from the point (2,3-4)


NCERT Solutions Maths - Inverse Trignometry

 Inverse Trignometry

Part 1



Part 2


Tricks 



NCERT Solutions Maths - Differential Equations

 NCERT Solutions  Maths - Differential Equations

How to find order and Degree of Differential Equations





Differential Equations of Homogeneous Equations


Factor Theorem of Differential Equations 



Ncert solutions of Maths -Application of Derivatives

 Ncert solutions of Maths -Application of Derivatives


Increasing and Decreasing function

how to find a given function is strictly increasing or not

01:20how to find a given function is strictly decreasing  or not

5:40how to find a given function is  neither strictly increasing or  strictly decreasing

07:45find intervals in which a given function is increasing and. interval for which it is decreasing





Part 1 Equation of Tangent and normal


Lagrange mean value theorem




Ncert Solutions of Maths Class 12 - Differentiabilty and Continuity

 Ncert Solutions of Maths - Differentiabilty and Continuity


Limit

Part 1 


Part 2


Part 3



Part 4


Differentiation

Part 1

Introduction





Part 2

How to differentiate Trigonometric functions







NCERT solutions Maths -Integration

 NCERT solutions Maths -Integration

What is the basic formula for differentiation and integration and with the help of this particular video I tried to explain what is the basic difference between differentiation and integration




Lecture 1

This particular lecturer explain the simple basic problems of NCERT solutions of class 12th exercise number 7.1 in this particular lecture this is the first part of integration and with this lecture you can easily start with your topic integration by your own so watch the lecture





Lecture 2


Lecture 3 



Lecture 4 



Lecture 5 






Tricks 



Probability

 NCERT solutions - Probability


Lecture 1 





Binomial distribution

Let X be a random variable which can take n values X1 X2 xn then binomial distribution we have
P(x) = nCx P^x Q^n-x

n is the total number of trials in an experiment

P is the probability of success in one trial

Q is the probability of failure in one trial

X is the number of trials for which probability is required

p+q=1


Probability distribution


Let X be  random variable which can take and values x1,x2,x3...xn  and let p1,p2.... be the respective probabilities then the probability distribution table is given as

X.            x1.             x2.           x3.        xn

P.            p1.             p2.          p3.          pn


Such that 
p1+p2+p3+.........pn= 1




Mean and variance of a probability distribution

Mean = £ xipi

It is also called expectation


Variance = £xi^2pi-(£xipi)^2











NCERT Solutions Class 12 Maths -Matrices and Determinants

Matrices and Determinant


Basic of Matrices


What are rows and column in a matrix and how a matrix can be represented

Types of matrices

What is row matrix,column matrices,identity matrices , null matrix ,diagonal matrices,empty matrices


Watch the lecture to know about above concepts



How to construct a matrix 

What is the general form of matrix away Matrix can be represented a matrix can be formed if it as given in the form of ij what is the row and column of a matrix




How to find unknown quantities in a matrix 





Addition and Subtraction of matrices

Lecture of the mattresses explain how to find unknown values of a mattresses how to add and subtract two matrices using simple arithmetic operations







How to find Transpose of a Matrix

The transpose of a matrix can be found by converting rows into column and columns into row it is denoted by A'






Multiplication of 2×2 Matrices

In this particular lecturer explain how will you multiply a matrix  which have two rows and two column



Multiplication of 3×3 Matrices

Multiplication of 3 cross 3 matrix is very important in this particular video explain how will we multiply 3 cross 3 matrix that is a matrix in which there are three rows and three columns watch this video




Trick to find inverse of matrices useful for competitive exams

In this particular attached video explain how to find the inverse of 3 cross 3 matrix in justice s and district is very useful in competitive exams to find the answer in a very less time




What is cofactor  Matrix 

Cofactor matrix is a part of matrix and this particular lecture I explain how will you find the cofactor matrix for a 3 × 3 matrix



Part 2





How to find the adjoint of any matrix

 what is the formula of finding the inverse of any matrix using the adjoint and determinant of the matrix in the particular lecture given below the topic explain does how to find the inverse of 2 ×2 matrix



Inverse of matrices

Inverse of matrix is very important topic in the mattresses in this particular lecture I explain the how to find the inverse of 3 cross 3 matrix using elementary operations
Moreover we can also lot in this video whether the solution of matrix is consistent or inconsistent


Inverse of matrices by elementry operations



How to find solutions of Matrices



What is inconsistent and consistent matrices


How to find inverse of matrix using adjoint method



How to find Determinant




Mock Test series on Determinants
Test 1



Test 2





Tricks for competitive exams





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