Showing posts with label 11-Maths. Show all posts
Showing posts with label 11-Maths. Show all posts

Friday, November 19, 2021

Ncert solutions class 11 Maths -- Statistics

 Ncert solutions class 11 Maths -- Statistics

Ex 15. 1


Lecture 1 


Mean deviation for ungrouped data


How to find the mean deviation for the following data

4,7,8,9,10, 12,13,17

Mean deviation for  grouped data

How to find the mean deviation for the following  grouped data

0--100.         4

100-- 200.     8

200-- 300.      9

300-- 400.     10 

Solution -
Lecture given below the contains  solution of both of the questions 




Lecture 2


What is coefficient of variation and how it is related with the standard deviation and variance

Que 
Coefficient of variation of two distribution are 60 and 70. The value of the standard deviations are 21 and 16 what are their arithmetic mean 

Solution



Lecture 3


How to find median for a grouped data

0-- 10                  6

10-- 20                8

20 --30.              14

30--40.               16 

40 --50.               4

50 --- 60.            2

Solution 



Lecture 4


How to find the mean deviation about the median

9,5,3,12,10,18,4,7,19,21,3

Solution





Lecture 6


How to find mean,  variance and standard deviation for a given data

6,7,10,12,13,4,8,12

Solution



Lecture 7


How to find mean and variance for the grouped data

0. ---. 10.           5

10 --- 20.           8

20 --- 30.          15

30 --- 40           16

40 --- 50            6


Solution




Tuesday, October 5, 2021

NCERT solutions class 11-Maths- Mathematical Induction

 NCERT solutions class 11-Maths- Mathematical Induction


Induction in one shot 

First principle of mathematical induction
Verification step
Actual verification of the proposition for the standing value I
Induction step
Assuming the proposition to be true for ke k is greater than or equal to I end and providing that it is true for all the value
 k + 1 which is next higher integer
Generalized step
Combine the above two steps




OR 

Let PNB statement involving natural number n to prove P (n) is true for all natural numbers we follow the following process

1 prove thatP(1) is true

2 Assume P(k) is true

3 using first and second prove that the statement is true for n= k+1 ie P(k+1) is true

This is first principle of mathematical induction


Watch the attach lecture in order to understand how to solve different questions


Second principle of mathematical induction

Second principle of mathematical induction following steps are used

1 prove that P (1)  is true

2 Assume p(n) is true for all natural numbers such that 2<=n<k

3 using one and two prove that P (k+1) is true


Most Important question
is  explained in the attached lecture which is generally came in examination



Multiple choice question


Que 1

2^3n -7n -1  is divisible by 

a) 64

b) 36

c) 49

d) 25

Ans (c)


Que 2

The greatest positive integer which divides
(n+2) (n+3)(n+4)(n+5)(n+6) for all n€N is 

a) 4

b) 120

c) 240

d) 24


Ans (b)


Que 3

The inequality n!>2^n-1 is true for

a) n>2

b) n€N

c) n>3

d) none of these


Ans (a)

Que 4

The remainder when 5^99 is divided by 13 is

a) 6

b) 9

c). 8

d) 10

Ans (c)


Que 5

If P(n)  is a statement n€N such that if P(k) is true P(k+1)  is true for k€N then P(n) is  true  for all

a) n

b) n>1

c) n>2

d) nothing can be said

Ans (d)


Que 6

If P(n) is a statement n€N such that if P(3) is true P(k+1) is true for k>= 3  then P(n) is true for all

a) n

b) n>= 3

c) n>4

d) none of these

Ans (b)

Que 7 

If P(n): 3^n <n! ,n€N then P(n) is true for 

a) n>= 6

b) n>= 7

c) n>= 3

d ) for all n


Ans (b)

Que 8

The number a^n - b^n is always divisible by

a) a. - b

b) a+ b

c) 2a - b

d) a - 2b

Ans (a)

Que 9

Let P(n) denotes the statement that n^2+n is odd It is seen that P(n) --- P(n+1) ,P(n) is true for all

a) n>1

b) n> 2

c) for n

d) none of these

Ans (d)


Que 10 

Let S(k) =1+3+5 .....(2k -1) = 3+ k ^2 which of the following is true

a) S(1) is true

b) S(k)=. S(k+1)

c) S(k)=! S(k+1)

d) principle of mathematical induction can be used to prove the formula

Ans (b)


Linear inequalities

 NCERT solutions class 11 -Maths -Linear  Inequations


Lecture 1



Lecture 2 



How to draw graph of linear inequality



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 8, 2021

Coordinate Geometry

 Coordinate Geometry 


What are cartesian system 

What is abscisa and ordinate

There are two lines cutting each other  perpendicularly 

The point at which they cut each other is called origin(0,0)

The horizontal axis is known as x axis  and vertical axis is known as y axis 

The points lying on x axis are  termed as abisccae and points lying on y axis are termed as ordinate so any point can be represented as  (x,y)

Eg (2,3)

Here 2 is termed as abisscae and 3 is termed as ordinate  




If (x,x')and (y,y') are two points then 

What is distance formula

√(x-x')^2+(y-y')^2


What is section formula

If (x,x')and (y,y') are two points and ratio is m:n

Formula for section formula(internally) is

X=(nx+mx')/(m+n)

Y= (ny+my')/(m+n)

Section formula (externally)

X=(nx-mx')/(m-n)

Y= (ny-my')/(m-n)


Midpoint formula

If (x,x')and (y,y') are two points 

X=(x+x')/2

Y=(y+y')/2

How we calculate area of triangle

Formula of area of triangle is

1/2{(x1(y2-y3)+x2(y3-y1)+x3(y1-y2)}


Three dimensional Geometry  lecture 




How to draw graph 




Lesson in one shot 




Wednesday, September 1, 2021

Trignometry Ratio and Equations

Ncert solutions for class 11 Maths -Trignometry Ratio and Equations


Lecture 1



Lecture 2


Lecture 3



Lecture 4


Lecture 5


Lecture 6


Lecture 7


Lecture 8


Lecture 9


Lecture 10



Lecture 11


Lecture 12



NCERT Solutions Maths - Measurement of Angles -Trignometry

 Angle and arc Length

Part 1


Part 2


Trigonometric Identities


Part 1 






Part 2


Trignometric Functions



Application of Trigonometry




Circles

 Ncert solutions  for Class 11 Maths -Conics - Circles


How to find equation of the circle with the radius and the centre is given and how to find equation of circle in the two endpoints of diameter are given in this particular we t explain what are the general form of equation of the circle




How to find the equation of tangent open a circle


Orthogonal circles


How to find number of tangents when circles each other externally or internally



 Tricks for jee iit engineering exam 

Trick 1



Trick 2

Trick 3




How to find radius of inscribed circle



Some questions on topic of circles











Ncert solutions class 11 Maths - parabola,Ellipse,Hyperbola

 NCERT solutions Maths Conics

Exersise 11.2 








Equation of parabola when focus and directrix is given


Equation of parabola when focus and vertex is given


Extra questions on parabola




Ellipse

Ex 11.3 







Hyperbola




For Iit Jee preparation





How to find whether given conic is parabola,hyperbola or  ellipse






Straight line and pair of straight lines

 Different forms of straight lines


How to find equation of parallel and perpendicular lines



How to find slope of a straight line


Questions on Straight line




Position of a point with respect to two points



How to find angle  between two lines

 How to find distance between two parallel lines

Class 11

Ex 10.2 part 1









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











Binomial Theorem

 Binomial Theorem

General formula of Binomial Theorem 


Topic covered in lecture given below

0:30

How to find value of 

nCr=n!/r!(n-r)!

12C3


Expand ---(2+3x)^2


How to find number of terms in a expansion


What is general term of binomial expansion


T(n+1) = nCr(p)^n- r (q)^r

5:44

How to find term independent of x in any binomial expansion

10:38

How to find pth term from end

If (x+y)^5  then find 3rd term from end

12:30

How to find middle term of expansion when power is odd or when power is even 

(x-2/x)^8 find middle term


(x-2/x) 9 find the middle term


15: 38 

How to find total number of  terms in an expansions 

16 :37 

How to find sum of coefficients in an binomial expansion



Some special Tricks

 if you are preparing for IIT and examinations because this video contains various questions which are came in the entrance examination and it contains tricks to solve different types of questions

Topic covered

00:00 relation between wavelength and work function

Work function of substance is 4 electron volt find wavelength of light which emits photoelectrons


03:15 Relation between stopping potential and wavelength

If wavelength of a light changes from 300 nm to 400 nm find decrease in stopping potential


06:17 relationship between Kinetic energy and wavelength

If a light of wavelength 260 nm is incident on a metal surface find change in kinetic energy if the value of threshold wavelength is 360 nm


08:48 work potential and threshold frequency


11:21 how to find photons emitted per second in photoelectric emission


13:23 how to find change in de broglie wavelength before and after collision of two particles




NCERT solutions - permutations and Combinations

 NCERT solutions - permutations and Combinations


Permutation is to find number of ways by which we can arrange any things 

Part 1



Lecture 2




Part 2







Part 3



Part 4




















Arithmetic Progression

 NCERT Solutions Maths - Sequences and Series 


Sample paper for Term 2 -2022

Que 1-- 

Find the first term and common difference and the 25th term for the series

-4,-5/2,-1,1/2,2............

Que. 2-- 

If nth term of arithmetic progression is. 
Tn = 4n-7
 find the 20th term of the series

Que 3--

Which term of the arithmetic progression 
4,9,14,19........
is 79

Que 4--- 

How many terms are there in the arithmetic progression

9,13,17,21...............97


Que 5 ----

Which term  of arithmetic progression is first negative term

45,42,39,36............

Que 6 ----

Find 8th term from the end of the arithmetic progression

4,9,14...................254


Que 7 ---

If (k- 3) (2k+1) and (4k+3) are three terms of arithmetic progression find the value of k


Que 8 ---

Find three numbers which are in AP whose sum is 15 and product is 80

Que 9 ---

Find the sum of the following series

a) 3,8,13,18...............  upto 15 terms 

b) 9,7,5,3 ..............upto 14 terms 

c) 33,37,41,..........101 


Que 10 ---

Find sum of all two digit numbers which are divisible by 3


Que 11--- 

The 4th term of an AP  is 22 and 15th term of an AP  is 66. Find the sum of series upto 8 terms

Que 12 ---

A  sum of rupees 2800 is to be used to award four prizes if each  price after the first is 200 less than the preceding price find the value of each prize 


Que 13---

200 logs are stacked in a way that there are 20 logs in bottom row 19 logs in the next row 18 log  is in the next row how many rows are formed and how many logs are there in the top row




Mock Test Series


Que 1
If the mth  term of an AP is 1/n and nth term is 1/m then sum of mn terms is 

a) 1/2(m n-1)

b) 1/2(mn+1)

c) mn+1

d) mn-1


Ans (b)

Que 2

The ratios  of sums of  m and n  terms of an AP is m^2:n^2 then the ratio of mth term and nth term is 

a) (2m+1):(2n+1)

b) m:n

c) ( 2m-1):(2n-1) 

d) none of these

Ans ( c) 

Que 3
The sum of the series 1+ 4/5 +7/5^2+10/5^3.....

a) 7/16

b) 5/16

c) 105/64

d) 35/16


Ans ( d) 


Que 4
If the roots of the equation
 x^3-12x^2+39x-28=0 are in AP then their common difference will be

a) 1

b) 2

c) 3

d) 4

Ans (c)

Que 5

If AM of two numbers is twice of their GM then ratio of  greatest number to smallest number is 

a) 7-4√3

b) 7+4√3

c) 21

d) 5

Ans (b)


Que 6
Let two numbers have arithmetic  mean 9 and geometric mean 4 then these numbers are the roots of the quadratic equation

a) x^2-18x-16=0


b) x^2-18x+16=0


c) x^2+18x-16=0


d)x^2+18x+16=0


Ans (b)


Que 7

The sum of all odd numbers between 1 and 1000 which are divisible by 3


a) 83367


b) 90000


c) 93660


d) none of these


Ans (a)


Que 8 

The sum of the series 1.2.3 + 2.3.4+ 3.4.5 ..

to n terms is 


a) n(n+1)(n+2)


.b) (n+1)(n+2)(n+3)


c) 1/4 n(n+1)(n+2)(n+3) 


d) 1/4  (n+1)(n+2)(n+3)


Ans (c)


Que 9

The sum of series 6+66+666+6666+... upto n  terms 

a) (10^n -1 -9 n+10)/81


b) 2(10^n- 1-9n+10)/27


c) 2(10^n -9 n+10)/27


d) none of these


Ans (b)


Que 10

If a,b,c,d,e,f are in AP then value of e-c will be

a) 2(c- a)


b) 2(f- d)


c) 2(d- c)


d) d - c


Ans (c) 

Recorded lectures on different topics of AP


How to find sum of series



Geometric Series


Arithmetic Series

Solution of Questions in attached lecture are 
Que 1---

 If An=n²+5 

Find

 a) first three terms


b) find 10th term


c) find n-1 th term


Que 2----


 Find 10th term if sum of n terms is 6n²+7


Que 3-----


Find 20th term if sum to n terms is 


3/2(3^n -1)





Question discussed in given video are 

This is basic video what is arithmetic series

Que 1---

For any given series 2,8,11

Find 20th term of series 

 Find sum of 20 terms 

Find nth term of series





Arithmetic Mean 

Questions discussed in attached lecture are
 
Que 1----
Find four arithmetic mean between 5 and 7

Que 2---- 

Find k if k+2,4k-6,3k-2 are in AP 


Que 3---

If pth term is q and qth term is p show that mth term of the series is p+q- m 


Que 4--

If mth term of series is 1/n and nth term of series is 1/ m find (a-d) 


Que 5---

If sum of three terms of AP series is 36 and its product is 1620 Find AP series 


Solution----




Questions covered in next lecture are 

1) Formula to Insert m arithmetic mean between a and b 

(b-a)/(m+1)

2)  Insert one arithmetic mean between 4 and 8 


3) Insert 4 arithmetic mean between 4 and 19 


4) Find arithmetic mean between (x+y) and 
(x-y)

5) There are n arithmetic mean between. 3 and 17 .The ratio of last mean to first mean is 3:1 find value of n

Solution



How to find sum of series


Que 1---

How to find sum of n terms of series whose kth term is 5k+1

Que 2--

If sum of n terms of an AP series is (pn+qn²) where p and q are constant find common difference

Que 3

If sum of n terms of series is 3n²+5n and its  mth term is 164. Find  value of m 


Solution ----



How to find ratio of terms if sum of ratio of two series is given


Que --
 If ratio between sum of n terms of two AP series is (7n+1):(4n+27). find ratio of 11th terms 





Geometric Series

Part 1
What is Geometric series

Let a be the first term and r be the common ratio the general term of GP series 
ar^n-1

General formula for Tth term from last in  GP series 

l(1/r)^n-1

Watch the attached lecture for GP series from basic to advance Topics covered in lecture are


Que 1 ----

If 3,9,27 ...... is a serie then

What is first term 

What is common ratio

How we find wether this is geometric series or not 


Que 2--

If 3,6,12 .... is a series then find its 100th term 


Que 3---

If 3,6,12...... 3072 

Then find 10th term from the last 


Que 4---

Which term of the GP is 128 for  the series

2, 1,1/2,1/4.....


Que 5 ---

If 3rd term of series is 24 and 6th term is 192 Find 10th term of the series


Que 6 --

Find the sum of series up to 7 terms if series is  2,6,18.......


Que 7---

Find sum of series upto 10 terms if series is 

4,2,1,1/2 .......


Que 8 --

Find sum of series of infinite terms 

5+55+555........




Special Type of Sum of Geometric Series
Mathematical Induction


Que discussed in lecture 

7+77+777.......

Solution--


Mathematical Induction

Basic video to solve different questions based on Mathematical Induction are 

Questions discussed in given lecture ---

Que 1 

Prove that 
1+3+5 ........n terms = m²

Que 2---

Let P(n) be a statement n²+n is an even integer Show that P(m) is true then it is also true for P(m+1) 

Que 3--

Let P(n) be a statement n²+n is an odd  integer Show that P(m) is true then it is also true for P(m+1) 

Que 4 

Prove that 3²n - 1 is divisible by 8  for all values of n 




Important and Tricky questions for competitive Exams 


Type 1--

Find sum of given series upto n terms 

0.3+0.33+0.333.........

Solution


Que -- 

Find 5th term if sum of n terms of series is 
n²-2n

Solution--



Ncert solutions Class 10 Maths -Quadratic Equations

Quadratic Equations Class 10 -- Maths 

Sample questions

 Term 2-- 2022


Que 1--- Find quadratic equation whose solution set is 
a) {2,-3}

b) { -3,2/5}

Que 2--. Find value of k for which x=3  is the solution of equation

(k+2)x²-kx+6=0


Que 3--- Find set of values of k for which the equation.  px²-5x+p=0  has real and equal roots

Que 4 -- Find set of values of k for which the equation.  2x²+kx+2=0  has real  roots

Que 5 -- Find set of values of k for which the equation.  kx²+2x+1=0  has distinct  roots

Que 6-- Find roots of equation 
             3(2x-1)². + 4(2x-1) -. 4 = 0

Que 7 -- What is the nature of roots of equation
15x²-11x+3=0

Que 8 -- If  one root of equation 
 4x²-2x+p- 4=0  is reciprocal of other then find the value of p

Que 9-- Which constant should be added to solve quadratic equation
4x²-√3x+5=0

Que 10 -- Solve for x 

(x+2)/(x+3).  = (2x-3)/(3x-7)



Mock Test Series


Que 1
The roots of the given equation is 

(p-q)x^2+(q-r)x+(r-p)=0

a) (p-q)/(r-p),1

b) (q-r) / (p-q),1

c) (r-p)/ (p-q),1

d) 1,(q-r)/(p-q)

Ans ( c)

Que 2
Number of real roots of the equation
(6-x)^4+(8-x)^4=16

a) 4

b) 2

c) 0

d) none of these

Ans (b)

Que 3

If the product of the roots of the equation
(a+1)x^2+(2a+3)x+(3a+4)=0 is 2 then the sum of roots is 

a) 1

b) -1

c) 2

d) -2

Ans (b) 

Que 4 

If the equation 2x^2+3x+5p =0 and x^2+2x+3p =0 have a common root then value of p is 

a) 0

b) -1

c) 0,-1

d) 2,-1

Ans (c)


Que 5

If p is one root of equation 4x^2+2x-1=0 then other root is 

a) 4p^3-3p

b) 4p^3+3p

c) p-1/2

d) 0

Ans  (a) 

Que 6

If one root is square of the other root of the equation
X^2+px+q=0 then relation between p and q 

a) p^3-(3p-1)q+q^2=0

b) p^3- (3p+1)q+q^2=0

c) p^3+ (3p-1)q+q^2=0

d) p^3-(3p+1)q+q^2=0

Ans (a)


Que 7
If the roots of the quadratic equation
x^2+ px+q=0 are tan 30°,tan 15° then 
value of 2+q-p is 

a) 3

b) 0

c) 1

d)2

Ans ( a) 

Que 8

If one root of the equation x^2+ax+12=0 is 4
while the equation x^2+ax+b=0 has equal roots then value of b 

a) 4/49

b) 49/4

c) 7/4

d) 4/7

Ans ( b)

Que 9
If x=✓1+✓1+✓1+....... then x is equal to

a) (1+√5)/2

b) (1-√5)/2

c) 1

d) none of these

Ans (a)


Que 10

The values of x satisfying |x-4|+|x-9|=5 is

a) x=4,9

b) 4<x<9

c) x>4 or x<9

d) none of these

Ans  (a)


Recorded Lectures on various topics on quadratic equations



Basic concepts

General form of quadratic equations

The general form of quadratic equations 

ax^2+bx+c=o

We can factorise a quadratic equations by two methods simply by prime factorization and discriminant method this particular video explain what is the discriminant method

Formula is b^2-4ac

Value of discriminant is positive then the roots of the quadratic equations are unequal and real

Value of discriminant is equal to zero then roots of the quadratic equations are equal and real

Value of discriminant is negative then roots of the quadratic equations are imaginary

Watch the educational lecture to understand how you will factorize the equation using that discriminant  method



Find square root by completing its square

How we add in 16a²-12a so that it becomes complete square 




How to factorise any quadratic equations using prime factorisation method

How to solve different questions based on age 

Moreover how to solve question based on topic speed and boat 




Questions on boat and stream



How to divide any polynomial before using Remainder theorem in order to factoriae any equation



Tricks for competitive exams 
This video contains questions which are tough and complex but we explain tricks to solve them in less than minutes






Complex Number

Complex Number

Multiple choice questions


Que 1

The value of (1+i)^4(1+1/i)^4 is 

a) 12

b) 2

c) 8

d) 16

Ans  (d)


Que 2

The value of x and y if 2+ (x+yi) = 3-i

a) 1,-1

b) -1,1

c) 1,1

d) -1,-1

Ans  (a)


Que 3

The value of z1/z2 where z1 = 2+3i and z2=1+2i

a) 8/5+1/5i

b) 8/5 - 1/5i

c) 1/5 -8/5i

d) none of these

Ans (b)

Que 4 

The conjugate of a complex number
 (2-3i)/(4-i) is 

a) 3i/4

b) (11+10i)/17

c) (11-10i)/17

d) (2+3i)/4i

Ans (c)

Que 5

If z= (1-√3i)/(1+√3i) then arg (z) is equal to 

a) π/3

b) 2π/3

c) (-2π/3)

d) none of these


Ans (c)


Que 6

The square root of (7+24i) are

a) +-(3+4i)

b) +-(3-4i)

c) +-(4+3i)

d) +-(4-3i)

Ans (c)


Que 7

If z be a complex number angle between z and iz is 

a) π

b) 0

c) (-π/2)

d) none of these

Ans (c)

Que 8

If |z+4|<=3 the greatest and least value of

|z+1| are

a) 6,-6

b) 6,0

c) 7,2

d) 0,-1

Ans  (b)

Que 9

If w is an imaginary cube root of unity then 
(1+w-w^2)^7 equals

a) 128w

b) -128 w

c) 128 w^2

d) -128w^2


Ans ( d )


Que 10 

If |(z+i)/(z-i)| = 3  then radius of the circle is 


a) 2/√21

b) 1/√21

c) √3

d) √21

Ans  (c) 

Sample Lecture- find polar form and argument of a complex number                                                                                                 



                                                                                                                                                                                                                                                                                                                           






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