Kishen, Author at NCERT Books

Conversation Between Shopkeeper/ Salesman and Customer [Simple Four Scenarios]

November 8, 2023November 8, 2023 by Kishen

Conversations Between Salesman and Customer about Clothes

Introduction: When you think about sales, what are you thinking about?

You’re imagining some people talking.

One person is a customer speaking about buying something, and they’re probably having lots of queries.

Another man is selling that same thing, answering questions and explaining why it’s worth the value.

Gather Information Regarding Basic English Skills and become proficient in the language and speak fluently with confidence. Try the Tips over here and Improve your English Writing and Speaking Skills.

Simple Conversations Between Salesman and Customer about Shoes, Clothes, Grocery, and Vegetables

The truth is that sales are likely about interacting with people. To work in a sales job, you’ll need to get to know your customers’ needs and wants, as well as how to give them exactly what they want to pay for.

So, if you plan to shine in sales, you should start with learning how to make conversation.

Once you get that potential buyer to talk to you, the final moment when you “close the deal” (sell something) isn’t very far away.

The whole article is about the conversation between a shopkeeper/ salesman and a customer at the time buying the following three items:

Shoes,
Clothes,
Grocery, and
Vegetables and fruits (this one involves the mediation of price)

The following are the examples or sample conversations on these three items.

1. Conversation while Buying Shoes

Salesman: Mam, how may I help you?

Customer: I’m exploring your branch for a pair of sports shoes.

Salesman: May I know the foot size that you do wear usually?

Customer: Nine.

Salesman: Eight for which brand?

Customer: Does the same size mean a distinctive thing for diverse brands? It should be standard across all brands.

Salesman: For a few brands, the same size can mean slightly distinguished fitting.

Customer: Is it? I thought differently. Anyhow, I wear a size Eight of Clarks, and as I need to buy the same brand, it shouldn’t be a problem.

Salesman: Yes, it doesn’t make any change in that scenario. Take a look at this section, which has some incredible new arrivals in shoes on sale.

(The customer first glimpses through the shoes on sale.)

Customer: The variety of shoes is more limited here. Because I’m purchasing shoes for at least several years, I would like to purchase something I want. I’ll go with new arrivals.

Salesman: The majority of the shoes on sale were sold out to customers within a day of the opening of the sale. That’s why you find far less collection there.

Salesman: Why don’t you try this model of shoes?

Customer: Not these. I’m looking for a more formal look, the ones with borders and predominantly decent form.

Customer: I like to give it a try with this one.

(The customer puts on the pair of shoes and walks few yards.)

Customer: This goes perfectly. I’ll buy this one in black—size eight.

Salesman: Anything else, Mam?

Customer: No, thanks.

Salesman: Then, may I get the billing done.

Customer: Sure.

2. Conversation while Buying Grocery

Customer: I need a dozen eggs, one pouch of buttermilk, two liters of double-toned milk, a liter of soybean cooking oil, a half kg of raw grounded coconut, two kilograms of Mung dal, and three flavoured yogurts.

(Once the shop assistant collected all these items from the shelves in the store, the shopkeeper makes the bill.)

Customer: How much will all these items cost?

Shopkeeper: 10 Dollars

Customer: What’s the cost of dozen eggs?

Shopkeeper: 1 dollar a dozen.

Customer: That’s a lot more than what you sold the last time.

Shopkeeper: Rates are increased compared to the previous week.

Customer: OK. Please give me a discount as I’m purchasing quite a few items in a row.

Shopkeeper: We rarely make any margins of profit on these particular items.

Customer: I know how much you make.

Shopkeeper: OK, give me rupees 9$

Customer: Please use this Jute bag to carry the items.

3. Conversation During the Clothes Purchase

Salesman: Mam, how may I help you?

Customer: I’m looking for casual and formal T-shirts and trousers.

Salesman: You can find a T-shirt for casual wear on that side of the showroom. Yet, formal wear is on the second floor.

Customer: All right, I’ll start with formal wear.

(The customer and the salesman proceed to the formal wear section which is on the second floor.)

Salesman: What size T-shirt do you prefer mam?

Customer: I wear 32″.

Salesman: You’ll find 32″ T-shirts on this particular shelf.

Customer: Can you please take out these three T-shirts? I want T-shirts without sleeves.

Salesman: None of these is with sleeves.

(The salesman takes the chosen three shirts out and unleashes them on the waist-high table, placed in between him and the customer.)

Customer: I like the plain, light colors in white and blue.

Salesman: These are nice and comfortable T-shirts. They would go well with dark-shaded trousers, especially those in shades of grey and black. You can take a look at the trouser section as well before you try all of them together.

(They then proceed to check on trousers on the same floor.)

Customer: Can you please show me the trousers in 28″ waist size?

Salesman: These shelves are for waist size 28″.

Customer: Can you please take this and this out?

(The salesman pulls the two trousers out and lines them on the table. The customer killed a few minutes deciding between them and then, ultimately selects one.)

Customer: Where can I try them? Is there any trial room on this floor?

Salesman: Right side of the staircase.

(The customer tries the selected T-shirts and the trousers and comes back.)

Customer: I’ll buy two T-shirts and black trousers.

Salesman: OK, mam. We can now proceed to the ground floor for casual wear.

(They move down to the ground floor and proceed to the section for casual wear.)

Customer: Show me the new arrivals for the summer collection.

Salesman: You’ll really like this. You can wear these casuals for office, shopping, etc. with much comfort in the summer as the color contrast is light in shades and the pattern is simple and elegant.

Customer: I like this one. Let me try this one out.

(The customer tries the selected casual wear and comes back.)

Customer: OK. Please pack this one as well. Now, I’ve two formal T-shirts, one pair of trousers, and one casual dress.

Salesman: Would you want to see anything matching for these selected clothes mam?

Customer: Not now. Thanks.

Salesman: No problem, mam. Our tailor can measure the length of your trousers and would stitch the bottom accordingly and trim off the excess. Meanwhile, I’ll make your billing done.

Customer: When can I collect the trousers?

Salesman: In one hour.

Customer: OK, Thanks.

4. Conversation while Buying Electronics

Salesman: Hello sir, welcome to our store.

Customer: Thank you.

Salesman: How can I help you, sir.

Customer: Actually, I am planning to buy an air conditioner, so could you please help me with that.

Salesman: Sure, sir. Could you please specify me the brand so that I can be more transparent with the details?

Customer: I am using Panasonic AC these days but I am not that satisfied with its services.

Salesman: You should try the LG air conditioner. It is already in high demand.

Customer: What kind of service does it offer?

Salesman: In LG air conditioner, you will get a two-year warranty. It also has five free services. The installation is free. The maintenance is also taken care by the company.

Customer: That’s perfect.

Salesman: Sir, where do you want to install the air conditioner?

Customer: In the living hall.

Salesman: How much Tonne is required?

Customer: 1.0 Tonne would be better.

Salesman: Okay sir.

Customer: Could you please tell me the price of this air conditioner and the mode of payment?

Salesman: It will cost you around 20000/- INR including service tax. And you can pay by card, cash or cheque.

Customer: Perfect, I would like to pay by card.

Salesman: Sure, sir. Let’s complete the formalities and proceed further.

Customer: Okay!

Maths Notes for class 11 – Free PDF Download

November 9, 2023November 8, 2023 by Kishen

Once finished with class 10th, most of the students get tensed as the upcoming year has tough subjects, especially maths. Maths is a subject which needs conceptual understanding rather than learning. The students passed out their class 10th by learning the theorems, concepts, or formulas will not qualify class 11th maths.

The fact is that maths of class 11th needs a lot of understanding of concepts. Most of its chapters will again be asked in class 12th. If the basics of class 11th maths are cleared, only then students can able to attend class 12th class. In brief, class 11 maths is the most significant as it is a beginning of your journey of complex maths.

Do you ever find out the reason why students scored fewer marks in maths class 11th? Why they joined tuition’s? Why they got depressed before exams? The solution is simple and accurate. The preparation of maths needs the thorough understanding of each concept which is possible only if you have maths notes.

Well, it has cleared that Maths notes for class 11th are the ultimate solution to get high scores. The students who skipped to take the help of these notes keep lacking their basics, hence scored fewer marks.

The fact is that everything has a solution. If you will try, only then, you will able to get it. The same criteria are used in maths subject. When you take the help of maths notes, only then you will able to understand the concepts. Without understanding the concepts, you will never able to attempt the maths questions.

Maths Notes for Class 11 – Free PDF Download Chapter wise

Do you know, in maths, many of the questions use the same formulas and equation, if you don’t understand the basics you always become unable to attempt the number of questions? In the other side, if you get a formula, you will able to solve numerous questions by using the same formula.

Maths is a subject which is different from others. It has the more practical part which is an advantage actually as students can able to score 100/100. But it is not possible in other subjects.

Why is the introduction of Maths notes for class 11th the best tactic?

When you get everything in well-defined, precise, and accurate form, you just need to follow it. It saves much of your time as well as other resources. The same thing relies upon when you get maths notes. These have all the chapters as per your syllabus commencing from chapter 1- Sets Revision Notes to Chapter 16- Probability Revision Notes.

It means you will not need to consider your maths notebooks as well as maths notes for preparing your maths, you just need to consider you maths notes only. It is enough to cover all the maths concepts of class 11th. Moreover, all the things are well-planned and designed over there. You never need to prepare another note. You just need to concentrate entirely on these maths notes.

The best time to downloading these notes of your class 11th maths as soon as possible as it will help you throughout the year. Once started preparing from it, you will start noticing the chance on yourself. You start getting the concepts as well as able to sort out most of the complex problems yourself. Even you don’t need to take tuition’s anymore.

You don’t need to call your parents or sibling for help. You will become a maths scholar within a few days of introducing these maths notes. It is glad to say that you can share these notes with your friends. Moreover, they can download these anytime like you as well.

Say goodbye to saying maths is a complex subject. It is rather an easy one if you download the maths notes which are specially designed for the students studying in class 11th. The procedure is very simple. If you will find any barrier while downloading, then you are free to comment.

Physics Formulas – Definition, Equations, Examples

November 9, 2023November 8, 2023 by Kishen

Learning Physics is all about implementing concepts & its derivations to solve the problems. Well, this article provides you exactly what you require to solve the physics problems. Physics Questions will always be challenging to students and it challenges your skills and physics knowledge. Some of the major tasks that students should face while solving the physics questions are Examine what numerical are given and asked in the problem, Applying the correct physics formula or equation, and Filling in the values and calculating properly.

In order to get success in these types of challenges, everyone wants to have an adequate understanding of physics formulas along with its concepts. Hence, we are providing a comprehensive list of physics formulas in this article, which will act as an intermediary reference at the time of solving physics problems. Also, you can make use of this physics equations list as a quick revision before an exam.

List of Physics Formulas & Equations

Physics is the most basic subject of all sciences, and also it seems very harder to master. Basically learning physics is nothing but studying the basic laws which lead our universe. One can master in physics if they understand the theories thoroughly, and also they can easily identify the relation between the quantities or numbers by which they can form the formulas, derive it and learn simply.

Students who are looking for physics formulas can grab the list from this page and utilize them to revise daily and before your exams. While memorizing the physics formulas, first of all, try to understand what a formula says and means, and then what physics relation it outlined. If you comprehend the physics topics underlying those formulas, deriving & memorize them will be easy for you. So, use the list of physics equations available over here and solve the basic physics problems very easily & quickly.

1) Average Speed Formula:

The average speed is the average of speed of a moving body for the overall distance that it has covered.

S = $ \frac{d}{t} $

Where,

S	Average speed
d	Total distance traveled
t	Total time is taken

2) Acceleration Formula:

Acceleration is defined as the rate of change in velocity to the change in time. It is denoted by symbol a.

a =$ \frac{v-u}{t} $

where,

a	Acceleration
v	Final Velocity
u	Initial Velocity
t	Time taken

3) Density Formula:

The density of material shows the denseness of it in a specific given area.

$ \rho = \frac{m}{V} $

Where,

ρ

ρ

Density

Mass of the body

The volume of the body

4) Newton’s Second Law:

According to Newton’s second law of motion, the force can be expressed by the product of mass and acceleration of the body.

F = m × a

Where,

F	Force
m	Mass of the body
a	Acceleration in velocity available

5) Power Formula:

The capacity to do some work is termed as Energy. The Energy spent to do work in a unit amount of time is termed as Power.

P=$ \frac{W}{t} $

Where,

P	Power
W	Work done
t	Time taken

6) Weight Formula:

Weight is not anything but the force which an object experiences due to gravity.

W=mg

Where,

W	Weight
m	Mass of the body
g	Acceleration due to gravity

7) Pressure Formula:

The pressure is defined as the amount of force applied per unit area of the object.

P=$ \frac{F}{A} $

Where,

P	Pressure
F	Force applied
A	Total Area of the object

8) Kinetic Energy formula:

Kinetic energy is the energy that is possessed by a body due to its state of motion.

E = $ \frac{1}{2}mv^2 $

Where,

E	Kinetic Energy
m	Mass of the body
v	The velocity with which the body is traveling

9) Ohms Law Formula:

Ohms law says that the current running through some conductor material is directly proportional to the potential difference between two endpoints of the conductor.

V= I × R

Where,

V	Voltage measured in Volts
I	Electric current flowing through the conductor in amperes.
R	The resistance of the material in ohms.

10) Frequency Formula:

Frequency is the revolutions completed per second or as the number of wave cycles.

f= $ \frac{V}{\lambda} $

Where,

Frequency of the wave

Velocity or wave speed

λ

λ

The wavelength of the wave

Final Words

This is the list of some of the important physics formulas that are used by all class students mostly to solve physics problems. Every branch of physics theory is replaced with countless formulas. If you grasp the underlying theory behind the formulas then physics will be easier for you. So, don’t just mug up the physics formulas mentioned here for the sake of exams, understand them, and relate them with every physics concept and be a creative student in your class.

If you need Maths and Chemistry Formulas list to solve problems easily and score better marks in the final board exams, then visit our site and get an updated list of formulas and equations for better preparation.

Frequently Asked Questions on Physics Formulas

1. Where can I find all the formulas of Physics?

You can discover all major and commonly used physics formulas from the above list provided over here and whenever you need them, you can use them for free within no time.

2. What are the major formulas in physics?

Basic Formulas in Physics are provided below for the sake of students. So, let us have a loot at them before solving the physics problems:

v=u+at
s=ut+1/2at^2
v^2-u^2=2as
F=ma
P=force/area

3. How Can I learn Scientific Formulas?

One of the finest ways to understand and learn the physics formulas is practicing & memorizing the derivation of the formula. Solve the example sums with the formula and read the formulas fluently in a day. Use the memory palace and jot down all the formulas in an understandable way and memorize them easily.

4. How to Memorize Physics Formulas?

There are four three steps you should follow to memorize all physics formulas. They are as under

Using Mnemonic Devices
Understanding Each Formula
Taking Care of Your Body

5. What are the tips to learn Physics Formulas?

Below are furnished top tips to quickly learn several Physics formulas:

Practice concentration
Relax your brain
Practice as much as you can
Minimize your reference checks
Understand the basic concept of the formula
Keep away with all distractions
Learn the derivation of every formula
Use memorizing tricks

CBSE Study Material for Class 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

November 9, 2023November 8, 2023 by Kishen

Study Materials for Class 1 to 12 Maths & Science (Physics, Chemistry & Biology), English, Hindi & All Subjects – Free PDF Download

Get free Online Study Materials for CBSE, ICSE, IIT JEE Main & Advanced, NEET, and Other Boards like NCERT Textbook Solutions, Syllabus, Revision Notes, Important Questions, Important Formulas, Previous Year Question Papers & Sample Papers with Solutions to help you to score well in your Examinations. By studying from the free exam materials, students can get exam-ready and ace any Competitive exams with ease.

CBSE Study Materials

CBSE Online Resources – PDF Download

CBSE Class 12 Study Materials

CBSE Class 12 Study Material – PDF Download

CBSE Class 11 Study Materials

https://drive.google.com/file/d/1TWPE3o8h4_WCzO0bKUcmF6tMIj5rWben/view?usp=sharing

CBSE Class 11 Study Material – PDF Download

CBSE Class 10 Study Materials

CBSE Class 10 Study Material – PDF Download

CBSE Class 9 Study Materials

CBSE Class 9 Study Material – PDF Download

CBSE Class 8 Study Materials

CBSE Class 8 Study Material – PDF Download

CBSE Class 7 Study Materials

CBSE Class 7 Study Material – PDF Download

CBSE Class 6 Study Materials

CBSE Class 6 Study Material – PDF Download

CBSE Class 5 Study Materials

NCERT Books for Class 5	NCERT Solutions for Class 5
Class 5 NCERT Math Magic NCERT Book Class 5 NCERT Science Books Looking Around Class 5 Our Environment NCERTBook Class 5 Marigold NCERT Book Class 5 Ibtedai Urdu – V NCERT Book	NCERT Solutions for class 5 Maths NCERT Solutions for class 5 EVS NCERT Solutions for class 5 Paryayana Adyayan NCERT Solutions for class 5 English NCERT Solutions for class 5 Hindi

CBSE Class 4 Study Materials

NCERT Books for Class 4	NCERT Solutions for Class 4
Class 4 Math Magic NCERT Book Class 4 NCERT Looking Around (EVS Book) Class 4 NCERT Marigold Book Class 4 NCERT Ibtedai Urdu – IV Book	NCERT Solutions for class 4 Maths NCERT Solutions for class 4 EVS NCERT Solutions for class 4 Paryayana Adyayan NCERT Solutions for class 4 English NCERT Solutions for class 4 Hindi

CBSE Class 3 Study Materials

NCERT Books for Class 3	NCERT Solutions for Class 3
Class 3 NCERT Math Magic Book Class 3 NCERT Looking Around Book Class 3 NCERT Marigold Book Class 3 NCERT Ibtedai Urdu – III Book	NCERT Solutions for class 3 Maths NCERT Solutions for class 3 EVS NCERT Solutions for class 3 Paryayana Adyayan NCERT Solutions for class 3 English NCERT Solutions for class 3 Hindi

CBSE Class 2 Study Materials

NCERT Books for Class 2	NCERT Solutions for Class 2
Class 2 NCERT Math Magic Book Class 2 NCERT Marigold Book Class 2 Raindrops NCERT Book Class 2 Ibtedai Urdu – II Book	Class 2 Maths NCERT Solutions Class 2 English NCERT Solutions Class 2 Hindi NCERT Solutions

CBSE Class 1 Study Materials

NCERT Books for Class 1	NCERT Solutions for Class 1
Class 1 Math Magic NCERT Books Class 1 Marigold NCERT Books Class 1 Raindrops NCERT Books Class 1 Ibtedai Urdu – I NCERT Books	Class 1 Maths NCERT Solutions Class 1 English NCERT Solutions Hindi NCERT Solutions Class 1 NCERT Solutions

KVS NCERT CBSE Worksheets for Class 1 to Class 12

KVS NCERT CBSE Unseen Passages for Class 1 to Class 12

CBSE Online study materials

CBSE board is an important board which runs most of the schools in the country. The schools which are affiliated to the Central Board of Secondary Education follows the same syllabus as per prescribed the board. All of those schools follows the NCERT curriculum, which are also in accordance to the other state boards of India. This is why we have prepared the study materials for classes 1 to 12, based on this syllabus and curriculum, so that students find it easy to learn and prepare for their exams.

Versionweekly is providing here the online learning materials for the students of standard 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, and 1. These materials can be used as a reference while preparing for exams. We are including here CBSE Syllabus for Maths, Science, Social Science, English and Hindi as per CBSE curriculum, solutions for all the chapters, revision notes to have a quick revision before exam, sample papers and previous year question will help to know the exam pattern and also important questions which they can practice regularly before exams. All these materials are prepared keeping in mind student’s benefits and learning capabilities. They have been created in such a way, that each and every student can understand them easily.

Benefits of Referring CBSE Online study materials

Some key benefits of studying from CBSE Online study materials are given in the points below.

It offers in-depth knowledge of the course curriculum in an easy manner.
The study material is designed strictly as per the latest CBSE syllabus and exam pattern.
The study material is created by experts after the depth analysis of the course structure.
It builds the basic concepts and clears students doubts.
The study material is provided in simpler language so that students can easily understand it.
The sample papers and question papers are provided for students practice.

The CBSE study material will help students in their studies from the beginning of the academic session to last-minute preparation. Also, students need to be consistent with their studies and practice different problems to perform outstandingly in the CBSE exam. Stay tuned to get the latest update on CBSE and other competitive exams.

Maths Formulas for Class 11 PDF Download Free | 11th Std Maths Formulae List

November 9, 2023November 8, 2023 by Kishen

Most of the Students feel Maths as a Difficult Subject and Quite Hard to Master. To help all such people we have come up with Maths Formulas for Class 11. The only way to get grip on the subject is through consistent practice and learn the complete Math Formulae of Class 11. Do your best in the exam by accessing the Formulae List of 11th Std Maths and become familiar with concepts. Apply the formulas in your problems and arrive at the solution easily.

Chapterwise Class 11 Maths Formulas List

Get a strong grip on the concepts by making the most out of the 11th Grade Math Formula Collection over here. Try to implement the formulas during your calculations and get the desired scores. You can download the 11th Class Maths Formulas prepared by subject expertise and rely on them whenever you need them. Just tap on the below available links and you will get related formulas all in one place making your job much simple.

Class 11 Sets Formulas
Class 11 Relations and Functions Formulas
Class 11 Trigonometric Functions Formulas
Class 11 Complex Numbers and Quadratic Equations Formulas
Class 11 Permutations and Combinations Formulas
Class 11 Binomial Theorem Formulas
Class 11 Sequence and Series Formulas
Class 11 Straight Lines Formulas
Class 11 Conic Sections Formulas
Class 11 Introduction to Three Dimensional Geometry Formulas
Class 11 Limits and Derivatives Formulas
Class 11 Statistics Formulas

Coordinate Geometry & Line Formula

Coordinate Geometry & Lines Formulas for Class 11
Distance Formula	$\left \| P_{1}P_{2} \right \|=\sqrt{\left ( x_{2}-x_{1} \right )^{2}+\left ( y_{2}-y_{1} \right )^{2}}$
Slope	$\large m=\frac{rise}{run}=\frac{\Delta y}{\Delta x}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$
Point-Slope Form	$y-y_{1}=m\left ( x-x_{1} \right )$
Point-Point Form	$y-y_{1}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\left ( x-x_{1} \right )$
Slope-Intercept Form	$y=mx+b$
Intercept-Intercept Form	$\frac{x}{a}+\frac{y}{b}=1$
General Form	$Ax+By+C=0$
Parallel & Perpendicular Lines	Parallel Lines $m_{1}=m_{2}$ Perpendicular Lines $ m_{1}m_{2}=-1$
Distance from a Point to a Line	$\large d=\frac{\left \| Ax_{0}+By_{0}+C \right \|}{\sqrt{A^{2}+B^{2}}}$

Algebra Formula

Algebra Formulas For Class 11
Distributive Property	$a\times \left ( b+c \right ) = a \times b\, +\, a \times c$
Commutative Property of Addition	$a\, +\, b\, =\, b\, +\, a$
Commutative Property of Multiplication	$a\,\times b\, =\, b\,\times a$
Associative Property of Addition	$a\, +\, \left ( b\, +\, c \right ) = \left ( a\, +\, b \right )\, +\, c$
Associative Property of Multiplication	$a\,\times \left ( b\,\times c \right ) = \left ( a\,\times b \right )\,\times c$
Additive Identity Property	$a\, +\, 0\, =\, a$
Multiplicative Identity Property	$a\, \times 1\, =\, a$
Additive Inverse Property	$a\,+\left ( -a \right )=0$
Multiplicative Inverse Property	$a \cdot \left ( \frac{1}{a} \right )=1$
Zero Property of Multiplication	$a\times \left ( 0\right )=0$

Trigonometric Formula

Trigonometry Class 11 Formulas
$\sin (-\theta ) = -\sin \theta$
$\cos (-\theta ) = \cos \theta$
$\tan (-\theta ) = -\tan \theta$
$ cosec (-\theta ) = -cosec \theta$
$\sec (-\theta ) = \sec \theta$
$\cot (-\theta ) = -\cot \theta$
Product to Sum Formulas
$\sin \, x \,\ sin \, y = \frac{1}{2}\left [ \cos\left ( x – y \right ) -\cos \left ( x+y \right ) \right ]$
$\cos\, x \, \cos\, y = \frac{1}{2}\left [ \cos \left ( x – y \right ) + \cos \left ( x+y \right ) \right ]$
$\sin\, x \, \cos\, y = \frac{1}{2}\left [ \sin\left ( x + y \right ) + \sin \left ( x-y \right ) \right ]$
$ \cos\, x \, \sin\, y = \frac{1}{2}\left [ \sin\left ( x + y \right ) – \sin\left ( x-y \right ) \right ]$
Sum to Product Formulas
$\sin\, x + \sin \, y = 2\, \sin \left ( \frac{x+y}{2} \right ) \cos \left ( \frac{x-y}{2} \right )$
$\sin\, x -\sin\, y = 2\, \cos \left ( \frac{x+y}{2} \right ) \sin \left ( \frac{x-y}{2} \right )$
$\cos \, x + \cos \, y = 2 \, \cos \left ( \frac{x+y}{2} \right ) \cos\left ( \frac{x-y}{2} \right )$
$\cos\, x -\cos\, y = – 2 \, \sin \left ( \frac{x+y}{2} \right ) \sin \left ( \frac{x-y}{2} \right )$

Maths Formulas For Class 11: Sets

A set is a well-collaborated collection of objects. A set consisting of definite elements is a finite set. Otherwise, it is an infinite set. You can find the essential symbols and properties for Sets below:

Symbol	Set
N	The set of all the natural numbers
Z	The set of all the integers
Q	The set of all the rational numbers
R	The set of all the real numbers
Z⁺	The set of all the positive numbers
Q⁺	The set of all the positive rational numbers
R⁺	The set of all the positive real numbers

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: $A\cup B$.
The intersection of two sets A and B are said to be contained elements that are common in both the sets. The intersection of A and B is denoted as: $A\cap B$.
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:
- (i) ${(A\cup B)}’={A}’\cap{B}’$
- (ii) ${(A\cap B)}’={A}’\cup{B}’$
If the finite sets A and B are given such that ${(A\cap B)}=\phi$, then: $n{(A\cup B)}=n(A)+n(B)$
If ${(A\cup B)}=\phi$, then: $n{(A\cup B)}=n(A)+n(B)-n(A\cap B)$

Class 11 Maths Formulas: Relations And Functions

An ordered pair is a pair of elements grouped together in a certain order. A relation R towards a set A to a set B can be described as a subset of the cartesian product A × B which is obtained by describing a relationship between the first of its element x and the second of its element y given in the ordered pairs of A × B.

The below-mentioned properties will surely assist you in solving your Maths problems.

A cartesian product A × B of two sets A and B is given by:
A × B = { $(a,b):a\epsilon A, b\epsilon B$ }
If (a , b) = (x , y); then a = x and b = y
If n(A) = x and n(B) = y, then n(A × B) = xy
A × $\phi$ = $\phi$
The cartesian product: A × B ≠ B × A
A function f from the set A to the set B considers a specific relation type where every element x in the set A has one and only one image in the set B.
A function can be denoted as f: A → B, where f(x) = y
Algebra of functions: If the function f: X → R and g: X → R; we have:
- (i) $(f + g) (x) = f (x) + g(x), x\epsilon X$
- (ii) $(f – g) (x) = f (x) – g(x), x\epsilon X$
- (iii) $(f.g)(x) = f (x) .g (x), x\epsilon X$
- (iv) $(kf) (x) = k ( f (x) ), x\epsilon X$, where k is a real number
- (v)$ \frac{f}{g}(x) = \frac{f(x)}{g(x)}, x\epsilon X, g(x)\neq 0$

Maths Formulas For Class 11: Trigonometric Functions

In Mathematics, trigonometric functions are the real functions which relate to an angle of a right-angled triangle forming some finite ratios of two side lengths. Find the important Maths formulas for Class 11 related to trigonometric functions below.

If in a circle of radius r, an arc of length l subtends an angle of θ radians, then $l = r × θ$.
Radian Measure = $\frac{\pi}{180}$ × Degree Measure
Degree Measure = $\frac{180}{\pi}$ × Radian Measure
$cos^2 x + sin^2 x = 1$
$1 + tan^2 x = sec^2 x$
$1 + cot^2 x = cosec^2 x$
$cos (2n\pi + x) = cos\: x$
$sin (2n\pi + x) = sin\: x$
$sin\: (-x) = -sin\: x$
$cos\: (-x) = -cos\: x$
$cos\:(\frac{\pi}{2}-x)=sin\:x$
$sin\:(\frac{\pi}{2}-x)=cos\:x$
$sin\: (x + y) = sin\: x\times cos\: y+cos\: x\times sin\: y$
$cos\: (x + y) = cos\: x\times cos\: y-sin\: x\times sin\: y$
$cos\: (x – y) = cos\: x\times cos\: y+sin\: x\times sin\: y$
$sin\: (x – y) = sin\: x\times cos\: y-cos\: x\times sin\: y$
$cos\:(\frac{\pi}{2}+x)=-sin\:x$
$sin\:(\frac{\pi}{2}+x)=cos\:x$
$cos\: (\pi-x) = -cos\: x$
$sin\: (\pi-x) = sin\: x$
$cos\: (\pi+x) = -cos\: x$
$sin\: (\pi+x) = -sin\: x$
$cos\: (2\pi-x) = cos\: x$
$sin\: (2\pi-x) = -sin\: x$
If there are no angles x, y and (x ± y) is an odd multiple of (π / 2); then:
- (i) $tan\:(x+y)=\frac{tan\:x+tan\:y}{1-tan\:x\:tan\:y}$
- (ii) $tan\:(x-y)=\frac{tan\:x-tan\:y}{1+tan\:x\:tan\:y}$
If there are no angles x, y and (x ± y) is an odd multiple of π; then:
- (i) $cot\:(x+y)=\frac{cot\:x\:cot\:y-1}{cot\:y+cot\:x}$
- (ii) $cot\:(x-y)=\frac{cot\:x\:cot\:y+1}{cot\:y-cot\:x}$
$cos\:2x=cos^2\:x-sin^2\:x=2\:cos^2\:x-1=1-2\:sin^2\:x=\frac{1-tan^2\:x}{1+tan^2\:x}$
$sin\:2x=2\:sin\:x:cos\:x=\frac{2\:tan\:x}{1+tan^2\:x}$
$sin\:3x=3\:sin\:x-4\:sin^3\:x$
$cos\:3x=4\:cos^3\:x-3\:cos\:x$
$tan\:3x=\frac{3\:tan\:x-tan^3\:x}{1-3\:tan^2\:x}$
Addition and Subtraction of sin and cos
- (i) $cos\:x+cos\:y=2\:cos\frac{x+y}{2}\:cos\frac{x-y}{2}$
- (ii) $cos\:x-cos\:y=-2\:sin\frac{x+y}{2}\:sin\frac{x-y}{2}$
- (iii) $sin\:x+sin\:y=2\:sin\frac{x+y}{2}\:cos\frac{x-y}{2}$
- (iv) $sin\:x-sin\:y=2\:cos\frac{x+y}{2}\:sin\frac{x-y}{2}$
Multiplication of sin and cos
- (i) $2\:cos\:x\:cos\:y=cos\:(x+y)+cos\:(x-y)$
- (ii) $-2\:sin\:x\:sin\:y=cos\:(x+y)-cos\:(x-y)$
- (iii) $2\:sin\:x\:cos\:y=sin\:(x+y)+sin\:(x-y)$
- (iv) $2\:cos\:x\:sin\:y=sin\:(x+y)-sin\:(x-y)$
$sin\: x = 0;\: gives\: x = n\pi,\: where\: n\: \epsilon\: Z$
$cos\: x = 0;\: gives\: x = (2n+1)\frac{\pi}{2},\: where\: n\: \epsilon\: Z$
$sin\: x = sin\: y;\: implies\: x = n\pi\:+(-1)^n\:y,\: where\: n\: \epsilon\: Z$
$cos\: x = cos\: y;\: implies\: x = 2n\pi\pm y,\: where\: n\: \epsilon\: Z$
$tan\: x = tan\: y;\: implies\: x = n\pi+y,\: where\: n\: \epsilon\: Z$

Class 11 Maths Formulas: Complex Numbers And Quadratic Equations

A number that can be expressed in the form a + ib is known as the complex number; where a and b are the real numbers and i is the imaginary part of the complex number.

Let z₁ = a + ib and z₂ = c + id; then:
- (i) z₁ + z₂ = (a + c) + i (b + d)
- (ii) z₁ . z₂ = (ac – bd) – i (ad + bc)
If there is a non-zero complex number; z = a + ib; where (a ≠ 0, b ≠ 0), then there exists a complex number $\frac{a}{a^2+b^2}+i\frac{-b}{a^2+b^2}$; denoted by $\frac{1}{z} or z^–1 is known as the multiplicative inverse of z; such that
(a + ib) [ \(\frac{a^2}{a^2+b^2}+i\frac{-b}{a^2+b^2}$ ] = 1 + i 0 = 1
For every integer k, i^4k = 1, i^4k+1 = i, i^4k+2 = -1, i^4k+3 = -i
The conjugate of the complex number is $\bar{z}=a-ib$
The polar form of the complex number z = x + iy is $r(cos\: \theta+i\:sin\:\theta)$; where $r=\sqrt{x^2+y^2}$ (the modulus of z)
$cos\:\theta =\frac{x}{r}$ and $sin\:\theta =\frac{y}{r}$ (θ is the argument of z)
A polynomial equation with n degree has n roots.
The solutions of the quadration equation ax² + bx + c = 0 are:
$x=\frac{-b\pm \sqrt{4ac-b^2i}}{2a}$ where a, b, c ∈ R, a ≠ 0, b² – 4ac < 0

Maths Formulas For Class 11: Permutations And Combinations

If a certain event occurs in ‘m’ different ways followed by an event that occurs in ‘n’ different ways, then the total number of occurrence of the events can be given in m × n order. Find the important Maths formulas for class 11 as under:

The number of permutations of n different things taken r at a time is given by ${}^{n}\textrm{P}{r}$ $=\frac{n!}{(n-r)!}$ where 0 ≤ r ≤ n
$n!=1\times 2\times 3\times …\times n$
$n!=n\times (n-1)!$
The number of permutations of n different things taken r at a time with repetition being allowed is given as: n^r
The number of permutations of n objects taken all at a time, where p₁ objects are of one kind, p₂ objects of the second kind, …., p_k objects of kth kind are given as: $\frac{n!}{p_{1}!\:p_{2}!\:…\:p_{k}!}$
The number of permutations of n different things taken r at a time is given by ${}^{n}\textrm{C}{r}$ $=\frac{n!}{r!(n-r)!}$ where 0 ≤ r ≤ n

Class 11 Maths Formulas: Binomial Theorem

A Binomial Theorem helps to expand a binomial given for any positive integral n.
$(a+b)^n={}^{n}\textrm{C}_{0}\:a^n+{}^{n}\textrm{C}_{1}\:a^{n-1}.b+{}^{n}\textrm{C}_{2}\:a^{n-2}.b^2+…+{}^{n}\textrm{C}_{n-1}\:a.b^{n-1}+{}^{n}\textrm{C}_{n}\:b^n$

The general term of an expansion (a + b)ⁿ is $T_{r+1}={}^{n}\textrm{C}_{r}\:a^{n-r}.b^r$
In the expansion of (a + b)ⁿ; if n is even, then the middle term is $(\frac{n}{2}+1)^{th}$ term.
In the expansion of (a + b)ⁿ; if n is odd, then the middle terms are $(\frac{n+1}{2})^{th}$ and $(\frac{n+1}{2}+1)^{th}$ terms

Maths Formulas For Class 11: Sequence And Series

An arithmetic progression (A.P.) is a sequence where the terms either increase or decrease regularly by the same constant. This constant is called the common difference (d). The first term is denoted by a and the last term of an AP is denoted by l.

The general term of an AP is $a_{n}=a+(n-1)\:d$
The sum of the first n terms of an AP is: $S_{n}=\frac{n}{2}[2a+(n-1)\:d]=\frac{n}{2}(a+l)$

A sequence is said to be following the rules of geometric progression or G.P. if the ratio of any term to its preceding term is specifically constant all the time. This constant factor is called the common ratio and is denoted by r.

The general term of an GP is given by: $a_{n}=a.r^{n-1}$
The sum of the first n terms of a GP is: S_{n}=\frac{a(r^n-1)}{r-1}\: or\: \frac{a(1-r^n)}{1-r}; if r ≠ 1
The geometric mean (G.M.) of any two positive numbers a and b is given by $\sqrt{ab}$

Class 11 Maths Formulas: Straight Lines

Slope (m) of the intersecting lines through the points (x₁, y₁) and x₂, y₂) is given by $m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{y_{1}-y_{2}}{x_{1}-x_{2}}$; where x₁ ≠ x₂
An acute angle θ between lines L1 and L2 with slopes m1 and m2 is given by $tan\:\theta =\left | \frac{m_{2}-m_{1}}{1+m_{1}.m_{2}} \right |$; 1 + m₁.m₂ ≠ 0.
Equation of the line passing through the points (x₁, y₁) and (x₂, y₂) is given by: $y-y_{1}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}(x-x_{1})$
Equation of the line making a and b intercepts on the x- and y-axis respectively is: $\frac{x}{a}+\frac{y}{b}=1$
The perpendicular distance d of a line Ax + By + C = 0 from a point (x₁, y₁) is: $d=\frac{\left | Ax_{1}+By_{1}+C \right |}{\sqrt{A^2+B^2}}$
The distance between the two parallel lines Ax + By + C₁ and Ax + By + C₂ is given by: d=$\frac{\left | C_{1}-C_{2} \right |}{\sqrt{A^2+B^2}}$

Maths Formulas For Class 11: Conic Sections

A circle is a geometrical figure where all the points in a plane are located equidistant from the fixed point on a given plane.

The equation of the circle with the centre point (h, k) and radius r is given by (x – h)² + (y – k)² = r²
The equation of the parabola having focus at (a, 0) where a > 0 and directrix x = – a is given by: y² = 4ax
The equation of an ellipse with foci on the x-axis is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
Length of the latus rectum of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is given by: $\frac{2b^2}{a}$
The equation of a hyperbola with foci on the x-axis is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
Length of the latus rectum of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is given by: $\frac{2b^2}{a}$

Class 11 Maths Formulas: Introduction To Three Dimensional Geometry

The three planes determined by the pair of axes are known as coordinate planes with XY, YZ and ZX planes. Find the important Maths formulas for Class 11 below:

The distance of two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is:
$PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$
The coordinates of a point R that divides the line segment joined by two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) internally as well as externally in the ratio m : n is given by:
$\left ( \frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n},\frac{mz_2+nz_1}{m+n} \right )\:and\:\left ( \frac{mx_2-nx_1}{m-n},\frac{my_2-ny_1}{m-n},\frac{mz_2-nz_1}{m-n} \right )$;
The coordinates of the mid-point of a given line segment joined by two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) are $\left ( \frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2} \right )$
The coordinates of the centroid of a given triangle with vertices (x₁, y₁, z₁), (x₂, y₂, z₂) and (x₃, y₃, z₃) are $\left ( \frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3},\frac{z_1+z_2+z_3}{3} \right )$

Maths Formulas For Class 11: Limits And Derivatives

A limit of a function at a certain point holds a common value of the left as well as the right hand limits, if they coincide with each other.

For functions f and g, the following property holds true:
- (i) $\lim\limits_{x \to a} \left [ f(x)\pm g(x) \right ]= \lim\limits_{x \to a}f(x) \pm \lim\limits_{x \to a}g(x)$
- (ii) $\lim\limits_{x \to a} \left [ f(x) .g(x) \right ]= \lim\limits_{x \to a}f(x) . \lim\limits_{x \to a}g(x)$
- (iii) $\large \lim\limits_{x \to a} \left [ \frac{f(x)}{g(x)} \right ] = \frac{\lim\limits_{x \to a}f(x)}{\lim\limits_{x \to a}g(x)}$
Standard Limits
- (i) $\lim\limits_{x \to a}\frac{x^n-a^n}{x-a}= n\:a^{n-1}$
- (ii) $\lim\limits_{x \to a}\frac{sin\:x}{x}=1$
- (iii) $\lim\limits_{x \to a}\frac{1-cos\:x}{x}=0$
The derivative of a function f at a holds as: ${f}'(a)=\lim\limits_{x \to a}\frac{f(a+h)-f(a)}{h}$
The derivative of a function f at a given point x holds as: ${f}'(x)=\frac{\partial f(x)}{\partial x}=\lim\limits_{x \to a}\frac{f(x+h)-f(x)}{h}$
For the functions u and v, the following holds true:
- (i) $(u\pm v)’=u’\pm v’$
- (ii) $(uv)’=u’v+uv’$
- (iii) $\left ( \frac{u}{v} \right )’=\frac{u’v-uv’}{v^2}$
Standard Derivatives
- (i) $\frac{\partial}{\partial x}(x^n)=nx^{n-1}$
- (ii) $\frac{\partial}{\partial x}(sin\:x)=cos\:x$
- (iii) $\frac{\partial}{\partial x}(cos\:x)=-sin\:x$

Class 11 Maths Formulas: Statistics

You will find the essential maths formulas for Class 11 of Statistics given below:

Mean Deviation for the ungrouped data:
- (i) $M.D.(\bar x)=\frac{\sum \left | x_i-\bar x \right |}{n}$
- (ii) $M.D.(M)=\frac{\sum \left | x_i-M \right |}{n}$
Mean Deviation for the grouped data:
- (i) $M.D.(\bar x)=\frac{\sum f_i|x_i-\bar x|}{N}$
- (ii) $M.D.(M)=\frac{\sum f_i|x_i-M|}{N}$
Variance and Standard Deviation for the ungrouped data:
- (i) $\sigma ^2=\frac{1}{N}\sum (x_i-\bar x)^2$
- (ii) $\sigma=\sqrt{\frac{1}{N}\sum (x_i-\bar x)^2}$
Variance and Standard Deviation of a frequency distribution (discrete):
- (i) $\sigma ^2=\frac{1}{N}\sum f_i(x_i-\bar x)^2$
- (ii) $\sigma=\sqrt{\frac{1}{N}\sum f_i(x_i-\bar x)^2}$
Variance and Standard Deviation of a frequency distribution (continuous):
- (i) $\sigma ^2=\frac{1}{N}\sum f_i(x_i-\bar x)^2$
- (ii) $\sigma=\frac{1}{N}\sqrt{N\sum f_ix_i^2-(\sum f_ix_i)^2}$
Coefficient of variation (C.V.) = $\frac{\sigma}{\bar x}\times 100$ ; where $\bar x\neq 0$

Maths Formulas for Class 9 PDF Free Download | Important 9th Grade Maths Formulae

November 9, 2023November 8, 2023 by Kishen

Students who feel Maths as Nightmare and difficult can use the Maths Formulas for Class 9 over here to understand the concepts better. In order to remove reluctance over the subject, we have curated the 9th Class Formulas for Maths in a simple way. The Maths Formulae Collection existing is prepared by subject experts adhering to the Latest Syllabus Guidelines. Nothing can stop you from scoring well in your exams if you get acquainted with all the 9th Grade Maths Formulas provided.

List of 9th Standard Maths Formulae

If you are clear with the logic behind the Formula Solving any Kind of Problem is quite easier. Have a glance at the Chapterwise 9th Class Maths Formula List and make the most out of them. Once you start analyzing the concepts carefully it becomes easier for you to grab the mathematical formulas. Implement these formulas in your problems and arrive at the solutions easily. Simply click on the topic and get all the related formulas in one place.