mathematics Archives – Mathemafia

State and Prove Pythagoras Theorem

Sunil Kumar — Tue, 11 Mar 2025 11:29:49 +0000

Pythagoras Theorem

In a right angled triangle, the square of hypotenuse is equal to the sum of squares of the other two sides.

In a right angled triangle, the square of hypotenuse is equal to the sum of squares of perpendicular and base of the triangle.

To Prove

\( \begin{aligned} &BC^2 =AB^2+AC^2\end{aligned}\)

Construction

\( \begin{aligned} &\text{Draw } AD \perp BC\end{aligned}\)

Proof

Part 1

\( \begin{aligned} &In \ \ \triangle ABC \text{ and } \triangle ADB \end{aligned}\)

\( \begin{aligned} &\angle {BAC} = \angle {ADB} = 90^{\circ}\\ &\angle B = \angle B \ \ \ \ \text{(Common)} \end{aligned}\)

By AA similarity criterion

\( \begin{aligned} &\triangle ABC \sim \triangle DBA \end{aligned}\)

Very Important: We write it as △ABC is similar to △DBA. We cannot write it similar to △ADB even though △DBA and △ADB are same triangles.

Why? Click here to read it or watch the video embedded below.

\( \begin{aligned} &\Rightarrow \frac{AB}{DB} = \frac{BC}{AB}\\\\ &\Rightarrow AB^2 = BC \times DB \ \ …(1)\end{aligned}\)

Part 2

\( \begin{aligned} &In \ \ \triangle ABC \text{ and } \triangle ADC \end{aligned}\)

\( \begin{aligned} &\angle {BAC} = \angle {ADC} = 90^{\circ}\\ &\angle C = \angle C \ \ \ \ \text{(Common)} \end{aligned}\)

By AA similarity criterion

\( \begin{aligned} &\triangle ABC \sim \triangle DAC \end{aligned}\)

\( \begin{aligned} &\Rightarrow \frac{BC}{AC} = \frac{AC}{DC}\\\\ &\Rightarrow AC^2 = BC \times DC \ \ …(2)\end{aligned}\)

Part 3

Adding (1) and (2)

\( \begin{aligned} AB^2+ AC^2 &= BC \times BD + BC \times DC \\ &=BC \times (BD + DC)\\ &=BC \times BC\\ &=BC^2\\ \end{aligned}\)

Hence proved.

Watch on YouTube

Why?

\( \begin{aligned} &\text{Why }\triangle ABC \sim \triangle DBA \text{ but } \triangle ABC \not\sim \triangle ADB\end{aligned}\)

When it comes to similarity and congruence of triangles, students always find it slightly confusing but let me tell you, it is as simple as Virat Kohli hits a six 🏏 at will. All you need is to possess good mathematics skills 😜 like Kohli has (in cricket of-course).

Ok … That’s not true. You don’t need to be as good as Kohli 😜 to understand this concept. 💃🕺 Wow!!! Nacho Nacho. 💃🕺 (If this song can win oscar, one day I will get the Fields Medal. 🏅 🏆 )

Explanation

We just proved that the triangles shown below are similar. The triangles are similar when all three corresponding angles are congruent in both the triangles. In this case:

\( \begin{aligned} \angle {BAC} &= \angle {ADB} = 90^{\circ}\\ \angle B &= \angle B \ \ \ \ \text{(Common)}\\ \Rightarrow \angle {ACB} &= \angle {DAB} \end{aligned}\)

In the picture displayed above you can clearly see mapping of the corresponding angles means the mapping of the corresponding vertices of the two triangles. So while writing the similarity expression, we must write the corresponding vertices in same order on both sides as shown in Mapping Pic.

Welldone Mathemafia 👏💯 !!!

Pytha: Oh my Goras!!! I can’t believe that you have explained it so Beautifully Colorfully.

Mathemafia: Wow! You struck out Beautifully. 💔

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Class 10 – Chapter 5 – Arithmetic Progression – Last Minute Formula Revision

Sunil Kumar — Thu, 06 Feb 2025 04:39:59 +0000

⏰ Last minute Formula Revision 🔥 Arithmetic Progression, Chapter 5, Class 10

This short video covers all the formulas of chapter 5 (Arithmetic Progression, AP Series) of class 10. It is recommended to watch this before your exam to quickly recall all the formulas of this chapter.

The tutorial is available in Hinglish + 9 different languages: English, French, German, Hindi, Spanish, Indonesian, Italian, Japanese, Portuguese

The post Class 10 – Chapter 5 – Arithmetic Progression – Last Minute Formula Revision appeared first on Mathemafia.

Class 10 – Chapter 2 – Polynomials – Last Minute Formula Revision

Sunil Kumar — Sun, 02 Feb 2025 09:06:02 +0000

⏰ Last minute Formula Revision 🔥 Polynomials, Chapter 2, Class 10

This short video covers all the formulas of chapter 2 (Polynomials) of class 10. It is recommended to watch this before your exam to quickly recall all the formulas of this chapter.

The tutorial is available in Hinglish + 9 different languages: English, French, German, Hindi, Spanish, Indonesian, Italian, Japanese, Portuguese

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Class 11 – Video Tutorials on Mathematics

Sunil Kumar — Mon, 25 Mar 2024 12:45:48 +0000

Here you will find the video tutorials for class 11 Mathematics.

Class 11 - Chapter 1 - Sets

This tutorial cover more than 50% percent of Sets, Chapter 1, of class 11 mathematics. The other half will be added once ready.

One-shot video

Class 11 - Chapter 2 - Relations (One-shot video)

This chapter has been split into 2 parts :

Relations
Functions

The reason is to cover this chapter in depth so that it should be helpful for students who are willing to appear in competitive exams as well.

Below embedded is a one-shot YouTube video that covers the Relations section in detail.

One-shot video

This is a one-shot video that covers the Functions section of chapter 2 (Relations and Functions) of class 11 in detail.

One-shot video

Class 11 - Chapter 3 - Trigonometric Functions

Trigonometry of class 11 is a very long and important topic. This chapter has been covered in multiple videos and the playlist of the same has been embedded below.

What is radian? What is the relationship between radians and degrees?

Example 1 and 2, Chapter 3, Class 11 | Trigonometric Functions | Radians to Degrees & vice versa

Example 3, Chapter 3, Class 11 | Trigonometric Functions | NCERT Maths

Example 4, Chapter 3, Class 11, Maths | Trigonometric Functions | NCERT, CBSE

Example 5, Chapter 3, Class 11, Maths | Trigonometric Functions | NCERT, CBSE

Question 1, 2 Exercise 3.1 Class 11 Maths | Chapter 3 (Trigonometric Functions) | NCERT

Question 3, Ex: 3.1, Ch 3, Class 11, NCERT Maths | Trigonometric Functions

Question 4, Ex: 3.1, Ch 3, Class 11, NCERT Maths | Trigonometric Functions

Question 5, Ex: 3.1, Ch 3, Class 11, NCERT Maths | Trigonometric Functions

Question 6 Exercise 3.1 Class 11 Maths | Chapter 3 (Trigonometric Functions) | NCERT

Question 7 Exercise 3.1 Class 11 Maths | Chapter 3 (Trigonometric Functions) | NCERT

🔥 Coordinates of a point on (1) Circle of radius r (2) Unit circle || Prove that sin^2x + cos^2x = 1

Formulas "sin(2npi + theta) = sin theta" and "sin[(2n+1)pi – theta] = sin theta"

Proof of (i) cos(- θ) = cos θ, (ii) cos(2npi + θ) = cos θ, (iii) cos(2npi – θ) = cos θ | Unit circle

sin(nπ)=0 and cos (2n+1) π/2 = 0 | Explained with unit circle

All School To College | ASTC Formula to remember the sings of trigonometric ratios in 4 quadrants

Prove that 🔥 cos (x + y) = cosx cosy – sinx siny 🔥 Full derivation by #mathemafia

Prove that 🔥 cos (x – y) = cosx cosy + sinx siny 🔥 Simple proof by #mathemafia

cos (pi/2 – x) = sin x 🔥 sin (pi/2 – x) = cos x 🔥 Solved trigonometric identities by #mathemafia

Prove that sin(x+y) = sinx cosy + cosx siny 🔥 💁👌😍 🔥 and sin(x-y) = sinx cosy – cosx siny

Prove that (i) sin(pi+x)=-sin x (ii) sin(pi-x)=sin x (iii) cos(pi+x)=-cos x (ii) cos(pi-x)=-cos x

tan(x + y)=(tan x + tan y) / (1 – tan x tan y) 🔥 & 🔥 tan(x – y)=(tan x – tan y) / (1 + tan x tan y)

cot(x + y)=(cot x cot y – 1)/(cot x + cot y) 🔥 & 🔥 cot(x – y) = (cot x cot y + 1)/(cot y – cot x)

Proof of sin 2x = 2 sinx cosx = 2 tan x /(1+tan²x) where x is not an odd multiple of pi/2 🔥

Proof of cos 2x = cos²x – sin²x = 2cos²x – 1 = 1 – 2sin²x = (1 – tan²x) /(1 + tan²x)

Proof of 🔥 tan 2x = 2 tan x /(1 – tan²x) 🔥 Class 11, Trigonometric Identities by #mathemafia

Proof of three trigonometric identities 🔥 sin 3x, cos 3x and tan 3x 🔥 by #mathemafia

Easy proof of 🔥 sin x + sin y 🔥 cos x + cos y 🔥 sin x – sin y 🔥cos x – cos y by #mathemafia

The post Class 11 – Video Tutorials on Mathematics appeared first on Mathemafia.

Class 10 | Chapter 14 | All Solutions | Probability

Sunil Syal — Fri, 17 Dec 2021 07:44:02 +0000

This playlist covers all the questions, concepts and examples of chapter 15 (Probability) of class 10, NCERT book. Click a thumbnail to watch the tutorial.

Watch complete chapter on YouTube

Probability – Understand Outcome and Event

Probability – What are Complementary Events

Example 1, 2, 3, 4 – Chapter 14 – Class 10 | Probability | NCERT Maths | CBSE

Example 5 – Chapter 14 – Class 10 | Probability | NCERT Maths | CBSE

Example 6 – Chapter 14 – Class 10 | Probability | NCERT Maths | CBSE

Example 7 – Chapter 14 – Class 10 | Probability | NCERT Maths | CBSE

Example 8 – Chapter 14 – Class 10 | Probability | NCERT Maths | CBSE

Example 9 – Chapter 14 – Class 10 | Probability | NCERT Maths | CBSE

Example 10 – Chapter 14 – Class 10 | Probability | NCERT Maths | CBSE

Example 11 – Chapter 14 – Class 10 | Probability | NCERT Maths | CBSE

Probability – Sure Event and Impossible Event

Example 12 – Chapter 14 – Class 10 | Probability | NCERT Maths | CBSE

Example 13 – Chapter 14 – Class 10 | Probability | NCERT Maths | CBSE

Probability – What are Elementary Events

Question 1 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 2 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 3, 4, 5 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 6 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 7 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 8, 9 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 10 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 11 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 12 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 13 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 14 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 15 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 16 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 17 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 18 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 19 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 20 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 21 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 22 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 23 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 24 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

Question 25 Exercise 14.1 Chapter 14 | Class 10 | Probability | NCERT Maths | CBSE

The post Class 10 | Chapter 14 | All Solutions | Probability appeared first on Mathemafia.

Class 10 | Chapter 13 | All Solutions | Statistics (WIP)

Sunil Syal — Fri, 17 Dec 2021 07:43:48 +0000

This playlist covers all the questions, concepts and examples of chapter 14 (Statistics) of class 10, NCERT book. Click a thumbnail to watch the tutorial.

Watch complete chapter on YouTube

Example 1 – Chapter 13 – Class 10 | Part 1 (Finding mean by Direct Method) Statistics | NCERT Maths

Example 1 – Chapter 13 – Class 10 | Part 2 (Assumed mean and step deviation Method) Statistics

Why Virat Kohli's Average is 98.67 in T20 World Cup 2022 | How to find batting average

Example 2 – Chapter 13 – Class 10 | Statistics | NCERT Maths | CBSE

Example 3 – Chapter 13 – Class 10 | Statistics | NCERT Maths | CBSE

What are class marks in statistics? Learn with simple examples.

Example 4 & Example 5 – Chapter 13 – Class 10 | Statistics | Introduction to Mode of Grouped data

Example 6 – Chapter 13 – Class 10 | Statistics | NCERT Maths | CBSE

Best tip to quickly remember formula of MODE (measure of central tendency)

🔥🔥 Question 4 – Exercise 13.2 | Class 10 | Chapter 13 | Statistics 💁👌😍 NCERT Maths | CBSE

Class 10, MEDIAN – Ex: 13.3 || Explained and solved in LIVE class

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Class 10 | Chapter 10 | All Solutions | Circles

Sunil Syal — Fri, 17 Dec 2021 07:41:58 +0000

This playlist covers all the questions, concepts and examples of chapter 10 (Circles) of class 10, NCERT book. Click a thumbnail to watch the tutorial.

Watch complete chapter on YouTube

Lengths of tangents from an external point are equal |Theorem 10.2| Class 10 |Circles | NCERT Maths

Example 1 – Chapter 10 – Class 10 | Circles | NCERT Maths | CBSE

Example 2 – Chapter 10 – Class 10 | Circles | NCERT Maths | CBSE

Example 3 – Chapter 10 – Class 10 | Circles | NCERT Maths | CBSE – Method 1

Example 3 – Chapter 10 – Class 10 | Circles | NCERT Maths | CBSE – Method 2 & 3

Theorem 9.7 Class 9 | Chapter 9 | Circles – Angle subtended by an arc at the centre | NCERT Maths

Equal Tangent Segments: Proving Congruence from an External Point

The post Class 10 | Chapter 10 | All Solutions | Circles appeared first on Mathemafia.

Class 10 | Chapter 8 | All Solutions | Introduction to Trigonometry

Sunil Syal — Fri, 17 Dec 2021 07:41:58 +0000

This playlist covers all the questions, concepts and examples of chapter 8 (Introduction to Trigonometry) of class 10, NCERT book. Click a thumbnail to watch the tutorial.

Watch complete chapter on YouTube

Trigonometry 💁👌😍 How to memorise values of Trigonometric ratios of some specific angles

Introduction to Trigonometry | Concepts and terms explained

Trigonometry Special 🔥 Relationship between sine and cosine of complementary angles

Example 1 – Chapter 8- Class 10 | Introduction to Trigonometry | NCERT Maths | CBSE

Example 2 – Chapter 8 – Class 10 | Introduction to Trigonometry | NCERT Maths | CBSE

🔥 Example 3 Chapter 8 Class 10 🔥 Introduction to Trigonometry | NCERT Maths | CBSE

Example 4 – Chapter 8- Class 10 | Introduction to Trigonometry | NCERT Maths | CBSE

Example 5 – Chapter 8- Class 10 | Introduction to Trigonometry | NCERT Maths | CBSE

Question 3, 4, 5 Exercise 8.1 | Class 10 | Chapter 8 |Introduction to Trigonometry |NCERT Maths

Trigonometry – How to derive values of all Trigonometric ratios for 45 degrees

Trigonometry – How to derive values of all Trigonometric ratios for 30, 60 degrees

Why sin 0 degree is 0 🔥 Class 10 – Chapter 8 👩‍🎓 Introduction to Trigonometry #mathemafia

Why sin 90 degree is 1 | Class 10 – Chapter 8 Introduction to Trigonometry | 90°

Example 6 – Chapter 8- Class 10 | Introduction to Trigonometry | NCERT Maths | CBSE

Example 7 – Chapter 8- Class 10 | Introduction to Trigonometry | NCERT Maths | CBSE

Example 8 – Chapter 8- Class 10 | Introduction to Trigonometry | NCERT Maths | CBSE

Trigonometric Ratios of Complementary Angles

Example 9, 10, 11 – Chapter 8 – Class 10 | Introduction to Trigonometry | NCERT Maths | (Old Book)

Question 1, 2 Exercise 8.3 | Class 10 |Chapter 8 | Introduction to Trigonometry |NCERT (old Book)

Question 6, 7 Exercise 8.3 | Class 10 |Chapter 8 | Introduction to Trigonometry |NCERT (old Book)

Basic Trigonometric Identities 🔥 Super easy to remember 🔥 Amazing tip #mathemafia

Example 9, 10 – Chapter 8 – Class 10 | Introduction to Trigonometry | NCERT Maths CBSE

Example 11 – Chapter 8 – Class 10 | Introduction to Trigonometry | NCERT Maths | CBSE

Tricky Example 12 – Chapter 8 – Class 10 | Introduction to Trigonometry | NCERT Maths | CBSE

2nd method 🔥 Question 4 (v) of Ex: 8.3, class 10 🔥 Introduction to #trigonometry || #class10math

Question 7, Exercise 8.1, Class 10 | Chapter 8 | Introduction to Trigonometry | NCERT Maths | CBSE

The post Class 10 | Chapter 8 | All Solutions | Introduction to Trigonometry appeared first on Mathemafia.

Class 10 | Chapter 9 | All Solutions | Some Applications of Trigonometry

Sunil Syal — Fri, 17 Dec 2021 07:41:45 +0000

This playlist covers all the questions, concepts and examples of chapter 9 (Some Applications of Trigonometry) of class 10, NCERT book. Click a thumbnail to watch the tutorial.

Watch complete chapter on YouTube

Example 1 – Chapter 9 – Class 10 | Some Applications of Trigonometry | NCERT Maths | CBSE

Example 2 – Chapter 9 – Class 10 | Some Applications of Trigonometry | NCERT Maths | CBSE

Example 3 – Chapter 9 – Class 10 | Some Applications of Trigonometry | NCERT Maths | CBSE

Example 4 – Chapter 9 – Class 10 | Some Applications of Trigonometry | NCERT Maths | CBSE

Example 5 – Chapter 9 – Class 10 | Some Applications of Trigonometry | NCERT Maths | CBSE

Example 6 – Chapter 9 – Class 10 | Some Applications of Trigonometry | NCERT Maths | CBSE

Example 7 – Chapter 9 – Class 10 | Some Applications of Trigonometry | NCERT Maths | CBSE

Angle of Elevation, Angle of Depression and Line of Sight

Very tricky Question 14, Ex 9.1, Class 10, Ch 9 | Some Applications of Trigonometry, NCERT Maths

मजेदार Question 15, Ex 9.1, Class 10, Ch 9 | Some Applications of Trigonometry | NCERT Maths | CBSE

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Class 10 | Chapter 12 | All Solutions | Surface Areas and Volumes

Sunil Syal — Fri, 17 Dec 2021 07:40:14 +0000

This playlist covers all the questions, concepts and examples of chapter 13 (Surface Areas and Volumes) of class 10, NCERT book. Click a thumbnail to watch the tutorial.

Watch complete chapter on YouTube

Example 1 – Chapter 12 – Class 10 | Surface Areas and Volumes | NCERT Maths | CBSE

Example 2 – Chapter 12 – Class 10 | Surface Areas and Volumes | NCERT Maths | CBSE

Example 3 – Chapter 12 – Class 10 | Surface Areas and Volumes | NCERT Maths | CBSE

Example 4 – Chapter 12 – Class 10 | Surface Areas and Volumes | NCERT Maths | CBSE

Example 5 – Chapter 12 – Class 10 | Surface Areas and Volumes | NCERT Maths | CBSE

Example 6 – Chapter 12 – Class 10 | Surface Areas and Volumes | NCERT Maths | CBSE

Example 7 – Chapter 12 – Class 10 | Surface Areas and Volumes | NCERT Maths | CBSE

Example 8 – Chapter 13 – Class 10 | Surface Areas and Volumes | NCERT Maths | CBSE

Example 9 – Chapter 13 – Class 10 | Surface Areas and Volumes | NCERT Maths | CBSE

Example 10 – Chapter 13 – Class 10 | Surface Areas and Volumes | NCERT Maths | CBSE

Example 11 – Chapter 13 – Class 10 | Surface Areas and Volumes | NCERT Maths | CBSE

Tips to remember Frustum of a cone Formulas | How to remember these formulas | कैसे याद रखें

Example 12 – Chapter 13 – Class 10 | Surface Areas and Volumes | NCERT Maths | CBSE

Example 13 – Chapter 13 – Class 10 | Surface Areas and Volumes | NCERT Maths | CBSE

Example 14 – Chapter 13 – Class 10 | Surface Areas and Volumes | NCERT Maths | CBSE

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