## (13 June) Separate NCERT Solutions of Inverse Trigonometric Functions Class 12 Maths & Other updates

Hey guys,

In videos mein sirf NCERT Solutions hain. Exercise 2.2 YouTube par bhi available hai. Baki sab kuch website par.

**PHYSICS**

Pichle kuch dino jis tarah aapne mujhe support diya hai, mere decisions ko support kiya hai, use dekh kar lag rha hai ki isi saal main Maths aur Physics ka syllabus Class 11 aur 12 dono ke liye aasaani se complete kar paunga. Main Physics tab se start karna chahta hun jab mujhe khud par bharosa ho ki main regular class 11 aur class 12 dono ke lectures upload kar paaunga. Taaki aapko bhi bharosa rahe ki syllabus pura ho jaayega. Isliye abhi puri jaan laga kar Maths ka syllabus end karne mein laga hun. Ummeed hain ki bahut jaldi complete ho jaayega. Main literally din aur raat kaam kar rha hun. Raat zayadatar website ko set karne mein lagata hun taki din mein website par lectures dekhte waqt aap disturb na ho. Jis raat website ko edit nahi karna hota us raat notes aur assignments par kaam karta hun. Taaki agle din sirf lectures record kar sakun.

**MATHS**

Aaj dopahar ko Class 12 ki second book ke lectures record karunga. First book ke jo topics pending hain un par thoda sa kaam aur karna hai. Isliye use ek do din baad record karna start karunga. Uske liye aapki help ki need bhi hogi. Second book record karna start kar dun fir aapse is baare mein aage discuss karta hun. Class 11 walon ko lag rha hoga ki main unke liye kaam nahi kar rha. Lekin aisa nahi hai. Class 11 ka kaafi had tak syllabus last year hi complete kar diya tha. Class 11 ke jo topics pending hain unka Class 12 mein bahut zyada use hai. Jaise Probability Chapter hai. Jahan par class 11 mein probability khatam hoga theek wahi se class 12 ka start hoga. Ab dikkat ye hai ki NCERT me Probability ke aise kuch topics hain jinhe 11th mein khul kar nahi explain kiya hain. Lekin 12th mein unke next level ke questions hain. To mujhe yahi decide karna hai ki un topics ko 11th ke end wale lectures mein explain karun ya 12th ke start wale lectures mein. Wo topics dono class ke liye zaruri hain. Is par main kaam kar rha hun. Aisi hi dikkat dusre chapters ke sath bhi hai. To 11th ke lectures mein thoda aur time lag sakta hai. Tab tak Class 11 ke students baaki chapters apne prepare karein. Sabhi lectures aapko website par aasaani se mil jaayenge. Aap befikar rahein, bache hue topics jaldi hi upload honge. Class 11 ka syllabus pehle khatam karna zaruri hai kyonki unke bina 12th ka syllabus bhi complete nahi ho paayega.

Aap sabhi ko bahut saara pyaar…

Mujhe aur mere decisions ko Support karne ke liye bahut bahut shukriya🙏

## (7 June)Two more lectures of Class 12 Inverse Trigonometry

Hey guys,

Wednesday sabse zyada busy din hota hain to kal lectures record karne ka mauka nahi mila. Isliye aaj ye do lectures record kiye hain.

Dono Class 12 Maths ke hain aur website par upload kar diye hain.

Lectures record to nahi kar paya kal, lekin website edit kar di hain… Class 10 ke jo chapters abhi tak record kiye hain wo sabhi wahan par available hain. Index ko bhi aur better bana diya hai.

Ab website par koi badi editing ki need nahi hai. Aapke feedback ke according Chhote chhote changes main karta rahunga. To ab lectures ke liye aur zyada time mil jaayega.

Aaj 2 lectures upload kiye hain… Jaldi hi speed badha kar 4 se 5 lectures daily upload karunga. I am trying my best…

Kal ek Class 11 Maths ka lectures bhi jarur upload hoga.

May God Bless you all,

Happy learning…

## (6 June)Class 11 Physics Lectures are available now…

Hey guys, https://www.ashishkumarletslearn.com -> W

ebsite par Class 11 Physics ke lectures arrange kar diye hain. Aur abhi tak jo notes banaye hain, wo bhi wahan download ke liye available hain… To aap website par ab Physics ke lectures dekh sakte ho aur notes download kar sakte ho… Jaldi hi Physics continue karunga 11th aur 12th dono ke liye…

-> Wednesday thoda busy hota hai mera, isliye aaj ka lecture ho sakta hai thoda late aaye but ab se roz lectures upload honge… Meri puri koshish hai ki roz ke 4-5 lectures upload karun…

-> Website par video quality aur speed ka option set karne ki koshish raat ko ki, lekin kaam nahi bana… Aaj raat ko fir se try karunga…

-> Website se related aur koi feedback ho to zarur batana, taaki aapko padhai karne mein asaani ho sake…

Aap aise hi mere initiative ko apna support dete rehna aur koshish karna ki website par hi lectures dekhna…

Shukriya..

Aap sabhi ko Eid Mubarak ho!😊

## (5 June)Inverse Trigonometric Functions Lecture 3 Part 1 available on the website

Class 12 Maths Chapter 2 Inverse Trigonometric Functions Lecture 3 Part 1 available on the website. Website Link: https://ashishkumarletslearn.com/ This lecture will be available on YouTube in a couple of days.

## NCERT Solutions for Class 11 Maths

Free education is the right of every student. Ashish Kumar – Let’s Learn is providing deep and detailed explanations of full syllabus, all important questions, all important examples and all NCERT solutions for Class 11 Maths through videos on YouTube Channel as well as Blogs and PDFs on website.

Students can learn through videos and blogs and can ask their doubts on Website’s Discussion panel or on YouTube’s Comments Page. Students will also be provided notes, assignments, books and various other educational resources in electronic forms like PDFs, Docs, mp4 etc., which will help them to prepare for CBSE Class 12 Board Exams but more importantly for their upcoming life’s adventures.

You can easily access all chapters with NCERT Solutions for class 11 maths on this page: https://www.ashishkumarletslearn.com/cbse/class-11/maths/

Following are summaries of chapter wise syllabus recommended by CBSE for Class 11 mathematics students with their YouTube as well as Website links.

Unit-I: Sets and Functions

1. Sets:

Sets and their representations. Empty set. Finite and Infinite sets. Equal sets. Subsets. Subsets of a set of real numbers especially intervals (with notations). Power set. Universal set. Venn diagrams. Union and Intersection of sets. Difference of sets. Complement of a set. Properties of Complement.

Ordered pairs. Cartesian product of sets. Number of elements in the Cartesian product of two finite sets. Cartesian product of the set of reals with itself (upto R x R x R). Definition of relation, pictorial diagrams, domain, co-domain and range of a relation. Function as a special type of relation. Pictorial representation of a function, domain, co-domain and range of a function. Real valued functions, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic and greatest integer functions, with their graphs. Sum, difference, product and quotients of functions.

Positive and negative angles. Measuring angles in radians and in degrees and conversion from one measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the identity sin2x+cos2x=1, for all x. Signs of trigonometric functions. Domain and range of trigonometric functions and their graphs. Expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple applications. Deducing identities. Identities related to sin2x, cos2x, tan2x, sin3x, cos3x and tan3x. General solution of trigonometric equations of the type siny = sina, cosy = cosa and tany = tana.

Unit-II: Algebra

Process of the proof by induction, motivating the application of the method by looking at natural numbers as the least inductive subset of real numbers. The principle of mathematical induction and simple applications.

Need for complex numbers, especially √ , to be motivated by inability to solve some of the quadratic equations. Algebraic properties of complex numbers. Argand plane and polar representation of complex numbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equations (with real coefficients) in the complex number system. Square root of a complex number.

Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation on the number line. Graphical solution of linear inequalities in two variables. Graphical method of finding a solution of system of linear inequalities in two variables.

Fundamental principle of counting. Factorial n. (n!) Permutations and combinations, derivation of formulae for and and their connections, simple applications.

History, statement and proof of the binomial theorem for positive integral indices. Pascal’s triangle, General and middle term in binomial expansion, simple applications.

Sequence and Series. Arithmetic Progression (A. P.). Arithmetic Mean (A.M.) Geometric Progression (G.P.), general term of a G.P., sum of n terms of a G.P., infinite G.P. and its sum, geometric mean (G.M.), relation between A.M. and G.M. Formulae for the following special sums.

Unit-III: Coordinate Geometry

10. Straight Lines:

Brief recall of two dimensional geometry from earlier classes. Shifting of origin. Slope of a line and angle between two lines. Various forms of equations of a line: parallel to axis, point -slope form, slope intercept form, two-point form, intercept form and normal form. General equation of a line. Equation of family of lines passing through the point of intersection of two lines. Distance of a point from a line.

## Unit-IV: Calculus

Derivative introduced as rate of change both as that of distance function and geometrically. Intuitive idea of limit. Limits of polynomials and rational functions trigonometric, exponential and logarithmic functions. Definition of derivative relate it to scope of tangent of the curve, derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions.

## आइये COORDINATE GEOMETRY शुरू करें!

COORDINATE = CO + ORDINATE

CO का मतलब होता है together और ORDINATE बना है order से |

GEOMETRY = GEO + METRY

GEO का मतलब होता है earth और METRY का मतलब to measure |

Geometry में हम अलग-अलग shapes के बारे में पढ़ते हैं | Generally, इसमें shapes की position या order पर ख़ास ध्यान नहीं दिया जाता | लेकिन Coordinate Geometry में हम उन shapes को उनकी proper positions और order के साथ पढ़ते हैं |

Mathematicians ने shapes की positions को define करने के लिए अलग-अलग तरह के systems बनाये हैं जिनमे दो सबसे ख़ास हैं, पहला Cartesian Coordinate System और दूसरा Polar Coordinate System. इस blog में हम Cartesian Coordinate System के बारे में जानेंगे |

CARTESIAN COORDINATE SYSTEM

ये system French Mathematician René Descartes के नाम पर रखा गया है | उसने ये system 1637 में दुनिया के आगे रखा था | इस system में mathematical shapes की positions और order को Straight Lines की मदद से represent किया जाता है | किसी shape की position को बताने के लिए कम से कम जितनी straight lines का इस्तेमाल किया जाता है, वो उतने ही dimension का system होता है | Example के लिए, जैसे की नीचे दिए diagram में दिख रहा हैं, एक rectangle के हर point की position को बताने के लिए दो perpendicular straight lines का इस्तेमाल किया जाता है इसलिए rectangle एक Two-Dimensional(2D) shape है और उस system को हम Two-Dimensional Cartesian Coordinate System कहते हैं |

हम जिस दुनिया में रहते हैं वो एक Three-Dimensional(3D) World है क्योंकि इसमें हर object की position को represent करने के लिए कम से कम तीन straight lines की ज़रूरत होती है | हालांकि, हम 1D और 2D दोनों के बारे में पहले पढेंगे | इसके दो reasons हैं, पहला ये कि 3D को इन दोनों के बिना समझना बहुत मुश्किल हो जाता है और दूसरा कि practical life में आने वाली कई 3D problems को अगर 1D या 2D में convert कर लिया जाए तो उनको solve करना काफ़ी आसान हो जाता है |

इस series के अगले blog में हम 1D और 2D के basics के बारे में पढेंगे |

इस Topic पर Video देखने के लिए इस link पर click करें: https://www.youtube.com/watch?v=xQ7zMyDOF78

Class 10 Mathematics के Chapter 7 पर based lectures और NCERT Solutions आपको इस link पर मिल जाएंगे : https://www.ashishkumarletslearn.com/cbse/class-10/maths/ncert-solutions/coordinate-geometry/

## More on Coordinate Geometry

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## More for Class 11 Maths

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## Why Online Learning is More Productive and Cost Effective?

We live in a world where technology is providing a way for online education to become more personalized. In today’s time nearly, everyone carries a connected device almost all the time. By learning using the new technologies in their day to day life become more equipped for success in all aspects as one can learn anything as per their convenient time. Online learning not only provide students the opportunity to plan their study time but help them to study and learn when they are at their peak energy, whether it’s an early morning, noon, evening or late at night.

As we all know that every student is different and learns differently, and with the help of technology, educators can accommodate unique learning styles by providing live examples of things, concepts through videos or animation which makes students’ foundation more stronger and education more interesting and easy for them.
While traditional education of the past focused on providing students with essential skills to turn them into skilled professionals, but in today’s time educators are more concerned with teaching students how to learn on their own. They are more concerned about making students learn independently with the help of online learning and they believe when they start learning online or by their own, there brain will become more empowered, experimenting with research and information. Here, the motive is teaching students to learn, think, explore and experiment.

Online learning is also a way cost effective than traditional learning. In traditional learning, one needs to spend thousands of rupees per month to attend a coaching class or hire special educator after paying huge amount of School fees. With all those amounts spends, busy classrooms, more students in one batch, not every student always get the chance to study exactly what he/she wants. There are many other expenses also that must be included in traditional learning i.e., transportation expense, additional fees for extra classes, extra notes other than books etc. whereas online learning not only saves huge amount of tuition fees but provide expert guidance, techniques, solutions, and help students to manage their busy schedule, learn the materials/notes that are available online, watching experts video to clear their concepts, and complete assignments on their own schedules and convenience.

When it comes to Internet or Google, where everything is available, one can access to the best, can watch top influencers/educators online video, find unique material from the top educators all around the world for their projects, notes and assignments. That’s where online education is more valuable than traditional education. As supposed, if I’m going to spend the kind of money that schools require for tuition, I want to make sure that I have the best possible lecturers or educators to provide me best education for my projects and notes whereas in an open environment, I can find the finest lecturer or educator online and absorb that information at my own pace. And as long as I have a connection to the Internet, I can do it for free.

One of the greatest achievement of online learning I feel is the enhanced entry for students, removing the limitation of travel to a specific location at an acknowledge time.

Also, helps one to learn anything at any place or time with the advancement of technologies, online resources and data.

As I always said” One should always seek to be an agent of positive change and progression for learners and students”.

With the help of technology, I started “FREE ONLINE CLASSES” on YouTube Channel (www.youtube.com/ashishkumarletslearn) for Maths & Physics Class IX, X, XI and XII CBSE.

## आइये Sequences and Series समझें!

इस blog में आपको Sequences and Series के बारे में पूरी जानकारी मिलेगी जो कि CBSE  Class 11 की NCERT के chapter 9 पर based हैं | साथ में आपको उन Video lectures के links भी मिल जाएंगे जिनमें मैंने इसके सारे concepts और एक-एक NCERT exercise question को काफी deeply और details के साथ explain किया है |

#### Meaning of Sequence and Series

सबसे पहले शुरुआत करते हैं Sequence के नाम के साथ | Sequence का हिंदी में meaning होता है “क्रम” यानि कि numbers या objects का ऐसा group जो किसी particular order में आ रहा हो | ये particular order सभी numbers या objects के साथ एक जैसा ही होता है | Sequence शब्द का इस्तेमाल हम maths में numbers के particular orders के लिए करते हैं और उन numbers को sequence की terms कहा जाता है | Practical life में इसका इस्तेमाल किसी भी object के साथ किया जा सकता है | Example के लिए:

1, 2, 3, 4, 5, 6, 7, … ये एक ऐसी sequence है जिसकी हर term के बीच में 1 का difference है |

5, 10, 20, 40, 80, 160, … ये एक ऐसी sequence है जिसकी हर अगली term, पिछली term का twice या दूगना है |

इसी तरह से अनगिनत तरीकों से sequence बनाई जा सकती है | sequence में हर term को हम उसकी position के हिसाब से “variableposition” की form में represent करते हैं | जैसे कि पिछले पहले example को ही लेते है;

1, 2, 3, 4, 5, 6, 7, …

इसमें अगर variable मैं “a” लेता हूँ तो सभी terms को इस तरह से लिखा जायेगा:

a1=1, a2=2, a3=3, a4=4, a5=5, a6=6, a7=7, ….

अब बात करते हैं series की | जब हम किसी sequence की सभी terms के बीच में addition का sign लगाते हैं तो उस sequence को series कहा जाता है | अब पिछली दो sequence को ही लेते हैं | अगर उन्हें इस तरह से लिखें:

1+2+3+4+5+6+7+…

5+10+20+40+80+160+..

तो ये series बन जाती हैं | Sequence और series में सिर्फ़ इतना सा ही फ़र्क नहीं हैं | series को हम  summations की form में आसानी से लिख सकते हैं |

#### General Term

अब बात करते हैं कि sequence और series की terms के साथ काम कैसे किया जाता है ? जैसे कि ऊपर बताया, हर terms के बीच में एक particular order होता हैं जिसकी वजह से हम हर term को एक common पहचान दे सकते हैं | एक ऐसी पहचान जिस से उस sequence या series में आने वाली किसी भी term को find किया जा सके | उस पहचान को ही General term कहा जाता है | फिर से पिछले दो examples लेते हैं, लेकिन इस बार terms की positions के साथ |

पहली sequence:

 1, 2, 3, 4, 5, 6, 7,… a1 a2 a3 a4 a5 a6 a7,…

अब इस sequence में जो term की position हैं वही term की value भी है, तो इसी common बात को देखते हुए इसकी general term होगी: an = n

दूसरी sequence को हमें थोडा modify करना होगा ताकि उनका common relation पता चल सके:

 5, 10, 20, 40, 80, 160,… 5(1), 5(2), 5(4), 5(8), 5(16), 5(32),… 5(2)0, 5(2)1, 5(2)2, 5(2)3, 5(2)4, 5(2)5,… a1 a2 a3 a4 a5 a6,…

इस sequence की हर term में 5(2) है और हर term में (2) की power term की position से एक कम है, तो इसकी general term होगी: an=5(2)n-1

General term का सबसे बढ़ा फ़ायदा ये है कि हम इससे उस sequence या series में आने वाली कोई भी term find कर सकते हैं | जैसे कि दूसरी sequence की general term को अगर लिया जाए:

an=5(2)n-1

अब अगर इसकी 10th term हमें find करनी है तो बस n कि जगह पर 10 रखना होगा;

a50=5(2)10-1=5(2)9=5(512)=2560

इसका मतलब इस sequence की 10th term 2560 है |

इस blog पर based video lecture देखने के लिए यहाँ click करें: https://www.youtube.com/watch?v=I_wWn6KN1P0

Class 10 Mathematics के Chapter 5 पर based lectures और NCERT Solutions आपको इस link पर मिल जायेंगे: https://www.ashishkumarletslearn.com/cbse/class-10/maths/ncert-solutions/arithmetic-progression/

Class 11 Mathematics के Chapter 9 पर based lectures और NCERT Solutions आपको इस link पर मिल जायेंगे: https://www.ashishkumarletslearn.com/cbse/class-11/maths/ncert-solutions/sequences-and-series

## More on Sequences & Series

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## More for Class 11 Maths

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