Math Olympiad, Indian Statistical Institute, Chennai Mathematical Institute and Institute of Mathematics and Applications aspirants will find useful mathematics in this blog. Visit www dot cheenta dot com (our official website).

Sunday 26 June 2011

3rd July 2011 PREVIEW (class XII evening)

The class will start precisely at 5 PM. The basic topics to be covered are:
  1. RMO 1997 paper
  2. Integral Calculus Concept Building continued (Apostol Ch.1)
  3. Passage 64 -66 from M.L.Khanna - 25 problems.
  4. Geometry sheet 1 completed
We will have diagnosis test and effect test.

Homework:
  1. Calculus M.L. Khanna Passage 64, 65, 66
  2. Geometry - exercises from challenges and thrills of pre-college mathematics (Ex. 3.4 to 3.6)
  3. RMO 1998

3rd July 2011 PREVIEW (class XI afternoon)

The class will start precisely at 1:30 PM. The basic topics to be covered are:
  1. RMO 1993 paper
  2. Trigonometry Problem Solving - 25 problems from passage 32 to 34 of M.L. Khanna
  3. Inequality Sheet 2 final part
We will have diagnosis test and effect test.

Homework:
  1. Calculus M.L. Khanna Trigonometrical Identities Complete (Trigonometry)
  2. Inequality - exercises from challenges and thrills of pre-college mathematics
  3. RMO 1994

Friday 24 June 2011

26th June 2011 PREVIEW (class XI afternoon)

The class will start precisely at 1:30 PM. The basic topics to be covered are:
  1. RMO 1992 paper
  2. Trigonometry Problem Solving - 25 problems from passage 32 to 34 of M.L. Khanna
  3. Inequality Sheet 2
We will have diagnosis test and effect test.

Homework:
  1. Calculus M.L. Khanna Passage 32 to 35 (Trigonometry)
  2. Inequality - exercises from challenges and thrills of pre-college mathematics
  3. RMO 1993

Sunday 19 June 2011

26th June 2011 PREVIEW (class XII evening)

The class will start precisely at 5 PM. The basic topics to be covered are:
  1. RMO 1996 paper
  2. Integral Calculus Concept Building continued (Apostol Ch.1)
  3. Passage 63 from M.L.Khanna - 25 problems.
  4. Geometry sheet 1 (Hall and Stevens till Theorem 22)
We will have diagnosis test and effect test.

Homework:
  1. Calculus M.L. Khanna Passage 64, 65, 66
  2. Geometry - exercises from challenges and thrills of pre-college mathematics (Ex. 3.1, 3.2, 3.3)
  3. RMO 1997

Saturday 18 June 2011

CHEENTA CALCULUS COURSE


(this course is a part of Cheenta Ganit Kendra's I.S.I. entrance preparation program, but it is also available as a standalone course)
  • Prerequisite: Trigonometry, Algebra, Coordinate Geometry
  • Books: M.L.Khanna's IIT Mathematics, Maron, Tarasov, Apostle, Piscunov
  • Duration:
    • Ground Work - 5 classes
    • Differential Calculus - 10 classes
    • Integral Calculus - 10 classes
  • Course Structure
    • The course consists of rigorous and thorough problem solving from M.L.Khanna's IIT Mathematics book (Passage 45 to 73 of 2012 edition). We spend 20 classes (or about 40 hours) on these 28 passages.
    • I.S.I. centric concept building and relevant problem solving is pursued in 5 classes (or 10 hours). In these classes we walk through the excellent works of Tarasov, Maron, Apostle, Piscunov

Thursday 16 June 2011

13th to 19th June 2011 - a summary of classes at Cheenta

Some of these classes are over. Some of them are about to come. Wondering how to use them? Follow the following easy steps:
  1. If the class has already been conducted, and you wish were present, do not worry. You can always request for a specific class. On the basis of your request we will come back with the session once more as soon as possible.
  2. A class is coming up and you want to attend it. But you are worried, 'I am not a regular student of that batch, will I be allowed?'. Feel free to request permission to attend the session. If the seats are not already filled up, we will happily allow your entry.
14th June (Tuesday) and 16th June (Thursday) - LIMIT, CONTINUITY CONCEPT SESSIONS

Special discussion seminar on Limit and Continuity. The sessions were designed to train students appearing for interview in I.S.I. Naturally the sessions were very interesting. We covered many hitherto less discussed topics like Bolzano Weierstrass Theorem, Axioms of Real Numbers (including Completeness Axiom), Cauchy's definition of limit of a sequence and function, idea of continuity, intermediate value theorem etc. Saikat cracked B.Math and I.M.A entrance this year. Along with him, a new entrant Megha experienced the sessions. The students and the faculty were thrilled by the depth of the classroom discussion. We have decided to put the session in the upcoming Cheenta journal on Advanced Precollege Mathematics (we have named it 'Leelavati')

17th June, Friday, INEQUALITY + TRIGONOMETRY (1st session)

We will discuss the basic mean-power inequality, introduce Bernoulli Inequality and also the intuitive idea of Xeno's paradox. The Trigonometry portion will define the 'angle' (but everybody knows that isn't it!). We will also go into the first layer of properties of triangle. RMO 1990 and IMO 1959 will also be under scanner

18th June Saturday, JUNIOR OLYMPIAD GEOMETRY SESSION

This is for the students of class 5-6. But even the experienced geometer of high school may benefit from the class. We will continue working on some of the constructions of geometrical figures pointing out ways to prove them analytically.

19th June Sunday (morning), JUNIOR OLYMPIAD GEOMETRY 2ND LAYER

This is for students of 7-8. Definitely advanced students of lower classes will also benefit from the class. We will face a marathon test for the first time. The syllabus is till 31st Theorem of Euclid (volume I). The students from higher classes are advised to take the test.

19th June Sunday (Afternoon), SENIOR SESSION (Class XI) on INEQUALITIES AND TRIGONOMETRY

This session is designed to achieve certain height in Inequalities. We work on Problem sheet 2 (covering Bernoulli, Cauchy Schwarz, Rearrangement and more). In Trigonometry we troubleshoot and solve till Problem set 33 of M.L. Khanna IIT Mathematics. We also work on RMO 1991.

19th June Sunday (Evening), SENIOR SESSION (Class XII) on GEOMETRY AND CALCULUS

We start off with the remaining portion of RMO 1995. Then we switch over to Geometry (Euclid School Geometry). We cover atleast till Theorem 22. Homework will be Riders from Challenges and Thrills of Pre College Mathematics. Calculus portion will experience two apparently different things. We reintroduce integration as a limit of a sum and discuss some problem solving techniques from M.L. Khanna. We also initiate discussion on Limit of a sequence.

Stay tuned for more mathematics at Cheenta Ganit Kendra.

Tuesday 7 June 2011

Toward the Bolzano Weierstrass theorem from scratch

Understand that mathematical systems are built on 'taken for granted' axioms (or accepted truths. These axioms are 'accepted' to be true without any proof. Any proof of any mathematical law is built on a set of axioms.

We will state one such axiom for real number system called the Completeness Axiom. From wikipedia we quote "The axiom states that every non-empty subset S of R that has an upper bound in R has a least upper bound, or supremum, in R"

Let us understand the different terms in the axiom or given truth:
  1. R is bold = This symbol denotes the set (that is collection) of real numbers.
  2. Real Number = collect all the fractions, integers, square-roots of non-square numbers and other irrational numbers like e, \(\pi\), and put them in one set. This makes the set of real numbers.
  3. subset of R = collection of some (distinct) numbers from the set of real numbers.
  4. upper bound = take the literal meaning
  5. least upper bound = again take the literal meaning
  6. supremum = another word for 'least upper bound'
Now we are set to proof a simple form of Bolzano - Weierstrass theorem. Understand that theorems are based on logical conclusions derived from axioms by reasoning. So we will use our Completeness Axiom to build the proof of the theorem.

Statement: If \(a_k\) is a monotone sequence of real numbers (e.g., if \(a_k ≤ a_{k+1}\), then this sequence has a finite limit if and only if the sequence is bounded.

Please note that this is NOT the original statement of the Bolzano Weierstrass theorem. But for the purpose of high school mathematics we can use this clone. Stay tuned and we will continue


Monday 6 June 2011

understanding limit and continuity - brief sketch of a Cheenta Session

The limit and continuity is considered as the most important 'idea' in Analysis (for our purpose 'calculus'). Unfortunately deep understanding of these to concepts are beyond the reach of school level mathematical culture of India. Cheenta presents a series of intensive classes that puts the knowledge thirsty student in the right track.
  1. Understanding monotonic and bounded sequences by definition and examples.
  2. Considering the optional existence of analytical expression (eg. series of primes) and graphical expression (Dirichlet Sequence).
  3. Considering the presence of monotonicity and boundedness in the same sequence (Xeno's paradox, eye of Horas). Considering the case of \((1+ \frac{1}{n})^n\) and  \((1+ \frac{1}{n})^{n+1}\)
  4. The epsilon definition of limit of a sequence
    1. The number a is said to be the limit of a sequence ( \(y_n\) ) if for any positive number \(\epsilon\) there is a real number N such that for all n > N the following inequality holds: | \(y_n\) - a | < \(\epsilon\).
  5. Some problems concerning trivial limit calculations.
  6. Some theorems related to limit of a sequence:
    1. If a sequence has a limit it is bounded.
    2. If a sequence is both bounded and monotonic, it has a limit. (Weierstrass)
    3. A convergent sequence has only one limit.
  7. A schematic approach to handle boundedness, monotonicity, convergency.
  8. Some more theorems on limit of a sequence:
    1. If a sequence (\(y_n\) ) and (\(z_n\) are convergent (we denote their limits by a and b respectively), a sequence (\(y_n + z_n\) is convergent too, it's limit being a+b.
    2. Converse of the above theorem is not necessarily true. Very important point and is illustrated by an example. (if sum is convergent, then both are convergent or both are divergent but not otherwise).
    3. \(\displaystyle\lim_{n\to\infty}c y_n = c \displaystyle\lim_{n\to\infty}y_n\)
  9. Understanding infinitesimal sequences (definition)
  10. Some theorems concerning infinitesimals
    1. If (\(y_n\)) is a bounded sequence and (\(\alpha_n\) is infinitesimal, then (\(y_n \alpha_n\) is infinitesimal as well.
    2. A sequence (\(y_n z_n\)) is convergent to ab if the sequences (\(y_n\)) and (\(z_n\)) is convergent to a and b respectively.
    3. If a sequence (\(y_n\) ) and (\(z_n\) are convergent (we denote their limits by a and b respectively when b is not 0), a sequence (\(\frac{y_n}{z_n}\) is convergent too, it's limit being \(frac{a}{b}\).
  11. Understand elimination, or addition, and any other change of a finite number of terms of a sequence do not affect either it's convergence or it's limit (if the sequence is convergent). We consider the case for the change of infinite terms as well.
  12. Finally we take up problem solving from I.A. Maron (testing convergence of sequences).
This discussion concludes the first session of Limit and continuity concept building.

Thursday 2 June 2011

M.L. Khanna IIT MATH tracked

We recommended this book as a high school math staple for ISI aspirants (note that I.S.I. entrance requires IIT MATH + OLYMPIAD MATH and M.L.Khanna takes care of the problem solving part of HIGH SCHOOL math only. For other parts refer to the books recommended in our book-list).
Edition 2012 has 131 passages each containing about 50 problems. We want to present discussion on each passage. It is a mammoth task. We will put contents to this note from time to time. Follow the active links. 

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