STUDY SYSTEMS

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STUDY SYSTEMS

• NANO---TECHNOLOGY OF EVOLUTION

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STUDY SYSTEMS

• 1.1-hour everyday
• practice of Pranayama
• for the Body-Intellect-
• Soul-Mind is to be
• enjoyed through! and
• through!

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STUDY SYSTEMS

• In a world based on advanced technology and
  evolving around the market, in which rapid change
  is the rule, we can hope to understand how
  changes are related to each other if we know what
  a variable is, know the meaning of function in
  mathematical sense, know how to derive
  interpret rates of changes. That is to say, in
  order to appreciate what our neighbour, our
  colleague, our media, or our government is saying
  , we must know some math what contributes to the
  development of this border/deeper kinda literacy!

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Study systems

• We need to be fit as possible, all physically,
  mentally, intellectually, spiritually to make the
  most of growth time---. Pranayama facilitates!
  Our key learning resources are inspiration
  concentration ----Pranayama facilitates!

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Study Systems

• Truly important is the work that we do on our
  own. Some work will be just for ourselves and
  may/ will not be seen/ assessed by others -

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Study Systems

• i. Doing practice
• examples problem
• solving

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Study systems

• ii. Working on deeper understanding

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Study systems

• iii. Reviewing /Cross -
• Referencing.

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Study Systems

• iv. Investigation / Research / Study of the
  works of different Mathematicians / Scientists

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Aryabhatta

• Indian Mathematician Aryabhatta

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Brahmagupta

• Great Mathematician

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Bhaskara

• Bhaskara was somewhat of a poet as were many
  Indian mathematicians at this time.

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John Napier

• John Napier was the Scottish Mathematician

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Reme Descartes
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Thales

• The seven sages a group of semi-legendary wise
  people from the Archaic age, often regarded as
  the founders of Greek philosophy, Thales was
  amongst them.

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Pythagoras

• Theorem of Pythagoras was invented by this great
  Mathematician

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Plato

• Wise man of the day Plato

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Euclid Archimedes

• Archimedes principle was invented by Euclid
  Archimedes

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Galileo

• A page from Galileos notebook

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Kepler

• Keplers second Law of Motion

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Pascal

• Animation Pascals Law.

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Newton

• Newton discovered gravity when he saw an apple
  from a tree

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Leibniz

• One of the great Mathematician

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Michael Rolle

• One of the Mathematician

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Lagrange

• Lagranges Polynomial

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DAlembert

• D'Alembert made other important contributions to
  mathematics. He suggested that the theory of
  limits be put on a firm foundation. He was one of
  the first to understand the importance of
  functions and he defined the derivative of a
  function as the limit of a quotient of
  increments. His ideas on limits led him to the
  test for convergence, known today as d
  Alembert's ratio test.

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Varigon

• When the midpoints of any quadrilateral are
  joined in order, they form a parallelogram.  This
  theorem is due to a French mathematician named
  Pierre Varignon . The resulting parallelogram is
  named in his honor

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Euler

• Euler was a mathematician

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2. Study Habits Formation/ Organisation

• Most folks know how to do the job better than
  what they are doing.But they are not able to do
  so.
• Most kids know how to get better marks than what
  they have been getting but they dont seem to be
  improving on that. Their action what they know
  do not go togetherthe rot cause being that they
  havent applied their knowledge/ understanding to
  form good habits---habits are cruces.

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2. Study Habits Formation/ Organization

• Know to your subconscious level the
  multiplication tables up to 25,squares
  squares-roots up to 20, cubes cube-roots up to
  15,and varied equivalences among per cent
  fractions ratios, and your math would be
  faster, youll be able to use the power of the
  subconscious to tackle the tricky problems/
  tests.

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Study Habits Formation/ Organisation

• By forming an the habit of doing an activity you
  are able to do it with complete ease almost
  automatically---true for both good bad habits.
  When you do an activity for the first time, you
  have to think of each step perform the
  activity. But when you keep doing the activity
  repetitively, you develop habit of doing that---

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2 Study Habits Formation/ Organisation

• For any test/ examination you have to be at your
  conceptual best during test/ examination hours.
  If you need to recall the theories/ concepts/
  formulae while solving the problems in the
  examination, when you will not be able to attempt
  sufficient number of questions to get you
  meritorious position, especially in a
  competitive examination, say IIT JEE--/
  concepts/ ideas to solve those problems should
  keep coming to your intellect.

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2 Study Habits Formation/
Organisation

• Habit formation uses the power of the
  subconscious most of our activities are dictated
  by the subconscious and so it is important to use
  its power. As you practice an activity become
  independent of conscious thinking during the
  performance of the activity , you achieve a state
  where the activity is performed automatically by
  the subconscious intellect.

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2 Study Habits Formation/ Organisation

• You need to develop the habit of understanding
  the theories well (especially, their derivation)
  thinking conceptualizing those theories ,
  applying them in solving conceptual problems
  continually developing better techniques/
  alternatives to solve the problemsbuilding
  theories.

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2 Study Habits Formation/ Organisation
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2 Study Habits Formation/ Organisation

• You can reduce the occurrence of silly mistakes
  by becoming careful in that---
• we first develop a habit , then it rules
  us.' It's better to start developing these
  creative habits at the earliest. Developing a new
  habit is easier compared to changing a
  well-formed habit.

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3 Absorption of Elements / Fundamentals.

• It is absolutely essential that you understand
  the basic principles of Mathematics, Sciences in
  so much as problems are based on certain
  well-articulated principles of Mathematics
  Science
• While studying a principle or solving a problem,
  pay attention to words used in sentences. Never
  consider any word redundant either in principles
  or in problems---word throw a lot of information
  about the formulation of theorem as well as
  situations of problems all laws, formulae,
  equations, transforms have many limitations.

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3 Absorption of Elements / Fundamentals

• Be absolutely sure about their limitations pore
  over between the lines the fine print/ small
  print, the large print, the bold print all the
  conditions/parameters allied in the law/
  theorem/formula/ concept/ theory/ ---also learn
  to deal with the situation where the laws,
  formulae, equations fail.

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3 Absorption of Elements / Fundamentals

• While practising if you are not able to solve a
  problem, dont rush for the provided solution.
  Apply your imagination further, may be youre
  just overlooking certain part of theory implied
  in the problem. Also think about the assumptions
  that can simplify the problems, reduce the
  calculations.

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3 Absorption of Elements / Fundamentals

• In Science Mathematics, a good diagram is a
  very valuable asset. Try drawing a clean proper
  diagram whenever required during the course of
  study or a problem. Also, think of drawing
  successive simplifications which will lead to
  precise understanding of either a theory or a
  problem.

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3 Absorption of Elements / Fundamentals

• The better better insight of a theory is gained
  while solving the stiffer stiffer
  problemsproblems-solving gives you the insight
  into the type of intellectual skills that are
  required in developing mastery over the subject.

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3 Absorption of Elements / Fundamentals

• In Mathematics, most of its concepts and the
  theoretical results are introduced as the basics.
  Theory in Math is developed in the abstract and
  stands on its own. Math makes you logical
  analytical, when you sift information draw
  reasonable inferences, as every succeeding step
  is drawn from the preceding by using sound logic/
  algorithm. Explore mathematical ideas and justify
  their reasoning .

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3 Absorption of Elements / Fundamentals

• Any Mathematical problem can be tackled by
  employing more than merely one logical or
  analytical approach as Problem X could be solved
  trigonometrically as well as by Vector Algebra,
  too.

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3.Absorption of Elements / Fundamentals

• The aim outcome of good teaching should be to
  develop the skill, intuition capacity in
  students to think of mathematical ideas without
  these being explained to them by somebody, and
  solve problems through methods not taught to
  them. This is possible only if the same is
  understood practiced by teachers both
  teachers students must become autonomous
  learners. A good teacher should be able to create
  curiosity in students.

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4 Knowledge vis-à-vis Information

• Aptitude is a function of time. Learning is a
  slow process it does demand that one spends
  time efforts to understand a concept/ topic/
  be able to apply. One learns best through an
  acquisition of basic cognitive process underlying
  all contents rather than contents.

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4 Knowledge vis-à-vis Information

• In Mathematics sciences one has to be more
  concerned with basic structures
• concepts than numerical drill---there is a
  world of difference between information
  knowledge.

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4 Knowledge vis-à-vis Information

• Information is making the student memorize the
  statements of law, say, Law of Logarithm, in
  45-minute period while knowledge is spending
  four periods or so in explaining the meaning of
  each every word in these statements of Law of
  Logarithmic Laws, deriving the laws making
  the Students apply these formulae in solving a
  number of practical problems.

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4)Knowledge vis-à-vis Information

• The proof, the derivation of the Law/ formula is
  really what Mathematics is all about, what
  Sciences are all about.
• CASE STUDY
• Lets have a close look at typical student
  from class VIII. He computes simple interest
  formula I(Pxrxn) /100. During the course of the
  year he solves more than 40 similar problems. He
  is merely computing the product of P,r and n and
  dividing it by 100. He then checks this number
  with the number given at the back of the book. If
  it does not tally it computes it again. All these
  efforts only ensure that the student has correct
  knowledge about how to compute simple interest.
  To gain the knowledge he must understand why this
  formula really computes the Simple Interest-----
  the proof, the derivation of the formula
  (theoretically) is realistically the crux, is
  really what Mathematics is all about.

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5 Functioning of / Sharpening the
Intellect.

• One should not accept any formula without proof
  similarly, one ought to learn the justification/
  rigour of each every geometric construction.
  For this, one ought to learn more about ones
  most important instrument the intellect/ brain
  ---its functioning. You have more than 1000
  volumes in your library, say. If you buy a new
  book arbitrarily place it somewhere then it
  will be very difficult for you to retrieve the
  book. You should certainly not keep a math book
  along the physics books.

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5 Functioning of / Sharpening the
Intellect.

• You should certainly not keep the math book along
  the physics book. Among the math books, you must
  separately store the books on Arithmetic,
  Algebra, Geometry, Probability, Calculas, etc.
  Among the Geometry books, it should be further
  classified to Euclidean Geometry, Coordinate
  Geometry, Differential Geometry, Algebraic
  Geometry, Fractal Geometry so on.

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5 Functioning of / Sharpening the
Intellect.

• In our brain we have an astonishing amount of
  information stored. So any new concepts we learn
  we must be extremely careful in keeping it in the
  correct place in the brain.
• Now how do we really do it- If you study a new
  idea in isolation then in a very short time you
  will forgot it . You must make a conscious effort
  to associate the new idea with a number of
  similar ideas already existing in the brain .

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5 Functioning of / Sharpening the
Intellect.

• Whenever you study a new concept / law, follow as
  many of the suggestions mentioned below-
• First try to think about how the discoverer of
  this result might have found this result.
• Try to prove this result independently. Who
  knows, your solution may be much simpler one.
• If you do not get the proof, just read a couple
  of lines of the proof with this starting point,
  try once again.

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5 Functioning of / Sharpening the
Intellect.

• Now relativistically you still have a lot of
  time, which you could utilize to understand the
  concept of Compound Interest derive the formula
  for the Amount in this case.
• You could create a number of formulae in the case
  of Compound Interest according as the interest is
  paid after each month or each day or even after
  each hour.

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5 Functioning of / Sharpening the
Intellect.

• d. Once the complete proof is understand, list
  all the theorems/ results, which you might have
  used in proving that theorem. Prove each of those
  theorems/ results.
• e. Find out the number of corollaries of that
  theorem.
• f. Find out the various ways in which that
  theorem could be generalized.
• g. Teach that theorem to your friends/ others.

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5 Functioning of / Sharpening the
Intellect.

• You could relate this concept / formula/ result
  to the population-growth problem and estimate the
  future population of our Bharat, that is, India
  or to the depreciation/ decay rateyoud truly
  appreciate exponential growth or decay
  application problem (11/n)n is similar to
• AP(1r/n)nt Pert , e incorporating the
  growth increase into the number ! This way you
  would start enjoying Mathematics/ Sciences.

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5 Functioning of / Sharpening the
Intellect

• Having mastered the concepts/ topics you could,
  now you may go through quickly all the problems
  in the textbook and identify that actually there
  are only 5 to 7 types of different problems in
  the concept. This way you would start developing
  the ability of sifting the essence from the
  gross, inculcate the habit of studying
  Mathematics/ Sciences / joyfully!.

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5 Functioning of / Sharpening the
Intellect

• Creativity and Innovation are not drilling and
  monotony. Discover the creativity innovation
  within yourself. Try to construct some new
  problems on the topic you have completed from the
  textbook.

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5 Functioning of / Sharpening the
Intellect

• Following are the reference books you should have
  in your personal library, besides reading as many
  books written by world-famous authors Polya,
  Sharygin, Coxeter, Yaglom, Martin Gardner, Ross
  Hansberger-
• a. Elementary Number Theory by David
• Burton.
• b. An Excursion in Mathematics---
• Bhaskaracharya Pratishthan.
• c. Mathematical circles A Russian Experience
  by Fomin.
• d. Challenge Thrill of Pre-college
  Mathematics by
• Krishnamurthy.
• e. Adventures in Problem-solving by Shailesh
  S.

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6 Fundamentals Mantras.

• i. Practice Pranayama regularly.
• ii. Link the topics/ chapters / units, i.e., make
  a chain of preceding succeeding topics / units
  in your study.
• iii. Go on reviewing what you go on studying---
• brushing up helps you remember the things
• with greater effectively.
• iv. Use your idle time to memorize formulae/ laws
  so as to utilize your free time judiciously.

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6 Fundamentals Mantras.

• v. Attempt to connect common life events with
  your studies , so as to analyze the events
  occurring everyday around you with scientific
  attitude----apple falling event arouses g in
  Sir Newton.
• iv. Learn the lessons by heart by with
• understanding.
• vii. Try tricky problems after small-time gaps.
  If
• problem has become just inconceivable for
• you, leave it for a while do something
  else,
• i.e., relax. Then come back to it.

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6 Fundamentals Mantras.

• viii. Handle the problem in three different
  phases
• a beginning, a development an end.
• ix. Of all the principles of time management,
• none is more basic than concentration. The
• amount of time spent on problem /topic is
  not
• what counts its the amount of
  uninterrupted
• / focused time, knowledge/ understanding/
• wisdom gained thereof. So your life style
  ought to be
• Pranayama-oriented.

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6 Fundamentals Mantras.

• x. Share your experiments / experiences with
  othersyour thoughts are further stimulated by
  doing so.
• xi. You can underline key points, number them in
  the margin and index them on the flyleaf.
• xii. Three or four difficult text are easier than
  one, for one
• book illuminates another get several books
  on the same
• subject.
• Xiii. In a relaxed frame of mind, you will be in
  a better
• position to identify the different links of
  various concepts
• in the same question and solve unknown
  problems.
• Pranayama enriches your EQ/ SQ.

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6 Fundamentals Mantras.

• xiv. The habit of referencing, i.e., consulting a
  
• dictionary and cross-references will make
• the learning more meaningful.
• xv. Organize your study, timetable-wise you
• do need a work-plan. Allow more time for
• the areas you are weak on.
• xvi.You will achieve more if you vary your
• revision, breaking up your study sessions
  into
• fairly shorter periods and challenging
  subjects
• frequently

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6. Fundamentals Mantras.

• xvii. Be scientific/ methodical in your
  approach. Let us take a very simple example of
  mathematics to explain it-
• Example X Prove that in acute-angled triangle
  ABC, a/sin Ab / sin Bc/ sin C.

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6 Fundamentals Mantras.

• (P). Start by drawing a diagram if it is
  relevant. In this case it is obviously essential,
  so begin with the triangle.
• (Q). Next write down any number of equations that
  are given and fill in on the diagram any
  information / facts you have in this case very
  little.
• Mark the standard notation for the elements of
  the triangle and also that although the equations
  in the problem are written as one line, these can
  be re-written as three separate equations.
• (R). Write down any other equations you may need
  add any constructions line to your diagram that
  may help with the solution. In this case the
  problem uses sine of an acute angle, so the
  sine function is probably going to be needed
• Sin y opposite side/hypotenuse of a right
  triangle y being acute .

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6 Fundamentals Mantras.

• (S). By drawing inside a perpendicular
• construction line from B to AC, you
• produce two right-angled triangles.
• (T). With the help of h you can now begin
• to formulate some relationship between the
  
• lengths a c and the angles A C. With
  the
• sine function you get
• sin Ah/c, sin C h/a.
• Your question is almost solved. A little
• algebra will show that c sin A c/ sin C.

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6 Fundamentals Mantras.

• (U). Have you finished the problem ? In
• this case you havent, since only one
• of the three equations has yet been
• proved. However, you can now go on
• to prove another of them in a similar
• way but with a different construction
• line. The fundamental point about
• answering Math problem is that you have
• to be algorithmic in your approach.

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6 Fundamentals Mantras.

• The summary of the method suggested in the
  Problem X above follows.
• (p) As with any other subjects, read the
  questions carefully, noting how many parts there
  are to it and thinking about it until you are
  quite sure you understand what you are being
  asked to do.
• (q) Draw any relevant diagram or graphs, using
  the information that is given to you in the
  problem. If a diagram is not needed, it may be
  useful to note down the information given to you.
• (r) Add to your diagram anything else that might
  be helpful. (construction lines, angles,
  assumptions, etc.).

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6 Fundamentals Mantras.

• (s) Write down the equation given others that
  you may need. Jot down relevant formulae as well.
  These steps are important, especially if you
  dont immediately see the way to solving a
  problem, since the very process of collecting and
  assembling the information often shows you how
  the problem can be tackled.

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6) Fundamentals Mantras.

• (t) Work through the questions stage-by-stage.
  The success in solving an early part of the
  problem may help you with another part later on
  in the question.
• (u) With problems it is very important that you
  show all the stages of your working fully
  clearly. If you arrive at end of question with
  the wrong answer, it may be that you have simply
  made a mistake somewhere. It will be easier for
  you to check for this find the mistake if you
  can actually see clearly what you have done. It
  also makes it possible for the teacher or
  examiner to see that, in spite of the error, you
  have understood the problem and know the methods
  of solving it, you may be awarded some credit.
• Thats trigonometric approach to solving
  Problem X.
• Alternatively, by

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6. Fundamentals Mantras.

• Vector Algebra, to prove
• a/sin A b/ sin Bc / sin C,
• you assume vectors a, b, c as
• marked in the figure. From the triangle
• Law of Addition Of Vectors, we have
• ab-c
• Or, abco----1
• On taking vector product with a, we get
• ax(abc) axo.
• A little vector algebra would yield
  axb-ax-c, as axa o
• This implies axbcxa----2 Vector Product
  Definition
• Similarly, on taking vector product with b in
  1,
• bx(abc)bxo----- bxc-bxa, as bxb o.
• Or, bxcaxb3 On combining 2 3,
• axbbxccxa------ axbbxccxa.
• This implies a b sin (pie-C)b c sin ( pie-A) ca
  sin (pie-B), on
• Applying vector product analysis, which yields a
  b sin C b c sin A c a sin B, by property of
  sine function. Now a little algebra would yield
  the desired result.

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6) Fundamentals Mantras.

• xviii. Noting.
• Note-Taking Note-Making Exercises are
  quite intellectual in nature. You cant take
  notes without thinking about the material you are
  dealing with, since you have to pick out
  important points summarize/ abstract them.
• Noting process helps to understand what you are
  trying to learn, besides fixing the material in
  your memory, and helping you remember where to
  find them again for effective revision.

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6 Fundamentals Mantras.

• Buzans have emphasized so scientifically on the
  why/ how of note-making/ taking process in their
  Radiant Thinning Mind Mapping.

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6. Fundamentals Mantras.

• In the Linear Layout of Notes, loose leaf-paper
  may be used instead of an exercise book, leaving
  spaces for addition later on loose sheets make it
  easier to add to your notes in terms of
  enrichment as much as blank spaces on each sheet
  of paper a closely-written page (layout-wise)
  may look very neat but it doesnt give instant
  clues to its contents. In Pattern layout of
  Notes, you may draw box-figures to inter-connect
  various themes, beginning at the center with the
  main topic.
• A spider-pattern giving you a memorable overall
  view of the concept/ topic/ subject (whole
  pattern) be supplemented by linear-layout notes
  where you feel more explanatory details are
  required.

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6. Fundamentals Mantras.

• CASE STUDY OF TOPPERS-
• While studying, they tried to catch on what
  lesson was about and then read intensively,
  poring over.
• The habit of consulting a dictionary and
  cross-reference made the learning more meaningful
  for them.
• They prepared their own notes from the texts/
  books they read, underlying the important points
  and truly practicing Radiant Thinking Mind
  Mapping.
• They reviewed the learnt material at regular
  intervals.
• They related/ correlated materials learnt in one
  subject or course to those learnt in others, and
  contemplated on practical application (s) of the
  concept/ theory.
• They studied different subjects each day
  according to a fixed routine at home.

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6. Fundamentals Mantras.

• The process of enjoying problem-solving
  Mathematical Scientific involves following
  steps
• (a) Be aware of all the Mathematical
  Scientific
• data vocabulary ( words phrases )/
  symbols/ conventions /scripts/ hierarchies with
  respect to Definitions, Theorems, Properties
  of Figures, Relations, etc. of all the
  previous current years of your studies---all
  noted in your daily diary for easy reference.
• (b) Study the Text contemplate on figuring
  out the step/ logic involved in the solved
  problem in the Text. Go on adding in your Diary
  the newly learnt concepts/ theories/ results/
  formulae/ vocabulary.
• (c) While solving a problem , sketch it, if
  possible, in terms of its all parameters, take
  care of signs, degrees or radians, or pi, or
  units/ dimensions----all be in the same scale.
  Youll be able to figure out the logic/ steps
  with the help of the knowledge/ understanding of
  the currents (data provided in the problem)
  and previous theories/ concepts / years wisdom
  to solve the problem. With the help of your
  imagination / brainstorming / presence of mind/
  maturity, youd arrive at the solution in stages.
• 
  
  

77
6. Fundamentals Mantras.

• xxi. Should you be scheduling your studying all
  scientifically/ spiritually practicing your
  Pranayama all regularly, processing-wise youll
  be on your progressive way to be able to write
  the National Standard Examination/ Indian
  National Olympiad in Math/ Physics/ Chemistry/
  Biology in November/ December, 2010 and onwards,
  followed by the International Olympiad, I.S.C.,
  P.M.T., P.E.T.,
• I.I.T., / J.E.E.,
• K.V.P.Y., S.C.R.A., N.A.D., B.I.T.S., S.A.T.,
  etc. in 2011 and onwards all successfully .

78
6. Fundamentals Mantras.

• xxii. Catch/ Understand Perceive the context of
  new ideas / concepts/ theories, derive /
  integrate them.

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7. What is Understanding

• I feel I understand something if, when, I can
• State it in my own words.
• Give examples of it.
• Recognize it in various guises circumstances.
• See connections between it other facts/ ideas.
• Make use of it various ways.
• Foresee some of its consequences.
• State its opposite or converse.

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7. What is Understanding

• h. Can search for innovative forms, for
  alternatives fresh resources.
• i. Form a concept build a theory derive/
  integrate the results.
• j. Use the ideas that are already in my mind
  in ten or fifteen different ways to come up with
  news ones
• k. Respond flexibility effectively to the
  unexpected
• l. Uncover newer knowledge enhance resources

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Closing note

• We concur with the view of Jamshedji Tata, the
  farsighted pioneer visionary of scientific
  research development in India when he says
  that-
• What advances a nation or community is not so
  much to prop up its weakest and most helpless
  members as to lift up the best most gifted so
  as to make them of the greatest service to the
  country. I prefer this constructively philosophy
  which seeks to educate develop the faculties of
  the best of your young men.

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Closing Note

• Start enjoying study/ exploration of the sublime
  Self the spectacular.
• Good Luck!