Posted by sreeram sudha on July 5, 2010
Matrices is an interesting topic in the field of Mathematics. A matrix is a rectangular arrangement of numbers , as horizontal rows and vertical columns , similar to the arrangement of elements in the Periodic Table . So each number of the matrix can be called as an element of the Matrix.
If the number of Rows in a matrix ( A ) is m and the number of Columns is n , then the said Matrix is of the Order m X n and is written as A = (aij)
There are different types of Matrices , like
1) Equal matrix : Two matrices are called as Equal matrices only if they are of the same order and their corresponding elements are equal.
2) Row Matrix : Has only one Row of elements.
3) Column matrix : Has only one Column of Elements.
4) Square matrix : The number of Rows and Columns are equal.
5)Zero or Null Matrix : Each element is Zero.
6) Diagonal matrix : All elements except the elements in the main Diagonal are
Zero..
7) Unit matrix : Each of its diagonal element is 1 and each of all other
or Elements is zero.
(Identity Matrix)
Addition and Subtraction of matrices :
Two matrices of the same order are added by adding their corresponding elements. The element in the 1st row and 1st column of the first matrix is to be added with the element in the first row and first column of the second matrix and placed in the 1st row and 1st column of the newly formed matrix..
Similarly , the other elements of the first Matrix are added to thier corresponding elements in the second matrix .
For example A and B are two matrices of the same order , then
A+B=B+A
But A-B is not equal to B-A
This is an important fact the student has to remember while working sums on Matrices.
Multiplication of Matrices :
The product AB of two matrices A and B is possible only when the number of columns in A is equal to the number of rows in B .
A is called the pre-multplier and B is called the post multiplier.
The student can follow the following steps while finding the product of two matrices A and B.
1) First multiply the elements of the 1st row of A by the corresponding elements of the 1st column of B and add. The number got after adding becomes the 1st element in the 1st row of the resulting matrix. AB
2)Multiply the elements of the 1st row of A by the corresponding elements of the 2nd column of B and add. This becomes the second element in the first row of the resulting matrix AB.
3)In the same manner , the remaining elements of the first row of the resulting matrix AB can be obtained.
4) Proceeding in the same way , the student can get the elements of the remaining rows of the resulting matrix. AB .
Transpose of a Matrix is another interesting fact about a matrix.
The transpose of a matrix A is A^T .The elements in A^T can be obtained by interchanging the rows and columns of the given matrix.
For example A = 1 2
5 9
then A^T = 1 5
2 9
Important fact about multiplication of Matrices :
If the order of matrix A is m X n and the order of matrix B is n X p then the product AB is possible and the order of the product matrix obtained is m X p , i.e number of rows of matrix A X number of columns of matrix B .
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Filed Under: ICSE
Posted by sreeram sudha on
Sums are an important feature in the syllabus of CBSE and ICSE. By following certain steps, the working of sums in these subjects can be made easy and interesting.
Steps to follow while doing calculations in Physics and Chemistry:
1) First write the data that is given. This will enable the student to be aware of the formula, the student has to use in solving the sum.
2) Then write the formula that can be used to solve the sum.
3) It is easier to work with fraction numbers than decimals. So, convert the decimal numbers into fraction numbers, in powers of ten and after arriving at the answer, convert the fraction numbers into corresponding decimal numbers corrected up to two decimal places.
Students following the above steps while doing sums in physics and chemistry will find better comfort level.
Not only sums, but there are other interesting features in these two subjects, Physics and Chemistry.
While working with sums in Mole Concept and Stoichiometry in chemistry, the students can follow certain methods to arrive at the correct solution.
1) First write the chemical equation and balance it, by making the number of atoms of each element the same on the reactants side and the products side.
2) Always remember that One molecule of any compound weighs the sum total of atomic weights of the molecule and One mole of any gas occupies a volume of 22.4 litres or 22,400 ml or 22,400 cubic cm at standard temperature and pressure .
3) While finding the molecular formula, the student has to remember
Molecular formula = n X Empirical formula and
The value of n = Molecular weight/Empirical Formula weight.
Sometimes, the molecular weight may not be provided in the sum, instead, Vapor density would have been provided. In such cases, use the formula
Molecular Weight = 2 X Vapour density.
In Physics, while doing sums, Correct Units have to be mentioned in the final answer.
While working the sums, necessary conversions have to be made so that the Units of all the components are in the same, either in S.I or C.G.S. units.
For example, if the Distance is in Kilometers, then the unit of speed has to be in Kilometer per Hour. If the unit of speed is provided in the sum as Meters per second, then it can be converted into Kilometers per hour by multiplying with 18/5. If the distance is in meters, speed in Kilometers per hour, then the speed can be converted into meters per second by multiplying with 5/18
1) The Formula has to be written first
2) Next, the units of the individual component in the formula have to be mentioned.
3) After necessary reductions, the final unit will be available.
For example Speed = Distance in meters/time in seconds or
Distance in kilometers and time in hours, then
Speed = meters/second or Kilometers/hour
so the unit of speed is meter per second or kilometer /hour. = m/sec or km/hour
V^2 – u^2 /a = 2S (v=final velocity, u = initial velocity, a = acceleration and S = (distance) (meters per second) ^2 – (Meters per second) ^2/ Meters per second square = meters ^2 X Seconds ^2/metres per seconds ^2 = Meters which is the unit of distance.
So writing the Formula for any sum is important to make the sum interesting and easy.
In the branch of organic chemistry, to identify Alkanes, Alkenes and Alkynes,
1) First write the number of carbon atoms in the unknown Hydrocarbon and call it n
2) Then write the number of hydrogen atoms in multiples of n.
3) The general formula for Alkane is CnH2n+2; Alkene is CnH2n; Alkyne is CnH2n-2. By substituting the values of n, the student can identify the unknown Hydrocarbon.
By following a few steps in any Topic in Physics and Chemistry, the student can work comfortably and find the subjects interesting and not monotonous.
Posted by TCYonline.com on December 1, 2008
LUDHIANA, Dec. 1 – Over 150 students from Ludhiana and its satellite towns like Khanna, Phillaur and Jagraon took Mock CBSE Mathematics examination for Class X here yesterday. This mock exercise was conducted by city’s ‘STEPS’, the schools division of Top Careers & You (TCY), at its Sarabha Nagar centre premises.
Keeping in view an ‘overwhelming’ response to this initiative by this premier education institution of Northern India, TCY has decided to conduct ten such mock exercises before the final board examination in February next.
The response was so huge to this mock test that some of the students could not have on-the-spot registrations and had to go back due to the ‘limited’ seats availability.
Five top schools of the city like Sacred Heart Senior Secondary School, Tagore Public School, BCM Arya Model School, Bhartiya Vidya Mandir School and Kundan Vidya Mandir School took part in this annual exercise by ‘STEPS’.
One of the salient features of this event is a provision for a threadbare online analysis of students’ performance by TCYonline.com, the online initiative of TCY. This would be followed by classroom analysis on December 7th.
“The students would be easily able to analyse their strengths and weaknesses following this detailed analysis by our experts. Not only that, they would know their rankings too. The idea is to help the students work on their grey areas,” according to ‘STEPS’ Sarabha Nagar Centre Head, Mr. Viji Raj.
It is worth noting that TCYonline.com is today the No. 1 Testing platform in the country today with over 2.6 lakh students’ registrations on its web-portal. Hundreds of free practice tests are available on the website www.tcyonline.com for the benefit of not only CBSE student but also for other school competitions and Olympiads being held in the country from time to time.
“This mock test would immensely help the first-timers for this board examination in helping them familiarize with the real exam situation. The main idea is shoot away their exam blues and prepare them well,” Mr. Viji Raj emphasized.
It may be recalled that ‘STEPS’ has been organizing Mock CBSE test annually for the students of class X in association with bodies such as Rotary Club and Bansi Lal Memorial Trust. Last year over 600 students from Ludhiana and its surrounding areas participated in the mock test.
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Posted by TCYonline.com on November 26, 2008
NEW DELHI, Nov. 26 – Recession or no recession, the modern Indian student always aspires for more – more knowledge, more money and more challenge. The just concluded CAT and the very thought of fat MBA pay packages spur him on to perform better each passing day.
Be it upcoming SNAP, FMS and XAT, or for that matter any college/school or competitive exam, the student’s Net gains have made him more confident after he has learnt how to wriggle out of difficult situations. Online testing being his new success mantra, all he has to do is benchmark his performance against the best in India and abroad, and raise his bar.
Today, we see a study shift from chalk-and-board to computers with the greater impact of education websites like TCYonline.com which have brought a revolution of sorts in the test prep sector. A smart student is supplementing his classroom education with online practice tests, which are taking precedence over traditional ‘bookish’ learning.
It is worth noting that enjoying the privilege of being India’s Number One Testing Platform, www.tcyonline.com has today over 2.70 lakh students in the age group of 13–32 years on its portal taking advantage of its various free resources — online tests ranging from CBSE to MBA to GRE and Medical/Engineering exams.
Talking of the CBSE, scores of practice tests are available on the website for the benefit of the school students. As a gesture of goodwill, like previous years, TCYonline.com will also make available number of pre board tests for the February 2009 exam.
Student benchmarking being the hallmark of this web-portal, its recently launched two testing tools – the ‘Challenge Zone’ and the ‘Test Generator’ – are making good rounds among the academic community.
Being the free website for the students, the portal has also found great favour with some schools, colleges and universities in the country. Of late, some academic institutions have shown keen interest in TCYonline.com’s elite testing platform.
TCYonline.com’s exhaustive and free material catering to all the major national and international exams coupled with technology-enabled tools is not only becoming a rage amongst students but is also catching the eye of educational institutions and other companies.
A huge student base and separate exam categories on the website provide a perfect platform for educational institutions to showcase their offerings to prospective students. With continuous product innovation and technological upgradation taking place every second, this platform offers a host of opportunities to education institutions in India and abroad, virtually on a platter.
While it is heartening to note that TCYonline.com is crossing the five-lakh registration mark during the next few months, the web-portal gives education bodies an opportunity to describe their programs, organize mock tests and virtual education fairs. Moreover, the students can gain by asking questions and interacting with the university representatives through an online chat facility.
TCYonline.com has in a way squeezed the whole world to such an extent that a now a student sitting in any part of the world, say in the US or Canada, can prepare for any international entrance exam like GRE or GMAT or TOEFL. Practice tests for all these entrance exams are available on the website free of cost.
All said and done, TCYonline.com has become the ‘testopedia’ of the modern student today. He can’t think of his life without it.
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