An algebraic structure

What is operation in mathematics languages?

An operation is like a function which when act on an element of a set changes its value or some thing like because to operate means to change, right !

Now, what is binary operation ?

Binary means which involves two.

So it is clear from the name that this operation involves two elements of a set.

I think you can now digest the below definition.

Binary operation is a function that assigns to each ordered pairs of elements of set, an element of that set .

Notation - *

operation between a and b is denoted by a*b, where a and b are elements of set.

Here ordered pairs talks about two elements of set, as you know pairs means two elements together but why there is “ ordered “ here.

Ordered pairs and not just pairs because here order matters, because a*b is different from b*a , it can be equal but not generally.

Duality -important results

Useful for both NET and GATE exam

1. If the primal problem has finite optimal feasible solution then it's dual has also optimal feasible solution and conversely.
2. If the primal problem has an unbounded objective function, then it's dual has no feasible solution and conversely.
3. The problem has finite optimal solution iff there is a feasible solution to it's dual problem.

Review of book - Csir - Net General Aptitude - A New outlook

Book Name
CSIR - NET GENERAL APTITUDE - A NEW OUTLOOK
Author- Christy Varghese a
Publisher- Lilly publishing house
This book has two sections-
Section 1.
Theory and questions of following topics
1. Numerical ability ( numbers, probability and arrangements, distance and directions)
2. daily life problems ( finding the X , average, monetary problems)
3. Geometry type problem's
( Geometry, mensuration and quantitative comparison)
4. Logical reasoning ( Data interpretation, observational ability, logical puzzles)
5. Typical problems ( calendar problem, clock problem, moving locomotive problem, series formation)
6. General science
Section 2
Previous year's exam question papers
From June 2012 to Dec 2017
( Mathematical sciences, chemical sciences, earth sciences, physical sciences, life sciences).
Each and every solution is properly solved in this book
Model question papers
Total 7 model question papers are given in the book ( along with answers ).
Review- If your part-A is very weak, you take much time in solving questions then you can purchase this book. It has full explanation of previous year papers
You can get guidance  and trick  to solve them.
Including both previous years papers and model papers there are enough papers to practice through.
Although I found this book somewhat expensive.
It's approximate price is Rs 330 ( may vary)
It is available in Amazon, Flipkart, Paytm, ebay .

IMPORTANT TOPICS WHICH COVER MAJOR PORTION OF QUESTION PAPER

UNIT – 1,2 and 3

Analysis-

Topics- countability, supremum ,infimum, convergence of sequences and series , limsup, liminf,

Bw Theorem, continuity , uniform continuity, differentiability , mean value theorem ,

sequence and series of functions, uniform convergence , monotonocity , function of bounded variation ,

directional derivative , metric space – compactness and connectedness.

Linear Algebra

vector space and its subspace, L.D , L.I , Basis and dimension , Linear Transformation and its matrix transformation , Matrices, Rank of determinant of matrices, system of linear equations, eigen values and eigen vectors, Diagonalisation, triangularization, Jordan canonical form, orthogonality , Quadratic forms.

Complex Analysis

Power series, Analyticity, Cauchy Riemann equations, Cauchy integral formula, Maximum modulus principle, open mapping theorem, Taylor and Laurent series , residues, conformal mappings ,Mobius Transformation.

Algebra

Solution of Congruence equations, Euler phi function, primitive roots, groups , subgroups- Normal , Quotient. Homomorphisms , Isomorphisms, cyclic , permutations and dihedral groups, class equations, sylow theorems, Maximal and prime ideals , finite fields , field extension , polynomial and Quotients rings, irreducibility, Euclidean domain , PID , UFD.

ODEs and PDEs

All topics

Numerical Analysis -

Newton Raphson method, Rate of convergence, Lagrange interpolation, Numerical differentiation, Picard method , Euler and modified Euler, RK methods.

Calculus of variations

Euler Lagrange equation , Necessary and sufficient conditions for extrema.

Linear Integral Equations

All topics

Csir Net/Jrf Mathematical sciences Question paper (2017-Dec ) Solutions

Booklet C - Solutions
For Question paper , click on the given link
https://drive.google.com/file/d/180749p_mMLjDFj48oR0L8kAgdcGn4tyj/view?usp=drivesdk
For questions
(1,2,3,6,7,10,11,12,13,14,15,16,21,22,23,24,25,27,29,32,36,38,39,40,44,45,46,47,48,62,66,69,72,73,74,92,98,99)

1. A person purchases two chains from a jeweller, one weighing 18 g made of 22 carat gold and another weighing 22 g made of 18 carat gold. which one of the following statements is correct ?
a. 22 carat chain contains 2/11 times more gold than 18 carat chain
b. 22 carat chain contains 1/11 times more gold than 18 carat chain
c.  Both chains contain the same quantity of gold
d. 18 carat chain contains 2/11 times more gold than 22 carat chain

 Solution 1

2. what is the volume of soil in an open pit of size 2 m * 2 m * 10 m ?
a. 40 m^3
b. 0.4 m^3
c. 0 m^3
d.  4 m^3

 Solution 2

3. what is the maximum number of cylinderical pencil of 0.5 cm diameter  that can be stood in a square shaped stand  of 5 cm * 5 cm inner cross section ?

a. 99

b. 121

c. 100

d. 105

Solution 3

6. The sum of two numbers is equal to sum of 11 and cube of 9 .The largest number is (5)^2 less than square of 25. What is the value of the sum of 24 percent of the smaller number and half of the largest number ?

a. 415

b. 400

c. 410

d. 420

Solution 6

    7: For which values of A and B is sin A = cot B ?
 
                       

   Solution 7

10 :  A rectangular flask of length 11 cm , width 8 cm and height 20 cm has water filled    up to height 20 cm has water filled up to height 5 cm . If 21 spherical marbles of radius 1 cm each are dropped in the flask, what would be the rise in water level ?

Solution 10

11 : The smallest square floor which can be completely paved with tiles of size 8 *6 without breaking ant tile , needs n tiles. Find n.

Solution 11:

 Solution 12

13. A plate of 5 m * 2 m  size with uniform thickness, weighing 20 kg, is perforated with 1000 holes of 5 cm  * 2 cm size. what is the weight of the plate(in kg) after perforation ?

Solution 13

14. Four small squares of side x are cut out of a square of 12 cm to make a tray by folding the edges. what is the value of x so that the tray has the maximum volume ?

Solution 14

                   
15. Three- quarters of a circle is shown in the figure: OA and OB are two radii perpendicular to each other. C is point on the circle.
Solution 15

16. A 2 m  long ladder is to reach a wall of height 1.75 m. The largest possible horizontal distance of the ladder from the wall could be

Solution 16

                         Solution 21

                       Solution 22

Solution 23

                        Solution 24

                   

                      Solution 27

                      Solution 29

                         Solution 32

                        Solution 36

                      Solution 38

                      Solution 39

                     Solution 40

                       Solution 44

                         Solution 45

                        Solution 46

                        Solution 47

                        Solution 48

                           Solution 66

                          Solution 64

Solution 69

Solution 72

Solution 73

Solution 74

Solution 92

Solution 98

Solution 99

Csir net/Jrf Dec - 2017

All the best to all Csir Net aspirants.

Whatever will be the situation, Give your best between 2 pm to 5 pm.

Keep hope until you finish your exam, it can be possible that one cannot be able to answer a single question  for the first 30 minutes but that doesn't mean it's all over !!

"The ball is still in your court".
Please remember these words,

With your positive thinking you can give your 100 %.

My tips in examination hall.

1)Take a small water bottle with you inside the examination hall, don't drink too much water before examination and always free yourself from any nature's call.

2) Enter at least 30 minutes so that you can relax on your seat and also fill the details accurately.

3) Most important, fill the answer key by confirming twice your answer and also don't leave it for the last rush hours, try to fill it parts by parts .

For example part A, part B, part C

You can change the sequence as per your convenience.

Csir net Dec 2017 Quiz
Question- why countable union of countable sets  is countable ?