topological space....

We have (f(x_n)) is bounded for all f E X', the dual of X

i.e (f(x_n)) is bounded for all f E X

Then we have to show that (||x_n||) is bounded. For this we use natural imbedding

x_n F_xn

By uniform bounded theorem, Fxn, is defined by

F_xn (f) = f(x_n)

Hence f(x_n) = {f(x_n)} is bounded for each f is equivalent to the assumption that {F_xn (f)} is bounded set for each f E X. Since (X, ||, ||) is a banach space, then from uniform bounded theorem {F_xn} is bounded set in X.

i.e. {||F||} is a bounded set. Since the natural imbedding preserves norms

||F|| = ||x_n|| for each x_n E X

Hence ||x_n||is bounded.

when solved with the condition y(0) =0

The differential equation would be

A meticulously sought after planning and proper & timely guidance can enable an alert individual to clear the tough UPSC examinations. According to the UPSC aspirants, half of the battle is won right after selecting the right optionals for the Union Public Service Commission (UPSC) selection examination. Provided that optionals in each IAS preliminary and IAS mains examination determine a candidate’s scoring pattern, questions asked in the final interview are based on the choice of optional subjects chosen. This makes the decision extremely difficult as well as critical.

There is a plethora of subjects to choose from, since students belong to varied streams of education, the subjects on offer include Science subjects that expands over Physics, Chemistry, Botany and Zoology; Humanities subjects that expands over History, Political Science, Public Administration, Geography and Sociology and languages such as Hindi, English, Urdu and Pali. UPSC also offers subjects such as Anthropology, Psychology and Philosophy. Additionally, the more number of subjects make it more difficult for students to make choices in optionals.

Optional subjects are chosen on the basis of a number of parameters, on one hand there are optionals that are considered to be scoring where as there are others that are favored for their relevance to either General Studies papers or Current Affairs and essay questions whereas there are many who choose subjects for their defined set of topics asked and comparatively short type answers. Students feel that this bit of planning makes their choice of subject a success as there are combinations that have common IAS syllabus which reduces their workloads and improves their chances of scoring in civil services examination.

Likewise botany and zoology are two complementary optionals that have common syllabus as these prove to be advantageous for their future. The IAS syllabus talks about emerging issues such as genetically modified (GM) foods, cloning, biotechnology and environmental sensitization. However, students should also show their aptitude skills for the subject they choose as they require a fair degree of specialized knowledge. While preparing for the optional, it is advised to study first for the IAS mains paper and then prepare for IAS preliminary as this provides students an edge and then allow them to specialize in certain scoring areas.

Appearing for IAS and IPS through combined Competitive Examination for Civil Services Exam and recruitment into Indian Forest Service is not so easy. UPSC is a self prevailing government organization that ascertains that the recruitments are conducted on the basis of impartial examination pattern and therefore the notification for the examination is published in the employment news or daily newspapers so that each and everyone knows about its happenings.
The competitive Examination is conducted as per the following three steps:

Step I- Preliminary Examination
IAS prelims is written objective type examination and is conducted in the month of June every year with two following papers:

Paper I- General Studies paper with maximum mark 150 against 150 questions. The exam expands over two hours and questions are asked from six topics which are Indian History, Constitution of India, Geography, Economics, Science and Current Affairs.

Paper II- The second paper is an optional of student's choice which may include Political Science, Psychology, Economics, Physics, and Economics etc. This exam has maximum marks 300 against 120 questions and is two hour long.

IAS prelims are very competitive and therefore only highest scorers qualify for the second stage that is the Main Examination. It is quite interesting to note that the qualifying percentage is 10% in preliminary examination.

Step II- Main Examination
IAS Mains is a written examination which is a subjective or essay type format. This is generally conducted in the month of November/December. IAS Mains expands over nine papers:

Paper I & II- Language papers with maximum marks 300 each. One paper is English which is mandatory and the other is any Indian language mentioned in the eighth schedule of Indian Constitution. Passing these papers with a certain minimum percentage is necessary.

Paper III- This is a subjective type paper with maximum marks 200. Student can opt to write in either English or any other Indian language mentioned in the eighth schedule of the Constitution.

Paper IV and V- General Studies with General Knowledge and Current Affairs and maximum marks of 300

Paper VI, VII, VIII and IX- Two optional subjects, with two papers thereby making it four papers with 300 each and total marks of 1200.

Those who pass IAS mains are further reduced with a ratio of 1:10 to direct students to the next level that is Personality Test.

Step III- Personality Test
This is the final elimination stage and as expected the competition is highest at this stage. Candidates are selected with a ratio of 1:2 and therefore with 10 vacancies, twenty candidates are called for interview. These candidates are tested on their personality, social traits, teamwork and leadership qualities and their aptitude and skills that are usually involved in work.

After the personality test, the candidates have to also undergo a mandatory training at Lal Bahadur Shastri National Academy of Administration in Mussorie as a probationary officer. This is the foundation course, after which specific job-oriented training is conducted separately by each service.

There are number of students who are allured by this rewarding career in the Electronics and Telecommunication Engineering field. With Maths and Physics being the required subjects in Class XII, admission in electronics and telecommunication engineering is done on the basis of entrance test, GATE. After completion in the same, one is offered openings in the public sector, defence, manufacturing industries, sales and marketing. Those willing to choose research and development or teaching or consultancy jobs need to pursue postgraduate qualifications. Graduate Aptitude Test for Engineering (GATE) is the qualifying test that has to be cleared by those looking for placement in postgraduate degree programmes. A degree in ME/MTech is two year courses and further specializations are offered through post graduate diplomas or research fellowships.

After pursuing 10+2 students must opt for acquiring technician’s qualifications through the 3 year diploma courses that are usually offered by the Technology Training Institutions. Then there is an option for studying for membership examinations of professional organizations or those diploma holders can carry out part time degree course for enhancing their qualification. There are numerous universities that are offering B.Sc Electronics course which can be followed by an MSc in the subject. Additionally, there are specialized B.Tech courses that are offered in varied fields such as Electronics & Computer Sc., Electronics & Communication, Electronics & Control, Electronics & Telecommunication, Electronics & Information Electronics & Power, Electronics & Instrumentation, Electronics & Telematics, Electronics & Power, Electronics & Electrical Communication and many more.

Career Avenues for Electrical and Telecommunication Engineering Students- Organization dealing with consumers, aircrafts and broadcast electronics, hospital and medical equipment, satellite, computers, marine navigation, audiovisual test and farm machinery offers opportunities for electronic /telecommunication engineers. The union or amalgamation of technology for telephones, computers, and cable, video and broadcast television requires proficient people skilled in various other disciplines. Engineers from computers, electronics telecommunications/communication have opportunities in a wide variety of companies.

Electronic and Telecommunication engineers are usually required in industries where the working environment varied significantly. Research based professions are carried out in laboratories and as far as fabrication is concerned, these are carried out in showroom. Engineers in Defence, Hospitals, and aeronautics and weather labs are subjected to severe working conditions. However, telecommunication engineers are highly in demand in the manufacturing industries and research and services sector. On the other hand, job involves desk work that expands over a lot of technical calculations, design and testing. Additionally, the field work requires hell lot of traveling and can be at times, physically demanding.