Maths

Maths

KARNATAKA EXAMINATIONS AUTHORITY

CET-2012 - Final Answer Keys

BANGALORE

Answer Keys

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Central Board of Senior Education

AIEEE 2012-Rescheduled Exam Papers Held On 29-April-2012

S.No. AIEEE
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Indian Institutes of Technology
Joint Entrance Examination, IIT-JEE - 2012 (Answer Key)

IIT-JEE -2012 : PAPER - I & II

CODE-0, 1, 2, 3, 4, 5, 6, 7, 8 & 9

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Indian Institute of Technology, Patna

PhD Admission - July 2012

Applications are invited for admission to the Doctor of Philosophy (Ph.D.) programme, starting in July 2012 in the following Departments. The areas of research in IIT Patna are as follows:

  • Computer Science and Engineering

  • Electrical Engineering

  • Mechanical Engineering

  • Chemistry

  • Mathematics

  • Physics

  • Humanities and Social Sciences

  • Inter-Disciplinary research

Category of Admission:

Regular and Full-Time

A student in this category works full-time for her/his PhD degree. They can be classified as

a) Institute Fellows
She/he receives assistantship from the Institute. The qualifying Degree for Financial Support is:

  • BE/ BTech/ MSc/ MA equivalent with valid GATE score above the prescribed cut off level/ NET /
    NBHM/ DBT-JRF qualification.
  • ME/ MTech or equivalent.

Age Limit: Please refer Eligibility Criteria for Admission into Ph.D. Programme

b) JRF/SRF Fellows
She/he receives fellowship from any Government recognized funding agencies, such as CSIR, UGC, DBT, etc.

Sponsored (Full -Time)

  • A student in this category is sponsored by a recognized R&D organization, academic institution, government organization or industry for doing research in the Institute on a full-time basis. The Institute does not provide any assistantship/fellowship to such a student.
  • Candidate in Sponsored category must be a regular employee of the sponsoring organization with at least one year of professional experience in the respective field.

Project Staff

This category refers to a student who is working on a sponsored project in the Institute and is admitted to the PhD Programme to work on a full-time or part-time basis. The remaining duration of the project at the time of admission should be at least two years. If the project gets completed before the student completes her/his PhD Programme, her/his category will be converted to that of SELF-FINANCED unless she/he is granted an assistantship/fellowship from the Institute or any other agency.

Employed & Part-Time

  • A student in this category is a professionally employed person (including the staff of IIT Patna), who pursues the PhD Programme while continuing the duties of her/his service. The Institute does not provide any assistantship/fellowship to such a student. PhD students registered in Employed and Part time category, the minimum residential requirement is one or two semester(s) depending on the completion of mandatory course work required for PhD students by the respective departments.
  • Candidate in Employed and Part-time categories must be a regular employee of the sponsoring organization with at least one year of professional experience in the respective field.

Distance requirement: The candidate must reside within 50 km radius from IIT Patna.

Eligibility Criteria

In all the disciplines, the upper age limit is 28 years (B.Tech./B.E./M.Sc./MA/MCA/MBA) and 32 years (M.Tech./M.E./M.S./M.Phil.) to be calculated from 30-04-2012 and is applicable only for candidates applying in Regular and Full time category (Institute Fellows). Upper age is relaxed upto 05 years in case of candidate belonging to Schedule Castes/Schedule Tribes, Women, Physically Handicapped, and OBC applicants.

PhD in Engineering
For admission to the PhD Programme in Engineering Department, a candidate must satisfy one of the following criteria:

  • Candidates having M.Tech./M.E. degree in a Engineering/Technology, with a minimum CPI of 6.5 or
    60% of marks.
  • Bachelor’s degree in Engineering/Technology (from any Institute other than
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WEST BENGAL JOINT ENTRANCE EXAMINATIONS BOARD

JEM 2012 - SYLLABUS

MATHEMATICS

Algebra

A.P., G.P., H.P. : Definitions of A. P. and G.P.; General term; Summation of first n-terms; A.M.and G.M.; Definitions of H.P. (only 3 terms) and H.M.; Finite arithmetico-geometric series.

Logarithms :Definition; General properties; Change of base.

Complex Numbers: Definition and properties of complex numbers; Complex conjugate; Triangle inequality; Square root of complex numbers; Cube roots of unity; D’Moivre’s theorem (statement only) and its elementary applications.

Quadratic Equations : Quadratic equations with real coefficients; Relations between roots and coefficients; Nature of roots; Formation of a quadratic equation, sign and magnitude of the quadratic expression ax2+bx+c (a,b,c are rational numbers and a≠0).

Permutation and combination : Permutation of n different things taken r at a time (r ≤ n). Permutation of n things not all different. Permutation with repetitions (circular permutation excluded). Combinations of n different things taken r at a time (r ≤ n). Combination of n things not all different. Basic properties.

Problems involving both permutations and combinations.

Principle of Mathematical Induction : Statement of the principle. Proof by induction for the sum of squares, sum of cubes of first n natural numbers, divisibility properties like 2 2n – 1 is divisible by 3 (n ≥ 1), 7 divides 32n+1+2 n+2(n ≥ 1).

Binomial theorem (positive integral index) :Statement of the theorem, general term, middle term, equidistant terms, properties of binomial co-efficients.

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All India Engineering Entrance Examinations (AIEEE)

About AIEEE:

All India Engineering/Architecture Entrance Entrance Examinations (AIEEE) is amongst the famous exams for admission in engineering/architecture bachelor courses throughout most of the leading engineering/architecture institutes across the country. Therefore AIEEE is the most popular engineering Entrance examination for BE/BTech aspiring students across the country. Every year a huge number of students from all over the country participate in this exam to get admission in the various reputed NIT’s and state level institutes. Last time nearly 9 lakhs students appeared for AIEEE exam.

This is given by large number of students from all over the country. There are large number of top level institutes and universities that approve AIEEE score. This all includes all the NITs National Institute of Technology. Although, the number of seats available for admission in all these institutes are fixed. So one must prepare very well. The competition is quite tough.

Eligibility Criteria:

The student should be passed in 12th class from a recognized board with Physics, Chemistry & Mathematics as compulsory subjects. Those who have to give the final year exams of the qualifying exam can also apply for AIEEE provided their result has to be out by the time of counseling.

Pattern:

Subject Combinations in Qualifying Examinations:

Course  Compulsory Subjects Any One of the Optional Subjects
B.E./B.Tech.* Physics & Mathematics
  • Chemistry
  • Bio-Technology
  • Computer Science
  • Biology
B. Arch./B. Planning** Mathematics with 50% marks in aggregate at 10+2 level  
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All India Engineering Entrance Examinations (AIEEE): 2012

About AIEEE:

All India Engineering/Architecture Entrance Entrance Examinations (AIEEE) is amongst the famous exams for admission in engineering/architecture bachelor courses throughout most of the leading engineering/architecture institutes across the country. Therefore AIEEE is the most popular engineering Entrance examination for BE/BTech aspiring students across the country. Every year a huge number of students from all over the country participate in this exam to get admission in the various reputed NIT’s and state level institutes. Last time nearly 9 lakhs students appeared for AIEEE exam.

This is given by large number of students from all over the country. There are large number of top level institutes and universities that approve AIEEE score. This all includes all the NITs National Institute of Technology. Although, the number of seats available for admission in all these institutes are fixed. So one must prepare very well. The competition is quite tough.

Eligibility Criteria:

The minimum academic qualification for appearing in AIEEE 2012 is that the candidate must have passed in final examination of 10+2 (Class XII) or its equivalent referred to as the qualifying examination (see Appendix – IX). Those appearing in 10+2 (Class XII) final or equivalent examination in 2012 may also appear in AIEEE 2012 provisionally.

Scheme of Examination:

Entrance examination would consist of two papers i.e. 1st paper consisting of three parts of Physics, Chemistry and Mathematics of equal weight age with objective type questions for B.E/B.Tech courses in offline/online mode and 2nd paper – consisting of Mathematics, Aptitude Test and Drawing for B. Architecture and B. Planning in offline mode. The Aptitude Test is designed to evaluate candidate’s perception, imagination, observation, creativity and architectural awareness.

Schedule of Examination:

  Subjects Type of Questions
Paper 1 Physics, Chemistry & Mathematics Objective type questions with equal weightage to Physics, Chemistry
& Mathematics
Paper 2
  • Mathematics – Part I
  • Aptitude Test – Part II &
  • Drawing Test – Part III
  • Objective type questions
  • Objective type questions
  • questions to test Drawing Aptitude
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CBSE: All India Engineering Entrance Examinations (AIEEE)
Syllabus: Mathematics - 2012

Unit 1 : Sets, Relations and Functions:

Sets and their representation; Union, intersection and complement of sets and their algebraic properties; Power set; Relation, Types of relations, equivalence relations, functions;. one-one, into and onto functions, composition of functions.

Unit 2 : Complex Numbers and Quadratic Equations:

Complex numbers as ordered pairs of reals, Representation of complex numbers in the form a+ib and their representation in a plane, Argand diagram, algebra of complex numbers, modulus and argument (or amplitude) of a complex number, square root of a complex number, triangle inequality, Quadratic equations in real and complex number system and their solutions. Relation between roots and co-efficients, nature of roots, formation of quadratic equations with given roots.

Unit 3 : Matrices and Determinants:

Matrices, algebra of matrices, types of matrices, determinants and matrices of order two and three. Properties of determinants, evaluation of determinants, area of triangles using determinants. Adjoint and evaluation of inverse of a square matrix using determinants and elementary transformations, Test of consistency and solution of simultaneous linear equations in two or three variables using determinants and matrices.

Unit 4 : Permutations and Combinations:

Fundamental principle of counting, permutation as an arrangement and combination as selection, Meaning of P (n,r) and C (n,r), simple applications.

Unit 5 : Mathematical Induction:

Principle of Mathematical Induction and its simple applications.

Unit 6 : Binomial Theorem and It's Simple Applications:

Binomial theorem for a positive integral index, general term and middle term, properties of Binomial coefficients and simple applications.

Unit 7 : Sequences and Series:

Arithmetic and Geometric progressions, insertion of arithmetic, geometric means between two given numbers. Relation between A.M. and G.M. Sum upto n terms of special series: S n, S n2, Sn3. Arithmetico – Geometric progression.

Unit 8 : Limit, Continuity and Differentiability:

Real - valued functions, algebra of functions, polynomials, rational, trigonometric, logarithmic and exponential functions, inverse functions. Graphs of simple functions. Limits, continuity and differentiability. Differentiation of the sum, difference, product and quotient of two functions. Differentiation of trigonometric, inverse trigonometric, logarithmic, exponential, composite and implicit functions; derivatives of order upto two. Rolle’s and Lagrange’s Mean Value Theorems. Applications of derivatives: Rate of change of quantities, monotonic - increasing and decreasing functions, Maxima and minima of functions of one variable, tangents and normals.

Unit 9 : Integral Calculus:

Integral as an anti - derivative. Fundamental integrals involving algebraic, trigonometric, exponential and logarithmic functions. Integration by substitution, by parts and by partial fractions. Integration using trigonometric identities. Evaluation of simple integrals of the type Integral as limit of a sum. Fundamental Theorem of Calculus. Properties of definite integrals. Evaluation of definite integrals, determining areas of the regions bounded by simple curves in standard form.

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Indian Institute of Space Science and Technology

IIST Ph.D. Programme – February 2012

Indian Institute of Space Science and Technology envisions the integration of Space Technology and Space Science educational programs with basic and applied research for meeting the national R&D requirements of science and technology in general and of the Indian Space Programme in particular. The institute provides a vibrant research atmosphere and offers doctoral and postdoctoral programmes.

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IIST: ISAT 2012 Mathematics Syllabus

PERMUTATIONS AND COMBINATIONS:

Fundamental principle of counting. Permutations and Combinations, derivation of formulae and their connections and simple applications.

MATHEMATICAL INDUCTION:

Principle of Mathematical Induction and its simple applications.

BINOMIAL THEOREM AND ITS SIMPLE APPLICATIONS:

Binomial theorem for positive integral indices, general term and middle term, properties of Binomial coefficients and simple applications.

SEQUENCES AND SERIES:

Arithmetic and Geometric progressions, insertion of arithmetic, geometric means between two given numbers. Relation between A.M. and G.M. Sum upto n terms of special series n, n2, n3. Arithmetico - Geometric sequence.

TRIGONOMETRY:

Trigonometric functions. Trigonometrical identities and equations. Inverse Trigonometric functions, their properties and applications.

COMPLEX NUMBERS AND QUADRATIC EQUATIONS:

Complex numbers as ordered pairs of reals. Representation of complex numbers in a plane. Argand plane and polar representation of complex numbers. Algebra of complex numbers, modulus and argument (or amplitude) of a complex number, square root of a complex number, triangle inequality. Quadratic equations in real and complex number system and their solutions. Relation between roots and coefficients, nature of roots, formation of quadratic equations with given roots.

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Indian Institute of Space Science and Technology

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Minimum Qualifying Mark for Ranking (MQMR) and Aggregate Cut-off : JEE-2011

Cutoff Marks - JEE-2011

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Indian Institute of Technology Ropar

Ph.D. Admission: 2011-2012

The Indian Institute of Technology Ropar (IIT Ropar) has invited applications from Indian citizens for Ph.D programs in various disciplines during second semester of academic year 2011-2012.

Eligibility criteria and areas of research for each department are given below:

Sciences:

Chemistry:

Eligibility Criteria: First Class (6.5 grade point out of 10) or 60% marks (55% marks for SC/ST) in M.Sc. or equivalent degree in Chemistry.
Candidates meeting this requirement must also fulfill one of the following additional requirements tenable for the current year:

  • Valid GATE score.

  • UGC/CSIR NET qualification

Area of Research: Organic Chemistry, Organometallics, Catalysis, Chemical Kinetics and Reaction Dynamics, Electronic Structure Calculations, Organic polymer synthesis.

Physics:

Eligibility Criteria: First Class (6.5 grade point out of 10) or 60% marks (55% marks for SC / ST) in M.Sc. Physics/B.Tech. Engineering Physics.
Candidates meeting this requirement must also fulfill one of the following additional requirements tenable for the current year:

  • Valid GATE score.

  • UGC/CSIR NET qualification.

Areas of Research: (1) Particle/Nuclear Physics, (2) Quantum Optics and Quantum Information.

Mathematics:

Eligibility Criteria: First class (6.5 grade point out of 10) or 60% marks (55% marks for SC/ST) in Master's degree in Mathematics, Statistics, Computer Science, Physical Sciences, Engineering Sciences or equivalent Master's Degree.
Candidates meeting this requirement must also fulfill one of the following additional requirements tenable for the current year:

  • Valid GATE score.

  • UGC-CSIR/NBHM or equivalent qualification.

Areas of Research: 1. Fluid Dynamics, 2. Mathematical Modeling, 3. Differential Equations, 4. Topology

Engineering:

Computer Science and Engineering:

Eligibility Criteria:
1. First class or equivalent Master's Degree in Engineering/Technology (55% marks for SC/ST) or
2. A first class Master's degree in Science (55 % marks for SC/ST) or a first class in Bachelor's degree in Engineering / Technology (55 % marks for SC/ST). Candidates meeting this requirement must also fulfill one of the following additional requirements:

  • Valid GATE score

  • CSIR/UGC/NBHM/DBT award.

Note: MCA degree will be considered as equivalent to a Master's degree in Science.

Areas of Research: The department is looking for excellent candidates willing to engage in multi-disciplinary research in systems and theory.

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Indian Institutes of Technology
Joint Entrance Examination, IIT-JEE - 2012

Mathematics Syllabus

Algebra: Algebra of complex numbers, addition, multiplication, conjugation, polar representation, properties of modulus and principal argument, triangle inequality, cube roots of unity, geometric interpretations.

Quadratic equations with real coefficients, relations between roots and coefficients, formation of quadratic equations with given roots, symmetric functions of roots.

Arithmetic, geometric and harmonic progressions, arithmetic, geometric  and harmonic means, sums of finite arithmetic and geometric progressions, infinite geometric series, sums of squares and cubes of the first n natural numbers.

Logarithms and their properties.

Permutations and combinations, Binomial theorem for a positive integral index, properties of binomial coefficients.

Matrices as a rectangular array of real numbers, equality of matrices, addition, multiplication by a scalar and product of matrices, transpose of a matrix, determinant of a square matrix of order up to three, inverse of a square matrix of order up to three, properties of these matrix operations, diagonal, symmetric and skew-symmetric matrices and  their properties, solutions of simultaneous linear equations in two or three variables.

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