IIT JEE (2012) (Matrices + Determinant + Function)

Size: px
Start display at page:

Download "IIT JEE (2012) (Matrices + Determinant + Function)"

Transcription

1 (+) PAPER B IIT JEE (01) (Matrices + Determinant + Function) TOWARDS IIT JEE IS NOT A JOURNEY, IT S A BATTLE, ONLY THE TOUGHEST WILL SURVIVE TIME: 60 MINS MAX. MARKS: 80 MARKING SCHEME In Section I (Total Marks: 4), for each question you will be awarded 3 marks if you darken ONLY the bubble corresponding to the correct answer and zero marks if no bubble is darkened. In all other cases, minus one (1) mark will be awarded. In Section II (Total Marks: 16), for each question you will be awarded 4 marks if you darken ALL the bubble(s) corresponding to the correct answer(s) ONLY and zero marks otherwise. There are no negative marks in this section. In Section III (Total Marks: 4), for each question you will be awarded 4 marks if you darken ONLY the bubble corresponding to the correct answer and zero marks otherwise. There are no negative marks in this section. In Section IV (Total Marks: 16), for each question you will be awarded marks for each row in which you have darkened ALL the bubble(s) corresponding to the correct answer(s) ONLY and zero marks otherwise. Thus, each question in this section carries a maximum of 8 marks. There are no negative marks in this section. NAME OF THE CANDIDATE CONTACT NUMBER L.K. Gupta (Mathematics Classes) FOR SOLUTIONS KINDLY VISIT (In latest Updates) 1

2 Section- I (Total Marks: 4) (Single Correct Answer Type) This section contains 8 multiple choice questions. Each question has four choices (a), (b), (c), (d) out of which ONLY ONE is correct. 1. The function f : N N (N is the set of natural numbers) defined by f ( n ) =n+3 is (a) surjective only (b) injective only (c) bijective (d) None of these. The function f(x) = sin log( x 1 x ) (a) even function (c) neither even nor odd is (b) odd function (d) periodic function 3. Let f :, 3 3 [0, 4] be a function defined as f(x) = 3 sin cos x x. Then f 1 ( x) is given by 1 x (a) sin 1 x (b) sin x (c) cos (d) None of these 3 4. If f is a function such that f (0) =, f (1) 3 and f ( x ) f ( x) f ( x 1) for every real x, then f (5) is (a) 7 (b) 13 (c) 1 (d) 5

3 5. { } The period of function x { /} + sinx 3 x cos x (where {x} denotes the fractional part of x) is (a) (b) 1 (c) 3 (d)none of these 6. f : N N where f(x) = x ( 1) x then f is (a) one-one and into (b) many-one and into (c) one-one and onto (d) many-one and onto 7. If matrix A is given by A = 6 11, then the determinant of 4 A 6A is (a) 006 (b) 005 ( 11) (c) (d) 004 ( 9) 8. The value of n n 1 n1 1 a 1 b 1 c a 1 1 b 1 c a 1 b 1 1 c + n1 1 a a 1 a 1 n 1 b 1 b b 1 n 1 c 1 c 1 c is equal to (a) 3 (b) 1 (c) 1 (d) none of these 3

4 Section II (Total Marks : 16) (Multiple Correct Answer (s) Type) This section contains 4 multiple choice questions. Each question has four choices (a), (b), (c), and (d) out of which ONE or MORE may be correct. 9. Suppose a 1,a, are real numbers, with a1 0. If a1, a, a 3, are in A.P., then a1 a a3 (a) A = a4 a5 a6 is singular a5 a6 a7 (b) the system of equations a1 x a y a3z 0, a4x a5 y a6z 0, a7x a8 y a9z 0has infinite number of solutions a1 ia (c) B = ia a1 10. The value of the determinant 6 i 3 6 is non-singular (where i = 1 ) i 3 6 i, where i 1, is 18 1 i 7 i (d) none of these (a) complex (b) real (c) irrational (d) rational cos sin Let A sin cos 0, then (a) A A A (b) A A (c) A A (d) A A I 1 1. Domain of (a) 3 3, 0 f x sin [ 4x ] is ([.] denotes the greatest integer function) (b) 3, 0 (c) 3 3, 0 0, (d) 3, 8 4

5 Section III (Total Marks : 4) (Integer Answer Type) This section contains 6 questions. The answer to each of the questions is a single-digit integer, ranging from 0 to 9.The bubble corresponding to the correct answer is to be darkened in the Answer sheet. 13. If f ( x ) = sin x sin x cos x cos x 3 3 and g 5 1, then (gof) (x) = 4. [ x] x [ x] 14. The least period of the function sin cos tan is, then the value of must be (where [.] denotes the greatest integer function) 8 y y 15. If f x,x = xy, then the value of f (60,48) f (80,48) f (13,5) must be 5x 16. If the period of the function cos ( nx ) sin is 3, then the number of integral n values of n must be. 17. Let n (A) = 4 and n (B) = 6, then the number of one-one functions from A to B is 357 +, find the value of. 008 r r If Ar, where r is natural number, then the value of Ar r 1 r 000 r1 must be. 5

6 Section IV (Total Marks : 16) (Matrix-Match Type) This section contains questions. Each question has four statements (a, b, c and d) given in Column I and four statements (p, q, r and s) in Column II. Any given statements in Column I can have correct matching with ONE or MORE statements(s) given in Column II. For example, if for a given question, statement B matches with the statements given in q and r, then for the particular question, against statement B, darken the bubbles corresponding to q and r in the ANSWER SHEET. 19. Column I : Function (a) sgn x n f ( x) {(sgn x) } ; x 0, n is an add integer x x (b) f ( x) 1 x e 1 (c) 0, If x isrational f(x) 1, If x isirrational (d) f ( x) max {tan x, cot x} Column II: Type of function (p) odd function (q) even function (r) neither odd nor even function (s) periodic 0. {.} denotes the fractional part function and [.] denotes the greatest integer function: Column I : (Function) Column II: (Period) (a) f(x) = e 4 cos x x [ x ] cos x (p) 1/3 (b) f(x) = cos { x} sin { x} (q) ¼ (c) f(x) = sin3 { x} tan [ x] (r) ½ (d) f(x) =3x [3x+a] b, where a, b R (s) 1 6

IIT JEE (2013) (Trigonometry and Algebra)

IIT JEE (2013) (Trigonometry and Algebra) L.K. Gupta (Mathematic Classes) www.pioneermathematics.com MOBILE: 985577, 4677 PAPER B IIT JEE () (Trigonometry and Algebra) TOWARDS IIT JEE IS NOT A JOURNEY, IT S A BATTLE, ONLY THE TOUGHEST WILL SURVIVE

More information

IIT JEE (2012) (Calculus)

IIT JEE (2012) (Calculus) L.K. Gupta (Mathematic Classes) www.pioneermathematics.com MOBILE: 985577, 4677 PAPER B IIT JEE (0) (Calculus) TOWARDS IIT JEE IS NOT A JOURNEY, IT S A BATTLE, ONLY THE TOUGHEST WILL SURVIVE TIME: 60 MINS

More information

IIT JEE (2011) PAPER-B

IIT JEE (2011) PAPER-B L.K.Gupta (Mathematic Classes) www.pioneermathematics.com. MOBILE: 985577, 4677 IIT JEE () (Integral calculus) TOWARDS IIT- JEE IS NOT A JOURNEY, IT S A BATTLE, ONLY THE TOUGHEST WILL SURVIVE TIME: 6 MINS

More information

IIT-JEE (2012) (Vector+3D+Probability) Solutions

IIT-JEE (2012) (Vector+3D+Probability) Solutions L.K. Gupta (Mathematic Classes) www.pioneermathematics.com MOBILE: 985577, 4677 PAPER -A IIT-JEE (0) (Vector+D+Probability) Solutions TOWARDS IIT- JEE IS NOT A JOURNEY, IT S A BATTLE, ONLY THE TOUGHEST

More information

(+1) PAPER -A IIT-JEE (2013) (Trigonomtery-1) Solutions

(+1) PAPER -A IIT-JEE (2013) (Trigonomtery-1) Solutions L.K. Gupta (Mathematic Classes) www.pioneermathematics.com MOBILE: 9877, 4677 IIT-JEE () (Trigonomtery-) Solutions (+) PAPER -A TOWARDS IIT- JEE IS NOT A JOURNEY, IT S A BATTLE, ONLY THE TOUGHEST WILL

More information

BRAIN TEASURES FUNCTION BY ABHIJIT KUMAR JHA EXERCISE I. log 5. (ii) f (x) = log 7. (iv) f (x) = 2 x. (x) f (x) = (xii) f (x) =

BRAIN TEASURES FUNCTION BY ABHIJIT KUMAR JHA EXERCISE I. log 5. (ii) f (x) = log 7. (iv) f (x) = 2 x. (x) f (x) = (xii) f (x) = EXERCISE I Q. Find the domains of definitions of the following functions : (Read the symbols [*] and {*} as greatest integers and fractional part functions respectively.) (i) f () = cos 6 (ii) f () = log

More information

RED. Name: Instructor: Pace Nielsen Math 290 Section 1: Winter 2014 Final Exam

RED. Name: Instructor: Pace Nielsen Math 290 Section 1: Winter 2014 Final Exam RED Name: Instructor: Pace Nielsen Math 290 Section 1: Winter 2014 Final Exam Note that the first 10 questions are true-false. Mark A for true, B for false. Questions 11 through 20 are multiple choice

More information

Solved Examples. Given are two sets A {1, 2, -2, 3} and B = {1, 2, 3, 5}. Is the function f(x) = 2x - 1 defined from A to B?

Solved Examples. Given are two sets A {1, 2, -2, 3} and B = {1, 2, 3, 5}. Is the function f(x) = 2x - 1 defined from A to B? Solved Examples Example 1: Given are two sets A {1, 2, -2, 3} and B = {1, 2, 3, 5}. Is the function f(x) = 2x - 1 defined from A to B? Solution : Out of all the ordered pairs, the ordered pairs which are

More information

RELATIONS AND FUNCTIONS

RELATIONS AND FUNCTIONS For more important questions visit : www.4ono.com CHAPTER 1 RELATIONS AND FUNCTIONS IMPORTANT POINTS TO REMEMBER Relation R from a set A to a set B is subset of A B. A B = {(a, b) : a A, b B}. If n(a)

More information

01 - SETS, RELATIONS AND FUNCTIONS Page 1 ( Answers at the end of all questions )

01 - SETS, RELATIONS AND FUNCTIONS Page 1 ( Answers at the end of all questions ) 0 SETS, RELATIONS AND FUNCTIONS Page ( ) Let R = { ( 3, 3 ) ( 6, 6 ) ( ( 9, 9 ) (, ), ( 6, ) ( 3, 9 ) ( 3, ), ( 3, 6 ) } be a relation on the set A = { 3, 6, 9, }. The relation ( a ) refleive and transitive

More information

CLASS XII CBSE MATHEMATICS RELATIONS AND FUNCTIONS 1 Mark/2 Marks Questions

CLASS XII CBSE MATHEMATICS RELATIONS AND FUNCTIONS 1 Mark/2 Marks Questions CLASS XII CBSE MATHEMATICS RELATIONS AND FUNCTIONS 1 Mark/ Marks Questions (1) Let be a binary operation defined by a b = a + b 3. Find3 4. () The binary operation : R R R is defined as a b = a + b. Find

More information

QUESTION BANK II PUC SCIENCE

QUESTION BANK II PUC SCIENCE QUESTION BANK II PUC SCIENCE I. Very Short answer questions. (x9=9). Define Symmetric relation. Ans: A relation R on the set A is said to be symmetric if for all a, b, A, ar b Implies bra. i.e. (a, b)

More information

FREE Download Study Package from website: &

FREE Download Study Package from website:  & SHORT REVISION (FUNCTIONS) THINGS TO REMEMBER :. GENERAL DEFINITION : If to every value (Considered as real unless otherwise stated) of a variable which belongs to some collection (Set) E there corresponds

More information

IIT JEE Maths Paper 2

IIT JEE Maths Paper 2 IIT JEE - 009 Maths Paper A. Question paper format: 1. The question paper consists of 4 sections.. Section I contains 4 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for

More information

[STRAIGHT OBJECTIVE TYPE] log 4 2 x 4 log. (sin x + cos x) = 10 (A) 24 (B) 36 (C) 20 (D) 12

[STRAIGHT OBJECTIVE TYPE] log 4 2 x 4 log. (sin x + cos x) = 10 (A) 24 (B) 36 (C) 20 (D) 12 [STRAIGHT OBJECTIVE TYPE] Q. The equation, ( ) +. + 4 4 + / (A) eactly one real solution (B) two real solutions (C) real solutions (D) no solution. = has : ( n) Q. If 0 sin + 0 cos = and 0 (sin + cos )

More information

Objective Mathematics

Objective Mathematics Multiple choice questions with ONE correct answer : ( Questions No. 1-5 ) 1. If the equation x n = (x + ) is having exactly three distinct real solutions, then exhaustive set of values of 'n' is given

More information

ISI B.STAT/B.MATH OBJECTIVE QUESTIONS & SOLUTIONS SET 1

ISI B.STAT/B.MATH OBJECTIVE QUESTIONS & SOLUTIONS SET 1 1 Blog: www.ctanujit.in Ph: +91-84053573 ISI B.STAT/B.MATH OBJECTIVE QUESTIONS & SOLUTIONS SET 1 1. How many zeros are at the end of 1000!? (a) 40 (b) 48 (c) 49 (d) None Ans:- (c) The number of two s is

More information

MockTime.com. (b) 9/2 (c) 18 (d) 27

MockTime.com. (b) 9/2 (c) 18 (d) 27 212 NDA Mathematics Practice Set 1. Let X be any non-empty set containing n elements. Then what is the number of relations on X? 2 n 2 2n 2 2n n 2 2. Only 1 2 and 3 1 and 2 1 and 3 3. Consider the following

More information

Instructions. 2. Four possible answers are provided for each question and only one of these is correct.

Instructions. 2. Four possible answers are provided for each question and only one of these is correct. Instructions 1. This question paper has forty multiple choice questions. 2. Four possible answers are provided for each question and only one of these is correct. 3. Marking scheme: Each correct answer

More information

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CORE COURSE. B.Sc. MATHEMATICS V SEMESTER. (2011 Admission onwards) BASIC MATHEMATICAL ANALYSIS

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CORE COURSE. B.Sc. MATHEMATICS V SEMESTER. (2011 Admission onwards) BASIC MATHEMATICAL ANALYSIS UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CORE COURSE B.Sc. MATHEMATICS V SEMESTER (2011 Admission onwards) BASIC MATHEMATICAL ANALYSIS QUESTION BANK 1. Find the number of elements in the power

More information

Functions. Definition 1 Let A and B be sets. A relation between A and B is any subset of A B.

Functions. Definition 1 Let A and B be sets. A relation between A and B is any subset of A B. Chapter 4 Functions Definition 1 Let A and B be sets. A relation between A and B is any subset of A B. Definition 2 Let A and B be sets. A function from A to B is a relation f between A and B such that

More information

(c) n (d) n 2. (a) (b) (c) (d) (a) Null set (b) {P} (c) {P, Q, R} (d) {Q, R} (a) 2k (b) 7 (c) 2 (d) K (a) 1 (b) 3 (c) 3xyz (d) 27xyz

(c) n (d) n 2. (a) (b) (c) (d) (a) Null set (b) {P} (c) {P, Q, R} (d) {Q, R} (a) 2k (b) 7 (c) 2 (d) K (a) 1 (b) 3 (c) 3xyz (d) 27xyz 318 NDA Mathematics Practice Set 1. (1001)2 (101)2 (110)2 (100)2 2. z 1/z 2z z/2 3. The multiplication of the number (10101)2 by (1101)2 yields which one of the following? (100011001)2 (100010001)2 (110010011)2

More information

MATHEMATICS. 61. If letters of the word KUBER are written in all possible orders and arranged as in a dictionary, then rank of the word KUBER will be:

MATHEMATICS. 61. If letters of the word KUBER are written in all possible orders and arranged as in a dictionary, then rank of the word KUBER will be: MATHEMATICS 61. If letters of the word KUBER are written in all possible orders and arranged as in a dictionary, then rank of the word KUBER will be: (A) 67 (B) 68 (C) 65 (D) 69 : Alphabetical order of

More information

are in c) A B (D) 2 = {4,5,6} by = {(4,4), (5,5), (6,6)} is (C) (B) 0 < (C) 0 = 8, = 5 = 8, = 8 (B) (D) (C) 2 +

are in c) A B (D) 2 = {4,5,6} by = {(4,4), (5,5), (6,6)} is (C) (B) 0 < (C) 0 = 8, = 5 = 8, = 8 (B) (D) (C) 2 + 1. If are in GP then AP GP are in HP 2. The sum to infinity of the series 1 3. The set B-A a subset of a) A c) A B b) B d)null set 4. The converse of the statement if 3 3 6 then I am the president of USA

More information

(iii) For each question in Section III, you will be awarded 4 Marks if you darken only the bubble corresponding to the

(iii) For each question in Section III, you will be awarded 4 Marks if you darken only the bubble corresponding to the FIITJEE Solutions to IIT - JEE 8 (Paper, Code 4) Time: hours M. Marks: 4 Note: (i) The question paper consists of parts (Part I : Mathematics, Part II : Physics, Part III : Chemistry). Each part has 4

More information

NATIONAL BOARD FOR HIGHER MATHEMATICS. M. A. and M.Sc. Scholarship Test. September 22, Time Allowed: 150 Minutes Maximum Marks: 30

NATIONAL BOARD FOR HIGHER MATHEMATICS. M. A. and M.Sc. Scholarship Test. September 22, Time Allowed: 150 Minutes Maximum Marks: 30 NATIONAL BOARD FOR HIGHER MATHEMATICS M. A. and M.Sc. Scholarship Test September 22, 2012 Time Allowed: 150 Minutes Maximum Marks: 30 Please read, carefully, the instructions on the following page 1 INSTRUCTIONS

More information

0 Sets and Induction. Sets

0 Sets and Induction. Sets 0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set

More information

MATHEMATICS. IMPORTANT FORMULAE AND CONCEPTS for. Final Revision CLASS XII CHAPTER WISE CONCEPTS, FORMULAS FOR QUICK REVISION.

MATHEMATICS. IMPORTANT FORMULAE AND CONCEPTS for. Final Revision CLASS XII CHAPTER WISE CONCEPTS, FORMULAS FOR QUICK REVISION. MATHEMATICS IMPORTANT FORMULAE AND CONCEPTS for Final Revision CLASS XII 2016 17 CHAPTER WISE CONCEPTS, FORMULAS FOR QUICK REVISION Prepared by M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist (Elect.),

More information

NATIONAL BOARD FOR HIGHER MATHEMATICS. M. A. and M.Sc. Scholarship Test. September 22, Time Allowed: 150 Minutes Maximum Marks: 30

NATIONAL BOARD FOR HIGHER MATHEMATICS. M. A. and M.Sc. Scholarship Test. September 22, Time Allowed: 150 Minutes Maximum Marks: 30 NATIONAL BOARD FOR HIGHER MATHEMATICS M A and MSc Scholarship Test September 22, 2018 Time Allowed: 150 Minutes Maximum Marks: 30 Please read, carefully, the instructions that follow INSTRUCTIONS TO CANDIDATES

More information

Practice Set for IIT JEE. Paper I

Practice Set for IIT JEE. Paper I Objective Questions I [Only one correct option] Practice Set for IIT JEE Paper I Q 1. The number of lines in the xy-plane, Whose distance from (-1, 2) is 2 and from (2, 6) is 3, is a. 2 b. 3 c. 4 d. None

More information

IIT-JEE-Mathematics-Screening 2001

IIT-JEE-Mathematics-Screening 2001 IIT-JEE-Mathematics-Screening 2001 SCREENING Time : Three hours Max. Marks : 100 Notations : R : set of real numbers. [x] : the greatest integer x. 1. Let f R R be a function defined by (x)=max { x,x3

More information

Paper Specific Instructions

Paper Specific Instructions Paper Specific Instructions. The examination is of 3 hours duration. There are a total of 60 questions carrying 00 marks. The entire paper is divided into three sections, A, B and C. All sections are compulsory.

More information

Introduction to Decision Sciences Lecture 6

Introduction to Decision Sciences Lecture 6 Introduction to Decision Sciences Lecture 6 Andrew Nobel September 21, 2017 Functions Functions Given: Sets A and B, possibly different Definition: A function f : A B is a rule that assigns every element

More information

A Short Review of Cardinality

A Short Review of Cardinality Christopher Heil A Short Review of Cardinality November 14, 2017 c 2017 Christopher Heil Chapter 1 Cardinality We will give a short review of the definition of cardinality and prove some facts about the

More information

AP Calculus. Limits, Continuity, and Differentiability

AP Calculus. Limits, Continuity, and Differentiability AP Calculus Limits, Continuity, and Differentiability Student Handout 016 017 EDITION Click on the following link or scan the QR code to complete the evaluation for the Study Session https://www.surveymonkey.com/r/s_sss

More information

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Fall

3 FUNCTIONS. 3.1 Definition and Basic Properties. c Dr Oksana Shatalov, Fall c Dr Oksana Shatalov, Fall 2014 1 3 FUNCTIONS 3.1 Definition and Basic Properties DEFINITION 1. Let A and B be nonempty sets. A function f from A to B is a rule that assigns to each element in the set

More information

AP Calculus AB Chapter 1 Limits

AP Calculus AB Chapter 1 Limits AP Calculus AB Chapter Limits SY: 206 207 Mr. Kunihiro . Limits Numerical & Graphical Show all of your work on ANOTHER SHEET of FOLDER PAPER. In Exercises and 2, a stone is tossed vertically into the air

More information

MockTime.com. (b) (c) (d)

MockTime.com. (b) (c) (d) 373 NDA Mathematics Practice Set 1. If A, B and C are any three arbitrary events then which one of the following expressions shows that both A and B occur but not C? 2. Which one of the following is an

More information

is equal to = 3 2 x, if x < 0 f (0) = lim h = 0. Therefore f exists and is continuous at 0.

is equal to = 3 2 x, if x < 0 f (0) = lim h = 0. Therefore f exists and is continuous at 0. Madhava Mathematics Competition January 6, 2013 Solutions and scheme of marking Part I N.B. Each question in Part I carries 2 marks. p(k + 1) 1. If p(x) is a non-constant polynomial, then lim k p(k) (a)

More information

Definition: Let S and T be sets. A binary relation on SxT is any subset of SxT. A binary relation on S is any subset of SxS.

Definition: Let S and T be sets. A binary relation on SxT is any subset of SxT. A binary relation on S is any subset of SxS. 4 Functions Before studying functions we will first quickly define a more general idea, namely the notion of a relation. A function turns out to be a special type of relation. Definition: Let S and T be

More information

SPECIMEN EXAMINATION 2014/2015 ADEDEX424. Access to Science - Mathematics 1. Dr. Anthony Cronin Dr. Anthony Brown. Time Allowed: 3 hours

SPECIMEN EXAMINATION 2014/2015 ADEDEX424. Access to Science - Mathematics 1. Dr. Anthony Cronin Dr. Anthony Brown. Time Allowed: 3 hours University College Dublin An Coláiste Ollscoile, Baile Átha Cliath SPECIMEN EXAMINATION 2014/2015 ADEDEX424 Access to Science - Mathematics 1 Dr. Anthony Cronin Dr. Anthony Brown Time Allowed: hours Instructions

More information

MATH 220 FINAL EXAMINATION December 13, Name ID # Section #

MATH 220 FINAL EXAMINATION December 13, Name ID # Section # MATH 22 FINAL EXAMINATION December 3, 2 Name ID # Section # There are??multiple choice questions. Each problem is worth 5 points. Four possible answers are given for each problem, only one of which is

More information

can only hit 3 points in the codomain. Hence, f is not surjective. For another example, if n = 4

can only hit 3 points in the codomain. Hence, f is not surjective. For another example, if n = 4 .. Conditions for Injectivity and Surjectivity In this section, we discuss what we can say about linear maps T : R n R m given only m and n. We motivate this problem by looking at maps f : {,..., n} {,...,

More information

REVIEW Chapter 1 The Real Number System

REVIEW Chapter 1 The Real Number System REVIEW Chapter The Real Number System In class work: Complete all statements. Solve all exercises. (Section.4) A set is a collection of objects (elements). The Set of Natural Numbers N N = {,,, 4, 5, }

More information

NATIONAL BOARD FOR HIGHER MATHEMATICS. M. A. and M.Sc. Scholarship Test. September 17, Time Allowed: 150 Minutes Maximum Marks: 30

NATIONAL BOARD FOR HIGHER MATHEMATICS. M. A. and M.Sc. Scholarship Test. September 17, Time Allowed: 150 Minutes Maximum Marks: 30 NATIONAL BOARD FOR HIGHER MATHEMATICS M. A. and M.Sc. Scholarship Test September 17, 2016 Time Allowed: 150 Minutes Maximum Marks: 30 Please read, carefully, the instructions that follow INSTRUCTIONS TO

More information

DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO SO

DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO SO DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO SO T.B.C. : P-AQNA-L-ZNGU Serial No.- TEST BOOKLET MATHEMATICS Test Booklet Series Time Allowed : Two Hours and Thirty Minutes Maximum Marks : 00

More information

GS-2013 (Mathematics) TATA INSTITUTE OF FUNDAMENTAL RESEARCH. Written Test in MATHEMATICS December 9, 2012 Duration : Two hours (2 hours)

GS-2013 (Mathematics) TATA INSTITUTE OF FUNDAMENTAL RESEARCH. Written Test in MATHEMATICS December 9, 2012 Duration : Two hours (2 hours) MTH GS-2013 (Mathematics) TATA INSTITUTE OF FUNDAMENTAL RESEARCH WrittenTestinMATHEMATICSDecember9,2012 Duration:Twohours(2hours) Name: Ref.Code: Pleasereadallinstructionscarefullybeforeyouattemptthequestions.

More information

Test Codes : MIA (Objective Type) and MIB (Short Answer Type) 2007

Test Codes : MIA (Objective Type) and MIB (Short Answer Type) 2007 Test Codes : MIA (Objective Type) and MIB (Short Answer Type) 007 Questions will be set on the following and related topics. Algebra: Sets, operations on sets. Prime numbers, factorisation of integers

More information

Fundamentals of Pure Mathematics - Problem Sheet

Fundamentals of Pure Mathematics - Problem Sheet Fundamentals of Pure Mathematics - Problem Sheet ( ) = Straightforward but illustrates a basic idea (*) = Harder Note: R, Z denote the real numbers, integers, etc. assumed to be real numbers. In questions

More information

Sets and Functions. MATH 464/506, Real Analysis. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Sets and Functions

Sets and Functions. MATH 464/506, Real Analysis. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Sets and Functions Sets and Functions MATH 464/506, Real Analysis J. Robert Buchanan Department of Mathematics Summer 2007 Notation x A means that element x is a member of set A. x / A means that x is not a member of A.

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #24 12/03/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #24 12/03/2013 18.78 Introduction to Arithmetic Geometry Fall 013 Lecture #4 1/03/013 4.1 Isogenies of elliptic curves Definition 4.1. Let E 1 /k and E /k be elliptic curves with distinguished rational points O 1 and

More information

Week 5: Functions and graphs

Week 5: Functions and graphs Calculus and Linear Algebra for Biomedical Engineering Week 5: Functions and graphs H. Führ, Lehrstuhl A für Mathematik, RWTH Aachen, WS 07 Motivation: Measurements at fixed intervals 1 Consider a sequence

More information

T ((x 1, x 2,..., x n )) = + x x 3. , x 1. x 3. Each of the four coordinates in the range is a linear combination of the three variables x 1

T ((x 1, x 2,..., x n )) = + x x 3. , x 1. x 3. Each of the four coordinates in the range is a linear combination of the three variables x 1 MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

More information

x 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7

x 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7 Linear Algebra and its Applications-Lab 1 1) Use Gaussian elimination to solve the following systems x 1 + x 2 2x 3 + 4x 4 = 5 1.1) 2x 1 + 2x 2 3x 3 + x 4 = 3 3x 1 + 3x 2 4x 3 2x 4 = 1 x + y + 2z = 4 1.4)

More information

WBJEE Answer Keys by Aakash Institute, Kolkata Centre

WBJEE Answer Keys by Aakash Institute, Kolkata Centre WBJEE - 08 Answer Keys by, Kolkata Centre MATHEMATICS Q.No. 0 A B C D 0 C D A B 0 B D A C 04 C B A B 05 C A C C 06 A C D C 07 B A C C 08 B *C,D C A 09 C D D B 0 D A C D B A B C C D A B B A A C 4 C C B

More information

2010 Maths. Advanced Higher. Finalised Marking Instructions

2010 Maths. Advanced Higher. Finalised Marking Instructions 00 Maths Advanced Higher Finalised Marking Instructions Scottish Qualifications Authority 00 The information in this publication may be reproduced to support SQA qualifications only on a noncommercial

More information

118 PU Ph D Mathematics

118 PU Ph D Mathematics 118 PU Ph D Mathematics 1 of 100 146 PU_2016_118_E The function fz = z is:- not differentiable anywhere differentiable on real axis differentiable only at the origin differentiable everywhere 2 of 100

More information

Mathematics Review for Business PhD Students

Mathematics Review for Business PhD Students Mathematics Review for Business PhD Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

More information

Functions as Relations

Functions as Relations Functions as Relations Definition Recall that if A and B are sets, then a relation from A to B is a subset of A B. A function from A to B is a relation f from A to B with the following properties (i) The

More information

MOST IMPORTANT QUESTIONS

MOST IMPORTANT QUESTIONS PART - Centurion Assignment By OP GUPTA INDIRA AWARD WINNER CLASS XII MATHEMATICS MOST IMPORTANT QUESTIONS Without Mathematics, there s nothing you can do Everything around you is Mathematics Everything

More information

(4) Using results you have studied, show that if x, y are real numbers,

(4) Using results you have studied, show that if x, y are real numbers, Solutions to Homework 4, Math 310 (1) Give a direct proof to show that if a, b are integers which are squares of integers, then ab is the square of an integer. Proof. We show that if a, b are integers

More information

Solutions to Homework Set 1

Solutions to Homework Set 1 Solutions to Homework Set 1 1. Prove that not-q not-p implies P Q. In class we proved that A B implies not-b not-a Replacing the statement A by the statement not-q and the statement B by the statement

More information

CLASS - X Mathematics (Real Number)

CLASS - X Mathematics (Real Number) CLASS - X Mathematics (Real Number) 1. 7 11 13 15 + 15is a (a) Composite number (c) Prime number (b) Whole number (d) None of these. For what least value of n a natural number, ( 4) n is divisible by 8?

More information

B Sc MATHEMATICS ABSTRACT ALGEBRA

B Sc MATHEMATICS ABSTRACT ALGEBRA UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc MATHEMATICS (0 Admission Onwards) V Semester Core Course ABSTRACT ALGEBRA QUESTION BANK () Which of the following defines a binary operation on Z

More information

More Books At www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com

More information

Midterm Preparation Problems

Midterm Preparation Problems Midterm Preparation Problems The following are practice problems for the Math 1200 Midterm Exam. Some of these may appear on the exam version for your section. To use them well, solve the problems, then

More information

NATIONAL INSTITUTE OF HOTEL MANAGEMENT, KOLKATA BUSINESS MATHEMATICS 3 rd Semester

NATIONAL INSTITUTE OF HOTEL MANAGEMENT, KOLKATA BUSINESS MATHEMATICS 3 rd Semester NATIONAL INSTITUTE OF HOTEL MANAGEMENT, KOLKATA BUSINESS MATHEMATICS 3 rd Semester Choose (tick) the appropriate from the following options given below. 1. Find the number of subsets of a set {x : x is

More information

Function Terminology and Types of Functions

Function Terminology and Types of Functions 1.2: Rate of Change by Equation, Graph, or Table [AP Calculus AB] Objective: Given a function y = f(x) specified by a graph, a table of values, or an equation, describe whether the y-value is increasing

More information

Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i

Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i Complex Numbers: Definition: A complex number is a number of the form: z = a + bi where a, b are real numbers and i is a symbol with the property: i 2 = 1 Sometimes we like to think of i = 1 We can treat

More information

JUST THE MATHS UNIT NUMBER DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) A.J.Hobson

JUST THE MATHS UNIT NUMBER DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) A.J.Hobson JUST THE MATHS UNIT NUMBER.5 DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) by A.J.Hobson.5. Maclaurin s series.5. Standard series.5.3 Taylor s series.5.4 Exercises.5.5 Answers to exercises

More information

Functions Functions and Modeling A UTeach/TNT Course

Functions Functions and Modeling A UTeach/TNT Course Definition of a Function DEFINITION: Let A and B be sets. A function between A and B is a subset of A B with the property that if (a, b 1 )and(a, b 2 ) are both in the subset, then b 1 = b 2. The domain

More information

Solutions to Assignment 1

Solutions to Assignment 1 Solutions to Assignment 1 Question 1. [Exercises 1.1, # 6] Use the division algorithm to prove that every odd integer is either of the form 4k + 1 or of the form 4k + 3 for some integer k. For each positive

More information

A. Incorrect! This equality is true for all values of x. Therefore, this is an identity and not a conditional equation.

A. Incorrect! This equality is true for all values of x. Therefore, this is an identity and not a conditional equation. CLEP-Precalculus - Problem Drill : Trigonometric Identities No. of 0 Instructions: () Read the problem and answer choices carefully () Work the problems on paper as. Which of the following equalities is

More information

Procedure for Graphing Polynomial Functions

Procedure for Graphing Polynomial Functions Procedure for Graphing Polynomial Functions P(x) = a nx n + a n-1x n-1 + + a 1x + a 0 To graph P(x): As an example, we will examine the following polynomial function: P(x) = 2x 3 3x 2 23x + 12 1. Determine

More information

Solutions to Homework Problems

Solutions to Homework Problems Solutions to Homework Problems November 11, 2017 1 Problems II: Sets and Functions (Page 117-118) 11. Give a proof or a counterexample of the following statements: (vi) x R, y R, xy 0; (x) ( x R, y R,

More information

FUNCTIONS. Note: Example of a function may be represented diagrammatically. The above example can be written diagrammatically as follows.

FUNCTIONS. Note: Example of a function may be represented diagrammatically. The above example can be written diagrammatically as follows. FUNCTIONS Def : A relation f from a set A into a set is said to be a function or mapping from A into if for each A there eists a unique such that (, ) f. It is denoted b f : A. Note: Eample of a function

More information

MATH 403 MIDTERM ANSWERS WINTER 2007

MATH 403 MIDTERM ANSWERS WINTER 2007 MAH 403 MIDERM ANSWERS WINER 2007 COMMON ERRORS (1) A subset S of a ring R is a subring provided that x±y and xy belong to S whenever x and y do. A lot of people only said that x + y and xy must belong

More information

PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES PRADEEP SHARMA. PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 1

PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES PRADEEP SHARMA. PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 1 PRADEEP SHARMA PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page Chapter Relation and Functions Mark Questions A relation R in a Set A is called..., if each element of A is related to every element

More information

NATIONAL BOARD FOR HIGHER MATHEMATICS. Research Awards Screening Test. February 25, Time Allowed: 90 Minutes Maximum Marks: 40

NATIONAL BOARD FOR HIGHER MATHEMATICS. Research Awards Screening Test. February 25, Time Allowed: 90 Minutes Maximum Marks: 40 NATIONAL BOARD FOR HIGHER MATHEMATICS Research Awards Screening Test February 25, 2006 Time Allowed: 90 Minutes Maximum Marks: 40 Please read, carefully, the instructions on the following page before you

More information

RED. Fall 2016 Student Submitted Sample Questions

RED. Fall 2016 Student Submitted Sample Questions RED Fall 2016 Student Submitted Sample Questions Name: Last Update: November 22, 2016 The questions are divided into three sections: True-false, Multiple Choice, and Written Answer. I will add questions

More information

1.7 Sums of series Example 1.7.1: 2. Real functions of one variable Example 1.7.2: 2.1 General definitions Example 2.1.3: Example 2.1.

1.7 Sums of series Example 1.7.1: 2. Real functions of one variable Example 1.7.2: 2.1 General definitions Example 2.1.3: Example 2.1. 7 Sums of series We often want to sum a series of terms, for example when we look at polynomials As we already saw, we abbreviate a sum of the form For example and u + u + + u r by r u i i= a n x n + a

More information

Solved Examples. (Highest power of x in numerator and denominator is ½. Dividing numerator and denominator by x)

Solved Examples. (Highest power of x in numerator and denominator is ½. Dividing numerator and denominator by x) Solved Examples Example 1: (i) (ii) lim x (x 4 + 2x 3 +3) / (2x 4 -x+2) lim x x ( (x+c)- x) (iii) lim n (1-2+3-4+...(2n-1)-2n)/ (n 2 +1) (iv) lim x 0 ((1+x) 5-1)/3x+5x 2 (v) lim x 2 ( (x+7)-3 (2x-3))/((x+6)

More information

Booklet Number: 2016 TEST CODE: DST. Objective type: 30 Questions Time: 2 hours

Booklet Number: 2016 TEST CODE: DST. Objective type: 30 Questions Time: 2 hours Booklet Number: 016 TEST CODE: DST Forenoon Objective type: 30 Questions Time: hours You are being given a separate Answer Sheet for answering all the questions. Write your Name, Registration Number, Test

More information

Math 110 Test # 1. The set of real numbers in both of the intervals [0, 2) and ( 1, 0] is equal to. Question 1. (F) [ 1, 2) (G) (2, ) (H) [ 1, 2]

Math 110 Test # 1. The set of real numbers in both of the intervals [0, 2) and ( 1, 0] is equal to. Question 1. (F) [ 1, 2) (G) (2, ) (H) [ 1, 2] Friday July 8, 00 Jacek Szmigielski Math 0 Test # Fill in the bubbles that correspond to the correct answers. No aids: no calculators, closed book. You are not permitted to consult with your fellow students

More information

Solutions to Practice Final

Solutions to Practice Final s to Practice Final 1. (a) What is φ(0 100 ) where φ is Euler s φ-function? (b) Find an integer x such that 140x 1 (mod 01). Hint: gcd(140, 01) = 7. (a) φ(0 100 ) = φ(4 100 5 100 ) = φ( 00 5 100 ) = (

More information

SEMESTER 1 EXAMINATION 2016/2017 MATH Access to Science, Engineering and Agriculture: Mathematics 1

SEMESTER 1 EXAMINATION 2016/2017 MATH Access to Science, Engineering and Agriculture: Mathematics 1 University College Dublin An Coláiste Ollscoile, Baile Átha Cliath SEMESTER 1 EXAMINATION 2016/2017 MATH00030 Access to Science, Engineering and Agriculture: Mathematics 1 Professor G. McGuire Dr. Anthony

More information

Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes

Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes These notes form a brief summary of what has been covered during the lectures. All the definitions must be memorized and understood. Statements

More information

AP Calculus AB Worksheet - Differentiability

AP Calculus AB Worksheet - Differentiability Name AP Calculus AB Worksheet - Differentiability MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The figure shows the graph of a function. At the

More information

Mathematic 108, Fall 2015: Solutions to assignment #7

Mathematic 108, Fall 2015: Solutions to assignment #7 Mathematic 08, Fall 05: Solutions to assignment #7 Problem # Suppose f is a function with f continuous on the open interval I and so that f has a local maximum at both x = a and x = b for a, b I with a

More information

IIT JEE MODEL QUESTIONS MATHS SEQUENCES AND SERIES

IIT JEE MODEL QUESTIONS MATHS SEQUENCES AND SERIES . If tan (A +B), tan B, tan (B + C) are in AP, then tan A, cot B,tan C are in (a) AP (b) GP (c) HP (d) None of these. Consider an acute-angled triangle ABC having P as its orthocentre. If the distance

More information

MAXIMA AND MINIMA CHAPTER 7.1 INTRODUCTION 7.2 CONCEPT OF LOCAL MAXIMA AND LOCAL MINIMA

MAXIMA AND MINIMA CHAPTER 7.1 INTRODUCTION 7.2 CONCEPT OF LOCAL MAXIMA AND LOCAL MINIMA CHAPTER 7 MAXIMA AND MINIMA 7.1 INTRODUCTION The notion of optimizing functions is one of the most important application of calculus used in almost every sphere of life including geometry, business, trade,

More information

Quick-and-Easy Factoring. of lower degree; several processes are available to fi nd factors.

Quick-and-Easy Factoring. of lower degree; several processes are available to fi nd factors. Lesson 11-3 Quick-and-Easy Factoring BIG IDEA Some polynomials can be factored into polynomials of lower degree; several processes are available to fi nd factors. Vocabulary factoring a polynomial factored

More information

1. Let g(x) and h(x) be polynomials with real coefficients such that

1. Let g(x) and h(x) be polynomials with real coefficients such that 1. Let g(x) and h(x) be polynomials with real coefficients such that g(x)(x 2 3x + 2) = h(x)(x 2 + 3x + 2) and f(x) = g(x)h(x) + (x 4 5x 2 + 4). Prove that f(x) has at least four real roots. 2. Let M be

More information

MATH 2112/CSCI 2112, Discrete Structures I Winter 2007 Toby Kenney Homework Sheet 5 Hints & Model Solutions

MATH 2112/CSCI 2112, Discrete Structures I Winter 2007 Toby Kenney Homework Sheet 5 Hints & Model Solutions MATH 11/CSCI 11, Discrete Structures I Winter 007 Toby Kenney Homework Sheet 5 Hints & Model Solutions Sheet 4 5 Define the repeat of a positive integer as the number obtained by writing it twice in a

More information

NATIONAL BOARD FOR HIGHER MATHEMATICS. M. A. and M.Sc. Scholarship Test. September 24, Time Allowed: 150 Minutes Maximum Marks: 30

NATIONAL BOARD FOR HIGHER MATHEMATICS. M. A. and M.Sc. Scholarship Test. September 24, Time Allowed: 150 Minutes Maximum Marks: 30 NATIONAL BOARD FOR HIGHER MATHEMATICS M. A. and M.Sc. Scholarship Test September 24, 2011 Time Allowed: 150 Minutes Maximum Marks: 30 Please read, carefully, the instructions on the following page 1 INSTRUCTIONS

More information

Mathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations

Mathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations Mathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations D. R. Wilkins Academic Year 1996-7 1 Number Systems and Matrix Algebra Integers The whole numbers 0, ±1, ±2, ±3, ±4,...

More information

3. Total number of functions from the set A to set B is n. 4. Total number of one-one functions from the set A to set B is n Pm

3. Total number of functions from the set A to set B is n. 4. Total number of one-one functions from the set A to set B is n Pm ASSIGNMENT CLASS XII RELATIONS AND FUNCTIONS Important Formulas If A and B are finite sets containing m and n elements, then Total number of relations from the set A to set B is mn Total number of relations

More information

Inverse Trigonometrical Functions 1. Properties of Inverse Trigonometrical Function. 1. The domain of sin x is [Roorkee Screening 1993] (a) (d)

Inverse Trigonometrical Functions 1. Properties of Inverse Trigonometrical Function. 1. The domain of sin x is [Roorkee Screening 1993] (a) (d) Inverse Trigonometrical Functions Basic Level Properties of Inverse Trigonometrical Function. The domain of [Roorkee Screening 99] ( ) [ ] ( 0 ) ( ). The range of [DCE 00] ( ) ( 0 ). cos equal to [Pb.

More information

(3) Let Y be a totally bounded subset of a metric space X. Then the closure Y of Y

(3) Let Y be a totally bounded subset of a metric space X. Then the closure Y of Y () Consider A = { q Q : q 2 2} as a subset of the metric space (Q, d), where d(x, y) = x y. Then A is A) closed but not open in Q B) open but not closed in Q C) neither open nor closed in Q D) both open

More information

AP Calculus Summer Packet

AP Calculus Summer Packet AP Calculus Summer Packet Writing The Equation Of A Line Example: Find the equation of a line that passes through ( 1, 2) and (5, 7). ü Things to remember: Slope formula, point-slope form, slopeintercept

More information