Er. L.K.Sharma - Weebly
(MAINS & ADVANCE)Er. L.K.Sharma
Er. L.K.Sharma an engineering graduate fromNIT, Jaipur (Rajasthan), {Gold medalist,University of Rajasthan} is a well known nameamong the engineering aspirants for the last 15years.HehasbeenhonoredwithBHAMASHAH AWARD two times for theacademic excellence in the state of Rajasthan.He is popular among the student community for possessing theexcellent ability to communicate the mathematical concepts inanalytical and graphical way.He has worked with many premiere IIT-JEE coaching institutesof Delhi, Haryana, Jaipur and Kota, {presently associated withTHE GUIDANCE, Kalu Sarai, New Delhi as seniormathematics faculty and Head of Mathematics department withIGNEOUS, Sonipat (Haryana)}. He has worked with DelhiPublic School, RK Puram, New Delhi for five years as a seniormathematics {IIT-JEE} faculty.Er. L.K.Sharma has been proved a great supportive mentor forthe last 15 years and the most successful IIT-JEE aspirantsconsider him an ideal mathematician for Olympiad/KVPY/ISIpreparation. He is also involved in the field of online teaching toengineering aspirants and is associated with www.100marks.inand iProf Learning Sols India (P) Ltd for last 5 years , as a seniormember of www.wiziq.com (an online teaching and learningportal), delivered many online lectures on different topics ofmathematics at IIT-JEE {mains & advance} level .“Objective Mathematics for IIT-JEE” authored by Er. L.K.Sharmahas also proved a great help for engineering aspirants and its e-bookformat can be downloaded from http://mathematicsgyan.weebly.com.
Contents1. Basics of Mathematics12. Quadratic Equations143. Complex Numbers204. Binomial Theorem295. Permutation and Combination326. Probability367. Matrices428. Determinants559. Sequences and Series6110. Functions6711. Limits7612. Continuity and Differentiability8013. Differentiation8614. Tangent and Normal9115. Rolle's Theorem and Mean Value Theorem9316. Monotonocity9517. Maxima and Minima9718. Indefinite Integral10119. Definite Integral10920. Area Bounded by Curves11421. Differential Equations118
22. Basics of 2D-Geometry12323. Straight Lines12524. Pair of Straight Lines12925. Circles13226. Parabola13827. Ellipse14228. Hyperbola14629. Vectors15230. 3-Dimensional Geometry16131. Trigonometric Ratios and Identities17032. Trigonometric Equations and Inequations17533. Solution of Triangle17734. Inverse Trigonometric Functions180
1. BASICS of MATHEMATICS{1} Number System :(i)Natural Numbers :The counting numbers 1,2,3,4,. are called Natural Numbers.The set of natural numbers is denoted by N.Thus N {1,2,3,4, .}.(ii)Whole Numbers :Natural numbers including zero are called whole numbers.The set of whole numbers, is denoted by W.Thus W {0,1,2, .}(iii)Integers :The numbers . 3, 2, 1, 0, 1,2,3,. are called integers and the set is denoted byI or Z.Thus I (or Z) {. 3, 2, 1, 0, 1, 2, 3.}(a)Set of positive integers denoted by I and consists of {1,2,3,.} called as set ofNatural numbers.(b)Set of negative integers, denoted by I and consists of {.,(c)Set of non-negative integers {0,1,2 .}, called as set of Whole numbers.(d)Set of non-positive integers {., 3, 2, 1} 3, 2, 1,0](iv)Even Integers :Integers which are divisible by 2 are called even integers.e.g.0, 2, 4.(v)Odd Integers :Integers, which are not divisible by 2 are called as odd integers.e.g. 1, 3, 5, 7.(vi)Prime Number :Let 'p' be a natural number, 'p' is said to be prime if it has exactly two distinct factors,namely 1 and itself.so, Natural number which are divisible by 1 and itself only are prime numbers (except 1).e.g.2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, .(vii) Composite Number :Let 'a' be a natural number, 'a' is said to be composite if, it has at least three distinctfactors.Note:(i)'1' is neither prime nor composite.(ii)'2' is the only even prime number.marahS(iii)Number which are not prime are composite numbers (except 1).82K.6L7(iv)'4' is the smallest composite number.Er. 81027 50589 801(viii) Co-prime Number :839Two natural numbers (not necessarily prime) are coprime, if thereH.C.F (Highest commonfactor) is one.e.g.(1,2), (1,3), (3,4), (3, 10), (3,8), (5,6), (7,8) etc.Mathematics Concept NoteIIT-JEE/ISI/CMIPage 1
These numbers are also called as relatively prime numbers.Note:(a)(b)Two prime number(s) are always co-prime but converse need not be true.Consecutive numbers are always co-prime numbers.(ix)Twin Prime Numbers :If the difference between two prime numbers is two, then the numbers are twin primenumbers.e.g.{3,5}, {5,7}, (11, 13}, {17, 19}, {29, 31}(x)Rational Numbers :All the numbers that can be represented in the form p/q, where p and q are integers andq 0, are called rational numbers and their set is denoted by Q.pThus Q { q : p,q I and q 0}. It may be noted that every integer is a rational numbersince it can be written as p/1. It may be noted that all recurring decimals are rationalnumbers .(xi)Irrational Numbers :There are real numbers which can not be expressed in p/q form. These numbers are calledirrational numbers and their set is denoted by Qc. (i.e. complementary set of Q) e.g. 2 ,1 3 , e, etc. Irrational numbers can not be expressed as recurring decimals.Note: e 2.71 is called Napier's constant and 3.14.(xii) Real Numbers :The complete set of rational and irrational number is the set of real numbers and is denotedby R. Thus R Q Qc.The real numbers can be represented as a position of a point on the real number line. Thereal number line is the number line where in the position of a point relative to the origin(i.e. 0) represents a unique real number and vice versa.Positive sideNegative side–3–2–101 2 23 Real lineAll the numbers defined so far follow the order property i.e. if there are two distinct numbersa and b then either a b or a b.Note:(a)Integers are rational numbers, but converse need not be true.(b)Negative of an irrational number is an irrational number.(c)Sum of a rational number and an irrational number is always an irrational number(d)(e)(f)marahe.g. 2 3S.82K.6L7The product of a non zero rational number & an irrational58be anEr. 8number027will5always10irrational number.9 801If a Q and b Q, then ab rational number, only if a 0.839sum, difference, product and quotient of two irrational numbers need not be airrational number or we can say, result may be a rational number also.Mathematics for IIT-JEEEr. L.K.Sharma9810277682/8398015058Page 2
(xiii) Complex Number :A number of the form a ib is called complex number, where a,b R and i 1 , Complexnumber is usually denoted by Z and the set of complex number is represented by C.Note : It may be noted that N W I Q R C.{2} Divisibility Test :(i)A number will be divisible by 2 iff the digit at the unit place of the number is divisibleby 2.(ii)A number will be divisible by 3 iff the sum of its digits of the number is divisible by 3.(iii)A number will be divisible by 4 iff last two digit of the number together are divisibleby 4.(iv)A number will be divisible by 5 iff the digit of the number at the unit place is either0 or 5.(v)A number will be divisible by 6 iff the digits at the unit place of the number is divisibleby 2 & sum of all digits of the number is divisible by 3.(vi)A number will be divisible by 8 iff the last 3 digits of the number all together aredivisible by 8.(vii)A number will be divisible by 9 iff sum of all it's digits is divisible by 9.(viii)A number will be divisible by 10 iff it's last digit is 0.(ix)A number will be divisible by 11, iff the difference between the sum of the digits ateven places and sum of the digits at odd places is 0 or multiple of 11.e.g. 1298, 1221, 123321, 12344321, 1234554321, 123456654321{3} (i)Remainder Theorem :Let p(x) be any polynomial of degree greater than or equal to one and 'a' be any realnumber. If p(x) is divided by (x a), then the remainder is equal to p(a).(ii)Factor Theorem :Let p(x) be a polynomial of degree greater than of equal to 1 and 'a' be a real number suchthat p(a) 0, then (x a) is a factor of p(x). Conversely, if (x a) is a factor of p(x), thenp(a) 0.(iii)Some Important formulae :(1)(a b)2 a2 2ab b2 (a b)2 4ab(2)(a b)2 a2 (a b)2(3)a2 b2 (a b) (a b)(4)(a b)3 a3 b3 3ab (a b)(5)(a b)3 a3 b3(6)(7)(8) 2ab b2 4abmaraha b (a b) 3ab(a b) (a b) (a b ab)S.82K.6L7a b (a b) 3ab (a b) (a b)(a b ab)r.E 81027 50589 801(a b c) a b c 2ab 2bc 2ca391 1 18 a b c 2abc 33 3ab(a b)33323222222Mathematics Concept NoteIIT-JEE/ISI/CMI2222abcPage 3
1[(a b)2 (b c)2 (c a)2 ]2(9)a2 b2 c2 ab bc ca (10)a3 b3 c3 3abc (a b c) (a2 b2 c2 ab bc ca) 1(a b c) [(a b)2 (b c)2 (c a)2 ]2(11)a4 b4 (a b)(a b) (a2 b2)(12)a4 a2 1 (a2 1)2 a2 (1 a a2) (1 a a2){4} Definition of indices :If 'a' is any non zero real or imaginary number and 'm' is the positive integer, then am a.a.a.a (m times). Here a is called the base and m is the index, power or exponent.(I) Law of indices :(1)a0 1,(2)a m (3)am n am an, where m and n are rational numbers(4)am n (5)(am)n amn(6)ap/q (a 0)1amq,aman(a 0), where m and n are rational numbers, a 0ap{5} Ratio & proportion :(i)Ratio :1. If A and B be two quantities of the same kind, then their ratio is A : B; which may bedenoted by the fractionA(This may be an integer or fraction )B2. A ratio may represented in a number of ways e.g.amana .where m, n, .bmbnbare non-zero numbers.3. To compare two or more ratio, reduced them to common denominator.4. Ratio between two ratios may be represented as the ratio of two integers e.g.ac::bdmaraSh ace82a .cKe.Ratios are compounded by multiplying them together i.e. L. . . 6 .b d f 277 bdf 8rE 810 5059 ratioIf a : b is any ratio then its duplicate ratio is a : b ; triplicate0is1a : b .etc.8983If a : b is any ratio, then its sub-duplicate ratio is a : b ; sub-triplicate ratio isa /badc / d bc or ad : bc. duplicate, triplicate ratio.5.6.7.a1/3 : b1/3 etc.Mathematics for IIT-JEEEr. L.K.Sharma9810277682/8398015058221/2331/2Page 4
(ii)Proportion :When two ratios are equal , then the four quantities compositing them are said to beproportional. Ifac , then it is written as a : b c : d or a : b :: c : dbd1.'a' and 'd' are known as extremes and 'b and c' are known as means.2.An important property of proportion : Product of extremes product of means.3.Ifa : b c : d, thenb : a d : c(Invertando)4.Ifa : b c : d, thena : c b : d(Alternando)5.Ifa : b c : d, thena bc d (Componendo)bd6.Ifa : b c : d, thena bc d (Dividendo)bd7.Ifa : b c : d, thena bc d (Componendo and Dividendo)a bc d{6} Cross Multiplication :If two equations containing three unknown area1x b1y c1z 0.(i)a2x b2y c2z 0.(ii)Then by the rule of cross multiplicationxyzb1c2 b2c1 c1a2 c2a1 a1b2 a2b1.(ii)In order to write down the denominators of x, y and z in (3) apply the following rule,"write down the coefficients of x, y and z in order beginning with the coefficients of y andrepeat them as in the diagram"b1c1a1b1b2c2a2b2Multiply the coefficients across in the way indicated by the arrows; remembering thatinforming the products any one obtained by descending is positive and any one obtained byascending is negative.{7} Intervals :Intervals are basically subsets of R and are commonly used in solving inequalities or infinding domains. If there are two numbers a,b R such that a b, we can define four typesof intervals as follows :Symbols Used(i) Open interval : (a,b) {x : a x b} i.e. end points are not included( ) or ] [marahS.82K.6L7r. also 1included27 058Eare0(ii) Closed interval : [a,b] {x : a x b} i.e. end points98 8015This is possible only when both a and b are finite.839(iii) Open-closed interval :(a,b] {x : a x b}( ] or(iv)Closed-open interval :[a,b) x : a x b}Mathematics Concept NoteIIT-JEE/ISI/CMI[]]][ ) or [ [Page 5
The infinite intervals are defined as follows :(i)(a, ) {x : x a}(ii)[a, ) {x : x a}(iii)( (iv)( ,b] {x : x b}(v)( , ) {x : x R} ,b) {x : x b}Note:(i)For some particular values of x, we use symbol { } e.g. If x 1, 2 we can writeit as x {1,2}(ii)If their is no value of x, then we say x (null set)Various Types of Functions :(i)Polynomial Function :If a function f is defined by f(x) a0 xn a1 xn 1 a2xn 2 . an 1 x anwhere n is a non negative integer and a0, a1, a2, ., an are real numbers anda0 0, then f is called a polynomial function of degree n. There are two polynomial functions, satisfying the relation; f(x).f(1/x) f(x) f(1/x),which are f (x) 1 xn(ii)Constant function :A function f : A B is said to be a constant function, if every element of A has thesame f image in B. Thus f : A B; f(x) c , x A, c B is a constant function.(iii)Identity function :The function f : A A define dby, f(x) x x A is called the identity function onA and is denoted by IA. It is easy to observe that identity function is a bijection.(iv)Algebraic Function :y is an algebraic function of x, if it is a function that satisfies an algebraic equationof the form, P0(x) yn P1(x) yn 1 . Pn 1 (x) y Pn(x) 0 where n is apositive integer and P0(x), P1(x) . are polynomials in x. e.g. y x is analgebraic function, since it satisfies the equation y2 x2 0. All polynomial functions are algebraic but not the converse. A function that is not algebraic is called Transcendental Function .(v)Rational Function :g(x)A rational function is a function of the form, y f(x) h(x) , where g(x) & h(x) arepolynomial functions.(vi)Irrational Function :An irrational function is a function y f(x) in which the operations of additions,substraction, multiplication, division and raising to a fractional power are usedmarahS.x x82KFor example y is an irrational function.6L72x xEr. 81027 5058(a) The equation f(x) g(x) is equivalent to the followingsystem9 8019f(x) g (x) & g(x) 08331/32(b)The inequationMathematics for IIT-JEEEr. L.K.Sharma9810277682/8398015058f(x) g(x) is equivalent to the following systemPage 6
f(x) g2(x) & f(x) 0 & g(x) 0(c)The inequationf(x) g(x) is equivalent to the following systemg(x) 0 & f(x) 0(vii)or g(x) 0&f(x) g2(x)Exponential Function :A function f(x) ax ex ln a (a 0, a 1, x R) is called an exponential function.Graph of exponential function can be as follows :Case - IFor a 1Case - IIFor 0 a 1f(x)f(x)(0,1)(0,1)x0x0(viii) Logarithmic Function : f(x) logax is called logarithmic function where a 0 anda 1 and x 0. Its graph can be as followsCase - IFor a 1Case - IIFor 0 a 1f(x)0f(x)(0,1)x0(1,0)xLOGARITHM OF A NUMBER :The logarithm of the number N to the base 'a' is the exponentindicating the power to which the base 'a' must be raised toobtain the number N. This number is designated as loga N.Hence :logaN x ax N , a 0 , a 1 & N 0If a 10 , then we write log b rather than log10 b .If a e , we write ln b rather than loge b .The existence and uniqueness of the number loga N follows from the properties of anexperimental functions .marahS.log N82K.identity : N , a 0 , a 1 & N 0a6L7Er. 81027 5058This is known as the FUNDAMENTAL LOGARITHMIC IDENTITY9 801NOTE :loga1 0(a 0 , a 1)839loga a 1(a 0 , a 1) andFrom the definition of the logarithm of the number N to the base 'a' , we have analog1/a a - 1Mathematics Concept NoteIIT-JEE/ISI/CMI(a 0 , a 1)Page 7
THE PRINCIPAL PROPERTIES OF LOGARITHMS :Let M & N are arbitrary posiitive numbers , a 0 , a 1 , b 0 , b 1 and is any real numberthen ;(i)loga (M . N) loga M loga N(ii)loga (M/N) loga M loga N(iii)loga M . loga Mlogb M (iv)NOTE : logba . logab 1 logba 1/logab.log a Mlog a b logba . logcb . logac 1xln a e ax logy x . logz y . loga z logax.PROPERTIES OF MONOTONOCITY OF LOGARITHM :(i)For a 1 the inequality 0 x y & loga x loga y are equivalent.(ii)For 0 a 1 the inequality 0 x y & loga x loga y are equivalent.(iii)If a 1 then loga x p 0 x ap(iv)If a 1 then logax p x ap(v)If 0 a 1 then loga x p x ap(vi)If 0 a 1 then logax p 0 x ap If the number & the base are on one side of the unity , then the logarithm is positive ; If thenumber & the base are on different sides of unity, then the logarithm is negative. The base of the logarithm ‘a’ must not equal unity otherwise numbers not equal to unity willnot have a logarithm & any number will be the logarithm of unity.n For a non negative number 'a' & n 2 , n N(ix)Absolute Value Function /Modulus Function :a a1/n. x if x 0The symbol of modulus function is f(x) x and is defined as : y x x if x 0yy –xy0 xxy x Properties of Modulus :For any a, b R(i) a 0(iii) a a, a (v)(vii)(ii) a a ab b a b a b Mathematics for IIT-JEEEr. L.K.Sharma9810277682/8398015058(iv)(vi)marahS.82 ab a b K.6L7Er. 81027 5058 a b a 9 b 018983 a a Page 8
(x)Signum Function :A function f (x) sgn (x) is defined as follows :Y 1 for x 0 0 for x 0f(x) sgn (x) 1 for x 0y 1 if x 0O (x) ; x 0(x)It is also written as sgn x 0; x 0 (xi)xy –1 if x 0y sgn x f(x) ; f(x) 0f(x)sgn f(x) 0; f(x) 0Greatest Integer Function or Step Up Function :The function y f(x) [x] is called the greatest integer function where [x] equals tothe greatest integer less than or equal to x. For example :for 1 x 0 ; [x] 1 ; for 0 x 1 ;[x] 0for 1 x 2 ; [x] 1 ; for 2 x 3;[x] 2and so on.Properties of greatest integer function :(a)x 1 [x] x(b)[x m] [x] m iff m is an integer.(c)[x] [y] [x y] [x] [y] 1(d)[x] [ x] (xii)Fractional Part Function :It is defined as, y {x} x [x].e.g. the fractional part of the number 2.1 is 2.1 2 0.1 and the fractional part of 3.7 is 0.3. 0 ;if x is an integer 1 ;otherwiseThe period of this function is 1 and graph of this function is as shown.y1y 1x–2–1012y (x)Properties of fractional part function(a){x m} {x} iff m is an integer(b){x} { x} Mathematics Concept NoteIIT-JEE/ISI/CMI0 ; if x is an integer1;otherwise3marahS.82K.6L7Er. 81027 50589 801839Page 9
Graphs of Trigonometric functions :(a)y sin xy [ 1,1]x R;y1 0 x –1(b)y cos xy [ 1,1]x R;y10 x –1(c)x R (2n 1) /2; n I;y tan xy Ry 0 (d)xx R n ; n I; y Ry cot xy Mathematics for IIT-JEEEr. L.K.Sharma9810277682/8398015058 0 amraxh S.82K.6L7Er. 81027 50589 801839Page 10
(e)x R n ; n I; y ( , 1,] [1, )y cosec xy1 0 x –1(f)x R (2n 1) /2; n I;y sec xy ( , 1,] [1, )y1 0 x –1Trigonometric Functions of sum or Difference of Two Angles :(a)sin (A B) sinA cosB cosA sinB(b)cos (A B) cosA cosB sinA sinB(c)sin2A sin2B cos2B cos2A sin(A B). sin(A B)(d)cos2A sin2B cos2B sin2A cos(A B). cos(A B)(e)tan A tanBtan (A B) 1 tan A tanB(f)cot A cot B 1cot (A B) cot B cot A(g)tan (A B C) tan A tanB tanC tan A tanB tanC1 tanA tanB tanB tanC tanC tanAFactorisation of the Sum or Difference of Two sines or cosines :(a)sinC sin D 2sinC DC Dcos22(b)sin C sinD 2cosC DC Dsin22a C DCm Drasinh 22 2S.K.68L77.58 :Er of02 &5CosinesTransformation of Products into Sum or DifferencesSines1089 801(a)2sinA cosB sin (A B) sin(A B)(b)2cosA sinB sin(A B) sin(A B)983(c)2cosA cosB cos(A B) cos(A B)(d)2sinA sinB cos(A B) cos(A B)(c)cosC cosD 2cosMathematics Concept NoteIIT-JEE/ISI/CMIC DC Dcos22(d)cosC cosD 2sin
mathematics {IIT-JEE} faculty. Er. L.K.Sharma has been proved a great supportive mentor for the last 15 years and the most successful IIT-JEE aspirants consider him an ideal mathematician for Olympiad/KVPY/ISI preparation. He is also involved in the field of online teaching to
184 vipul sharma 167. 185 sudhakar singh 168. 186 arushi anthwal 169. 187 nikita sharma 170. 188 sanjeev kumar 171. 189 devashish maharishi 172. 190 suman 173. 191 govind kant sharma 174. 192 manoj sharma 175. 193 shilpi satyapriya satyam 176. 194 mr d k sharma 17
1 11606592 jmi16009977 layba noor naseem ahmed 2 19805911 jmi16038210 rumana parvin azizur rahaman . 13 11623828 jmi16013846 gajraj sharma dalchand sharma 14 11620997 jmi16038811 abhishek gairola kishor gairola 15 21304829 jmi16021946 saurabh kant sharma dr saket prasad sharma
BEHAVIOR BASED SAFETY BY RAJNIKANT SHARMA BSc, BE(FIRE), PDIS, MBA (HR) MANAGER (FIRE & SAFETY) R.K.SHARMA . Contests & Awards R e pr i m a n ds R eg ul at io ns Policies Committees & Councils Slogans Safety Training Safety Meetings Fewer Accidents Traditional Safety Model 4/20/2015 R.K. SHARMA . Please think . . . How many of the above .
3 www.understandquran.com ‡m wQwb‡q †bq, †K‡o †bq (ف ط خ) rُ sَ _ْ یَ hLbB َ 9 آُ Zviv P‡j, nv‡U (ي ش م) اْ \َ َ hLb .:اذَإِ AÜKvi nq (م ل ظ) َ9َmْ أَ Zviv uvovj اْ ُ Kَ hw ْ َ Pvb (ء ي ش) ءَ Cﺵَ mewKQy ءٍ ْdﺵَ bِّ آُ kw³kvjx, ¶gZvevb ٌ یْ"ِKَ i“Kz- 3
PROF. P.B. SHARMA Vice Chancellor Delhi Technological University (formerly Delhi College of Engineering) (Govt. of NCT of Delhi) Founder Vice Chancellor RAJIV GANDHI TECHNOLOGICAL UNIVERSITY (State Technical University of Madhya Pradesh) 01. Name: Professor Pritam B. Sharma 02. Present Position: Vice Chancellor Delhi Technological University (formerly Delhi College of Engineering) Bawana Road .
1. Elementary Organic Spectroscopy By Y.R.Sharma 2. Principles of Physical Chemistry By Puri, Sharma, Pathania 3. Principles of Inorganic Chemistry By Puri, Sharma
R. K. Sharma, Manavi Yadav and Radhika Gupta, Chapter 5, Water Quality and Sustainability in India: Challenges and Opportunities, 2016, pp 183-206, Chemistry and Water, Edited By: Satinder Ahuja. R.K. Sharma, Shikha Gulati and
Gasification” by B.K.Sharma and Sujan Saha, proceeding in the International Seminar- Global Coal-2005 (New Delhi). “ Reverse micro emulsion mediated sol-gel synthesis of lithium silicate nanoparticles under ambient conditions: Scope for CO2 sequestration” by R.B.Khomane, B.K.Sharma
