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Mca Book

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HOW TO PREPARE FOR MCA ENTRANCE EXAMINATIONS * Preparation for MCA entrances takes 6-7 months with a daily and consistent study routine of 56 houts. The more dedicated time you spend on your preparation, more is the chance of your success. To on a safer side, it is better to start preparation 1 year ahead. * Revisit Mathematics, covering entire syllabus. You need to be essentially good at Higher Maths, as most of the MCA entrance tests comprise Maths, Statistics and basics of Computers, * Brush up computer fundamentals; as MCA entrances lay heavily on the test of computer fundamentals, * Kris advisable to selecta few institutes of interest and prepare for a Particular pattern of exams and the respective syllabus * Use the dedicated MCA entrance stud) Preparation. '¥ materials only, for a focused * Analyze your strong and weak aress, and work acco gly. * Once you have covered the syllabus, start taki raki actual testing conditions. nh mosh ssicloen os * As the exam is purely of objecti ‘asa Purely jective nature, work on your speed and Scanned by CamScanner Section \ Mathematics Sets and Cartesian Product of Sets 1 Set In our mathematical language, everything in this universe whether living or non-living is ealled an object. If we consider a collection of objects given in such a way that itis possible to tell beyond doubt, whether a given object is in the collection under consideration or not, then such a collection of objects is called a well-defined collection of objects. (a) Definition Any Collection of well defined objects is called a set, By ‘well-defined objects’, we mean that given a set and an object, it must be possible to decide whether or not the dbject belongs to the set. ‘The objects in set are called its members or elements. (b) Notations Sets are usually denoted by capital leters A,B, C etc. and their elements by small letters a, b, cet. Let A’ be any set of objects and let a be a member of A, then we write a © A and read it as ‘a belongs to A’ or ‘a is an element of A? or ‘a is member of A’. Ifa is not an object of Avthen we write @ ¢ A and read as ‘a does not belong to A’ or ‘as not an element of A? or ‘a is not member of A’. e841 The collection of first five prime numbers is a set containing the elements 2, 3, 5,7, 11. e842 The collection of good cricket players of India és not 4a set, since the term “good player is vague and it is not well-defined.” 1 Set Builder Form In this method, instead of listing all elements of a set, we ‘write the set by some special property or properties satisfied by all elements and write it a5, ‘A= (x: Pa) = (al x has the property PO) and read it as ‘A is the set of all elements x such that x has the property P’. The symbol *: or ‘I’ stands for ‘such tha” eg, If A= (1.2.3.4). then we can write, ee N:x<5) 1 Empty Set ‘A set consisting of no clement is called an empty set or null set or void set and is denoted by the symbol or { }. egal (xixeN,3ex<4 42 (xxeR, P+1=0=9 ‘A set which has atleast one element is called a nonempty set The set (0) & not an empty seta it contains the element O (0) + The set (6) is not a null set. tis a set connaning one element e | Singleton Set A set consisting of only one element is called a singleton set e.g The set (3) isa singleton set. | Finite Set |A set in which the process of counting of elements surely comes to an end is called a finite set ‘eg.. | Ser ofall persons on the earth. egu2 (xixel.xisa factor of 1000) 1 Infinite set A set which is not finite is called an infinite set. In other words, a set in which the process of counting of elements does not come to an end is called an infinite set 11 Set of all points in a plane. 2 (xeQ:0yeA, then A and B are equal. e841 (4,8, 10} = (8,4, 10) (The order in which the elements of a set is also immaterial] Scanned by CamScanner € ' 4-1 CHAPTER » Sets and Cartesian Product of Sets en? (24,7) 12 464,72) {re repetition of clements of sti also immaterial] 1 Equivalent Sets ‘Two finite sets A and B are said to be equivalent, ree st so equvet bt euler ned rot be equal Equivalence of two sets is denoted by the symbol *~'. “Thus if and B are equivalent sets, we write A ~ B which is read a8 ‘A fs equivalent to B’. egyA=(1,2,3),B= (2,35). thenn@)=3, n@)=3. ‘Since, n(A)=n(B), therefore A and B are equivalent ‘and we write A ~ B. f 1 Subset ‘The set Bis sad to be subset of set A, if every element of set B is also an clement of set A. Symbolically we write it 3, BoAorADB. @ BS Ais read as B is contained in A or B is subset of A. Gi) AD Bis read as A contains B or B is a subset of A. Evidently if A and B are two sets such that xe B = XA, then Bis subset ofA, The symbol stands foe ‘implies’. We reat sx belongs to B implies that x belongs to A. eg, Let =(1,2,3,4} ;B=(1, 2,4). Here, B is a subset of A, 1 Proper Subset A set B is said to be a proper subset of set A, if every clement of set B isan element of A whereas every element of Ais not an clement of B. We write it as BC A and read it as “B isa proper subset of A’. Ths, Bis a proper subset of A, if every clement of B isan element of A and there is atleast one element in A which is notin B. Observe that AG Ae, every set is a subset of itself but not a proper subse. egy LetA=(1,2,3}; B= (1,2), then BOA | Power Set ‘The set formed by all the subsets of a given set A is called the power set of , itis usually denoted by P(A), es A= (1,23), them PCA) = (6 (1), (2), (3) 1.2), {2,3}, (3,1), (12.39) Some Results on Subsets @ Every set is a subset of itself G The empty set is a subset of every set. Gi) The total umber of subset of infinite set containing nn element in 2. 1 Comparable Sets “Two sets A and B are said to be comparable, if one of them is a subset ofthe other ie, either AS Bor BSA. e.g The sets {1, 2,3) and (1, 2,3,4, 5} are comparable sets. 1 Universal Set In any discussion in set theory, there always happens 10 be 4 set that contains all the sets under consideration i., itis a super set of each of the given sets, Such a set is called the universal set and is denoted by U. egufA= (24,5), B= (13,5) C= (35,710), D= {2,4,8, 10} and U= (1, 2,3,4.5,6,7.8, 10, 1} be the given sets. Here, the sets A, B, C, D are subsets ofthe set U. Hence, U is a universal st. 1 Operations on Sets ‘Now, we introduce some operations on sets to construct new seis from the given below @ Union of sets The union of two sets A and B, denoted by AU Bis the set of all those elements, each one of iscither in or in Bor in both A and B. ‘Thus, AUB=(x:xeAorxe B} Clearly, xe AUB => xeAorxe B Ani, x€AUB= xe Aorxe B In the figure, the shaded part represents AUB. is evident that AC AUB, BC AUB. eg, A= (1, 2,3) and B= (1,3, 5,7), then AVB=[(1, 2,3, 5,7) (@ Intersection of sets The intersection of two sets A and B, denoted by AMBis the set of all elements, ‘common to both A and B. Thus, AN B=(x:xe A and xe B} Clearly, xe ANB = xeAandzeB And, x€ANB => xeAandreB CO) In the figure, the shaded part represents ANB. It is evident that ANBCA, ANB CB. Scanned by CamScanner CHAPTER» Sets and Cartesian Product of Sets | 5 (Go. Disjoin sets TWwo sets A and B are suid tobe disjoint, if ANB=6if ANB > then A and B are said 10 be intersecting or overapping set. eal U = (x 1,23, 45.6), B= (7, 8,9, 10,11) and (c:xisavowel), 6,8, 10, 12,14), then A ane Bare dsfoine (2 xls consonant). avty fig Mand Geare taerpenting 2s, 1 Some Results on Complementation CG) ‘The following results aze the direction consequences of the definition ofthe complement of the set. ou » lO ee @ Walrus zeq)=0; difference A ~ B isthe set of all those elements of A Bete nadhfeeDniadlwtl ‘which do not belong to B, @ AY = (eeu sxe) =(re Us xe A= Thus, A-Bs= [x:xeA mde) Gi) And alzeU xe An ed :2€A=6 Chay, £¢A~B= xed andxe B. 7 1 Laws of Algebra of Sets GX) keno Oe (@ AUA=A () AnA=A entity Laws Inthe figure, the shaded part represents A ~ B. () AUg=A © AnuU=A ‘Similarly, the difference B — A is the set of all those elements of B that do not belong to A i.e, sxe BandxeA) i) Commutative Laws @ AUB=BUA —(b) ANB=BNA (iv) Assoctative Laws q (@ (AUB)UC= AU(BUO) Gs) annce anton) z (9) Distributive Laws In the figure, the shaded part represents B — A. (@) AUBNO}= (ALB) (ALO) 1,3,5,7, 9} and B= (2, 3,5, 7, 1), then (0) AM(BUC)=(AN BULAN) 2, 11). ; (a) De Morgan’s Laws (®) Symmetric difference of two sets The symmetric (@ (AUBY=A'nB’ (An By =4'UB" difference of two sets A and B denoted by A A Bis the set (A~B)U(B-A). s (A-B)U(B-A) 1 Points to Remember xix€ANB) @ n(AVB)=n(A)+n(B) @®| Gi) (AUB)=n(A)+n(B)=n(AB) (ii), AUB=(A-B)U(AMB)U(B- A) a6 Gv) n(AUB)=n(A~B)+n(ANB)+n(B—A) ‘The shaded part represents A A B. (W) n(A)=n(A-B)+n(ANB) eg. A= (1, 2, 5,7, 9} and B= (2, 3,5,7, 1) (s)_ n(B)=n(B-A)+n(ANB) on i (1,990 211) (vii), n(AUBUC)=n(A)+n(B)+n(C)—2(ANB) (i) Complement ofa set Let U be the universal st and =n(BAC)=n(CNA)+N(ANBNO) ACU, then the complement of A, denoted by A’ or (vii) Tf A, B, C are finite sets, then U~Ais defined as A’=(x:x€U and x¢ A) n(AUBUC)=n(AOB'NC)+n(A'NBAC) Cleat, xe’ #9 x6 A. $n(A’ AB’ C) +n(A’ABNC)+M(ANB’AC) The shaded part represents A’. $n(AMBC) tna’ OBC’) Scanned by CamScanner Ro "Gl CHAPTER» Sets and Cartesian Product of Sots 1 Ordered Pair ‘Two elements a and b, listed in a specified order, form an ‘ordered pair, denoted by (a, b). In an ordered pair (a, regarded asthe first element and b the second element. Ttis evident from the definition that @ @b)#¢,a) ©) @.)=(6,d,ilfa=6,b=d I Cartesian Product of Sets Let A and B be two non-empty sets. The cartesian product of A and B denoted by A x B is defined as the set of all ordered paits (a,b), where a A and be B. ‘Symbolially, AXB= ((a, b);a€ A and be B) £8 (0) Suppose A= (1,2, 3) and B= (x, y} (0.9.42, 289.8 D) (0, G24 3), D2. DE Note that, if A B, then AxB# BXA, e.g, (ii) If there are three seti A, B, Cand ae A, beB, ‘© €C, then we form an ordered triplet (a,b c). The set of all ordered triplets (2, 6, c) is called the cartesian product of these seis A, B and C. te, AXBXC= (a,b,c): a€ A; BEB, c€C) IGraph of Ax B graph of A x Bis the st of all points in the plane by the ordered pais of A x B. ‘Some Results on Cartesian Products of Sets Let A and B be two non-empty subsets of R. Thea, eguLet A =(1,2,3) B= (3,4, 5), then AxB= (1,9 (4,1, 5,3) 2,4) 2,5) G3. G.4.B,5)) 0 @ Ax@UO=AxXDVAKO () AX(BAC)=(AxB)(AXC) ( Ax@-Q=AxB)-AxO Gi) (AKB)A(CxD)=(ANC)x(BOD) (iy) If Ag Band CoD, then (AXC)¢ (BXD) ( If Ac Bithen AXA=(AXB)O(BXA) (i) If A and B are non-empty subsets, then AxB=BxA@A=B. (vi), IFA and B are two non-empty sets having n elements in common, then Ax B and Bx A have n? elements in 1. The set of imeligen stadents in cls is (@) all set (0) a singleton se (© a fine se (2 not a well-defined collection 2 Which ofthe following isthe empty se? (@)_ (x: xis areal number and x7 -1=0} SY (ins real number and ¥?+1=0) (©) (x:.xis areal number and x? -: (@) (e: xis a real number and x? = x42) 3. The set A={rixe R, a" =16 and 2x=6) equals we ©) (14,3,4) © Bb © 4, Ifa set A has m elements, then the total number of subsets of A is @ nm wo are @ rm © 5. The number of proper subsets of the set (1, 2,3) is o3 wv 6. “Given the ses then AU(BAC) is @ 6) © (1.245) =(4,5,6), 08 11,2.3,4) © (1,2,3,4,5,6) common, Exercise i 1 i hk at onxea| xeR) ®) AmB (@) None of these xe Ryla-I21) and AUB=R-D, then the set Dis @ [rstexsay Y [e:1sx<2] © [e:lsxs7 (@) None of these 9% Ifthe sets A and B are defined as A=»): y=e%, x6 R); B=((x, y): yaa, xe Rh, then @ Boa ) ACB fh Ana} @ AuB=A 10. “Let n(U)=700, n(4)= 200, n(B)=300 and (ANB) =100, thea n(4° 05°) is equal to @ «0 ©) 0 fF @ 0 | Scanned by CamScanner 11, In a town of 10000 families it was found that 40% family buy newspaper A, 20% buy newspaper B and 10% families buy newspaper C, 5% families buy A and 2B, 3% buy Band Cand 4% buy A and C, 1 2% families buy all the three newspapers, then number of families which buy A only is @ 3100 48) 3300 (©) 2900 (@ 1400 12, Ina city 20% of the population travels by car, 50% travels by bus and 10% travels by both ear and bus, Then, persons travelling by car or bus is (@) 80% ©) 40% ey 60% (@) 70% 13. In a class of 55 students, the number of sudents studying different subjects are 23 in Mathematics, 24in Physics, 19 in Chemistry, 12 in Mathematics and Physis, 9 in Mathematics and Chemistry, 7 in Physics and Chemistry and 4 in all the three subjects. The numberof students who have taken exactly one subject @ 6 9 7 BY Ail ofthese 14. IEA, B and C are aly three sets, then Ax (BU Cis equal to AY ARB)L(AXC) (0) ALB) XAVO (© AXBAXC — @) Noneof these 18, IFA, B and C are any three sets, then A = (BU ©) is equalto A) A-BDUA-Q — &) A-B)NA-O © @-Huc @ @-Bac 16. IfA,B and Care non-empty sets, then (A~B) U (BA) equals @ AuB-B @) A-anB SS GUB-—AnB)— @ ANBDVALD 17. “If A=(2,4,5), B=(7, 89), then m(AXB) is equal to @ 6 479 ©3 @o 18, If the set A has p elements, B has q clements, then the ‘number of elements in A x Bis @ pra (b) pegs a] @ 19. “If A=(a,6},B={ed),C=(d.e), then {(ac).(ad),(aye),(6,e)s(b.d).(b,2)} is equal to @ ANB UC) (0) AUBNC) ATAKBLO) @ AxBO) 20. “Th P,Q and Rare subsets of ast A, then Rx (FU O'F iseqyalto (RxPORXD) MOF URKO)NRXP) ©) (RxP)U(RxQ) (d) None of these 24 Uaedensted ton themes o) > presen oOo © bexea) 22, 23. 27. 28. 29. 31. 32. 34. 35. 36. CHAPTER « Sets and Cartesian Produet of Sets 5 7 Az (x:x x) represents @) (0) ort) ou) @) (2) 16 0-frca-4nhosye Wen @ 06g Ai 160 © 260 @ 20 Which sets the subset of al given sets? @) (1.2.3.4) () {1} © 0) 0 Let $=(0,15,4,7). Then, the total number of subsets of Sis (@ 6 oy 2 © 40 (@ 20 ‘The pumber of non-empy subsets ofthe set (1, 2,3, 4) is fay 1S ~ (b) 14 “@ 16 . @17 ‘The smallest set A such that A U (1,2} = (1.2,3,5.9) is @ (2.3.5) A) (3,5,9) © (25,9) (@) None of these AAB=B,then (@) Ace DY BCA © A=0 (@®) B=9 IFA and Bare wo ses, then AUB= AO il @ Ace ©) BoA AS AaB (@) None of these Let and B be two sets, Then, (@) AUBG AGB anc AuB (©) ANB=AUB (d) None of these = (Gy): y=et xe R),, Bal(xy):y =e" xe R). Then, BS ANB &) AnBee © AuB=R (@) None ofthese Hf A= (2,3,4,8,10),B={3,4,5, 10.12), 45,6, 12, 4), then (AAB)U (AMO isequlto (3,4, 10} (b) {2,8, 10} (©) (45,6) (@) (3,5, 14) fie TEA and Bare any two sets, then A 6 (A.W By; equal to aA Oy © 4° © # ICA, B, C be tres sets such that A UB =A U C and ADB=ANGthen Let =(a, b,c], B [b,c d), C= (a,b, dhe}, them AQ BUOis tab, ch ©) (bad) (0) {a b.deb @ le IEA pm B are sets, then Am (B-ADis ° OA OB (@) None of these Scanned by CamScanner 8 | CHAPTER « Sets and Cartesian Product of Sots 37. IrAand Bare two ses, then AM(AUBY' is equal 10 mB 4 (a) None ofthese 38. “Let U = (,2,3,4,5,6,7,8,9,10), A=(1,2,5),B=(6,7}, thea ANB’ is # (@ B OYA © A @s 89, IFA sany set, then . (@) Ava" @) ava () ANA’ * @) None of these 40. If N, =lan:ne N}, then Ny, is equal to ©», ON ON lor, ©) Mp 41. If aN ={ax: xe N), then the set 3V.47N is SON +) 10N © 4N ‘ (@ None of these 42. The shaded region in the given figure is , ( @ Angug © Aven QY © An@-O) cA A-BOQ 43. fA and B are two sets, then (A — B) U (B= A) U AnBisequalto AuB © Ans OA @r 44. Let A and B be two sets, then {AUBYU(4’B) is eaus)to Ay () A © B (@) None of these 45. Let U be the universal set and AUBUC = ((A-B)UB-C)U(C-A)F is equal to @ AuBuc @) AUwac) JS AnBoc @ An@uey 46. If m(A)=3, n(B)=6 and ACB. Then, the number of elements in AUB is equal to (3 w9 6 (@) None of these 47. “Leta and B be two sets such that (A) =0.16, n(B)=0.14, n(AUB) =0.25 ‘Then, n(A™B) is equal to (@ 03 ©) 05 (©) 00s (@) None of these IFA and B are disjoint, then (AUB) is equal to (@) mA) &) me) ©) mA)+n(ey (@ n(A)-n(B) = Then, 49. 51. 52, 53, 55. 56. 57. 58, IEA and B ate not disjoint sets, then m(AWB) is equa) to (a) n(A)+n(B) (0) n(A)+n(B)—n(AMB) (©) mA) +n(B)+n(AB) (a) n{A)n(B) ©) nfA)~n(B) In a battle 70% of the combatants lost one eye, 80% aq ear, 75% an arm, 85% a leg, lost all the four Tien ‘The minimum value of xis @ 10 © 12 © 15 (d) None of these Out of 800 boys in a school, 224 played crickey, 240 played hockey and 336 played basketball Of tay total, 64 played both basketball and hockey; 80 playeg cricket and basketball and 40 played cricket ang hhockey; 24 played all the thee games. The number ef boys who did not play any game is (@ 128 () 216 (© 240 @ 160 A survey shows that 63% of the Americans like cheewe whereas 76% like apples. If x% of the Americans like both cheese and apples, then @) x=39 (@) x=63 © 39 vu x isnot eal but ris real +. No value of xis possible, @ 7 =16 => a 2x: => xa3 ‘There, is no value of x which satisfies both the above equations. Thus, A (c) Number of subsets of A="C, #"C, +..4°C, (©) Number of proper subsets of the set {1,2!3)=2?-2=6, (b) BNC=(4), 2. AU(BNC) = (1,2,3,4} (given) (© Since, y=,y=-x meet when —x= =2 x7 =-1, which does not give any real value of Hence, ans=$ . © Aa(x:xeR-le rel) Ba{x:xeR:x-1S-1 of x-12 1) = (erxeR:xS00rx22) 2 AUB=R-D, where D= (x:x€ R1Sx<2} (© Since, y=eF and y=. do not meet for any xe R “ AnB=6 (© mA BS) =nf(A UB)] = MU) m(AUB) = n(U)=[n(A)+n(B)=n(A0B)) = 700 ~ [200 + 300 ~ 100] =300 (b) n(A) = 40% of 10000 = 4000 ‘n(B) = 20% of 10000 = 2000 ‘n(C) = 10% of 10000 = 1000 (AM B)= 5% of 10000 = 500 mB. =3% of 10000 = 300 n(C A) = 4% of 10000 = 400 (ABC) =246 of 10000 = 200 We want to find (A 0 BF 0 C) =nlAN Bucy) =11A)-nlA BVO! =n(A)—nlA NB) VAN nA) - IA 9B) +n. 0) =n B.C) = 4000 — [500 + 400 ~ 200] = 4000-700 = 3300, n(C)=20, n(B) = 80, (EM B) Now, n(CUB)n(G)+ n(B) — (7B) =20 #50 -10=60 Hence, required number of persons = 60% © 13, 4 15. 16. 17. 18. 19. 20. 21. 22, 23. Answer and Solutions nt \PAO)= We have to find n(M NPAC), n(PAM' AC), n(CaM’oP) Now, n(M AP'AC)=nIM A(PLCY) (M)— nl (POO) (M)~nf(M OP) OOD] (M)— n(M 0 P)— n(M 0 0) + nM APOC) -12-9+4=27-21=6 nPAM'AC)=nlPAM UO) P)—nfP (MUO) (P)—n(PAM)U(POC)] nP)=n(P 0M) — WP.) +P MAO) =24-12-744=9 nCOM'nP) =n(C)—n(CP)=n(C OM) +n(COPAM) =19-7-944=23-16=7 Ikis distributive law. Itis De Morgan's law. (-B)UB-A)=(AVB)-AOB) AX B= (2,1), 2,8), (2,9) (4.1.4, 844.9, 575, 8),5.9)} n(AxB) = n(A)-n(B)=3%3=9 n(AxB)= pq BUC= (c,d) (de) = (6,4, e} + AXBUO=(a,b} x (6.de} =(@, 0) (a, (a, 0), (6,0). (6.4, Goh (ab) Rx LOY = RAPE NON] = RX(POQ) = (RXPIO(RXO) = (RXQ)(RXP) (@ Itis fundamental concept. (b) Itis fundamental concept. 1 r0te24e wyem yy (b) Since, * eam be Ge ycanbeD) (@) Null set is the subset of all given sets. (b) $= {0,1,5,4,7) , then total number of subsets of Sis 2". Hence, 232 (@) The number of non-empty subsets = 2" -1 = (b) Given, AU(1.2)= Hence, A () Since, AnB=3, Scanned by CamScanner 29. 31. 32. 33. 34. 35. 36. 37. 39. 41. (©) Let xe A= xe AUB = re ANB (:AcAva (:AUB= ANB) = xe And xeB => xe 8 AGB Similarly, xe B=> xe A °.BCA Now, ACB,BEA = A=B ©) AMBCACAUB 2 AnBcAUa @ vy > Malkeesayel +A and B meet on(0, 1), « ANB => (@) AMB=2,3,4.8,10}913, 4, 5.10, 12) = (34,10), ANC=(4) ABN) =(,410) (@) AQ(AUB)=A (AgBuA) (©) Itis obvious. (@) BUC=(a,b,6.4,¢) 1 AN(BUC)=(,b, e} Ua, Bye, dye} esb.c} Set, y=" will meet, when e* = (sxe B-AS x64) (9) An(AUBy=An(a'nB) (oauBy = AaB] =(ANAIMB’ (by associative law) (®) Itis obvious. (©) Nyy = M5 (eS and 7 are relatively prime numbers) se N:xisa multiple of3} [ce N :xisa multiple of 7) [x€ is a multiple of 3 and 7} (xe Nis a multiple of 3 and 7) = (xe N xis amultple of 21) =21N (2) His obvious. (@) From Venn-Euler’s diagram, & (A~B)UB~AYU(ANB)= AUB (a) From Venn-Euler's diagram, 2 AUBYUA'NB) =A) 45. 46. 47. 48. ss 51. 52. CHAPTER « Sets and Cartesian Product of Sets 11 vol) Sie Clearly, ((A-B)U(B-Qu(C-AY =ANBAC (©) Since, AGB, -AUB=B So, {AUB)=n(B)=6 (©) AUB)=n(A)+n(B)=n(ArB) 025=0.16+0.14—nANB) = n(AmB)=0.30-025 = 0.05 © si 2A and B are digjoint ANB=6 (AB) =0 (A) +n(B)—n(ANB) -n(A)+n(B)~0 = n(A)+0(B) (0) AUB) =n(A)+n(B)-n(A0B) (@) Minimum value of n= 100-(30-+20+25+15) (00-90 n(C)=224,n(#) = 240, n(B) n(H OB) = 64, (BAC) =80 n(H AC) =40, n(COHB)=24 (CH! OB) =nl(CUHUBYT =n(U)-n(CUH UB) = 800-[n(C) +aC#)+n(B)—n(H 0) =n(H 0B)=n(COB)+n(CAH OB) =800— 224.+240+336-64-80—40-+24] =800-640=160 (©) Let A denote the set of Americans who like cheese and let B denote the set of Americans who like apples. Let population of American be 100. ‘Then, n(A)=63,n(B)=76 Now, n(AUB)=n(A)+n(B)—n( AB) =63+76-n(ANB) M(AUB)+0( ANB) =139 => n(AMB)=139-n( AUB) But n(AUB) S100 =n(AUB) 2-100" 139-n(AUB)2139-100=39 n(AMB) 239 Le, 39Sn(ANB) Again, ANBGA ANB CB = n(AnB)sn(A)=63 and (AB) Sn(B)=76 m1 nANB)S63 ‘Then, 39< (ANB) $63 395x563 © A) Ai) Scanned by CamScanner 7 12 | CHAPTER » Sets and Cartesian Product of Sets 53. 54. 55. (@) Let n(P)=Number of teachers in Physics n((f)= Number of teachers in Maths n(PUM)=() +n) —M(P OBE) 2=n(P)+12—4 = n(P)=12 (a) Let B, H, F denote the sets of members who are on the basketball team, hockey team and football team respectively. ‘Then we are given n(B)= 2,n(H) = 26,n(F) = nHOB)=14, nH F)=15, nF OB) =12 and n(BOH NE) =8 We have to find n(BUH UF), To find this, we use the formula n(BUHUF) =n(B)+n(H+n(F) -2(BOH)= nH VF)~n(FB) +n(BoHF) Hence, n(BUH UF) = (21+26+29)~(4415412) +8 =43, ‘Thus, these are 43 members in all, @ n(M)=55,n(P)=67,n(M UP)=100 Now, n(M GP) =n(M)#n(P)—n(M AP) 100=55+67—n(M oP) & n(M MP)=122-100=22 Now. (P only) =a(P)—n(M AP) =67~22=45 (© Ingeneral, AxBe BxA AXB=BXA is true, ifA=B (©) From De Morgan's law, (Am BY’ @ Aq =[x:x€ Aand xe BY) = ANB (2) Itis obvious. (@) From De Morgan’s law, A~(BNC)=(A-B)U(A-C) (©) From distributive law, ANBLO)=(ANB)UIANC) ©) A-B=(1) and B-C= (4) (A~B)x(B-C) ={(1,4)) (@) Itis obvious © AvB=(1, 23,8); Ane = (3) (AVB)X(40B) =((1,3),(2,3),,3),,)) © A-B=(3},A0B=(2,5} (4-B)x(ANB)=(6,2),8,5)) 66. 67. 69. 70. © © () @ @ @ o 74. 5. 76. @ © (A) =4,, n(B) N(AYen(B)xn(C) = n(AXBXC) Araneta = neha Given, seis ((@,)):2a* +36? =35,a,be 2) We can see that, 2(42)* +3(43)? =35 and 2¢84)? 341)? =35 2,3), (2,3), 2, -3),(-2,3) 4, Ds =D), 1), © 4, 1) are 8 elements of the set. n=8 obvious. Itis obvious. Let the original set contains (2n+1) elements, then subsets of this set containing more than » elements, ice, subsets containing (n+1) elements, (n+2) elements, .. (2n-+1) elements, +. Required number of subsets BEC tC, MC, C4 = CG ag, age ig a ticg tC, 46, Last aDyaae = [arn] = 512 Itis obvious (4,8,12,16,20,24,...) 6,12,18, 24, 30,.. HAC B={12,24,..) (Malone) = m(M)—2M AC)~n(M AP) +m PAC) (xx is a multiple of 12) AB A=(ANB)U(A-B) is comect, (3) is false. 2 (and @)aretue. —a*B 4 Nanay nl(AxB) (Bx A)) =nl(AOB)XBOA))= m4 B)-n(Be A) =MAMB)-n( AB) = (9999) = 992 (© MAUB)=n(A)+m(B)=n(48)=1249-4 217 Now, nl(AUB)'}=2U)—n(4U8) = 20-1723 Scanned by CamScanner 1 Matrices A rectangular arrangement of numbers in rows and columns, is called a matrix. Such a rectangular arrangement of uimbers is enclosed by small () or big [ } brackets. Generally 1 matrix is represented by a capital letter A, B, C, ny ete, and its elements are represented by small lewers a, b,c, % y ete. Following are some examples of a matrix a(t hall Spe} ons E=(5] 1 Order of Matrix A matrix which has m rows and n columns is called a matrix of order m xn and it represented By Aan 9F A=[2j)nxn Itis obvious to note that « matrix of order m x n contains mn elements, Every row of such a matrix contains m elements and every column contains m elements. | Representation of a Matrix ‘A matrix of order m x nis generally expressed as ay ay ay ay Gait Gn yy Gm OF A= Laylyag F= 152, ony OF A=[Oilang F= 1,2 0M From this representation, a matrix Ais waiten 38 ap clear that (i, Jth element of CHAPTER | Matrices and Determinan, I Types of Matrices (@ Row matrix If in a matrix, there is only one ro, itis called a row matrix. ‘Thus, A = [4p iS a row matrix, if m= ‘equal to [1, 3, 5] is a row matrix of order 1 x 3. i) Column matrix If in a matrix, there is only one colang then it is called a column matrix. Thus, A = [4j]joq,i8 @ column matrix, if m 1 eBoy | is column matrix of order 3 x 1. 5 Gii) Square matrix If number of rows and numbed ‘columns in a matrix are equal, then itis called a square mit Thus, A= [4c i8 @ square matrix, if m= is a square matrix of order 3x3 Gv) Singleton matrix If in a matrix, there is 061) element, then itis called singleton matrix. Thus, A = [4j]uq i8 a singleton matrix, if m=: €-8., (4), [2], (6), [-5] are singleton matrices. (¥) Null or zero matrix If in a matrix, all the elem 37 nmterermainy ce 6} ig zero, then iti called a zero matrix and it is generally Scanned by CamScanner rs, A= [alee i8 8 280 matrix, fo for all i and j 000 ag, | 0 0 0| 188-2210 mattis of order 3x 3. 000 (vp Dingonal matrix If all elements except the principal agonal na square MAU are zero, i is ealled a diagonal rjus, a square matrix A = [aj] is a diagonal matr agen 50 0 eg.{0 6 0] is. diagonal matcx of order 3 x 3, which 007, also can be denoted by dia [5, 6, 7] f BS oe (vi) Sealar matrix If all the elements ofthe diagonal of a cigonel mati are equal, its called a scalar matix. "Thus, a square matrix A = [ay] is scalar matrix, if , ied y= Ye, ia jo Where K is a constant 10 0 egq| 0-1 0| is ascalar matrix. 0 0-1 (sii) Unit matrix If all elements of principal diagonal in a gna! mauix are 1, then itis called a nit matrix, A unit matrix f oer is denoted by J, “Thus, a square matrix A = [ay] is a unit mateix, if gg), Maneolar matrix A square matix (a) is sai to be Wangular, if each element above or below the principal ‘agonal is zero, then it is of two types. (® Upper triangular matrix A square matrix (ay) is called the upper triangular matrix, if ay = 0, when i>j. 425 £8,|9 6 7) is upper triangular matrix of order 3 x 3. O04 CHAPTER = Matrices and Determinants {147 (©) Lower triangular matrix A square matrix (ay) 58 called the lower triangular marx, if a, =O, when isj they are of same order and their corresponding elements are equal , [i & See (3) Singular matrix Matrix A is said to be singular ‘matrix, if its determinant [Al =0, otherwise non-singular matrix le, If delAl=0 => singular and detlAl #0 => non-singular 1 Addition and Subtraction of Matrices If A=Lo]oen and B=[2j]ayq af€ two matrices of the ‘same order, then their sum A + B is a matrix whose each clement is the sum of corresponding element. he, A+ B= lay + by] man ‘Similarly, their subtraction A ~ Bis defined as A=B=[ay~bjlwer 7 go tA=|4 2 3 2| and B= ft 746 3+2] [13 5 A+B=|4+5 2+4/=/9 6] and 243 147, 5 8. 7-6 3-2) Pat Abe [4-5 2-4)2 [1 2 231-7] [a + Scanned by CamScanner 148 4 CHAPTER » Matrices and Determinants Properties of Addition of Matrices If A, B and C aze matrices of same order, then AB=B+A (Commutative fa) Gi) A+B)+C=A4B+O (Associative law) Gil) A4+0= 0+ A=A, where O is zero matrix which is audive identity ofthe matrix. Gy A+ (CA) =0=(CA)+A, where (-A) is obtained by changing the sign of every element of A which is additive inverse of the matrix. AsB=A+C] pranceal > BE (Cancellation law) iy Trace (A # B) = Trace (A) * Trace (B) 1 Scalar Multiplication of Matrices Let A= [ain BE 8 matrix and k be'a number, then the matrix which is obtained by multiplying every element of A by kes clled scalar multiplication of A by k and it denoted by kA. Thus, A= [aloe =? FA= (lg 4 2 20 10 we A=\3 5),then SA=]15 25 67. 30 35, Properties of Scalar Multiplication of Matrices rele toca enh Gi) (m+ mA=mA+ nA (iii) m(nA) = (mnyA = n (mA) mf SHI} i [2 nr @L2 O42 samo (5 SHE O-r2 O12 Vasa and ~446 - tat. b=? “Hence, (isthe comet option, ange 2 [lf] me off a ih then ais @. 2 sa saver rf] @o 2 3-2 3a-3)_[1 30-3) 2 uis-[5p Salle Now, by equality of two matrices, we have a=2 ja-3=3 = Hence, (b) is the correct option. Example 3. 1[X and ¥are two matrices such that rr? 4 rt ca coral ee maer acer J} wh) of; Jolt domes, woe ‘subtracting Ba. i) from Bq. (we get 32] [1 2 Fel ols 4 3-1 2-2) [2 4 ocr SELL] . rtf ‘VE | [4 aL 2 2, Hence, (0 is dhe correct option Example 4 A matrix A = [aj] of order 2 x 3 whose element ot this 23 4 Mls 4 3. (@) None of these Solution @y is the element of th row and th column of matrix} & @y=14 + @y=14283, ay =143=4 = 241=3, an 425) + dy #24325 “fs ahbty Hence, (a) isthe correct option 1g “1 2] | =|3 2]andB=| 0 s| and avB-D-? 2s a ual (ero matris), then D matrix wit be or 0 2 0 2 @(37 a3 Y 02 Ss. s6 OF 7) @[23 5 6 5 Example 517 o Bn eter NT rene ere ey Scanned by CamScanner ef] wool HEHE a 3-2-6) fo a [310-€ eclloo pede StI-f Hence, (0) isthe corect option, 132 gnample 6 I A=|2_ & 5| is @ singular matrix, then k is vain HD @-1 Os ©4 @-8 sotton Ais singular matic => 140 1 3 . bh & 5-0 kb 2 3 1(&= 10) 32-20) 4264-48) =0 - Tk+56=0 = k=-8 Hence, (isthe core option. I Multiplication of Matrices IEA and B be any two matrices, then their product AB will ‘te defined only when number of columns in A is equal to the sunber of rows in B. If A=[4j yxy and B=[b, yep, then their Fodut AB=C-=[,), will be matrix of order Xp , where (AB), = 142 I 23 q{adB=|2 1 [ 144-2421 1-244-242+ * |a143-241-1 2:243-241- [4 CHAPTER » Matrices and Determinants {149 @ Properties of Matrix Multiplication IF A,B and C are three matrices such that their product is defined, then + ABBA (Genereily not commutative) + (AB) C=A (BC) (Associative law) + IA=A= Al, where Lis identity matsix for matrix ‘multiplication. + A@+O=AB+AC +) WAB=AC = B=C (Cancellation law is not applicable) + IfAB=0, then it does not mean that A~=0 or B= 0, again product of two non-zero matrix may be 2270 mats. + Trace (AB) = Trace (BA) (Distributive taw) Gi) Positive Integral Powers of a Matrix ‘The positive integral powers of a matrix A are defined only ‘when A isa square matrix. Also, then A?=A:A, A? =A Also, for any postive integers m,n ° ata sat «ata acaye © M=nmst AP =z, where A is @ square matrix of order n. aA Example 7 1/ matric | cose Caan cos oP P+! (@) None of these on pepal! |) 9 ~ancoray saan rel Haan | 0 -uni2)) tne) 0 Elen or Lf tanray =1an(@/2) eos sing 2U-P\ sing cose Scanned by CamScanner ee 150 | CHAPTER » Matrices and Determinants 1 tam(o/2)]fooso al “Lranigr2y 1 ind cons c0s9-+tan(6/2)sing ue isteril ~1an(g/2)c0s@+ sing tan(9/2)sing +eoso 1 2sin®(Q/2) +2sin5(Q22) n(9/2){208(G/2)~1}+ 2sing/ 2809/2) ~2sin(/2}e0s(9/2) + Sea tan 2) 2sin/2cos(g/2}1+(1~28n2(472) jo “et ap fang) Hence, (bis the coneet option 2-1 2 Example 8 fa = =4A-nl =0, then n is Jeo 8 case 3 Solution Hence, (b) is the correct option, ] @. 231) Example 9 17 112]]0 4 2|} 1/0, shen she value ofx is 032 @-t wo Solution ‘The LHS of the equation F H[2e 4+ 9-25-5]= dred @2 Thos, = xed Hence, (ais the comet option _[co10 sino Example 10 20 -[ mal then the value of £(0)-£@) & @ EO) © FON) © Fa) Ea-py Solution £(c- EB) sin ) cosB. cos sin + sinceou incsin + cosa =sinB [scm -( melo cos —cosasinB. a+) = [-singce#B) costar+P), Hence, (¢) is the correct option. Eo sinta+B) crm. yae[? 2) ene me (a) 22 15 ©-10 7 seo hte os OO RENEW EA tty cunts 7 6-4f7J-3 Hence, (b) is the correct option ome of IEE @ =1 @)x @x=3,y=2 @x=2y=3 Solution The given matrix equation can be written as xt2y)_[5 EEE) = x4 2y=Sand 2x +y=4 = xehy=2 Hence, (b) is the correct option, I Transpose of Matrix If we interchange the rows and columns of a matrix A, thet the matrix so obtained is called the transpose of A and iti denoted by AT or At or A’, From this definition, it is ‘obvious to note that (@) Order of A ism xn => order of ATis n xm © A=, Vij Properties of Transpose of Matrix TFA, B are matrices of suitable order, then @ Area i) Asay oat gar Gi) (A~ByF = at gt Go Gay? kar ©) (ABY = pt ar hy ADT = AP AP At | 8 AY Lay re Scanned by CamScanner ool mee ET -7 28428) [1 0} “2 748 JLo 1 Hence, (a) is the eorret option, Example 14 17 A: 4 a CM aaeaote off) off f]of] omcven ws a -6 al fs 9 i (ABY lk 4] Hence, (0) isthe comect option, soon Bf I Symmetric and Skew-Symmetric Matrix ‘Symmetric matrix A square matrix A=[a,] is called sgmmetic mati, if 4, =a, forall i,j or AT=A sf e. iad Sa 2S a a Syme aes) fatimam nimber of diferent clement ia a symupetic maint is | nit) Skew-Symmetric matrix A square matrix A=[a,] is called skew-symmetric matrix, if ay =—aj., for all i, j or Nana a Ong eg, |-h 0 Ff -e -f 0. a giao Seine wi ee mtn yg a oe hs ei geaaieuny whine CHAPTER » Matrices and Determinants $ 151 Properties of Symmetric and Skew-Symmetric Matrix (@ IEA is a square matrix, then A+ AT, AAT, ATA are symmetric matrices while A ~ AP is skew-symmetric matrices. (ii) TEA and B are two symmetric matrices, then (8) A#B,AB + BA are aso symmetric matrices. (b) AB ~ BA isa skew-symmetsic mauix. (©) AB is a symmetric matrix, when AB = BA. IFA and B are two skew-symmetric matrices, then (@) AB, AB~ BA are skew-symmetric matrices. (©) AB + BAisa symmetric matix. jv) IFA is a skew-symmetric matrix and C is a column ‘matrix, then CT AC is a zero mati. (0) Every square matrix A can unequally be expressed as sum of a symmetric and skew-symmetric matrix £e. -[husarfLe-a] bere 5 4=[ 7] enema pa ole] ol S| 1 9] 0 sit ofa a] ol, (ava yantcoL aa Solution Let A=B+C, where 2 respectively symmetric and skew-symmetric par of A. Now, 04 Hence, () is the corect option 1 Determinant of a Matrix Ca If A=|a, a3 a| be a square matrix, then its determinant, denoted by IAI or det (A) is defined as. Ca yay Scanned by CamScanner 182.4 CHAPTER » Matrices and Determinants Properties of Determinant of a Matrix G) lAlexist > Ais a square matrix, (i) LABI= ALIBI (a LATI=1A1 AL = Al, if A is a square matsx of order () IEA and B are square matrices of same order, then 14B! =1BA\. (i) IFA is skew-syinmetic matrix of odd onde, then Al =0 (i) TEA = ding (44,23, then LAL ii) IAP =1AM ne N J Adjoint of a Matrix If every clement of a square matrix A be replaced by its cofactor in IAI, then the transpose of the matrix s0 obtained is called the adjoint of A and itis denoted by, dj (A) ‘Thus, fA = aj] bea square matix and Fy be the cofactor of ay in Al, then aaj (4)= (FP > (ed) y= Fy Fa Fa \ 4 Fa Fam Fad [Fin Fox Fm | WPropefties of Adjoint of Matrix If A, B are square matrices of order n and J, is corresponding unit matix, then © A (Gd) A)=IA1 I, = (ad AYA {Thus, A (adj) is avays a scalar matrix} Pad Al= 1a! pads) =t04 (G9) ad (aj AY 1A ft (1) a (AP) = (05 A)? (si) adj (AB) = adj B) (ad A) (ity adj (A") = (adj AY", me (oi) adj KA) = HO! (ad A), Kee in adj «) adj 0=0 (a Ais symmetric = adj (A) is also symmetio, saeonal =? adj (A) is also diagonal, iy A is diagonal ci ic) 0 mp, ay Ai tan fan Aissingular => 10d) AI=O 103 ‘her lad dA i eu ample 16 A=]? 1 2 day Bam ie E 02 os m6 2 iio 103 sotuion 14i=[2 1 1]=2 Jo 02 asad (adj A= LAE LA Co Hete,ney, == 16 Hence, (b) isthe correct option. 135 Example 17 if A. [ 5 seer mtn p13 “4-20 “M4 oe | 2 *] of: 2 “| rey 2 44 4 wo 4-2] os in > (0 ow of tee a 4 Mons ey py -4 Solution aj (4) [= “2 «| «|| Suen “2 4-4] | a 4] Hens, (3) isthe comet option, oo 9 hen a yi eg 100 16/1 1 9 hid © Nove of these “QHQHwg oj 2 o| 2 rc 98 a) mtn g Scanned by CamScanner 200) flog 2 2 ofat6|1 1 0 p22) trad, ce (0 te cones on =8| inverse Matrix ' and B are two matrices such that AB BA, then ge iverse ofA and its denoted by A! Ths, sal @ AB=I=BA we may note from above property (i) of adjoint in f1AT #0, then vi adi (A) (adj A) Tar TAT ea my ial 4 pus, Aenists <> [ALO (i By = BA" women (adj 4°) = (adj AY diet ew (i) Wl Gy slat (0) A= diag (a42,.0,) = AM! = ding (ot 474) (iil) A is symmetric = AW is also symmetric. (@) Ais diagonal, [Al # 0 => “Vis also diagonal. (0) Als scalar matsix => 4-1 is also scalar matrix. Gi) A is triangular, [Al #0 =9 A“ is also triangular. apie 19 tose mais of [3 ae wf Jo 1? Jot 3] of] Selon Lt the given matrix is A, then Al =-8 wis yf? T[23 sol Ty eek 23 im mals a eae, (a) is tho conect option. ‘CHAPTER » Matrices and Determinants | 153 aay [ot Example 20. 4=[2 “1 j], ae] of ont at =A hen aa of *]1 9 sequal to 2 2 wows of 7] [i us “U3 v3 -V3) Ols 16 @ | v6 “ip jie a ae Fan gee eae, (6 isthe comet option. 1 Some Important Cases of Matrices (i Orthogonal matrix A square matrix A is called orthogonal, if AAT idempotent matrix, if A= A AA iy rin mic Asem Ae ‘ involutory matrix, fA?=2 or A Go Nite mi A sq mae Al nilpotent matrix, if there exist a P ¢ N such that A” = 0 d ces Ae[t S] sre (®) Hermition. matrix A square matrix A is hermition Scanned by CamScanner 154 f CHAPTER « Walrices and Determinants (vi) Skew-Hermition matrix A square matrix A is skew-Hermition, fA =—A® Le, ay=— a, i,j (vi) Period of a matrix If for any matrix A, AM! = A, then k is called period of matrix (where k is a least positive integer) eg Haba and 2+ (taints [2 2) oe a HPO) se isa dif tiation of matris 2,0 #9) annette (ix) Submatrix Let A be m x mattin, then a matrix obtained by leaving some rows or columns of both of A is called a sub-matrix of 4. 9 Rank of a matrix. A number r is said to be the rank of am xn matrix A, if (@ Every square submatrix of onder (r+ 1) of more is singular and (B) There exists atleast one square submatrix of order +r which is non-singular. ‘Thus, the rank of matrix is the order of the highest order ‘non-singular submatrix, 129 Example 21 The rank of matric A=|4 5 6| is 345 @2 os ws Solution We have, IAI = 0 therefore (A is less than 3, we observe ws 16 oa [3 9] ea msn Hence, r(A)=2. Hence, (a) is the correct option. vas esr) ‘© Element 1 Determinants seat ; ‘nan algebraic or numerical Expression is eg sv mses meen ira Ismet a determinant ofthat expresso, Forexangyg§ ay expression aby ~ 6 feelaosavaa [PS © OS Joab bate 0 joo aoe carein @ @+e+2—3abe — (b) a*b—be (@) AP © GP oo @ @abec . © HP (@) None of these 2S. Ihab +642 a2 -a? ob ae G4a%)x U+8%)x (1+c%)x| 18. Uf} ab 6 be |=ka*B%c?, then & is equal to and f(3)=|(1+a)x L+b2)x (L+e*x| thenfiii ac be -e Gta )x (4b%)x (eeyx w-4 2 4 polynomial of degree o4 os @ 3 © 2 ©1 @o aoa 26. The value of the determinant 19. If a +6 +e" =0 such that 0 Bee on op then the value of is 5 8, OTB isequal to @ 0 (b) abe BF 9 (© -abe (@) None of these @ Obed ® a9. ore kat fee © -2 B42 20. If] 6 (cay? BF |= kabela+b+e), then a7 fn? 888 sink oe (a+b) i ae Teos?8 — cos@ |=, thensind is si the value of k is 45in481+4sin 49| “1 wt equal to a 2. @ in ©2 O-n 1 @ a EEE Scanned by CamScanner 4 piowing sem of equations 3x-2y+2=0, a5 peisent, 427—Aen00H lon oer nya ec fe B iseaual to - m2 @! @s © jereasech- and x-3y+2=0 has 777 sauion fr kis equal 10 a @) 0 6 } @2 fraayrenOden ay ~B2=0, 839420 has ge ero soliton, then sequal to fel wo a @-3 41, Tenanbe ofsotion ofthe equations, etyez 0 3 4y— )x=3y+z=0 is ©) 1 wo 2 (@) infinite "sins cose sa. HF AG)=|n! sin cos], then the value of aa @ Eicon at xe0is @-l wo @. (© dependent of a 33. the system of equations, x+2y—3z=1, (k +3): (2k+I)x+z=0is inconsistent, then the value of kis @-3 ® 12 @o @2 ‘A. the system of equations, x—ky-z=0, kr-y-2=0 ad x+y-z=0 has a non-zero solution, then the possible value of k are. @ -1,2 & 1,2 or @-11 $8. Uf g.0.,0).4d ye ate in GP and a, >O for each i then the value of the determinant loge, logayas A=) l0ga,.6 1Ogayey 1084p.40| is equal to NoBGni2 IOBGer¢ HORA ys| 2 (@ None of these logan, @1 oo 36. 37, 38. 39. 40. a. 42. CHAPTER » Matrices and Determinants {161 ‘The system of linear equations, x+y+2=2, Qety—z=3, 3x+2y+ke = 4 has unique solution, if @ 20 () -leke1 © 2eke2 (@) k=0 ‘The system of equations Amy a2, Ixy 42m =-6 and 35,4, tay =-18 has, (@) no solution (€) infinite sotution (b) exactly one solution (@) None of these inant of ax? +2br-+cis negative, be ath brke 0 (@) positive (b) (ac—B (ax? +2bx+e) (©) negative @o If-xis a positive integer, then xl @#D! G+)! (HD! (+2)! G43)! (x+2)! +3)! +4) @) 2eder4D! ©) Axdx+ DIG)! © ANyir4H! (@ None of these If the system of linear equations, x+2ay+az=0, x43by+be=0, xtdeytez=Ohas a solution, then a.bs¢ (@) arein AP (©) are in HP ‘The system of equations axtyteso-l xtoyteza-t xtytor=a-1 has no solution, if «is a-| is equal to (©) areinGP (@) satisfy a+2b43¢=0 re 3 © -2 @) either-2 ort tae 2) aye « @ -2 (b) 2 © -4 @4 it Ae|t 1] den at is equal to of ten a ie an of *| 14 ® ( Hl Scanned by CamScanner 162 44. 45. 46. 47. 49. 50, 51. 52. | CHAPTER « Matrices and Determinants =, sand ony it © dr2.B~0 — @) An0.020 © A200 8-0 td) Nosed ce fr 3 ave Yiwemavic|2 4 {ising ten 2 ineg ps0] y @ (b) 4 o2 @ 4 ab taf Jon 40, ton he num (a) 2 3 od @ 5 ao It alt 5 then for what value of 4,4? = 0 fa) 0 ) +1 ©) @t 200 If A=/0 2 0|,then 4? is equal to 002 (b) 104 (@) 324 Which of the following is incorrect @ AB? =(A4 By A~B) () (ATI =A © (AB)" = A"B", where A, B commute (@) (A-DU+A)=0 6 A? =7 ‘Matrix theory was introduced by (b) Cayley-Hamilton (@) Newton (©) Cauchy @ Euclid ‘Which one of the following is not true (2) Matrix addition is commutative (©) Matrix addition is associative (©) Matrix multiplication is commutative (@) Matrix multiplication is associative Choose the correct answer. (2) Every identity matrix is scalar matrix (b) Every scalar matrix isan identity matrix (©) Every diagonal matrix is an identity matrix (@) A square matrix whose each element is 1, is an identity matrix 10 Lo welt 8 are! Fs is following holds for all n21, (by the principal of ‘mathematical induction) (@) AT =nA+(n-DI (© At =nd-(n-D (0) AN=2"' At (a= @ AN=2"4-(n-DI 34. 55. 56. 57. 58, 59. 60. 61. 62, 63. | ingular matrices, then, ) AB @ (aay! (o) (ABY = AB 43 “Adjoint ofthe max ¥=|1 0 1 (iy 443 @N (b) 2N ( -" (@) None of they From the following find the correct relation (a) (ABy=4'8" ©) (AB) =H 1 @) (ABy opp, © ae 1) (ABY" = pip, cose sing * 6 ~ and A adj (A) =| woth Be ain is equal to @o 1 (©) sinctcose (@ cosa Ia matic A is such that 34° 424454405 its inverse is (@) -GA7+2A+52) (0) 34? 42445) (©) 34*-2A-51 (2) None of these IFA and B are square matrices of the same order, ty (a) (ABY = A’B’ () (ABy=284" (©) AB=0;if |Al=Gor!Bl=0 (@ AB=O:if A=1orB Which of the following is not true? (@) Every skew-symmetric matrix of odd orders non-singular (©) If determinant of a square matrix is non-e itis non-singular (©) Adjoint of symmetric matrix is symmetic (d) Adjoint of a diagonal matrix is diagonal adj (AB) —(adj By(adj A) is equal to @) adjA-agje oT @o (@) None of these t-rd 427 let A=|2 1 3) ana Gqe=|-s 0 a).t! rug 1-2 3, is the inverse of matrix A, then ais 62 22 @ -2 For any 2x2 Mmawix A, if =['> Oh ee 1 f ACadjA)=| 9 oh 1A1 is equal @o © 2 ov Scanned by CamScanner 10 0 o 1 ol, oo4 shyt poked wine sale Ry thon pl alas (@4) @ 610) Ore A 2 Pa 2 ) 6-1) @) 6-1) vi . a-(, i and Q=PAPT, then peg)? is equal to 1 2005" @lo 1 Bal 1 | @ 1 © lpn 1 0 2005 6 Aisa nit matix of order, then A(adjA) is (a) zero matrix (b) row matrix (@) unit matrix (@) None of these TPA and B are square matrices of order 3 such that , then 13AB I is equal to @-9 (& -81 © -27 @ 81 (& Ae and BO are nxn matrix such that AB = then (@) det(A)=0 or det(B)=0 and det (B)=0 aet(B) #0 w [82 5] 18 72. . @ at=Bt p be 6. War pbeqerand|p+a qth 2 abr 1 is equal to 74. 7S. CHAPTER « Matrices and Determinants ) 163 on Bo I n#3b and 1, 0,0 are the cube roots of unity, then 1 ot oF a=|o% 1 ot | asthe value ort @o oo ow @. For postive numbers x,y and z the numerical value of 1 tog, y log, 2| the determinant [log,x 1 log, 2| is logex loy 1 @ 0 wd (©) oe, 92 (@) None of these If x=ey tbe, yaazter, r= bray (where x Yo 2 a6 not all zer0) have a solution other than x=0, ¥ 2=0, then a, band ¢ are connected by the relation (@ at 4btac? +3abe=0 (b) ab +b? +e +2abe (oe) 4b ee + 2b (@) 4b 422 —be-ca—ab=1 If Al denotes the value of the determinant of the square ‘matrix A of order 3, then |~2A lis equal to (@ 1A (b) 8IAl (© Al (8) None of these If the system of equations, art y-+2=0, xtby+2=0 and x+ytez=0, where a,bc#1 has a non-trivial solution, then the vale of 14 +h is Ta be @ - 0 ©1 (@) None of these IEA is a matrix of order 3 and | A! 8, then Jadj Al is equal to @ m2 ©? .@% Scanned by CamScanner L 2 3. 4 ‘Answer and Solutions ep eet 0430) [040 42 gy. te ey arti] bb eal am * @ oe arse a+6b| | 2% ‘| e [ib ofby RAR RIOB-K 8 at = ee (Bh, bee bk ponding, a—byb-eN(e-a) 1 " tex 158 6 {i tte Tre @ acti 5 feo iga=tmasc=c) © 11 ie 1s e atx oF (E26r6+ Le a] fro 0 @ pr Aan ‘eres @ lo of 1/-|1+0+0? of 1 ate —% jo? 1 | [itoro? 1 o te 4 2 raft x 1 ie 0 o oF ye 14x =|0 o 1 101 oO 1 @ Mohek re the a-x cb aan mF ao @}e bx «fe = (243s? = 0 4=0,0,-3 bo a ex rea be aebrene "2 7 @|b xe a |=0 => latbte-x b-x © a xtb Jatbte-x a os lob © = (rtatbtoll xte a [= a-Laht bx = ¢ ah 7 1 oa xb (904646) (by hypothesis) > #=-(atb+0) or 1{(6-x)(c-x)-a?}~c{e-x-a) is one of the root of the equation. +bla-b+x)=0 b b-c ¢ by expanding the detrniant 8. ©) A=@-a)b-a)Ja a-b b or PCa? +6? +62) +(ab+be tea) = cca a or #-(Se!)-HS a) =0 an; =(a~e)'/a a lq q) 0 by Gt {rattec=0=carseo?=0 cee Va a be 2 Le sE eba0s Yara} % @ |Ub Bog Ve or x28 25a! ° ‘ 1 @ abe lat 2. The solution is x=0 or ‘abe|! 2 abc] = abe] , #20 1? abe) abe] Vet Scanned by CamScanner (etytalztx 2 x ety yz sty y 2 (by RAR +R+R) raid s(xtyt2)|x 2% (by G 9G -G) yz a lety tod -9)- Ge) 49-9} a (rtyta(z-9? > kel 37 rid 12 @ [2 x 2J-0- 49/2 * 2| i 7 6 2 76% (by RR +R +R) => (14 9){(a2-1)-@x-14) 402-720) > (e+ 9) (a? 9x4 14) =0 > (xt 9MR-DUE-D) = Hence, the other two roots are x= 2.7- 13, (a) Splitting the determinant into two determinants, We lag lad ged b o|+abclt & 2°? 1c A re = atabesiiaDyb-O1(e- =O Because a, b ¢ ate different, he Sh 0 cannot be zero, Hence, option (2) Labe=0, is ‘comrect. 14, (b) Since, itis an identity in 2 so satisfied by every ‘CHAPTER » Matrices and Determinants 1 165 value of % Now, put 2=0 in the given equation, we have 0-413 te|1 2 ~4]=-12430=18 340 1a bee b cto 1c asp | 11 bee| =(a+b+o)f1 1 etal (9G+G) 11 aso i =0 ase) 16. (4) Muliplying by a/R, by and R, by , we have , me we aba Lathe? abe be+ab ‘abe |" abc abe actbe be 1 ab+ac| jac 1 be+ab] Jab 1 ac+be! be 1 Zab fac 1 Zab] jab 1 Eab| be 11 = abe-Eablea 1 1 jab 1 ab aatd| be bate| jaa+b bate 0 abc (by G9G+G) =abe| 2G) @ ae ab aatd [bc bate ‘ 0. 0 ~(aa? +2ba+e) by RIK-AR-K) faa? + ate)~0} bac +2ba.+€)~9} (by expanding along G.) al =P -acy(ao? +2040 ‘thus, £0, either B? -ae=0 aot +2hate=0 or fe, ab and c inGPor ant #25046 =0 Scanned by CamScanner 1185 7 CHAPTER » Matrices and Determinants no ab ae © 18. @ fad 8 te . ae be -< ~ ! att =tebeyabe] tt 1]eae'eenes to sabe = tac? (given) => kad 1p 11 tel Applying CC, =, and C9 0, — lea dod 19. | 1 bo 1 06 Onexpanding wert. R ab+bet cat abe =, “ 4b et e9 ie Given, = Aetstao = abeterca=o = debe {from Ea. ()} 20. (6) Operate C; +6, -G.C,-»—C, and take out 2+b+¢ from C, as wells from C to get +e) an-b-e a~bae| A=@tb+e)' | B ctay 0 0 atb-e| (Cperate & + R~ RR) he 20th = (atb+e)| Be cta-b oO 0 atb~e Crete 6-566, a 6464469 be 0 0 = latb+cP| erg & Sand =(a+b +c} 2bel(a+b)(e+a)—be)) =2abe(a+b-+e) 21. (b) Given, angles ofa triangle = A, Band C, We know that as A +B + C=", therefore A+B=n-¢ oF c0s(A+8) =cos(n—C). or cos Acos B—sin Asin B. c0sC 0s vy 08 ACOs + 208 =n Aa sin(A+B)=sinGe—C) = sing and expanding the given determinant, we gy 5 ge ntl- cba? -o a ¢ ab aw? =0'o|-) © bo? ~c a ca? ab al aba =@l-b c ble -w'[5 ¢ ple -c a | cae 25. 0) Applying c, PG+G46, 1 +6 yx FOS py a2y 1 46%) A+e*)x| (+e*)x| I+e7x ‘ (rat eee +cte2e0) Applying RR, — Rw» RA 1 +b) d+ery =)0 ans 0 Dens 20, eM, deste off) 9 = (xP Inx Scanned by CamScanner 2.0 0, a 1. (©) Ithas a non-zero solution, if pp sped 0 ott -aja’—b 1 t}=0 ab 1 ae wae (624-4 898. FG =G) andthe aking sor common (O'=a") from 2rd column and (ee?) om 3ndeolama} sin'@ sin? Lreos?0 cost 4singd 1+4sindo 1+sin"@ cos? sin’ (using CG, 96,-C,,6, 9G-G) 1 0 sinto -1 1 cost@ |=0 0-1 1+4sin4o| 2 (© Thesystem of equations has infinitely many (non tivial solution, if A=0 i 3 a -14 15]=0 123 342-30) 046-2) +1(-30+14) =0 aes 2(1+2sin 46) = 0=> sin 48: 2 > > 1k - 3 -k 13] > 6k +6=0=k=1 (@) The given system of homogeneous equations has a nonzero solution, if A=0. 1 4-1 ie, [3 -a -3|=20-6=0 ie, if a=-3 131 (©) The given system of homogeneous equations has 144i as)3 4 - 13] =7-16+540 ‘There exists only one trivial solution. (4-3) -4G+1)- 1-944) 32, CHAPTER » Matices and Determinants | 187 ) = [One . nt sin(o4 2) 33. (2) For the equation to be inconsistent 123 0 0 k43 +101 123 300 ood ‘So, that system is inconsistent for For non-trivial solution 1k k -1 -l}-0 = ae Ifris the common ratio, then a, = ar" forall n21 =05k=-3 and D,= #0 34. @ > 35. ©) = loga, = loga, +(n—Dlogr =AHH-DR, where log, = A and logr=R ‘Thus, in A, on applying GCG and 6, 9C,-C,,we obtain C, and Cyare identical ‘Thos, =0 36. (a) The given system of equations has a unique rid solution, if|2 1 -I]#0= #0. 32k Scanned by CamScanner 166 1 CHAPTER » Matrices and Determinants bent p-[s -1 2 we oft [-1-2}-16-3}4+13+3]=0 2-11 and Baf-6 1D sd 38, 39, (1=2)-1(-36+6)+ 16-18) =-6430- Aso, Dy =0:D,=0 So, the system is consistent (D=D, =D, =D, =0) de system has infinite soation. a b axtb) be, Bete Jactb bene 0 Applying —> Ry—2R,—Ry, we get | © Lea-| ab arte A=|bc brte 100 ~(ax* + 2br+0}) A= (6? ac) (ax +2bx +e) Now, #?-ae<0 and @>0 = Discriminant of ax? +-2bx+c is ve and a>0 > (@P +2br+e)>0 forall xeR 3 A=@-acy(ar* +2e+0)<0 © A=leryr (e421 C43) Ke+2)1 G43 ebay 1 Gt) @+2@4n rakes (42) (x43 (x42) 1 (+3) (r+4y(x43) Applying &, > R,~R,, Ra (RyRy), we get Ot ara) “Hermes 1 mn 1 43) rtaerssy (on simplification) the altemate, 1 GD! a = 221+)! (242)1 ‘Trick Put x= 1 and then match (Ry RRR o a b bea 2e-b c-b (b-ay2e~b)=0 tony 0 a,b care in HP. 41. (©) For no solution or infinitely many solutions ail 1 @ 1/=00=1a=-2 Lia But for a=1, clearly there are infinitely may solutions and when we put = ~2 in given syne of equations and adding them together, LHS # RHS ie., No solution. 42, (@) M?-2M-1,=0 ray. 2] pa 2a) 71 oy cael As 3ay"[o a> > falls SH, the = 5-2 8-20] [1 4 8-24 13-32] lo 1] = S-2=1,8-20=0,13-3021 = 2=4, which satisfies all the three equations soe TL af OP Lf 44. @ Since, AB=0, sevenif A* Oand B20. 13 As? 48.) Themauix|2 4 8 |i 35 10 13 Ase 24 8 gular, if 2042) 125,24 Scanned by CamScanner a Pla BP ; aoa? ab ab?) Wedd? eOand A* =O. forall n22, a Te ‘] o peene -all-1 (as given) 1g. (0) Since, (A+BXA~B) By matrix distribution law, = A -AB+BA-B = A? BP > BA-AB=0=> BA= AB 50. (b) Students should remember it. 51, (©) tis property of matrix multiplication, 52. (a) We know that every identity matrix is a scalar matrix. awed eb] AEH ractecore[) fea nd 54 (@) Weknow tha, if and B are non-singular matrices ofthe same orders, then (AB)"' = BA, 4 3-8 5. @ Teeofacorsof N=] 1 0 1 | are 56. 57. (CHAPTER « Matrices and Determinants } 169 43 se agy=] 10 1 i=n 443 (b) tis obvious. ote a(S me -sina cosa ha cea [fre ae 2 alta}e Te wap of mati of ct cosa sing! singe , cose cosa sina]feosa ~sina’ ina cosa.||sina cosa. [eae a] (as given) > kel (@ 342424? +5A+1=0=9 1 =-34?-24" ~5A = 34-24-51 > GA? +2445) () (ABy’= BA, by definition. (@) Every skew-symmetric matrix of odd order is singular. So, option (a) is incorrect. (©) (ABy! = BtAT aadj(AB) _ adj(B)_adj() 1ABl~1BI LAL = adi(AB)—adj(B)-adj(A)=0 422 (@) Given, |-5 0 «|=1047 123, 42 a - 10 0 0) =J-5 0 aff2 1 0 0 0 raga 0 0 0) (Equating the clement of 2nd row and Ist column) > Stas0>a=5 (b) Weave, aesiay=[") al 0 10 10 ; oe a0aiey=10h, t]eo" ~ and ck catia) vai Aadja)=1AUL Scanned by CamScanner y 170 1 CHAPTER « Matrices and Determinants ++ From Eqs. ) and (i), we get 1 0 of fo 9 °) 106 valde as 24) lo 10 v4 Jao 0 lo oa S6=lterd * (6,11) satisty the relation 65. @) If = papr PTQ™*P = apt gt p =#P"O™p — geprgimp =A™ PT Op) =4a™pr (ps) (Q= PAP" = OP = pay = 4mms ams _[1 2008 > of a 66. (2) IFA isa singular maui of order n, then A(OiA)= (adj A)A = 0 = 2er0 ment 67. (a) Weknow tat, if 4, Bare n square matics, then 1ABI=1A1IBi 68. (@) 420 nd B20 * AB=0 Hence, det (A)=0 or det (B)=0 Pob e¢ 69. ©) d=|p+a 945 2cln9 ob Applying Ry —> RR, Pbe =]a ¢ cso ao, Applying Ry > R—R and >, —@ Po ob ¢ a-p q-b 0 [a Ja=p 0 re (by equality of matrices) (as PPT =1) wv Pag (On expansion, we get p(q-bXr~C)—B(a~ pXr~e) (4 -bXa~ = (p-aNlg~0Xr-0) pi, b [ea (q-) es = (p~anq-BNr-0) A a peagebrec 70. (a) Applying GC; +C,+C,, we gt 1+@" +0" "a A=|1+o"+o0" 1 gt Ow" 1 140°+0" =0, ifn is not muliple ot) 1 tos.y tox, 2 71.) fiog,x 1 tog, z Joa.e lony 1 © (les, log, 2)—log,y(log, x—Iop, i, ++log, (log, xlog, yi = (-D-A-Iog, ylog, 2)+ (log, zlog, x~1 Colog, y-log, 22! 72. (©) The system of homogeneous equations tebe =0 ot ytar=0 betay— Ee ks Scanned by CamScanner ‘as the system of equations has a non-trivial so © it solution 1 isa 1=b" ie 1 erp 75. @ Wetwon, aaiiaele ial 0 . 0 0 SAL alice ot ole (B28 WAL 00 oem Oo 0 IAN a ab Ne-D-Md-ae-n -1l-a)(b-1)=0 : JAl-adjlAl= VAP > P or adjlAl= 8 = (2 = 2 a Scanned by CamScanner 1 Relation — ita Sty aid ‘ ‘Thus, if Ris a relation fro A to B, then R B. Tf an element a € A is associated with b € B under the function f then the element bis called the J image of « or the value of f at a It is denoted by the symbols bas a) or fra bor (a bef ‘Also, ais called the pre-image of b under f. Example 1. f.Z +2 f(a) =3r— 4 is a faction, while MeN Nf) = ax ~ 4 is no a faneton Because image of 1 e N under f does not exist in N. Scanned by CamScanner 174-1 CHAPTER = Relations and Functions le oN, F(a)= VF is not a function because ce pena ets Sata Reicks y Whether (14> Bis fneton or not tet the flowing 4 Bnistence of fimage of very clement of di the st. 2 Uniuencs of fmage of every lament ofthese 1 Function as a Set of Ordered Pairs ‘Afunctonf: A > Bean be expressed as a st of ordered air in which frst element is the member of A and second Element isthe member of B. Hence, fis «st of ordexed pais (a. 8) such that (ais an clement of A G@bis an element of B Gdn two ordered pies off have the same frst element Gv)every member ofA is first element of one of the ordre pais off. 4:A~ Bisa subse of Ax B, His expressed inthe form of ordered pits a follows 17 ((@ b=F0),0€ Amd be B) Example 3. A=(0,1,2),thenf= (0.3). (49,0) na fection from Ato H bat = (0, 4 (2, )} and b= (0, (0,4) (1,5) 13) i ot a finton fom AN. Example 4: N+ N, fie) = 2x-+ 1, them the terms of ordered prs $= {0,3 23), 7), od 1 Domain, Codomain arid Range of a Function Suppose that fs a function from A to B, ie, f:A > B, then set A is called the domain and set B is called the codoman, 1 Also, the set ofall images of the elements of A is called the range of f and itis denoted by f(A), Therefore (A) = (fa)! a A) CB. I fis expressed in ferms of ordered pairs, then set of first element of ordered Pairs off willbe domain and set of second elements of these ordered pairs will be range off, ie, Domain of f= (r1(3,9)< f) Range of f= (y 1G, ef) The Set Ys also called the codomain off, clea, SOsY. x nae given figure find the domain, cody, a : ee ba : + Domain of f ={P. qr. 5) 4 codomain of f = Cl 2,3, 4,5) Range of f = (1, 2, 3) ae domain, codomain and range of function. Domain of f= Solution Codomain of f= W and Range = (3,5,_) Example 7 ff: Z>Zf@) =k then ‘Sokaion Domain of f= codomain of f= Z and Range of f= (0, 1. 2,3, 0) I Identity Function ‘The function f: R — R is called an identity function f() =x, Vx R. The domain of this ‘identity function ist and its range is also R, 1 Equal Function ‘Two function fand g are said to be equal funtion, only it © domain of = domain of (i codomain of f= cod lomain of @ Fe) 2 = 802. V x6 their common domain 162), B= (10,13), f:4-+B, BAB, acy Example 8 74 Scanned by CamScanner y 6 5 te id he dana of ston 1 aay javier F73 geneninet=(5") je 10 Finds dont of hereto 0) = VTE wl e pex2d lee won ponle reco l Find the domain ofthe function vo Fara given function to be defined w(t jee 2! se-P26 22-5465 5 G@-De-DS0 exe) ample 12. Find domain of (3) = logy +2°) adn (6) =logy( +29) exists, if 14.2> 0 (9+) > 0, where (= +29 fs always putin ss D 0 site 21 xe} het ‘Tins, domain of above function f(x) is (-1, =). Number of Functions (or Mapping) from oY Vet {3% oo Aq) (Kee elements) tod ip 3o Syren Jy) (Les, m elements) Ther, each element in domain x; (¢ = 1, 2, 3, comesponds m images fixoy x m2 t Thus total number of function from X to ¥ é = nxnx.m times =n" + (Number of elements in in Master emesin doin CHAPTER « Relations and Functions { 175 1 Representation and Testing for a function @) Mapping It show the graphical aspect of the relation ‘of the elements of X withthe elements of ¥. @ fixoy x ©) fixry \_ pL ——\ © h:xXo¥ x In the above given mapping rule f, and f, shows a function because each element of X is associated with ‘unique element of ¥. Whereas f, is not a funetion because in {fa clement ¢ is associated with two elements of Y. Algebraic method tshow the relation between the ‘elements of two sets in the ftom of two variables x and y where x is independent variable and y is dependent variable. If X and ¥ be two given sets. X= (1, 2,3), ¥= 15.7.9) then f:X9 Yy=fla)=2+3 (Gi) In the form of ordered pairs A function f:X— ¥ can be expressed as a set of ordered! pars in which first element of every ordered pair is a member of X and second element is the member of ¥. So, fis a set of order pairs (a, b) such that @) ais an clement of X (©) disan element of ¥ (©) Two ordered pairs should not have the same first clement. (iv) Vertical line test for a function If we are given a graph ‘of the relation. Then, we can check whether the given relation is. function or not, IFitis possible to draw a vertical line which ‘cuts the given curve at more than one point, then given relation is not a function and when this vertical ine means line parallel to y-axis cuts the cucve at only one point, then itis function. Figure represents a function, Scanned by CamScanner 176 1 CHAPTER « Relations and Functions % | Classification of Function @ Constant function Ifthe range of a function f consists ‘of only one number, then fis called a constant function. 0.6) reac egaLet fc) =: @ Identity function The function defined by f(x) Wax R, is called the identity function, Where ¢ is a constant number. a 6 x 2 Modulus function The function defined by F@)=1x is called a modulus >) Signum function The function defined by 8h aon rao [when « rol nese ere 9. when 200) EE 22 is called signum function, (©) Greatest integer function This function ig fx), where [2] = greatest integer less than oF equa nh, y — «Ld Important Identities + fds. (Ohisis always true) + Gitex Properties of Greatest Integer Function @ [x)=x, holds, if x is an integer. @) [e+ 1]= 13] + £ if Fis an integer. © b431=b+Dhif be} 4+ 6) <1 =] +D1+ Lif (2) + (y} 21 © WE EA then 27 TE (21S 4 then Ga) < 47 ~ ff xe integer [el-1,ifx @ integer () Fraction part function (x) denotes fractional pate Its equal to x~ [x]. *& 27}207, £32)a0y° 8)" n= R Range =(0, 1) (A Teigonomet Scanned by CamScanner

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