MATHEMATICS (860) 2. To develop the ability to apply the knowledge and understanding of Mathematics to unfamiliar situations or to new problems.

Transcription

1 MATHEMATICS (860) Aims: 1. To enle cndidtes to cquire knowledge nd to develop n understnding of the terms, concepts, symols, definitions, principles, processes nd formule of Mthemtics t the Senior Secondry stge.. To develop the ility to pply the knowledge nd understnding of Mthemtics to unfmilir situtions or to new prolems. 3. To develop n interest in Mthemtics. 4. To enhnce ility of nlyticl nd rtionl thinking in young minds. 5. To develop skills of - () Computtion. () Logicl thinking. (c) Hndling strctions. (d) Generlizing ptterns. (e) Solving prolems using multiple methods. (f) Reding tles, chrts, grphs, etc. 6. To develop n pprecition of the role of Mthemtics in dy-to-dy life. 7. To develop scientific ttitude through the study of Mthemtics. A knowledge of Arithmetic, Bsic Alger (Formule, Fctoriztion etc.), Bsic Trigonometry nd Pure Geometry is ssumed. As regrds to the stndrd of lgeric mnipultion, students should e tught: (i) To check every step efore proceeding to the next prticulrly where minus signs re involved. (ii) To ttck simplifiction piecemel rther thn en lock. (iii) To oserve nd ct on ny specil fetures of lgeric form tht my e oviously present. 11

2 CLASS XI The syllus is divided into three sections A, B nd C. Section A is compulsory for ll cndidtes. Cndidtes will hve choice of ttempting questions from EITHER Section B OR Section C. There will e one pper of three hours durtion of 100 mrks. Section A (80 Mrks): Cndidtes will e required to ttempt ll questions. Internl choice will e provided in three questions of four mrks ech nd two questions of six mrks ech. Section B/ Section C (0 Mrks): Cndidtes will e required to ttempt ll questions EITHER from Section B or Section C. Internl choice will e provided in two questions of four mrks ech. S.No. UNIT TOTAL WEIGHTAGE SECTION A: 80 Mrks 1. Sets nd Functions Mrks. Alger 34 Mrks 3. Coordinte Geometry 8 Mrks 4. Clculus 8 Mrks 5. Sttistics & Proility 8 Mrks SECTION B: 0 mrks 6. Conic Section 1 Mrks 7. Introduction to Three Dimensionl Geometry 4 Mrks 8. Mthemticl Resoning 4 Mrks OR SECTION C: 0 Mrks 9. Sttistics 6 Mrks 10. Correltion Anlysis 6 Mrks 11. Index Numers & Moving Averges 8 Mrks Totl 100 Mrks 1

3 1. Sets nd Functions (i) Sets SECTION A Sets nd their representtions. Empty set. Finite nd Infinite sets. Equl sets. Susets. Susets of set of rel numers especilly intervls (with nottions). Power set. Universl set. Venn digrms. Union nd Intersection of sets. Prcticl prolems on union nd intersection of two nd three sets. Difference of sets. Complement of set. Properties of Complement of Sets. (ii) Reltions & Functions Ordered pirs, Crtesin product of sets. Numer of elements in the crtesin product of two finite sets. Crtesin product of the set of rels with itself (upto R x R x R). Definition of reltion, pictoril digrms, domin, co-domin nd rnge of reltion. Function s specil type of reltion. Function s type of mpping, types of functions (one to one, mny to one, onto, into) domin, co-domin nd rnge of function. Rel vlued functions, domin nd rnge of these functions, constnt, identity, polynomil, rtionl, modulus, signum, exponentil, logrithmic nd gretest integer functions, with their grphs. Sum, difference, product nd quotient of functions. Sets: Self-explntory. Bsic concepts of Reltions nd Functions - Ordered pirs, sets of ordered pirs. - Crtesin Product (Cross) of two sets, crdinl numer of cross product. Reltions s: - n ssocition etween two sets. - suset of Cross Product. - Domin, Rnge nd Co-domin of Reltion. - Functions: - As specil reltions, concept of writing y is function of x s y = f(x) Introduction of Types: one to one, mny to one, into, onto. - Domin nd rnge of function. - Sketches of grphs of exponentil function, logrithmic function, modulus function, step function nd rtionl function. (iii) Trigonometry Positive nd negtive ngles. Mesuring ngles in rdins nd in degrees nd conversion from one mesure to nother. Definition of trigonometric functions with the help of unit circle. Truth of the identity sin x+cos x=1, for ll x. Signs of trigonometric functions. Domin nd rnge of trignometric functions nd their grphs. Expressing sin (x±y) nd cos (x±y) in terms of sinx, siny, cosx & cosy nd their simple pplictions. Deducing the identities like the following: tn (x ± y) = cot(x ± y)= tn x± tn y 1 tn xtn y cot xcot y 1 coty± cotx sinα ± sin β =sin 1 (α ± β )cos 1 (α β ) cosα + cos β = cos 1 (α + β ) cos 1 (α - β ) cosα - cos β = - sin 1 (α + β ) sin 1 (α - β ) Identities relted to sin x, cosx, tn x, sin3x, cos3x nd tn3x. Generl solution of trigonometric equtions of the type siny = sin, cosy = cos nd tny = tn. Properties of tringles (proof nd simple pplictions of sine rule cosine rule nd re of tringle). Angles nd Arc lengths - Angles: Convention of sign of ngles. - Mgnitude of n ngle: Mesures of Angles; Circulr mesure. - The reltion S = rθ where θ is in rdins. Reltion etween rdins nd degree.,

4 - Definition of trigonometric functions with the help of unit circle. - Truth of the identity sin x+cos x=1 - NOTE: Questions on the re of sector of circle re required to e covered. Trigonometric Functions - Reltionship etween trigonometric functions. - Proving simple identities. - Signs of trigonometric functions. - Domin nd rnge of the trigonometric functions. - Trigonometric functions of ll ngles. - Periods of trigonometric functions. - Grphs of simple trigonometric functions (only sketches). NOTE: Grphs of sin x, cos x, tn x, sec x, cosec x nd cot x re to e included. Compound nd multiple ngles - Addition nd sutrction formul: sin(a ± B); cos(a ± B); tn(a ± B); tn(a + B + C) etc., Doule ngle, triple ngle, hlf ngle nd one third ngle formul s specil cses. - Sum nd differences s products sinc + sind = C + D sin cos C D, etc. - Product to sum or difference i.e. sinacosb = sin(a + B) + sin(a B) etc. Trigonometric Equtions - Solution of trigonometric equtions (Generl solution nd solution in the specified rnge). - Equtions expressile in terms of sinθ =0 etc. - Equtions expressile in terms i.e.. Alger sinθ = sin α etc. - Equtions expressile multiple nd su- multiple ngles i.e. sin θ = sin α etc. - Liner equtions of the form cosθ + sinθ = c, where nd, 0 - Properties of Δ c + Sine formul: = = c ; sin A sin B sin C Cosine formul: + c cos A =, etc c Are of tringle: = 1 c sin A, etc Simple pplictions of the ove. (i) Principle of Mthemticl Induction Process of the proof y induction, motivting the ppliction of the method y looking t nturl numers s the lest inductive suset of rel numers. The principle of mthemticl induction nd simple pplictions. Using induction to prove vrious summtions, divisiility nd inequlities of lgeric expressions only. (ii) Complex Numers Introduction of complex numers nd their representtion, Algeric properties of complex numers. Argnd plne nd polr representtion of complex numers. Squre root of complex numer. Cue root of unity. - Conjugte, modulus nd rgument of complex numers nd their properties. - Sum, difference, product nd quotient of two complex numers dditive nd multiplictive inverse of complex numer. 14

5 - Locus questions on complex numers. - Tringle inequlity. - Squre root of complex numer. - Cue roots of unity nd their properties. (iii) Qudrtic Equtions Sttement of Fundmentl Theorem of Alger, solution of qudrtic equtions (with rel coefficients). Use of the formul: ± x = 4c In solving qudrtic equtions. Equtions reducile to qudrtic form. Nture of roots Product nd sum of roots. Roots re rtionl, irrtionl, equl, reciprocl, one squre of the other. Complex roots. Frming qudrtic equtions with given roots. NOTE: Questions on equtions hving common roots re to e covered. Qudrtic Functions. Givenα, β s roots then find the eqution whose roots re of the formα 3, β 3, etc. Cse I: > 0 Cse II: < 0 Rel roots Complex roots Equl roots Rel roots Complex roots, Equl roots Where is the coefficient of x in the equtions of the form x + x + c = 0. Understnding the fct tht qudrtic expression (when plotted on grph) is prol. 15 Sign of qudrtic Sign when the roots re rel nd when they re complex. Inequlities - Liner Inequlities Algeric solutions of liner inequlities in one vrile nd their representtion on the numer line. Grphicl representtion of liner inequlities in two vriles. Grphicl method of finding solution of system of liner inequlities in two vriles. Self-explntory. - Qudrtic Inequlities Using method of intervls for solving prolems of the type: x + x A perfect squre e.g. x 6x Inequlities involving rtionl expression of type f( x) et.c to e covered gx ( ) (iv) Permuttions nd Comintions Fundmentl principle of counting. Fctoril n. (n!) Permuttions nd comintions, derivtion of formule for n P nd n C nd r r their connections, simple ppliction. Fctoril nottion n!, n! =n (n-1)! Fundmentl principle of counting. Permuttions - n P r.. - Restricted permuttion. - Certin things lwys occur together.

6 - Certin things never occur. - Formtion of numers with digits. - Word uilding - repeted letters - No letters repeted. - Permuttion of like things. - Permuttion of Repeted things. - Circulr permuttion clockwise counterclockwise Distinguishle / not distinguishle. Comintions - n C r, n C n =1, n C 0 = 1, n C r = n C n r, n C x = n C y, then x + y = n or x = y, n+1 C r = n C r-1 + n C r. - When ll things re different. - When ll things re not different. - Mixed prolems on permuttion nd comintions. (v) Binomil Theorem History, sttement nd proof of the inomil theorem for positive integrl indices. Pscl's tringle, Generl nd middle term in inomil expnsion, simple pplictions. Significnce of Pscl s tringle. Binomil theorem (proof using induction) for positive integrl powers, i.e. (x + y ) n = -1 Cx 0 + Cx 1 y Cn y. n n n n n n Questions sed on the ove. (vi) Sequence nd Series Sequence nd Series. Arithmetic Progression (A. P.). Arithmetic Men (A.M.) Geometric Progression (G.P.), generl term of G.P., sum of first n terms of G.P., infinite G.P. nd its sum, geometric men (G.M.), reltion etween A.M. nd G.M. Formule for the following 3 specil sums n, n, n. Arithmetic Progression (A.P.) - T n = + (n - 1)d n - S n = { + ( n 1) d} - Arithmetic men: = + c - Inserting two or more rithmetic mens etween ny two numers. - Three terms in A.P. : - d,, + d - Four terms in A.P.: - 3d, - d, + d, + 3d Geometric Progression (G.P.) - T n = r n-1, n r ( 1) Sn =, r 1 S ; r 1 r < 1 Geometric Men, = c - Inserting two or more Geometric Mens etween ny two numers. - Three terms re in G.P. r,, r -1 - Four terms re in GP r 3, r, r -1, r -3 Arithmetico Geometric Series Identifying series s A.G.P. (when we sustitute d = 0 in the series, we get G.P. nd when we sustitute r =1 the A.P). Specil sums n, n n Using these summtions to sum up other relted expression. 3. Coordinte Geometry (i) Stright Lines Brief recll of two dimensionl geometry from erlier clsses. Shifting of origin. Slope of line nd ngle etween two lines. Vrious forms of equtions of line: prllel to xis, pointslope form, slope- intercept form, two-point form, intercept form nd norml form. Generl eqution of line. Eqution of fmily of lines pssing through the point of intersection of two lines. Distnce of point from line. Bsic concepts of Points nd their coordintes., 3 16

7 The stright line - Slope or grdient of line. - Angle etween two lines. - Condition of perpendiculrity nd prllelism. - Vrious forms of eqution of lines. - Slope intercept form. - Two point slope form. - Intercept form. - Perpendiculr /norml form. - Generl eqution of line. - Distnce of point from line. - Distnce etween prllel lines. - Eqution of lines isecting the ngle etween two lines. - Eqution of fmily of lines - Definition of locus. - Eqution of locus. (ii) Circles Equtions of circle in: - Stndrd form. - Dimeter form. - Generl form. - Prmetric form. Given the eqution of circle, to find the centre nd the rdius. Finding the eqution of circle. - Given three non colliner points. - Given other sufficient dt for exmple centre is (h, k) nd it lies on line nd two points on the circle re given, etc. Tngents: - Condition for tngency - Eqution of tngent to circle 4. Clculus (i) Limits nd Derivtives Derivtive introduced s rte of chnge oth s tht of distnce function nd geometriclly. Intuitive ide of limit. Limits of polynomils nd rtionl functions trigonometric, exponentil nd logrithmic functions. Definition of derivtive relte it to scope of tngent of the curve, Derivtive of sum, difference, product nd quotient of functions. Derivtives of polynomil nd trigonometric functions. Limits - Notion nd mening of limits. - Fundmentl theorems on limits (sttement only). - Limits of lgeric nd trigonometric functions. - Limits involving exponentil nd logrithmic functions. NOTE: Indeterminte forms re to e introduced while clculting limits. Differentition - Mening nd geometricl interprettion of derivtive. - Derivtives of simple lgeric nd trigonometric functions nd their formule. - Differentition using first principles. - Derivtives of sum/difference. - Derivtives of product of functions. Derivtives of quotients of functions. 5. Sttistics nd Proility (i) Sttistics Mesures of dispersion: rnge, men devition, vrince nd stndrd devition of ungrouped/grouped dt. Anlysis of frequency distriutions with equl mens ut different vrinces. Men devition out men nd medin. Stndrd devition - y direct method, short cut method nd step devition method. NOTE: Men, Medin nd Mode of grouped nd ungrouped dt re required to e covered. 17

8 (ii) Proility Rndom experiments; outcomes, smple spces (set representtion). Events; occurrence of events, 'not', 'nd' nd 'or' events, exhustive events, mutully exclusive events, Axiomtic (set theoretic) proility, connections with other theories studied in erlier clsses. Proility of n event, proility of 'not', 'nd' nd 'or' events. Rndom experiments nd their outcomes. Events: sure events, impossile events, mutully exclusive nd exhustive events. - Definition of proility of n event - Lws of proility ddition theorem. 6. Conic Section SECTION B Sections of cone, ellipse, prol, hyperol, point, stright line nd pir of intersecting lines s degenerted cse of conic section. Stndrd equtions nd simple properties of prol, ellipse nd hyperol. Conics s section of cone. - Definition of Foci, Directrix, Ltus Rectum. - PS = epl where P is point on the conics, S is the focus, PL is the perpendiculr distnce of the point from the directrix. (i) Prol e =1, y = ±4x, x = 4y, y = -4x, x = -4y, (y -β) =± 4 (x - α), (x - α) = ± 4 (y - β). - Rough sketch of the ove. - The ltus rectum; qudrnts they lie in; coordintes of focus nd vertex; nd equtions of directrix nd the xis. - Finding eqution of Prol when Foci nd directrix re given, etc. - Appliction questions sed on the ove. 18 (ii) Ellipse x y - + = 1, e< 1, = (1 e ) - ( x α) ( y β) + = 1 - Cses when > nd <. - Rough sketch of the ove. - Mjor xis, minor xis; ltus rectum; coordintes of vertices, focus nd centre; nd equtions of directrices nd the xes. - Finding eqution of ellipse when focus nd directrix re given. - Simple nd direct questions sed on the ove. - Focl property i.e. SP + SP =. (iii) Hyperol - x y = 1, e> 1, = ( e 1) ( x α) ( y β) - = 1 - Cses when coefficient y is negtive nd coefficient of x is negtive. - Rough sketch of the ove. - Focl property i.e. SP - S P =. - Trnsverse nd Conjugte xes; Ltus rectum; coordintes of vertices, foci nd centre; nd equtions of the directrices nd the xes. Generl second degree eqution x + hxy + y + gx + fy + c = 0 - Cse 1: pir of stright line if c+fgh-f -g -ch =0, - Cse : c+fgh-f -g -ch 0, then represents prol if h =, ellipse if h <, nd hyperol if h >. Condition tht y = mx + c is tngent to the conics, generl eqution of tngents, point of contct nd locus prolems.

9 7. Introduction to three-dimensionl Geometry Coordinte xes nd coordinte plnes in three dimensions. Coordintes of point. Distnce etween two points nd section formul. - As n extension of -D - Distnce formul. - Section nd midpoint form 8. Mthemticl Resoning Mthemticlly cceptle sttements. Connecting words/ phrses - consolidting the understnding of "if nd only if (necessry nd sufficient) condition", "implies", "nd/or", "implied y", "nd", "or", "there exists" nd their use through vriety of exmples relted to the Mthemtics nd rel life. Vlidting the sttements involving the connecting words, Difference etween contrdiction, converse nd contrpositive. Self-explntory. SECTION C 9. Sttistics - Comined men nd stndrd devition. - The Medin, Qurtiles, Deciles, Percentiles nd Mode of grouped nd ungrouped dt. 10. Correltion Anlysis Definition nd mening of covrince. Coefficient of Correltion y Krl Person. If x - x, y - y re smll non - frctionl numers, we use r = ( x - x)( y - y) ( x - x) ( y - y) If x nd y re smll numers, we use 1 xy x y r = N 1 1 x x y y N N ( ) ( ) Otherwise, we use ssumed mens A nd B, where u = x-a, v = y-b r = 1 uv - ( u)( v) N 1 1 u ( u) v ( v) N N Rnk correltion y Spermn s (Correction included). 11. Index Numers nd Moving Averges Index Numers - Price index or price reltive. - Simple ggregte method. - Weighted ggregte method. - Simple verge of price reltives. - Weighted verge of price reltives (cost of living index, consumer price index). Moving Averges - Mening nd purpose of the moving verges. - Clcultion of moving verges with the given periodicity nd plotting them on grph. - If the period is even, then the centered moving verge is to e found out nd plotted. 19

10 The syllus is divided into three sections A, B nd C. CLASS XII Section A is compulsory for ll cndidtes. Cndidtes will hve choice of ttempting questions from EITHER Section B OR Section C. There will e one pper of three hours durtion of 100 mrks. Section A (80 Mrks): Cndidtes will e required to ttempt ll questions. Internl choice will e provided in three questions of four mrks ech nd two questions of six mrks ech. Section B/ Section C (0 Mrks): Cndidtes will e required to ttempt ll questions EITHER from Section B OR Section C. Internl choice will e provided in two questions of four mrks ech. S.No. UNIT TOTAL WEIGHTAGE SECTION A: 80 MARKS 1. Reltions nd Functions 1 Mrks. Alger 14 Mrks 3. Clculus 40 Mrks 4. Proility 14 Mrks SECTION B: 0 MARKS 5. Vectors 6 Mrks 6. Three - Dimensionl Geometry 8/10 Mrks 7. Applictions of Integrls 6/4 Mrks OR SECTION C: 0 MARKS 8. Appliction of Clculus 8 Mrks 9. Liner Regression 6 Mrks 10. Liner Progrmming 6 Mrks Totl 100 Mrks 130

11 SECTION A 1. Reltions nd Functions (i) Types of reltions: reflexive, symmetric, trnsitive nd equivlence reltions. One to one nd onto functions, composite functions, inverse of function. Binry opertions. Reltions s: - Reltion on set A - Identity reltion, empty reltion, universl reltion. - Types of Reltions: reflexive, symmetric, trnsitive nd equivlence reltion. Binry Opertion: ll xioms nd properties Functions: - As specil reltions, concept of writing y is function of x s y = f(x). - Types: one to one, mny to one, into, onto. - Rel Vlued function. - Domin nd rnge of function. - Conditions of invertiility. - Composite functions nd invertile functions (lgeric functions only). (ii) Inverse Trigonometric Functions Definition, domin, rnge, principl vlue rnch. Grphs of inverse trigonometric functions. Elementry properties of inverse trigonometric functions. - Principl vlues. - sin -1 x, cos -1 x, tn -1 x etc. nd their grphs. - sin -1 1 x = cos 1 x 1 = tn x. 1 x - sin -1 1 x= cosec 1 ; sin -1 x+cos -1 π x= nd x similr reltions for cot -1 x, tn -1 x, etc.. Alger ( 1 1 ) ( 1 1 ) sin x ± sin y = sin x y ± y x cos x ± cos y = cos xy y x x+ y similrly t n x + tn y = tn, xy < 1 1 xy x y t n x tn y = tn, xy > 1 1+ xy - Formule for sin -1 x, cos -1 x, tn -1 x, 3tn -1 x etc. nd ppliction of these formule. Mtrices nd Determinnts (i) Mtrices Concept, nottion, order, equlity, types of mtrices, zero nd identity mtrix, trnspose of mtrix, symmetric nd skew symmetric mtrices. Opertion on mtrices: Addition nd multipliction nd multipliction with sclr. Simple properties of ddition, multipliction nd sclr multipliction. Noncommuttivity of multipliction of mtrices nd existence of non-zero mtrices whose product is the zero mtrix (restrict to squre mtrices of order upto 3). Concept of elementry row nd column opertions. Invertile mtrices nd proof of the uniqueness of inverse, if it exists (here ll mtrices will hve rel entries). (ii) Determinnts Determinnt of squre mtrix (up to 3 x 3 mtrices), properties of determinnts, minors, co-fctors nd pplictions of determinnts in finding the re of tringle. Adjoint nd inverse of squre mtrix. Consistency, inconsistency nd numer of solutions of system of liner equtions y exmples, solving system of liner equtions in two or three vriles (hving unique solution) using inverse of mtrix. 131

12 - Types of mtrices (m n; m, n 3), order; Identity mtrix, Digonl mtrix. - Symmetric, Skew symmetric. - Opertion ddition, sutrction, multipliction of mtrix with sclr, multipliction of two mtrices (the comptiility) E.g. 0 = AB( sy) ut BA is 1 1 not possile. - Singulr nd non-singulr mtrices. - Existence of two non-zero mtrices whose product is zero mtrix. - Inverse (, 3 3) 1 A = Mrtin s Rule (i.e. using mtrices) 1x + 1y + c 1z = d 1 x + y + c z = d 3x + 3y + c 3z = d 3 A = 1 3 AX = B X = A AdjA A c1 d1 c = B d c d 3 3 B x X = y z Prolems sed on ove. NOTE 1: The conditions for consistency of equtions in two nd three vriles, using mtrices, re to e covered. NOTE : Inverse of mtrix y elementry opertions to e covered. Determinnts - Order. - Minors. - Cofctors. - Expnsion. - Applictions of determinnts in finding the re of tringle nd collinerity. - Properties of determinnts. Prolems sed on properties of determinnts. 3. Clculus (i) Continuity, Differentiility nd Differentition. Continuity nd differentiility, derivtive of composite functions, chin rule, derivtives of inverse trigonometric functions, derivtive of implicit functions. Concept of exponentil nd logrithmic functions. 13 Derivtives of logrithmic nd exponentil functions. Logrithmic differentition, derivtive of functions expressed in prmetric forms. Second order derivtives. Rolle's nd Lgrnge's Men Vlue Theorems (without proof) nd their geometric interprettion. Continuity - Continuity of function t point x =. - Continuity of function in n intervl. - Alger of continues function. - Removle discontinuity. Differentition - Concept of continuity nd differentiility of x, [x], etc. - Derivtives of trigonometric functions. - Derivtives of exponentil functions. - Derivtives of logrithmic functions. - Derivtives of inverse trigonometric functions - differentition y mens of sustitution. - Derivtives of implicit functions nd chin rul - e for composite functions. - Derivtives of Prmetric functions. - Differentition of function with respect to nother function e.g. differentition of sinx 3 with respect to x 3. - Logrithmic Differentition - x x Finding dy/dx when y = x. - Successive differentition up to nd order.

13 NOTE 1: Derivtives of composite functions using chin rule. NOTE : Derivtives of determinnts to e covered. L' Hospitl's theorem. - 0 form, form, 0 0 form, form 0 etc. Rolle's Men Vlue Theorem - its geometricl interprettion. Lgrnge's Men Vlue Theorem - its geometricl interprettion (ii) Applictions of Derivtives Applictions of derivtives: rte of chnge of odies, incresing/decresing functions, tngents nd normls, use of derivtives in pproximtion, mxim nd minim (first derivtive test motivted geometriclly nd second derivtive test given s provle tool). Simple prolems (tht illustrte sic principles nd understnding of the suject s well s rel-life situtions). Eqution of Tngent nd Norml Approximtion. Rte mesure. Incresing nd decresing functions. Mxim nd minim. - Sttionry/turning points. - Asolute mxim/minim - locl mxim/minim - First derivtives test nd second derivtives test - Point of inflexion. - Appliction prolems sed on mxim nd minim. (iii) Integrls Integrtion s inverse process of differentition. Integrtion of vriety of functions y sustitution, y prtil frctions nd y prts, Evlution of simple integrls of the following types nd prolems sed on them. Definite integrls s limit of sum, Fundmentl Theorem of Clculus (without proof). Bsic properties of definite integrls nd evlution of definite integrls. Indefinite integrl - Integrtion s the inverse of differentition. - Anti-derivtives of polynomils nd functions (x +) n, sinx, cosx, sec x, cosec x etc. - Integrls of the type sin x, sin 3 x, sin 4 x, cos x, cos 3 x, cos 4 x. - Integrtion of 1/x, e x. - Integrtion y sustitution. - Integrls of the type f ' (x)[f (x)] n, f ( x). f( x) - Integrtion of tnx, cotx, secx, cosecx. - Integrtion y prts. - Integrtion using prtil frctions. f ( x) Expressions of the form when g( x) degree of f(x) < degree of g(x) E.g. x+ A B = + ( x 3)( x+ 1) x 3 x+ 1 x+ A B C = + + ( x )( x 1) x 1 x 1 x ( ) 133

14 ( x x + 1 = + 3)( x 1) Ax + B C + x + 3 x 1 When degree of f (x) degree of g(x), e.g. x + 1 3x + 1 = 1 x + 3x + x + 3x + Integrls of the type: dx dx px + q px + q,,, dx dx x ± x x c x ± + + x + x + c nd ± x dx, x dx, x + x + c dx, ( px + q) x + x + c dx, integrtions reducile to the ove forms. dx, cos x+ sin x dx dx dx,, + cos x + sin x cos x+ sin x+ c ( cos x + sin x) dx, ccos x+ dsin x dx cos x+ sin x+ c 1± dx, + x4 1 x dx, x tn xdx, cot xdx etc. Definite Integrl - Definite integrl s limit of the sum. - Fundmentl theorem of clculus (without proof) - Properties of definite integrls. - Prolems sed on the following properties of definite integrls re to e covered. ( x) dx = f f ( t) dt ( x) dx = f f ( x) dx c ( x) dx = f ( x) dx + f f ( x) dx where < c < f ( x) dx = f ( + x) dx 0 0 f ( x) dx = f ( x) dx f( x) dx, if f( x) = f( x) f ( x) dx = 0 0 0, f( x) = f( x) f ( x) dx,if f is n even function f ( x) dx = 0 0,if f is n odd function (iv) Differentil Equtions Definition, order nd degree, generl nd prticulr solutions of differentil eqution. Formtion of differentil eqution whose generl solution is given. Solution of differentil equtions y method of seprtion of vriles solutions of homogeneous differentil equtions of first order nd first degree. Solutions of liner differentil eqution of the type: dy +py= q, dx where p nd q re functions of x or constnts. dx + px = q, where p nd q re dy functions of y or constnts. - Differentil equtions, order nd degree. - Formtion of differentil eqution y eliminting ritrry constnt(s). - Solution of differentil equtions. - Vrile seprle. - Homogeneous equtions. dy - Liner form + Py = Q where P nd Q re functions dxof x only. Similrly for dx/dy. c 134

15 - Solve prolems of ppliction on growth nd decy. - Solve prolems on velocity, ccelertion, distnce nd time. - Solve popultion sed prolems on ppliction of differentil equtions. - Solve prolems of ppliction on coordinte geometry. NOTE 1: Equtions reducile to vrile seprle type re included. NOTE : The second order differentil equtions re excluded. 4. Proility Conditionl proility, multipliction theorem on proility, independent events, totl proility, Byes theorem, Rndom vrile nd its proility distriution, men nd vrince of rndom vrile. Repeted independent (Bernoulli) trils nd Binomil distriution. - Independent nd dependent events conditionl events. - Lws of Proility, ddition theorem, multipliction theorem, conditionl proility. - Theorem of Totl Proility. - Bye s theorem. - Theoreticl proility distriution, proility distriution function; men nd vrince of rndom vrile, Repeted independent (Bernoulli trils), inomil distriution its men nd vrince. SECTION B 5. Vectors Vectors nd sclrs, mgnitude nd direction of vector. Direction cosines nd direction rtios of vector. Types of vectors (equl, unit, zero, prllel nd colliner vectors), position vector of point, negtive of vector, components of vector, ddition of vectors, multipliction of vector y sclr, position vector of point dividing line segment in given rtio. Definition, Geometricl Interprettion, properties nd ppliction of sclr (dot) product of vectors, vector (cross) product of vectors, sclr triple product of vectors As directed line segments. - Mgnitude nd direction of vector. - Types: equl vectors, unit vectors, zero vector. - Position vector. - Components of vector. - Vectors in two nd three dimensions. - iˆ, ˆ, j kˆ s unit vectors long the x, y nd the z xes; expressing vector in terms of the unit vectors. - Opertions: Sum nd Difference of vectors; sclr multipliction of vector. - Section formul. - Tringle inequlities. - Sclr (dot) product of vectors nd its geometricl significnce. - Cross product - its properties - re of tringle, re of prllelogrm, colliner vectors. - Sclr triple product - volume of prllelepiped, co-plnrity. NOTE: Proofs of geometricl theorems y using Vector lger re excluded. 6. Three - dimensionl Geometry Direction cosines nd direction rtios of line joining two points. Crtesin eqution nd vector eqution of line, coplnr nd skew lines, shortest distnce etween two lines. Crtesin nd vector eqution of plne. Angle etween (i) two lines, (ii) two plnes, (iii) line nd plne. Distnce of point from plne. - Eqution of x-xis, y-xis, z xis nd lines prllel to them. - Eqution of xy - plne, yz plne, zx plne. - Direction cosines, direction rtios. - Angle etween two lines in terms of direction cosines /direction rtios. - Condition for lines to e perpendiculr/ prllel. Lines - Crtesin nd vector equtions of line through one nd two points. - Coplnr nd skew lines.

16 - Conditions for intersection of two lines. - Distnce of point from line. - Shortest distnce etween two lines. NOTE: Symmetric nd non-symmetric forms of lines re required to e covered. Plnes - Crtesin nd vector eqution of plne. - Direction rtios of the norml to the plne. - One point form. - Norml form. - Intercept form. - Distnce of point from plne. - Intersection of the line nd plne. - Angle etween two plnes, line nd plne. - Eqution of intersection of plne two through plnes the i.e. P 1 + kp = Appliction of Integrls Appliction in finding the re ounded y simple curves nd coordinte xes. Are enclosed etween two curves. - Appliction of definite integrls - re ounded y curves, lines nd coordinte xes is required to e covered. - Simple curves: lines, circles/ prols/ ellipses, polynomil functions, modulus function, trigonometric function, exponentil functions, logrithmic functions 8. Appliction of Clculus SECTION C Appliction of Clculus in Commerce nd Economics in the following: - Cost function, - verge cost, - mrginl cost nd its interprettion - demnd function, - revenue function, - mrginl revenue function nd its interprettion, - Profit function nd rekeven point Rough sketching of the following curves: AR, MR, R, C, AC, MC nd their mthemticl interprettion using the concept of mxim & minim nd incresing- decresing functions. Self-explntory NOTE: Appliction involving differentition, integrtion, incresing nd decresing function nd mxim nd minim to e covered. 9. Liner Regression - Lines of regression of x on y nd y on x. - Sctter digrms - The method of lest squres. - Lines of est fit. - Regression coefficient of x on y nd y on x. - xy yx = r, 0 xy yx 1 - Identifiction of regression equtions - Angle etween regression line nd properties of regression lines. - Estimtion of the vlue of one vrile using the vlue of other vrile from pproprite line of regression. Self-explntory 10. Liner Progrmming Introduction, relted terminology such s constrints, ojective function, optimiztion, different types of liner progrmming (L.P.) prolems, mthemticl formultion of L.P. prolems, grphicl method of solution for prolems in two vriles, fesile nd infesile regions(ounded nd unounded), fesile nd infesile solutions, optiml fesile solutions (up to three non-trivil constrints). Introduction, definition of relted terminology such s constrints, ojective function, optimiztion, dvntges of liner progrmming; limittions of liner progrmming; ppliction res of liner progrmming; different types of liner progrmming (L.P.) prolems, mthemticl formultion of L.P prolems, grphicl method of solution for prolems in two vriles, fesile nd infesile regions, fesile nd infesile solutions, optimum fesile solution.