CBSE-XII-2015 EXAMINATION. Section A. 1. Find the sum of the order and the degree of the following differential equation : = 0
|
|
- Ezra Short
- 6 years ago
- Views:
Transcription
1 CBSE-XII- EXMINTION MTHEMTICS Pper & Solution Time : Hrs. M. Mrks : Generl Instruction : (i) ll questions re compulsory. There re questions in ll. (ii) This question pper hs three sections : Section, Section B, Section C. Section. Find the sum of the order nd the degree of the following differentil eqution : dy y d y dy d y y order of eqution degree of eqution hence sum of order nd degree. Find the solution of the following differentil eqution : ( y ) y ( ) dy By ( y ) y ( ) dy y dy y Integrting, we get y C y y cosθ sin θ. If, then for ny nturl number n, find the vlue of Det ( n ). sin θ cosθ cosθ sin θ Q sin θ cosθ Now. cos θ sin θ cos θ sin θ sin θ cos θ sin θ cos θ cos θ sin θ sin θcosθ cosθ sin sin θ cos θ θ Similrly we cn sy tht n cosnθ sin nθ sin nθ cosnθ sin θcosθ cos θsin θ dy Code : //, CP Tower, Rod No., IPI, Kot (Rj.), Ph: - Website : Emil: info@creerpointgroup.com /
2 Now det ( n cosnθ ) sin nθ cos nθ sin nθ CBSE-XII- EXMINTION sin nθ cosnθ. Find vector of mgnitude which is perpendiculr to both of the vectors î ĵ kˆ nd b î ĵ kˆ. Let c be ny vector perpendiculr to nd b then c {( î ĵ kˆ ) ( î ĵ kˆ )} i j k c i j k c ( b) Now vector of mgnitude in the direction of c is given by mgnitude. ĉ (i j k) 9 i j k. Find the ngle between the lines y z nd y z. The eqution of stright lines re y z y z nd The given lines re prllel to the vectors i j k nd bi j k (i j k) nd b (i j k) Now ngle between these two vectors is given by cos θ. b b (i j k).(i j k) cos θ 9 9 cos θ 9 Hence both lines re perpendiculr.. In tringle OC, if B is the mid-point of side C nd O, OB b, then wht is OC? If O is origin then position vector of point, B nd C re, b nd w.r.t. origin then O, OB bnd OC c c nd if c divide B in the rtio :, CP Tower, Rod No., IPI, Kot (Rj.), Ph: - Website : Emil: info@creerpointgroup.com /
3 CBSE-XII- EXMINTION then O OB OC b OC Section B cos. Evlute : sin cos I sin sin I sin I I. ( sin I. π Let tn t sec I π. sec ) sin tn t π I. (tn t ) π π I I π π I π sin cos. Evlute : sin sin cos I (sin cos ) I sin cos () (sin cos ) Let sin cos t (cos sin ) t, t, CP Tower, Rod No., IPI, Kot (Rj.), Ph: - Website : Emil: info@creerpointgroup.com /
4 CBSE-XII- EXMINTION t π, t I t I t l n t we know tht I l n ln I ln I ln ln c 9. Let î ĵ kˆ, b î ĵ kˆ nd c î ĵ kˆ. Find vector d which is perpendiculr to both nd b nd c. d. Given tht d is perpendiculr to both nd b then d is given by d λ( b ) i j d λ k () d λ{i( ) j( ) k( )} d λ{i j k} () gin given tht c. d λ( î ĵ kˆ ).( î ĵ kˆ ) λ( ) 9λ λ Hence from eqution () the resultnt vector dis d (i j k). Find the shortest distnce between the following lines : r ( î ĵ kˆ ) λ( î ĵ kˆ ) r ( î ĵ kˆ ) µ( î ĵ kˆ ) Find the eqution of the plne pssing through the line of intersection of the plnes y z nd y z y z 9 nd is prllel to the line. From the given equtions, we observe tht first line psses through î ĵ kˆ nd prllel to α î ĵ kˆ nd second line psses through b î ĵ kˆ, CP Tower, Rod No., IPI, Kot (Rj.), Ph: - Website : Emil: info@creerpointgroup.com /
5 CBSE-XII- EXMINTION nd prllel to β î ĵ kˆ Becuse both lines re prllel then shortest distnce between both lines re given by (b ) α SD α () where b î ĵ kˆ î ĵ kˆ î ĵ kˆ () (i j k) (i j k) SD 9 i k 9 9 Eqution of plne through the intersection of the given plnes y z nd y z 9 is y z λ( y z 9)...(i) ( λ) ( λ)y ( λ)z ( 9λ) y s per this plne is prllel to the stright line z y z Then norml of plne is perpendiculr to stright line b b c c ( λ) ( λ) ( λ) λ λ λ λ λ λ Now from eqution () the eqution of plne is y z ( y z 9) Multiply by we get y z y z 9 9y z. mn tkes step forwrd with probbility. nd bckwrd with probbility., Find the probbility tht t the end of steps, he is one step wy from he strting point. Suppose girl throws die. If she gets or, she tosses coin three times nd notes the number of 'tils'. If she gets,, or, she tosses coin once nd notes whether 'hed' or 'til' is obtined. If she obtined ectly one 'til', wht is the probbility tht she threw,, or with the die? Prob. of step forwrd p. Prob. of step bckwrd q. So by binomil distribution (p q) C.p C.p.q C.p.q C.p.q C.p.q C.q Prob. for mn so tht he is one-step wy from strting point will be C.p.q C.p.q.(.).(.).(.).(.).(.).(., CP Tower, Rod No., IPI, Kot (Rj.), Ph: - Website : Emil: info@creerpointgroup.com /
6 / MTHEMTICS CBSE-XII- EXMINTION, CP Tower, Rod No., IPI, Kot (Rj.), Ph: - Website : Emil: info@creerpointgroup.com.(.).(.) (.). (.). E E E is the event tht on throwing dice or comes So ) (E P E is the event tht on throwing dice,,, comes So ) (E P If is event tht on throwing coin only one til. Then for Prob. of ectly one til if he got,,, or is be P(E /) ) E ).P( / P(E ) E ).P( / P(E ) E ).P( / P(E P(/E ), P(/E ) P(/E ).... Using elementry row opertions (trnsformtions), find the inverse of the following mtri : If, B, C, then clculte C, BC nd ( B) C. lso verify tht ( B) C C BC. Given tht we write I So operte R R we get
7 / MTHEMTICS CBSE-XII- EXMINTION, CP Tower, Rod No., IPI, Kot (Rj.), Ph: - Website : Emil: info@creerpointgroup.com operte R R R, we get 9 operte R R R nd R R R we get operte R R R nd R R R 9 Hence 9, B, C Now C 9 () BC () Now (B) C
8 Now dd () nd () 9 C BC ( B) C Now from () & () ( B) C C BC Hence proved CBSE-XII- EXMINTION () (). Discuss the continuity n differentibility of the function f() in the intervl (, ). f() Defined this function we get < f() > Now test continuity t nd t f() LHL t f( ) RHL t f( ) lim ( h) h lim h Q f() f( ) f( ) So f() continuous t Now t f() LHL t lim f( h) RHL t h h lim h lim f( h) lim ( h) h Q f() f( ) f( ) hence f() continuous t So we cn sy tht f() continuous in (, ) < Differentibility Q f() > f() & f() t f ( h) f () L f () lim h h ( h) lim h h f ( h) f () Rf () lim h h lim h h Lf () Rf (), CP Tower, Rod No., IPI, Kot (Rj.), Ph: - Website : Emil: info@creerpointgroup.com /
9 CBSE-XII- EXMINTION Hence f() not differentible t So we cn sy tht f() not differentible in (, ) d y. If (cos t t sin t) nd y (sin t t cos t), then find. Given tht (cos t t sin t) Diff. w.r. to t ( sin t t cos t sin t) t cost...() gin y (sin t tcos t) D.w.r. to t dy [cost cost t sin t] dy t sin t...() dy dy / then Now from () & () / dy t sin t t cost dy tn t gin diff. w.r. to d y sec t. Put from () y y d d cos t. sec t t t cost. If ( b) e y/, then show tht d y dy y Given tht ( b)e y/ e y/ () b Diff. w.r. to dy e y/ y b ( b) From (), CP Tower, Rod No., IPI, Kot (Rj.), Ph: - Website : Emil: info@creerpointgroup.com 9 /
10 CBSE-XII- EXMINTION dy b y b ( b) dy b y () b gin differentite w.r. to d y dy dy ( b).b b ( b) d y b d y ( b) dy y d y dy y Hence proved Now from equ. () sin cos. Evlute : ( sin ) Evlute : ( )( ) sin cos I ( sin ) sin cos I sin sin Let t cos sin I t I ln t c sin I ln c I ln c sin I ( )( ) I ( )( ), CP Tower, Rod No., IPI, Kot (Rj.), Ph: - Website : Emil: info@creerpointgroup.com /
11 CBSE-XII- EXMINTION ( )( ) I ( )( ) ( )( ) I ( )( ) I ( )( ) I I I () where I Let t I t I ln t C I ln ( ) C () Now I ( )( ) () Let ( )( ) B C () ( ) B( ) C( ) ( )( ) Compre numerte ( ) B( ) C( ) t t C C C t B C B B From (). ( )( ) I ( )( ) I ln( ) ln( ) tn () Now from equ (), () nd () I ln( ) ln ln( ) I ln( ) ln tn C. There re fmilies nd B. There re men, women nd children in fmily, nd men, women nd children in fmily B. The recommended dily mount of clories is for men, 9 for women, for children nd grms of proteins for men, grms for women nd grms for children. Represent the bove informtion using mtrices. Using mtri multipliction, clculte the totl requirement, CP Tower, Rod No., IPI, Kot (Rj.), Ph: - Website : Emil: info@creerpointgroup.com /
12 CBSE-XII- EXMINTION of clories nd proteins for ech of the fmilies. Wht wreness cn you crete mong people bout the blnced diet from this question? The members of the two fmilies cn be represented by the mtri m w c F B nd the recommended dily llownce of clories nd proteins for ech member cn be represented by clories proteins m R w 9 c The totl requirement of clories nd proteins for ech of the two fmilies is given by the mtri multipliction FR B B Hence fmily required clories nd gm proteins nd fmily B requires clories nd gm proteins.. Evlute : tn tn π π Q tn tn Q tn tn π So tn tn tn tn tn tn tn tn tn tn tn tn tn tn tn tn tn, CP Tower, Rod No., IPI, Kot (Rj.), Ph: - Website : Emil: info@creerpointgroup.com /
13 CBSE-XII- EXMINTION 9. Using properties of determinnts, prove tht b c Let b ( b)(b c)(c )( b c). c b c b b c c b c b c b c operte R R R R R R we get b c b c b c (b )(b (c )(c b) c) (b ) (c ) b c b c operte R R R (b ) (c ) b c c b c c (b ) (c ) (b c)( b c) c c (b ) (c ) (b c) ( b c) Hence proved Section C. n urn contins red nd blck blls. Two blls re rndomly drwn, without replcement. Let represent the number of blck blls drwn. Wht re the possible vlues of X? Is X rndom vrible? If yes, find the men nd vrince of X., CP Tower, Rod No., IPI, Kot (Rj.), Ph: - Website : Emil: info@creerpointgroup.com /
14 Red Blck CBSE-XII- EXMINTION (i),, (ii) yes, is rndom vrible it vries from to urn. No.of bll P() blck R () P() P(R ) P R B (b) P() P(R ) P R P(B ) P R B B (c) P() P(B ) P B (iii) Men ( ) P P P (iv) Vrince P ()( ) P(). P(). P() 9 () () () () (). Solve the following liner progrmming problem grphiclly. Minimise z y subject to the constrints y y y, y The objective function is z y The constrints re y () y () y (), y (), CP Tower, Rod No., IPI, Kot (Rj.), Ph: - Website : Emil: info@creerpointgroup.com /
15 CBSE-XII- EXMINTION (, ) B C(, ) D(, ) E(, ) The shded unbounded region is the fesible region of the given LPP. The vertices of the unbounded fesible. Region re B(, ) C(, ) nd D(, ) nd E(, ) t B(, ) z t C(, ) z t D(, ) z t E(, ) z The minimum of,,, is. We drw the grph of the inequlity y < The corresponding eqution is y < nd this psses through L, nd M, The open hlf plne of y < is shown in the figure. This hlf plne hs no point in common with the fesible region Minimum vlue of z nd occurs when nd y. Determine whether the reltion R defined on the set R of ll rel numbers s R {(, b) :, b R nd b S, where S is the set of ll irrtionl numbers}, is refleive, symmetric nd trnsitive. Let R R nd * be the binry opertion on defined by (, b) * (c, d) ( c, b d). Prove tht * is commuttive nd ssocitive. Find the identity element for * on. lso write the inverse element of the element (, ) in. Let R be the given reltion nd defined s R {(, b) :, b, R nd b s} where S is set of irrtionl number then (i) Refleive : Let R then R S So R therefore R is refleive (ii) symmetric Let, b R nd Let, b then Rb S, CP Tower, Rod No., IPI, Kot (Rj.), Ph: - Website : Emil: info@creerpointgroup.com /
16 CBSE-XII- EXMINTION So Rb Q S So br/ Q Rb br/ So R is not symmetric (iii) Trnsitive : Let, b nd c Now Rb S brc S nd Rc S R/ c Q Rb & brc R/ c Therefore reltion R is not trnsitive hence reltion nd is refleive but not symmetric nd not trnsitive. Let (, b), (c, d), (e, f) be ny three elements of (i) Commuttive : Given tht (, b) * (c, d) ( c, b d) (c, d b) (c, d) * (, b) Q (, b) * (c, d) (c, d) * (, b) Therefore * is commuttive (ii) ssocitive : By the given definition (c, d) * (e, f) (c e, d f) then (, b) * {(c, d) * (e, f)} (, b) * (c e, d f) (, b) * {(c, d) * (e, f)} ( c e, b d f) () Now (, b) * (c, d) ( c, b d) then {(, b) * (c, d)} * (e, f) ( c, b d) * (e, f) ( c e, b d f) () hence by () nd () (, b) * {(c, d) * (e, f)} {(, b) * (c, d)} * (e, f) So * is ssocitive (iii) Let (, y) be the identity element in (, b) * (, y) (, b) by definition (, b y) (, b) nd b y b nd y hence (, ) be the identity element of * Now let (p, q) be the inverse element of (, ) in of opertive * then by definition of inverse we sy tht (, ) * (p, q) (, ) ( p, q) (, ) p nd q p nd q so inverse element is (, ). Tngent to the circle y t ny point on it in the first qudrnt mkes intercepts O nd OB on nd y es respectively, O being the centre of the circle. Find the minimum vlue of (O OB). dy. Show tht the differentil eqution ( y) y is homogeneous nd solve it lso., CP Tower, Rod No., IPI, Kot (Rj.), Ph: - Website : Emil: info@creerpointgroup.com /
17 CBSE-XII- EXMINTION Find the differentil eqution of the fmily of curves ( h) (y k) r, where h nd k re rbitrry constnts.. Find the eqution of plne pssing through the point P(,, 9) nd prllel to the plne determined by the points (,, ), B(,, ) nd C(,, ). lso find the distnce of this plne from the point. P(,, 9) (, y, z ) ( ) C (,, ) (,, ) B (,, ) Eqution of the plne which is pssing through the point P is Where, n B C (It's norml of the plne) Q B OB O b î ĵ kˆ C OC O c î kˆ î ĵ kˆ n B C î( ) ĵ( ) kˆ ( ) î ĵ kˆ r. n. n Eqution of plne is r.( î ĵ kˆ ) ( î ĵ 9 kˆ ).( î ĵ kˆ ) r.( î ĵ kˆ ) y z y z () Distnce between the point nd the plne () is () ( ) () ( ). If the re bounded by the prbol y nd the line y m is sq. units, then using integrtion, find the vlue of m., CP Tower, Rod No., IPI, Kot (Rj.), Ph: - Website : Emil: info@creerpointgroup.com /
1. If * is the operation defined by a*b = a b for a, b N, then (2 * 3) * 2 is equal to (A) 81 (B) 512 (C) 216 (D) 64 (E) 243 ANSWER : D
. If * is the opertion defined by *b = b for, b N, then ( * ) * is equl to (A) 8 (B) 5 (C) 6 (D) 64 (E) 4. The domin of the function ( 9)/( ),if f( ) = is 6, if = (A) (0, ) (B) (-, ) (C) (-, ) (D) (, )
More informationMATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC
FIITJEE Solutions to AIEEE MATHEMATICS PART A. ABC is tringle, right ngled t A. The resultnt of the forces cting long AB, AC with mgnitudes AB nd respectively is the force long AD, where D is the AC foot
More informationJEE(MAIN) 2015 TEST PAPER WITH SOLUTION (HELD ON SATURDAY 04 th APRIL, 2015) PART B MATHEMATICS
JEE(MAIN) 05 TEST PAPER WITH SOLUTION (HELD ON SATURDAY 0 th APRIL, 05) PART B MATHEMATICS CODE-D. Let, b nd c be three non-zero vectors such tht no two of them re colliner nd, b c b c. If is the ngle
More informationLinear Inequalities: Each of the following carries five marks each: 1. Solve the system of equations graphically.
Liner Inequlities: Ech of the following crries five mrks ech:. Solve the system of equtions grphiclly. x + 2y 8, 2x + y 8, x 0, y 0 Solution: Considerx + 2y 8.. () Drw the grph for x + 2y = 8 by line.it
More informationPrerna Tower, Road No 2, Contractors Area, Bistupur, Jamshedpur , Tel (0657) ,
R rern Tower, Rod No, Contrctors Are, Bistupur, Jmshedpur 800, Tel 065789, www.prernclsses.com IIT JEE 0 Mthemtics per I ART III SECTION I Single Correct Answer Type This section contins 0 multiple choice
More informationBoard Answer Paper: October 2014
Trget Pulictions Pvt. Ltd. Bord Answer Pper: Octoer 4 Mthemtics nd Sttistics SECTION I Q.. (A) Select nd write the correct nswer from the given lterntives in ech of the following su-questions: i. (D) ii..p
More informationA LEVEL TOPIC REVIEW. factor and remainder theorems
A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of + 7+. ( mrks) +. ( mrks). Use lgebric division
More informationSCIENCE ENTRANCE ACADEMY PREPARATORY EXAMINATION-3 (II P.U.C) SCHEME 0F EVALUATION Marks:150 Date: duration:4hours MATHEMATICS-35
JNANASUDHA SCIENCE ENTRANCE ACADEMY PREPARATORY EXAMINATION- (II P.U.C) SCHEME F EVALUATION Mrks:5 Dte:5-9-8 durtion:4hours MATHEMATICS-5 Q.NO Answer Description Mrk(s)!6 Zero mtri. + sin 4det A, 4 R 4
More informationTime : 3 hours 03 - Mathematics - March 2007 Marks : 100 Pg - 1 S E CT I O N - A
Time : hours 0 - Mthemtics - Mrch 007 Mrks : 100 Pg - 1 Instructions : 1. Answer ll questions.. Write your nswers ccording to the instructions given below with the questions.. Begin ech section on new
More informationMathematics Extension 1
04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen
More informationMathematics Extension 2
00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Extension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors m be used A tble of stndrd
More informationCET MATHEMATICS 2013
CET MATHEMATICS VERSION CODE: C. If sin is the cute ngle between the curves + nd + 8 t (, ), then () () () Ans: () Slope of first curve m ; slope of second curve m - therefore ngle is o A sin o (). The
More informationMathematics Extension 2
00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Etension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors my be used A tble of stndrd
More informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationCBSE 2013 ALL INDIA EXAMINATION [Set 1 With Solutions]
M Mrks : Q Write the principl vlue of CBSE ALL INDIA EXAMINATION [Set With Solutions] SECTION A tn ( ) cot ( ) Time Allowed : Hours Sol tn ( ) cot ( ) tn tn cot cot cot cot [Rnge of tn :,, cot : ], [ 5
More informationKEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a
KEY CONCEPTS THINGS TO REMEMBER :. The re ounded y the curve y = f(), the -is nd the ordintes t = & = is given y, A = f () d = y d.. If the re is elow the is then A is negtive. The convention is to consider
More informationFORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81
FORM FIVE ADDITIONAL MATHEMATIC NOTE CHAPTER : PROGRESSION Arithmetic Progression T n = + (n ) d S n = n [ + (n )d] = n [ + Tn ] S = T = T = S S Emple : The th term of n A.P. is 86 nd the sum of the first
More informationMathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
More information6.2 CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS
6. CONCEPTS FOR ADVANCED MATHEMATICS, C (475) AS Objectives To introduce students to number of topics which re fundmentl to the dvnced study of mthemtics. Assessment Emintion (7 mrks) 1 hour 30 minutes.
More informationLog1 Contest Round 3 Theta Individual. 4 points each 1 What is the sum of the first 5 Fibonacci numbers if the first two are 1, 1?
008 009 Log1 Contest Round Thet Individul Nme: points ech 1 Wht is the sum of the first Fiboncci numbers if the first two re 1, 1? If two crds re drwn from stndrd crd deck, wht is the probbility of drwing
More informationPART - III : MATHEMATICS
JEE(Advnced) 4 Finl Em/Pper-/Code-8 PART - III : SECTION : (One or More Thn One Options Correct Type) This section contins multiple choice questions. Ech question hs four choices (A), (B), (C) nd (D) out
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationOn the diagram below the displacement is represented by the directed line segment OA.
Vectors Sclrs nd Vectors A vector is quntity tht hs mgnitude nd direction. One exmple of vector is velocity. The velocity of n oject is determined y the mgnitude(speed) nd direction of trvel. Other exmples
More informationR(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of
Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of
More informationYear 12 Mathematics Extension 2 HSC Trial Examination 2014
Yer Mthemtics Etension HSC Tril Emintion 04 Generl Instructions. Reding time 5 minutes Working time hours Write using blck or blue pen. Blck pen is preferred. Bord-pproved clcultors my be used A tble of
More informationMATHEMATICS PAPER & SOLUTION
MATHEMATICS PAPER & SOLUTION Code: SS--Mtemtis Time : Hours M.M. 8 GENERAL INSTRUCTIONS TO THE EXAMINEES:. Cndidte must write first is / er Roll No. on te question pper ompulsorily.. All te questions re
More informationMATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Note: This question paper consists of three sections A,B and C. SECTION A
MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. TIME : 3hrs M. Mrks.75 Note: This question pper consists of three sections A,B nd C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS. X = ) Find the eqution
More informationMH CET 2018 (QUESTION WITH ANSWER)
( P C M ) MH CET 8 (QUESTION WITH ANSWER). /.sec () + log () - log (3) + log () Ans. () - log MATHS () 3 c + c C C A cos + cos c + cosc + + cosa ( + cosc ) + + cosa c c ( + + ) c / / I tn - in sec - in
More informationUS01CMTH02 UNIT Curvature
Stu mteril of BSc(Semester - I) US1CMTH (Rdius of Curvture nd Rectifiction) Prepred by Nilesh Y Ptel Hed,Mthemtics Deprtment,VPnd RPTPScience College US1CMTH UNIT- 1 Curvture Let f : I R be sufficiently
More informationπ = tanc 1 + tan x ...(i)
Solutions to RSPL/ π. Let, I log ( tn ) d Using f () d f ( ) d π π I log( tnc d m log( cot ) d...(ii) On dding (i) nd (ii), we get +,. Given f() + ), For continuit t lim " lim f () " ( ) \ Continuous t.
More informationSUBJECT: MATHEMATICS ANSWERS: COMMON ENTRANCE TEST 2012
MOCK TEST 0 SUBJECT: MATHEMATICS ANSWERS: COMMON ENTRANCE TEST 0 ANSWERS. () π π Tke cos - (- ) then sin [ cos - (- )]sin [ ]/. () Since sin - + sin - y + sin - z π, -; y -, z - 50 + y 50 + z 50 - + +
More informationSession Trimester 2. Module Code: MATH08001 MATHEMATICS FOR DESIGN
School of Science & Sport Pisley Cmpus Session 05-6 Trimester Module Code: MATH0800 MATHEMATICS FOR DESIGN Dte: 0 th My 06 Time: 0.00.00 Instructions to Cndidtes:. Answer ALL questions in Section A. Section
More informationEXPECTED ANSWERS/VALUE POINTS SECTION - A
6 QUESTION PPE ODE 65// EXPETED NSWES/VLUE POINTS SETION - -.... 6. / 5. 5 6. 5 7. 5. ( ) { } ( ) kˆ ĵ î kˆ ĵ î r 9. or ( ) kˆ ĵ î r. kˆ ĵ î m SETION - B.,, m,,, m O Mrks m 9 5 os θ 9, θ eing ngle etween
More informationIndefinite Integral. Chapter Integration - reverse of differentiation
Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the
More informationPre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs
Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
More informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
More informationChapter 2. Vectors. 2.1 Vectors Scalars and Vectors
Chpter 2 Vectors 2.1 Vectors 2.1.1 Sclrs nd Vectors A vector is quntity hving both mgnitude nd direction. Emples of vector quntities re velocity, force nd position. One cn represent vector in n-dimensionl
More informationFINALTERM EXAMINATION 9 (Session - ) Clculus & Anlyticl Geometry-I Question No: ( Mrs: ) - Plese choose one f ( x) x According to Power-Rule of differentition, if d [ x n ] n x n n x n n x + ( n ) x n+
More informationProblem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:
(x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationThe use of a so called graphing calculator or programmable calculator is not permitted. Simple scientific calculators are allowed.
ERASMUS UNIVERSITY ROTTERDAM Informtion concerning the Entrnce exmintion Mthemtics level 1 for Interntionl Bchelor in Communiction nd Medi Generl informtion Avilble time: 2 hours 30 minutes. The exmintion
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
More informationLoudoun Valley High School Calculus Summertime Fun Packet
Loudoun Vlley High School Clculus Summertime Fun Pcket We HIGHLY recommend tht you go through this pcket nd mke sure tht you know how to do everything in it. Prctice the problems tht you do NOT remember!
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)
HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time llowed Two hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions
More informationMath 426: Probability Final Exam Practice
Mth 46: Probbility Finl Exm Prctice. Computtionl problems 4. Let T k (n) denote the number of prtitions of the set {,..., n} into k nonempty subsets, where k n. Argue tht T k (n) kt k (n ) + T k (n ) by
More informationOptimization Lecture 1 Review of Differential Calculus for Functions of Single Variable.
Optimiztion Lecture 1 Review of Differentil Clculus for Functions of Single Vrible http://users.encs.concordi.c/~luisrod, Jnury 14 Outline Optimiztion Problems Rel Numbers nd Rel Vectors Open, Closed nd
More informationChapter 1: Fundamentals
Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,
More informationLesson-5 ELLIPSE 2 1 = 0
Lesson-5 ELLIPSE. An ellipse is the locus of point which moves in plne such tht its distnce from fied point (known s the focus) is e (< ), times its distnce from fied stright line (known s the directri).
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More informationQUESTION PAPER CODE 65/1/2 EXPECTED ANSWERS/VALUE POINTS SECTION - A. 1 x = 8. x = π 6 SECTION - B
QUESTION PPER CODE 65// EXPECTED NSWERS/VLUE POINTS SECTION - -.. 5. { r ( î ĵ kˆ ) } ( î ĵ kˆ ) or Mrks ( î ĵ kˆ ) r. /. 5. 6. 6 7. 9.. 8. 5 5 6 m SECTION - B. f () ( ) ( ) f () >, (, ) U (, ) m f ()
More informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time)
HIGHER SCHOOL CERTIFICATE EXAMINATION 999 MATHEMATICS UNIT (ADDITIONAL) Time llowed Three hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions re of equl vlue
More informationLecture 3 Gaussian Probability Distribution
Introduction Lecture 3 Gussin Probbility Distribution Gussin probbility distribution is perhps the most used distribution in ll of science. lso clled bell shped curve or norml distribution Unlike the binomil
More informationMATHEMATICS (Part II) (Fresh / New Course)
Sig. of Supdt... MRD-XII-(A) MATHEMATICS Roll No... Time Allowed : Hrs. MATHEMATICS Totl Mrks: 00 NOTE : There re THREE sections in this pper i.e. Section A, B nd C. Time : 0 Mins. Section A Mrks: 0 NOTE
More informationCoimisiún na Scrúduithe Stáit State Examinations Commission
M 30 Coimisiún n Scrúduithe Stáit Stte Exmintions Commission LEAVING CERTIFICATE EXAMINATION, 005 MATHEMATICS HIGHER LEVEL PAPER ( 300 mrks ) MONDAY, 3 JUNE MORNING, 9:30 to :00 Attempt FIVE questions
More informationPARABOLA EXERCISE 3(B)
PARABOLA EXERCISE (B). Find eqution of the tngent nd norml to the prbol y = 6x t the positive end of the ltus rectum. Eqution of prbol y = 6x 4 = 6 = / Positive end of the Ltus rectum is(, ) =, Eqution
More informationAlg. Sheet (1) Department : Math Form : 3 rd prep. Sheet
Ciro Governorte Nozh Directorte of Eduction Nozh Lnguge Schools Ismili Rod Deprtment : Mth Form : rd prep. Sheet Alg. Sheet () [] Find the vlues of nd in ech of the following if : ) (, ) ( -5, 9 ) ) (,
More information( β ) touches the x-axis if = 1
Generl Certificte of Eduction (dv. Level) Emintion, ugust Comined Mthemtics I - Prt B Model nswers. () Let f k k, where k is rel constnt. i. Epress f in the form( ) Find the turning point of f without
More informationForm 5 HKCEE 1990 Mathematics II (a 2n ) 3 = A. f(1) B. f(n) A. a 6n B. a 8n C. D. E. 2 D. 1 E. n. 1 in. If 2 = 10 p, 3 = 10 q, express log 6
Form HK 9 Mthemtics II.. ( n ) =. 6n. 8n. n 6n 8n... +. 6.. f(). f(n). n n If = 0 p, = 0 q, epress log 6 in terms of p nd q.. p q. pq. p q pq p + q Let > b > 0. If nd b re respectivel the st nd nd terms
More information15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions )
- TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the
More informationS56 (5.3) Vectors.notebook January 29, 2016
Dily Prctice 15.1.16 Q1. The roots of the eqution (x 1)(x + k) = 4 re equl. Find the vlues of k. Q2. Find the rte of chnge of 剹 x when x = 1 / 8 Tody we will e lerning out vectors. Q3. Find the eqution
More information3. Vectors. Vectors: quantities which indicate both magnitude and direction. Examples: displacemement, velocity, acceleration
Rutgers University Deprtment of Physics & Astronomy 01:750:271 Honors Physics I Lecture 3 Pge 1 of 57 3. Vectors Vectors: quntities which indicte both mgnitude nd direction. Exmples: displcemement, velocity,
More informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
More informationPre-Calculus TMTA Test 2018
. For the function f ( x) ( x )( x )( x 4) find the verge rte of chnge from x to x. ) 70 4 8.4 8.4 4 7 logb 8. If logb.07, logb 4.96, nd logb.60, then ).08..867.9.48. For, ) sec (sin ) is equivlent to
More informationBig idea in Calculus: approximation
Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:
More informationHAND BOOK OF MATHEMATICS (Definitions and Formulae) CLASS 12 SUBJECT: MATHEMATICS
HAND BOOK OF MATHEMATICS (Definitions nd Formule) CLASS 12 SUBJECT: MATHEMATICS D.SREENIVASULU PGT(Mthemtics) KENDRIYA VIDYALAYA D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA.
More information/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2
SET I. If the locus of the point of intersection of perpendiculr tngents to the ellipse x circle with centre t (0, 0), then the rdius of the circle would e + / ( ) is. There re exctl two points on the
More informationMathematics Higher Block 3 Practice Assessment A
Mthemtics Higher Block 3 Prctice Assessment A Red crefully 1. Clcultors my be used. 2. Full credit will be given only where the solution contins pproprite working. 3. Answers obtined from reding from scle
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More informationSpace Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.
Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)
More informationJUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 6 (First moments of an arc) A.J.Hobson
JUST THE MATHS UNIT NUMBER 13.6 INTEGRATION APPLICATIONS 6 (First moments of n rc) by A.J.Hobson 13.6.1 Introduction 13.6. First moment of n rc bout the y-xis 13.6.3 First moment of n rc bout the x-xis
More informationAP Calculus Multiple Choice: BC Edition Solutions
AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this
More information2A1A Vector Algebra and Calculus I
Vector Algebr nd Clculus I (23) 2AA 2AA Vector Algebr nd Clculus I Bugs/queries to sjrob@robots.ox.c.uk Michelms 23. The tetrhedron in the figure hs vertices A, B, C, D t positions, b, c, d, respectively.
More informationCalculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
More informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More informationINTRODUCTION TO LINEAR ALGEBRA
ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & / ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR
More information1. If y 2 2x 2y + 5 = 0 is (A) a circle with centre (1, 1) (B) a parabola with vertex (1, 2) 9 (A) 0, (B) 4, (C) (4, 4) (D) a (C) c = am m.
SET I. If y x y + 5 = 0 is (A) circle with centre (, ) (B) prbol with vertex (, ) (C) prbol with directrix x = 3. The focus of the prbol x 8x + y + 7 = 0 is (D) prbol with directrix x = 9 9 (A) 0, (B)
More informationContents. Latest CBSE Sample Paper Solution to Latest CBSE Sample Paper Practice Paper 2... Solution to Practice Paper 2...
Contents Ltest CBSE Smple Pper -7 Solution to Ltest CBSE Smple Pper -7 Prctice Pper Solution to Prctice Pper 8 Prctice Pper 9 Solution to Prctice Pper Unsolved Prctice Pper 7 Unsolved Prctice Pper 7 Unsolved
More informationJune 2011 Further Pure Mathematics FP Mark Scheme
. June 0 Further Pure Mthemtics FP 6669 Mrk dy 6x dx = nd so surfce re = π x ( + (6 x ) dx B 4 = 4 π ( 6 x ) + 6 4 4π D 860.06 = 806 (to sf) 6 Use limits nd 0 to give [ ] B Both bits CAO but condone lck
More informationChapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
More information(6.5) Length and area in polar coordinates
86 Chpter 6 SLICING TECHNIQUES FURTHER APPLICATIONS Totl mss 6 x ρ(x)dx + x 6 x dx + 9 kg dx + 6 x dx oment bout origin 6 xρ(x)dx x x dx + x + x + ln x ( ) + ln 6 kg m x dx + 6 6 x x dx Centre of mss +
More informationMathematics Extension Two
Student Number 04 HSC TRIAL EXAMINATION Mthemtics Etension Two Generl Instructions Reding time 5 minutes Working time - hours Write using blck or blue pen Bord-pproved clcultors my be used Write your Student
More informationSAINT IGNATIUS COLLEGE
SAINT IGNATIUS COLLEGE Directions to Students Tril Higher School Certificte 0 MATHEMATICS Reding Time : 5 minutes Totl Mrks 00 Working Time : hours Write using blue or blck pen. (sketches in pencil). This
More informationEdexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks
Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1
More information1.9 C 2 inner variations
46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for
More informationIMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB
` K UKATP ALLY CE NTRE IMPORTANT QUESTIONS FOR INTERMEDIATE PUBLIC EXAMINATIONS IN MATHS-IB 7-8 FIITJEE KUKATPALLY CENTRE: # -97, Plot No, Opp Ptel Kunt Hud Prk, Vijngr Colon, Hderbd - 5 7 Ph: -66 Regd
More information2008 Mathematical Methods (CAS) GA 3: Examination 2
Mthemticl Methods (CAS) GA : Exmintion GENERAL COMMENTS There were 406 students who st the Mthemticl Methods (CAS) exmintion in. Mrks rnged from to 79 out of possible score of 80. Student responses showed
More information( ) Straight line graphs, Mixed Exercise 5. 2 b The equation of the line is: 1 a Gradient m= 5. The equation of the line is: y y = m x x = 12.
Stright line grphs, Mied Eercise Grdient m ( y ),,, The eqution of the line is: y m( ) ( ) + y + Sustitute (k, ) into y + k + k k Multiply ech side y : k k The grdient of AB is: y y So: ( k ) 8 k k 8 k
More informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
More informationSULIT /2 3472/2 Matematik Tambahan Kertas 2 2 ½ jam 2009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING
SULIT 1 347/ 347/ Mtemtik Tmbhn Kerts ½ jm 009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 009 MATEMATIK TAMBAHAN Kerts Du jm tig puluh minit JANGAN BUKA KERTAS
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More informationx = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b
CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick
More informationTOPPER SAMPLE PAPER - 5 CLASS XI MATHEMATICS. Questions. Time Allowed : 3 Hrs Maximum Marks: 100
TOPPER SAMPLE PAPER - 5 CLASS XI MATHEMATICS Questions Time Allowed : 3 Hrs Mximum Mrks: 100 1. All questions re compulsory.. The question pper consist of 9 questions divided into three sections A, B nd
More information1.) King invests $11000 in an account that pays 3.5% interest compounded continuously.
DAY 1 Chpter 4 Exponentil nd Logrithmic Functions 4.3 Grphs of Logrithmic Functions Converting between exponentil nd logrithmic functions Common nd nturl logs The number e Chnging bses 4.4 Properties of
More informationHW3, Math 307. CSUF. Spring 2007.
HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem
More informationThomas Whitham Sixth Form
Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos
More informationThe discriminant of a quadratic function, including the conditions for real and repeated roots. Completing the square. ax 2 + bx + c = a x+
.1 Understnd nd use the lws of indices for ll rtionl eponents.. Use nd mnipulte surds, including rtionlising the denomintor..3 Work with qudrtic nd their grphs. The discriminnt of qudrtic function, including
More informationMath 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions
Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,
More information