PhysicsAndMathsTutor.com
|
|
- Lisa Hamilton
- 5 years ago
- Views:
Transcription
1 PhysicsAdMathsTutor.com
2 physicsadmathstutor.com Jue The fuctio f is defied by (a) Show that f:, > f( ) =, > 1. 1 (4) (b) Fid f 1 (). (3) The fuctio g is defied by g: + 5, R. 1 4 (c) Solve fg() =. (3) 6 *3494B060*
3 physicsadmathstutor.com Jauary The fuctios f ad g are defied by f: + l, R, g: e, R. (a) Prove that the composite fuctio gf is gf: 4e 4, R. (4) (b) I the space provided o page 19, sketch the curve with equatio y = gf(), ad show the coordiates of the poit where the curve cuts the y-ais. (1) (c) Write dow the rage of gf. (1) d (d) Fid the value of for which [gf ( )] = 3, givig your aswer to 3 sigificat d figures. (4) 18 *N3495A0180*
4 physicsadmathstutor.com Jauary 006 Questio 8 cotiued *N3495A0190* 19 Tur over
5 physicsadmathstutor.com Jue For the costat k, where k > 1, the fuctios f ad g are defied by f: l ( + k), > k, g: k, R. (a) O separate aes, sketch the graph of f ad the graph of g. O each sketch state, i terms of k, the coordiates of poits where the graph meets the coordiate aes. (5) (b) Write dow the rage of f. k (c) Fid fg i terms of k, givig your aswer i its simplest form. 4 (1) () The curve C has equatio y = f(). The taget to C at the poit with -coordiate 3 is parallel to the lie with equatio 9y = + 1. (d) Fid the value of k. (4) 18 *N3581A0184*
6 physicsadmathstutor.com Jue 006 Questio 7 cotiued *N3581A0194* 19 Tur over
7 physicsadmathstutor.com Jauary The fuctio f is defied by f: l(4 ), < ad. (a) Show that the iverse fuctio of f is defied by f : e 1 1 ad write dow the domai of f 1. (b) Write dow the rage of f 1. (4) (1) (c) I the space provided o page 16, sketch the graph of y =f 1 (). State the coordiates of the poits of itersectio with the ad y aes. (4) The graph of y = + crosses the graph of y =f 1 () at = k. The iterative formula is used to fid a approimate value for k. 1 = e, = 0.3 (d) Calculate the values of 1 ad, givig your aswers to 4 decimal places. (e) Fid the value of k to 3 decimal places. () () 14 *N3583A0144*
8 physicsadmathstutor.com Jauary 007 Questio 6 cotiued 16 *N3583A0164*
9 physicsadmathstutor.com Jue The fuctios f ad g are defied by 1 f : l( 1),, >, g:, 3, 3. (a) Fid the eact value of fg(4). (b) Fid the iverse fuctio f 1 (), statig its domai. () (4) (c) Sketch the graph of y = g( ). Idicate clearly the equatio of the vertical asymptote ad the coordiates of the poit at which the graph crosses the y-ais. (3) (d) Fid the eact values of for which 3 =3. (3) 10 *N6109A0104*
10 physicsadmathstutor.com Jue 007 Questio 5 cotiued *N6109A0114* 11 Tur over
11 physicsadmathstutor.com Jauary The fuctios f ad g are defied by 3 f: 1, 3 g: 4, > 0, -1 (a) Fid the iverse fuctio f. () (b) Show that the composite fuctio gf is gf : (4) (c) Solve gf ( ) = 0. () (d) Use calculus to fid the coordiates of the statioary poit o the graph of y = gf(). (5) *H6315RB04*
12 physicsadmathstutor.com Jauary 008 Questio 8 cotiued Q8 END (Total 13 marks) TOTAL FOR PAPER: 75 MARKS 4 *H6315RB044*
13 physicsadmathstutor.com Jue The fuctio f is defied by ( ) f: 1 1, 3. 3 > 3 (a) Show that 1 f( ) =, > (4) (b) Fid the rage of f. (c) Fid f 1 (). State the domai of this iverse fuctio. () (3) The fuctio g is defied by g: 3,. (d) Solve fg( ) = 1. 8 (3) 1 *N30745A014*
14 physicsadmathstutor.com Jue 008 Questio 4 cotiued *N30745A0134* 13 Tur over
15 physicsadmathstutor.com Jauary The fuctios f ad g are defied by (a) Write dow the rage of g. (b) Show that the composite fuctio fg is defied by fg : + 3e, R. (c) Write dow the rage of fg. d (d) Solve the equatio fg e ( ) ( + ). d = (1) () (1) (6) 1 *H3113A018*
16 physicsadmathstutor.com Jauary 009 Questio 5 cotiued *H3113A0138* 13 Tur over
17 5. physicsadmathstutor.com Jue 009 y O B A Figure Figure shows a sketch of part of the curve with equatio y = f(),. 1 The curve meets the coordiate aes at the poits A(0,1 k) ad B ( l k,0 ), where k is a costat ad k > 1, as show i Figure. O separate diagrams, sketch the curve with equatio (a) y = f( ), (3) (b) y = 1 f ( ). () Show o each sketch the coordiates, i terms of k, of each poit at which the curve meets or cuts the aes. Give that f( ) = e k, (c) state the rage of f, (1) (d) fid 1 f ( ), (3) (e) write dow the domai of 1 f. (1) 14 *H3464A0148*
18 physicsadmathstutor.com Jue 009 Questio 5 cotiued *H3464A0158* 15 Tur over
19 physicsadmathstutor.com Jue The fuctio f is defied by 8 f( ) = 1 +, ( + 4) ( )( + 4) 4, 3 (a) Show that f( ) = (5) The fuctio g is defied by e 3 g( ) =, e, l e (b) Differetiate g( ) to show that g( ) = (e ) (3) (c) Fid the eact values of for which g( ) = 1 (4) *H3464A08*
20 physicsadmathstutor.com Jue 009 Questio 7 cotiued *H3464A038* 3 Tur over
21 physicsadmathstutor.com Jue (a) Write dow si i terms of si ad cos. (b) Fid, for 0 < <, all the solutios of the equatio cosec 8cos = 0 (1) givig your aswers to decimal places. (5) 6 *H3464A068*
22 physicsadmathstutor.com Jauary The fuctio f is defied by f: 3 5,, 5 (a) Fid 1 f ( ). (3) y 4 1 O Figure The fuctio g has domai 1 8, ad is liear from ( 1, 9) to (, 0) ad from (, 0) to (8, 4). Figure shows a sketch of the graph of y = g(). (b) Write dow the rage of g. (c) Fid gg(). (d) Fid fg(8). (e) O separate diagrams, sketch the graph with equatio (1) () () (i) y = g( ), (ii) y = 1 g ( ). Show o each sketch the coordiates of each poit at which the graph meets or cuts the aes. (4) 1 (f) State the domai of the iverse fuctio g. (1) 18 *H35404RA0188*
23 physicsadmathstutor.com Jauary 011 Questio 6 cotiued *H35404RA0198* 19 Tur over
24 physicsadmathstutor.com Jue The fuctio f is defied by f : 4 l ( + ),, 1 (a) Fid f 1 ( ). (b) Fid the domai of f 1. (3) (1) The fuctio g is defied by g: e, (c) Fid fg (), givig your aswer i its simplest form. (3) (d) Fid the rage of fg. (1) 8 *P38159A084*
25 physicsadmathstutor.com Jauary The fuctio f is defied by ( ) f: ,, R > (a) Show that f ( )= 1 (4) (b) Fid f 1 ( ) (3) (c) Fid the domai of f 1 (1) g( )= l + 1 ( ) (d) Fid the solutio of fg ( )= 1 7, givig your aswer i terms of e. (4) 16 *P40084A0164*
26 physicsadmathstutor.com Jauary 01 Questio 7 cotiued *P40084A0174* 17 Tur over
27 physicsadmathstutor.com Jue The fuctios f ad g are defied by f : e +, g: l, 0 (a) State the rage of f. (b) Fid fg( ), givig your aswer i its simplest form. (c) Fid the eact value of for which f( + 3) = 6 (d) Fid f 1, the iverse fuctio of f, statig its domai. (1) () (4) (3) (e) O the same aes sketch the curves with equatio y = f( ) ad y = f 1 ( ), givig the coordiates of all the poits where the curves cross the aes. (4) 0 *P40686RA003*
28 physicsadmathstutor.com Jue 01 Questio 6 cotiued *P40686RA013* 1 Tur over
29 physicsadmathstutor.com Jauary h( ) = , 0 5 ( + 5)( + ) (a) Show that h( ) = + 5 (4) (b) Hece, or otherwise, fid h( ) i its simplest form. (3) y y = h() O Figure Figure shows a graph of the curve with equatio y (c) Calculate the rage of h( ). = h( ). (5) *P41486A08*
30 physicsadmathstutor.com Jauary 013 Questio 7 cotiued *P41486A038* 3 Tur over
31 physicsadmathstutor.com Jue 013 (R)
32 physicsadmathstutor.com Jue 013 (R)
33 physicsadmathstutor.com Jue The fuctio f has domai 6 ad is liear from (, 10) to (, 0) ad from (, 0) to (6, 4). A sketch of the graph of y = f() is show i Figure 1. y 10 O 6 Figure 1 (a) Write dow the rage of f. (b) Fid ff(0). (1) () The fuctio g is defied by g : 4 + 3, 5, (c) Fid g 1 () (3) (d) Solve the equatio gf() = 16 (5) 4 *P43016A043*
34 physicsadmathstutor.com Jue 013 Questio 7 cotiued *P43016A053* 5 Tur over
35 Core Mathematics C3 Cadidates sittig C3 may also require those formulae listed uder Core Mathematics C1 ad C. Logarithms ad epoetials e l a = a Trigoometric idetities si ( A ± B ) = si A cos B ± cos A si B cos( A ± B ) = cos A cos B m si A si B ta A ± ta B ta ( A ± B ) = ( A ± B ( k + ) 1m ta A ta B A + B A B si A + si B = si cos A + B A B si A si B = cos si A + B A B cos A + cos B = cos cos A + B A B cos A cos B = si si 1 π ) Differetiatio f() ta k sec cot cosec f( ) g( ) f () k sec k sec ta cosec cosec cot f ( )g( ) f( )g ( ) (g( )) 6 Edecel AS/A level Mathematics Formulae List: Core Mathematics C3 Issue 1 September 009
36 Edecel AS/A level Mathematics Formulae List: Core Mathematics C Issue 1 September Core Mathematics C Cadidates sittig C may also require those formulae listed uder Core Mathematics C1. Cosie rule a = b + c bc cos A Biomial series 1 ) ( 1 r r b b a r b a b a a b a = + K K ( N) where )!!(! C r r r r = = < = + r r r 1, ( 1 1 ) ( 1 ) ( 1 1 ) ( 1 ) ( 1 K K K K R) Logarithms ad epoetials a b b a log log log = Geometric series u = ar 1 S = r r a 1 ) ( 1 S = r a 1 for r < 1 Numerical itegratio The trapezium rule: b a y d 1 h{(y 0 + y ) + (y 1 + y y 1 )}, where a b h =
37 Core Mathematics C1 Mesuratio Surface area of sphere = 4π r Area of curved surface of coe = π r slat height Arithmetic series u = a + ( 1)d S = 1 (a + l) = 1 [a + ( 1)d] 4 Edecel AS/A level Mathematics Formulae List: Core Mathematics C1 Issue 1 September 009
PhysicsAndMathsTutor.com
PhysicsAdMathsTutor.com physicsadmathstutor.com Jue 005 4. f(x) = 3e x 1 l x, x > 0. (a) Differetiate to fid f (x). (3) The curve with equatio y = f(x) has a turig poit at P. The x-coordiate of P is α.
More informationphysicsandmathstutor.com
physicsadmathstutor.com 5. Solve, for 0 x 180, the equatio 3 (a) si( x + 10 ) =, 2 (b) cos 2x = 0.9, givig your aswers to 1 decimal place. (4) (4) 10 *N23492B01028* 8. (a) Fid all the values of, to 1 decimal
More informationphysicsandmathstutor.com
physicsadmathstutor.com 8. The circle C, with cetre at the poit A, has equatio x 2 + y 2 10x + 9 = 0. Fid (a) the coordiates of A, (b) the radius of C, (c) the coordiates of the poits at which C crosses
More informationphysicsandmathstutor.com
physicsadmathstutor.com 1. Fid the coordiates of the statioary poit o the curve with equatio y = 1. (4) Q1 (Total 4 marks) *N349B038* 3 Tur over 10. Figure 1 y 8 y = + 5 P R Q O Figure 1 shows part of
More information3 Show in each case that there is a root of the given equation in the given interval. a x 3 = 12 4
C Worksheet A Show i each case that there is a root of the equatio f() = 0 i the give iterval a f() = + 7 (, ) f() = 5 cos (05, ) c f() = e + + 5 ( 6, 5) d f() = 4 5 + (, ) e f() = l (4 ) + (04, 05) f
More informationphysicsandmathstutor.com
physicsadmathstuto.com 2. Solve (a) 5 = 8, givig you aswe to 3 sigificat figues, (b) log 2 ( 1) log 2 = log 2 7. (3) (3) 4 *N23492B0428* 3. (i) Wite dow the value of log 6 36. (ii) Epess 2 log a 3 log
More informationphysicsandmathstutor.com
physicsadmathstuto.com physicsadmathstuto.com Jue 005. A cuve has equatio blak x + xy 3y + 16 = 0. dy Fid the coodiates of the poits o the cuve whee 0. dx = (7) Q (Total 7 maks) *N03B034* 3 Tu ove physicsadmathstuto.com
More informationCALCULUS BASIC SUMMER REVIEW
CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=
More informationEDEXCEL STUDENT CONFERENCE 2006 A2 MATHEMATICS STUDENT NOTES
EDEXCEL STUDENT CONFERENCE 006 A MATHEMATICS STUDENT NOTES South: Thursday 3rd March 006, Lodo EXAMINATION HINTS Before the eamiatio Obtai a copy of the formulae book ad use it! Write a list of ad LEARN
More informationVICTORIA JUNIOR COLLEGE Preliminary Examination. Paper 1 September 2015
VICTORIA JUNIOR COLLEGE Prelimiary Eamiatio MATHEMATICS (Higher ) 70/0 Paper September 05 Additioal Materials: Aswer Paper Graph Paper List of Formulae (MF5) 3 hours READ THESE INSTRUCTIONS FIRST Write
More informationphysicsandmathstutor.com
physicsadmathstuto.com physicsadmathstuto.com Jue 005 5x 3 3. (a) Expess i patial factios. (x 3)( x ) (3) (b) Hece fid the exact value of logaithm. 6 5x 3 dx, givig you aswe as a sigle (x 3)( x ) (5) blak
More informationIYGB. Special Extension Paper E. Time: 3 hours 30 minutes. Created by T. Madas. Created by T. Madas
YGB Special Extesio Paper E Time: 3 hours 30 miutes Cadidates may NOT use ay calculator. formatio for Cadidates This practice paper follows the Advaced Level Mathematics Core ad the Advaced Level Further
More informationCATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination MATHEMATICS 9740/01
CATHOLIC JUNIOR COLLEGE Geeral Certificate of Educatio Advaced Level Higher JC Prelimiary Examiatio MATHEMATICS 9740/0 Paper 4 Aug 06 hours Additioal Materials: List of Formulae (MF5) Name: Class: READ
More informationPhysicsAndMathsTutor.com
PhysicsAdMthsTutor.com PhysicsAdMthsTutor.com Jue 009 4. Give tht y rsih ( ), > 0, () fid d y d, givig your swer s simplified frctio. () Leve lk () Hece, or otherwise, fid 4 d, 4 [ ( )] givig your swer
More informationPhysicsAndMathsTutor.com
PhysicsAdMthsTuto.com PhysicsAdMthsTuto.com Jue 009 3. Fid the geel solutio of the diffeetil equtio blk d si y ycos si si, d givig you swe i the fom y = f(). (8) 6 *M3544A068* PhysicsAdMthsTuto.com Jue
More informationMATH2007* Partial Answers to Review Exercises Fall 2004
MATH27* Partial Aswers to Review Eercises Fall 24 Evaluate each of the followig itegrals:. Let u cos. The du si ad Hece si ( cos 2 )(si ) (u 2 ) du. si u 2 cos 7 u 7 du Please fiish this. 2. We use itegratio
More informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
More informationNATIONAL JUNIOR COLLEGE SENIOR HIGH 1 PROMOTIONAL EXAMINATIONS Higher 2
NATIONAL JUNIOR COLLEGE SENIOR HIGH PROMOTIONAL EXAMINATIONS Higher MATHEMATICS 9758 9 September 06 hours Additioal Materials: Aswer Paper List of Formulae (MF6) Cover Sheet READ THESE INSTRUCTIONS FIRST
More informationMathematics: Paper 1
GRADE 1 EXAMINATION JULY 013 Mathematics: Paper 1 EXAMINER: Combied Paper MODERATORS: JE; RN; SS; AVDB TIME: 3 Hours TOTAL: 150 PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This questio paper cosists
More informationMathematics Extension 1
016 Bored of Studies Trial Eamiatios Mathematics Etesio 1 3 rd ctober 016 Geeral Istructios Total Marks 70 Readig time 5 miutes Workig time hours Write usig black or blue pe Black pe is preferred Board-approved
More informationTEMASEK JUNIOR COLLEGE, SINGAPORE JC One Promotion Examination 2014 Higher 2
TEMASEK JUNIOR COLLEGE, SINGAPORE JC Oe Promotio Eamiatio 04 Higher MATHEMATICS 9740 9 Septemer 04 Additioal Materials: Aswer paper 3 hours List of Formulae (MF5) READ THESE INSTRUCTIONS FIRST Write your
More informationGRADE 12 SEPTEMBER 2015 MATHEMATICS P1
NATIONAL SENIOR CERTIFICATE GRADE 1 SEPTEMBER 015 MATHEMATICS P1 MARKS: 150 TIME: 3 hours *MATHE1* This questio paper cosists of 10 pages, icludig a iformatio sheet. MATHEMATICS P1 (EC/SEPTEMBER 015) INSTRUCTIONS
More informationMATHEMATICS (Three hours and a quarter)
MATHEMATICS (Three hours ad a quarter) (The first fiftee miutes of the eamiatio are for readig the paper oly. Cadidates must NOT start writig durig this time.) Aswer Questio from Sectio A ad questios from
More informationCalculus 2 Test File Spring Test #1
Calculus Test File Sprig 009 Test #.) Without usig your calculator, fid the eact area betwee the curves f() = - ad g() = +..) Without usig your calculator, fid the eact area betwee the curves f() = ad
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE 1 MATHEMATICS P1 NOVEMBER 016 MARKS: 150 TIME: 3 hours This questio paper cosists of 9 pages ad 1 iformatio sheet. Please tur over Mathematics/P1 DBE/November 016 INSTRUCTIONS
More informationMath III-Formula Sheet
Math III-Formula Sheet Statistics Z-score: Margi of Error: To fid the MEAN, MAXIMUM, MINIMUM, Q 3, Q 1, ad STANDARD DEVIATION of a set of data: 1) Press STAT, ENTER (to eter our data) Put it i L 1 ) Press
More information7.) Consider the region bounded by y = x 2, y = x - 1, x = -1 and x = 1. Find the volume of the solid produced by revolving the region around x = 3.
Calculus Eam File Fall 07 Test #.) Fid the eact area betwee the curves f() = 8 - ad g() = +. For # - 5, cosider the regio bouded by the curves y =, y = 3 + 4. Produce a solid by revolvig the regio aroud
More informationAP Calculus BC Review Applications of Derivatives (Chapter 4) and f,
AP alculus B Review Applicatios of Derivatives (hapter ) Thigs to Kow ad Be Able to Do Defiitios of the followig i terms of derivatives, ad how to fid them: critical poit, global miima/maima, local (relative)
More informationCARIBBEAN EXAMINATIONS COUNCIL CARIBBEAN SECONDARY EDUCATION EXAMINATION ADDITIONAL MATHEMATICS. Paper 02 - General Proficiency
TEST CODE 01254020 FORM TP 2015037 MAY/JUNE 2015 CARIBBEAN EXAMINATIONS COUNCIL CARIBBEAN SECONDARY EDUCATION CERTIFICATE@ EXAMINATION ADDITIONAL MATHEMATICS Paper 02 - Geeral Proficiecy 2 hours 40 miutes
More informationPhysicsAndMathsTutor.com
PhysicsAndMathsTutor.com physicsandmathstutor.com June 2005 1. (a) Given that sin 2 θ + cos 2 θ 1, show that 1 + tan 2 θ sec 2 θ. (b) Solve, for 0 θ < 360, the equation 2 tan 2 θ + secθ = 1, giving your
More informationFor use only in Badminton School November 2011 C2 Note. C2 Notes (Edexcel)
For use oly i Badmito School November 0 C Note C Notes (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets For use oly i Badmito School November 0 C Note Copyright www.pgmaths.co.uk
More informationMATHEMATICS 9740 (HIGHER 2)
VICTORIA JUNIOR COLLEGE PROMOTIONAL EXAMINATION MATHEMATICS 970 (HIGHER ) Frida 6 Sept 0 8am -am hours Additioal materials: Aswer Paper List of Formulae (MF5) READ THESE INSTRUCTIONS FIRST Write our ame
More informationMathematics Extension 2
004 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etesio Geeral Istructios Readig time 5 miutes Workig time hours Write usig black or blue pe Board-approved calculators may be used A table of stadard
More informationChapter 2 The Solution of Numerical Algebraic and Transcendental Equations
Chapter The Solutio of Numerical Algebraic ad Trascedetal Equatios Itroductio I this chapter we shall discuss some umerical methods for solvig algebraic ad trascedetal equatios. The equatio f( is said
More informationPhysicsAndMathsTutor.com
PhysicsAdMthsTuto.com PhysicsAdMthsTuto.com Jue 009 7. () Sketch the gph of y, whee >, showig the coodites of the poits whee the gph meets the es. () Leve lk () Solve, >. (c) Fid the set of vlues of fo
More information(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)
. f() = 4 cosec 4 +, where is in radians. (a) Show that there is a root α of f () = 0 in the interval [.,.3]. Show that the equation f() = 0 can be written in the form = + sin 4 Use the iterative formula
More informationMathematics Extension 2
009 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etesio Geeral Istructios Readig time 5 miutes Workig time hours Write usig black or blue pe Board-approved calculators may be used A table of stadard
More informationEdexcel GCE Further Pure Mathematics FP1 Advanced/Advanced Subsidiary
Cetre No. Cadidate No. Surame Sigature Paper Referece(s) 6667/0 Edexcel GCE Further Pure Mathematics FP Advaced/Advaced Subsidiary Moday 28 Jauary 203 Morig Time: hour 30 miutes Materials required for
More informationCalculus 2 Test File Fall 2013
Calculus Test File Fall 013 Test #1 1.) Without usig your calculator, fid the eact area betwee the curves f() = 4 - ad g() = si(), -1 < < 1..) Cosider the followig solid. Triagle ABC is perpedicular to
More informationCoimisiún na Scrúduithe Stáit State Examinations Commission
M 9 Coimisiú a Scrúduithe Stáit State Examiatios Commissio LEAVING CERTIFICATE EXAMINATION, 006 MATHEMATICS HIGHER LEVEL PAPER 1 ( 00 marks ) THURSDAY, 8 JUNE MORNING, 9:0 to 1:00 Attempt SIX QUESTIONS
More informationphysicsandmathstutor.com
physicsadmathstuto.com physicsadmathstuto.com Jauay 2009 2 a 7. Give that X = 1 1, whee a is a costat, ad a 2, blak (a) fid X 1 i tems of a. Give that X + X 1 = I, whee I is the 2 2 idetity matix, (b)
More informationDe Moivre s Theorem - ALL
De Moivre s Theorem - ALL. Let x ad y be real umbers, ad be oe of the complex solutios of the equatio =. Evaluate: (a) + + ; (b) ( x + y)( x + y). [6]. (a) Sice is a complex umber which satisfies = 0,.
More informationHigher Course Plan. Calculus and Relationships Expressions and Functions
Higher Course Pla Applicatios Calculus ad Relatioships Expressios ad Fuctios Topic 1: The Straight Lie Fid the gradiet of a lie Colliearity Kow the features of gradiets of: parallel lies perpedicular lies
More informationHKDSE Exam Questions Distribution
HKDSE Eam Questios Distributio Sample Paper Practice Paper DSE 0 Topics A B A B A B. Biomial Theorem. Mathematical Iductio 0 3 3 3. More about Trigoometric Fuctios, 0, 3 0 3. Limits 6. Differetiatio 7
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P FEBRUARY/MARCH 03 MARKS: 50 TIME: 3 hours This questio paper cosists of pages, 3 diagram sheets ad iformatio sheet. Please tur over Mathematics/P DBE/Feb.
More informationNATIONAL SENIOR CERTIFICATE EXAMINATION MATHEMATICS P1 SEPTEMBER 2016 GRADE 12
NATIONAL SENIOR CERTIFICATE EXAMINATION MATHEMATICS P SEPTEMBER 06 GRADE MARKS: 50 TIME: 3 HOURS This questio paper cosists of 9 pages ad iformatio sheet, Please tur over Mathematics/P September 06 INSTRUCTIONS
More informationPhysicsAndMathsTutor.com
PhysicsAdMthsTuto.com 5. () Show tht d y d PhysicsAdMthsTuto.com Jue 009 4 y = sec = 6sec 4sec. (b) Fid Tylo seies epsio of sec π i scedig powes of 4, up to d 3 π icludig the tem i 4. (6) (4) blk *M3544A08*
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P ADDITIONAL EXEMPLAR 008 MARKS: 50 TIME: hours This questio paper cosists of 9 pages, diagram sheets ad a -page formula sheet. Mathematics/P DoE/Additioal
More informationNANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS
NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER. There will be oe 2-hour paper cosistig of 4 questios.
More informationMID-YEAR EXAMINATION 2018 H2 MATHEMATICS 9758/01. Paper 1 JUNE 2018
MID-YEAR EXAMINATION 08 H MATHEMATICS 9758/0 Paper JUNE 08 Additioal Materials: Writig Paper, MF6 Duratio: hours DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO READ THESE INSTRUCTIONS FIRST Write
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE 1 MATHEMATICS P1 FEBRUARY/MARCH 011 MARKS: 150 TIME: 3 hours This questio paper cosists of 8 pages, 3 diagram sheets ad 1 iformatio sheet. Please tur over Mathematics/P1
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P FEBRUARY/MARCH 009 MARKS: 50 TIME: 3 hours This questio paper cosists of 0 pages, a iformatio sheet ad 3 diagram sheets. Please tur over Mathematics/P DoE/Feb.
More information*X203/701* X203/701. APPLIED MATHEMATICS ADVANCED HIGHER Numerical Analysis. Read carefully
X0/70 NATIONAL QUALIFICATIONS 006 MONDAY, MAY.00 PM.00 PM APPLIED MATHEMATICS ADVANCED HIGHER Numerical Aalysis Read carefully. Calculators may be used i this paper.. Cadidates should aswer all questios.
More informationFINALTERM EXAMINATION Fall 9 Calculus & Aalytical Geometry-I Questio No: ( Mars: ) - Please choose oe Let f ( x) is a fuctio such that as x approaches a real umber a, either from left or right-had-side,
More informationFind a formula for the exponential function whose graph is given , 1 2,16 1, 6
Math 4 Activity (Due by EOC Apr. ) Graph the followig epoetial fuctios by modifyig the graph of f. Fid the rage of each fuctio.. g. g. g 4. g. g 6. g Fid a formula for the epoetial fuctio whose graph is
More informationGRADE 12 LEARNER SUPPORT PROGRAMME
Provice of the EASTERN CAPE EDUCATION Steve Vukile Tshwete Educatio Comple Zoe 6 Zwelitsha 5608 Private Bag X003 Bhisho 5605 REPUBLIC OF SOUTH AFRICA CHIEF DIRECTORATE CURRICULUM MANAGEMENT GRADE LEARNER
More information(5x 7) is. 63(5x 7) 42(5x 7) 50(5x 7) BUSINESS MATHEMATICS (Three hours and a quarter)
BUSINESS MATHEMATICS (Three hours ad a quarter) (The first 5 miutes of the examiatio are for readig the paper oly. Cadidate must NOT start writig durig this time). ------------------------------------------------------------------------------------------------------------------------
More informationHonors Calculus Homework 13 Solutions, due 12/8/5
Hoors Calculus Homework Solutios, due /8/5 Questio Let a regio R i the plae be bouded by the curves y = 5 ad = 5y y. Sketch the regio R. The two curves meet where both equatios hold at oce, so where: y
More informationPresentation of complex number in Cartesian and polar coordinate system
a + bi, aεr, bεr i = z = a + bi a = Re(z), b = Im(z) give z = a + bi & w = c + di, a + bi = c + di a = c & b = d The complex cojugate of z = a + bi is z = a bi The sum of complex cojugates is real: z +
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P FEBRUARY/MARCH 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, diagram sheet ad iformatio sheet. Please tur over Mathematics/P DBE/Feb.
More informationLIMITS AND DERIVATIVES
Capter LIMITS AND DERIVATIVES. Overview.. Limits of a fuctio Let f be a fuctio defied i a domai wic we take to be a iterval, say, I. We sall study te cocept of it of f at a poit a i I. We say f ( ) is
More informationConsortium of Medical Engineering and Dental Colleges of Karnataka (COMEDK) Undergraduate Entrance Test(UGET) Maths-2012
Cosortium of Medical Egieerig ad Detal Colleges of Karataka (COMEDK) Udergraduate Etrace Test(UGET) Maths-0. If the area of the circle 7 7 7 k 0 is sq. uits, the the value of k is As: (b) b) 0 7 K 0 c)
More informationFUNCTIONS (11 UNIVERSITY)
FINAL EXAM REVIEW FOR MCR U FUNCTIONS ( UNIVERSITY) Overall Remiders: To study for your eam your should redo all your past tests ad quizzes Write out all the formulas i the course to help you remember
More informationTopic 5 [434 marks] (i) Find the range of values of n for which. (ii) Write down the value of x dx in terms of n, when it does exist.
Topic 5 [44 marks] 1a (i) Fid the rage of values of for which eists 1 Write dow the value of i terms of 1, whe it does eist Fid the solutio to the differetial equatio 1b give that y = 1 whe = π (cos si
More informationCreated by T. Madas SERIES. Created by T. Madas
SERIES SUMMATIONS BY STANDARD RESULTS Questio (**) Use stadard results o summatios to fid the value of 48 ( r )( 3r ). 36 FP-B, 66638 Questio (**+) Fid, i fully simplified factorized form, a expressio
More information2018 YEAR 6 PRELIMINARY EXAMINATION MATHEMATICS 9758/01
RAFFLES INSTITUTION 018 YEAR 6 PRELIMINARY EXAMINATION MATHEMATICS 9758/01 Paper 1 Total Marks: 100 hours Additioal materials: Aswer Paper List of Formulae (MF6) READ THESE INSTRUCTIONS FIRST Write your
More informationEton Education Centre JC 1 (2010) Consolidation quiz on Normal distribution By Wee WS (wenshih.wordpress.com) [ For SAJC group of students ]
JC (00) Cosolidatio quiz o Normal distributio By Wee WS (weshih.wordpress.com) [ For SAJC group of studets ] Sped miutes o this questio. Q [ TJC 0/JC ] Mr Fruiti is the ower of a fruit stall sellig a variety
More informationMATH CALCULUS II Objectives and Notes for Test 4
MATH 44 - CALCULUS II Objectives ad Notes for Test 4 To do well o this test, ou should be able to work the followig tpes of problems. Fid a power series represetatio for a fuctio ad determie the radius
More informationThis paper consists of 10 pages with 10 questions. All the necessary working details must be shown.
Mathematics - HG Mar 003 Natioal Paper INSTRUCTIONS.. 3. 4. 5. 6. 7. 8. 9. This paper cosists of 0 pages with 0 questios. A formula sheet is icluded o page 0 i the questio paper. Detach it ad use it to
More informationFurther Methods for Advanced Mathematics (FP2) WEDNESDAY 9 JANUARY 2008
ADVANCED GCE 7/ MATHEMATICS (MEI) Furter Metods for Advaced Matematics (F) WEDNESDAY 9 JANUARY 8 Additioal materials: Aswer Booklet (8 pages) Grap paper MEI Eamiatio Formulae ad Tables (MF) Afteroo Time:
More informationC3 papers June 2007 to 2008
physicsandmathstutor.com June 007 C3 papers June 007 to 008 1. Find the exact solutions to the equations (a) ln x + ln 3 = ln 6, (b) e x + 3e x = 4. *N6109A04* physicsandmathstutor.com June 007 x + 3 9+
More informationLIMITS AND DERIVATIVES NCERT
. Overview.. Limits of a fuctio Let f be a fuctio defied i a domai wic we take to be a iterval, say, I. We sall study te cocept of it of f at a poit a i I. We say f ( ) is te epected value of f at a give
More information(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is
Calculus BC Fial Review Name: Revised 7 EXAM Date: Tuesday, May 9 Remiders:. Put ew batteries i your calculator. Make sure your calculator is i RADIAN mode.. Get a good ight s sleep. Eat breakfast. Brig:
More information( ) 2 + k The vertex is ( h, k) ( )( x q) The x-intercepts are x = p and x = q.
A Referece Sheet Number Sets Quadratic Fuctios Forms Form Equatio Stadard Form Vertex Form Itercept Form y ax + bx + c The x-coordiate of the vertex is x b a y a x h The axis of symmetry is x b a + k The
More informationSection A assesses the Units Numerical Analysis 1 and 2 Section B assesses the Unit Mathematics for Applied Mathematics
X0/70 NATIONAL QUALIFICATIONS 005 MONDAY, MAY.00 PM 4.00 PM APPLIED MATHEMATICS ADVANCED HIGHER Numerical Aalysis Read carefully. Calculators may be used i this paper.. Cadidates should aswer all questios.
More informationM06/5/MATHL/HP2/ENG/TZ0/XX MATHEMATICS HIGHER LEVEL PAPER 2. Thursday 4 May 2006 (morning) 2 hours INSTRUCTIONS TO CANDIDATES
IB MATHEMATICS HIGHER LEVEL PAPER DIPLOMA PROGRAMME PROGRAMME DU DIPLÔME DU BI PROGRAMA DEL DIPLOMA DEL BI 06705 Thursday 4 May 006 (morig) hours INSTRUCTIONS TO CANDIDATES Do ot ope this examiatio paper
More informationGRADE 12 JUNE 2017 MATHEMATICS P1
NATIONAL SENIOR CERTIFICATE GRADE 1 JUNE 017 MATHEMATICS P1 MARKS: 150 TIME: 3 hours *MATHE1* This questio paper cosists of 11 pages, icludig a iformatio sheet. MATHEMATICS P1 (EC/JUNE 017) INSTRUCTIONS
More informationExponential and Trigonometric Functions Lesson #1
Epoetial ad Trigoometric Fuctios Lesso # Itroductio To Epoetial Fuctios Cosider a populatio of 00 mice which is growig uder plague coditios. If the mouse populatio doubles each week we ca costruct a table
More informationMETRO EAST EDUCATION DISTRICT
METRO EAST EDUCATION DISTRICT COMMON PAPER GRADE 1 MATHEMATICS P1 SEPTEMBER 018 MARKS: 150 TIME: 3 hours This questio paper cosists of 10 pages ad 1 iformatio sheet. INSTRUCTIONS AND INFORMATION Read the
More informationContinuous Functions
Cotiuous Fuctios Q What does it mea for a fuctio to be cotiuous at a poit? Aswer- I mathematics, we have a defiitio that cosists of three cocepts that are liked i a special way Cosider the followig defiitio
More informationRegent College Maths Department. Core Mathematics 4 Trapezium Rule. C4 Integration Page 1
Regent College Maths Department Core Mathematics Trapezium Rule C Integration Page Integration It might appear to be a bit obvious but you must remember all of your C work on differentiation if you are
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P FEBRUARY/MARCH 009 MARKS: 50 TIME: 3 hours This questio paper cosists of 0 pages, a iformatio sheet ad diagram sheets. Please tur over Mathematics/P DoE/Feb.
More informationMAT136H1F - Calculus I (B) Long Quiz 1. T0101 (M3) Time: 20 minutes. The quiz consists of four questions. Each question is worth 2 points. Good Luck!
MAT36HF - Calculus I (B) Log Quiz. T (M3) Time: 2 miutes Last Name: Studet ID: First Name: Please mark your tutorial sectio: T (M3) T2 (R4) T3 (T4) T5 (T5) T52 (R5) The quiz cosists of four questios. Each
More informationAlgebra 1 Hour Final Exam Review Days
Semester Fial Eam Review Packet Name Algebra 1 Hour Fial Eam Review Days Assiged o Assigmet 6/6 Fial Eam Review Chapters 11 ad 1 Problems 54 7 6/7 Fial Eam Review Chapters 10 Problems 44 5 6/8 Fial Eam
More informationGRADE 12 NATIONAL SENIOR CERTIFICATE MATHEMATICS P1 PREPARATORY EXAMINATION 2008
GRADE NATIONAL SENIOR CERTIFICATE MATHEMATICS P PREPARATORY EXAMINATION 008 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, a formula sheet ad diagram sheet. Mathematics/P DoE/Preparatory
More informationA.1 Algebra Review: Polynomials/Rationals. Definitions:
MATH 040 Notes: Uit 0 Page 1 A.1 Algera Review: Polyomials/Ratioals Defiitios: A polyomial is a sum of polyomial terms. Polyomial terms are epressios formed y products of costats ad variales with whole
More informationMATHEMATICAL METHODS
8 Practice Exam A Letter STUDENT NUMBER MATHEMATICAL METHODS Writte examiatio Sectio Readig time: 5 miutes Writig time: hours WORKED SOLUTIONS Number of questios Structure of book Number of questios to
More informationPrecalculus MATH Sections 3.1, 3.2, 3.3. Exponential, Logistic and Logarithmic Functions
Precalculus MATH 2412 Sectios 3.1, 3.2, 3.3 Epoetial, Logistic ad Logarithmic Fuctios Epoetial fuctios are used i umerous applicatios coverig may fields of study. They are probably the most importat group
More informationGRADE 12 JUNE 2016 MATHEMATICS P1
NATIONAL SENIOR CERTIFICATE GRADE 1 JUNE 016 MATHEMATICS P1 MARKS: 150 TIME: 3 hours *MATHE1* This questio paper cosists of 14 pages, icludig a iformatio sheet. MATHEMATICS P1 (EC/JUNE 016) INSTRUCTIONS
More informationBITSAT MATHEMATICS PAPER III. For the followig liear programmig problem : miimize z = + y subject to the costraits + y, + y 8, y, 0, the solutio is (0, ) ad (, ) (0, ) ad ( /, ) (0, ) ad (, ) (d) (0, )
More informationMATH 1A FINAL (7:00 PM VERSION) SOLUTION. (Last edited December 25, 2013 at 9:14pm.)
MATH A FINAL (7: PM VERSION) SOLUTION (Last edited December 5, 3 at 9:4pm.) Problem. (i) Give the precise defiitio of the defiite itegral usig Riema sums. (ii) Write a epressio for the defiite itegral
More informationPrice per tonne of sand ($) A B C
. Burkig would like to purchase cemet, iro ad sad eeded for his costructio project. He approached three suppliers A, B ad C to equire about their sellig prices for the materials. The total prices quoted
More informationMath 142, Final Exam. 5/2/11.
Math 4, Fial Exam 5// No otes, calculator, or text There are poits total Partial credit may be give Write your full ame i the upper right corer of page Number the pages i the upper right corer Do problem
More informationGRADE 12 JUNE 2016 MATHEMATICS P2
NATIONAL SENIOR CERTIFICATE GRADE 1 JUNE 016 MATHEMATICS P MARKS: 150 TIME: 3 hours *MATHE* This questio paper cosists of 11 pages, icludig 1 iformatio sheet, ad a SPECIAL ANSWER BOOK. MATHEMATICS P (EC/JUNE
More informationCircular Functions (Trigonometry)
Circular Fuctios (Trigoometry) Circular fuctios Revisio Where do si cos ad ta come from? Uit circle (of radius ) cos is the coordiate si is the y coordiate si ta cos all are measures of legth. Remember
More informationGRADE 12 SEPTEMBER 2015 MATHEMATICS P2
NATIONAL SENIOR CERTIFICATE GRADE SEPTEMBER 05 MATHEMATICS P MARKS: 50 TIME: 3 hours *MATHE* This questio paper cosists of 3 pages icludig iformatio sheet, ad a SPECIAL ANSWERBOOK. MATHEMATICS P (EC/SEPTEMBER
More informationMETRO EAST EDUCATION DISTRICT NATIONAL SENIOR CERTIFICATE GRADE 12 MATHEMATICS PAPER 1 SEPTEMBER 2014
METRO EAST EDUCATION DISTRICT NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS PAPER SEPTEMBER 04 MARKS: 50 TIME: 3 hours This paper cosists of 7 pages ad a iformatio sheet. GR Mathematics- P MEED September
More informationMATHEMATICS. 61. The differential equation representing the family of curves where c is a positive parameter, is of
MATHEMATICS 6 The differetial equatio represetig the family of curves where c is a positive parameter, is of Order Order Degree (d) Degree (a,c) Give curve is y c ( c) Differetiate wrt, y c c y Hece differetial
More information