# Prerequisite Knowledge Required from O Level Add Math. d n a = c and b = d

Size: px
Start display at page:

Download "Prerequisite Knowledge Required from O Level Add Math. d n a = c and b = d"

Transcription

1 Prerequisite Knowledge Required from O Level Add Mth ) Surds, Indices & Logrithms Rules for Surds. b= b = b = b = ( ) = = = 5. + b n = c+ d n = c nd b = d Cution: + +, - Rtionlising the Denomintor (two tpes) () E.g. (b) E.g = = (5+ ) (+ ) (+ ) = = = (5 ) (5 )(5+ ) (5 ) 3 Rules for Indices Sme bse = = Power Sme inde ( ) n = = () =( )n Zero inde = Negtive inde = ( ) - n = ( )n Frctionl inde = ( )

2 Rules for Logrithms For log to be defined, both nd must be positive. Logrithm log is written s lg Logrithm log is written s ln Product of Logrithm log = log + log Quotient of Logrithm log log log = Power Lw r log = r log Chnge of Bse log logc b b= log c Log of to n bse log = 0 Log of number to the sme bse s number log = Eponentil Grphs The grphs of = b re shown below for different rnge of vlues of & b. NOTE: MUST DRAW THE ASYMPTOTES FOR THESE CASES!

3 Logrithmic Grphs The grphs of = ln (b + c) re shown below for different rnge of vlues of & b. ) Qudrtic Equtions Qudrtic Eqution: + b + c = 0 Recll Qudrtic Formul: = ± ) For roots α nd β b) Useful: b Sum of roots = α + β = Product of roots = αβ = c Eqution is: α (sum of roots) + (product of roots) = 0 OR: ( α)( β ) = 0 α + β = ( α + β ) αβ 3

4 Discriminnt & tpe of roots D = b 4c Coefficient of > 0. hs min. vlue Coefficient of < 0. hs m. vlue Rel & distinct roots b 4c> 0 Rel & repeted (equl) roots b 4c= 0 No rel roots b 4c< 0 Question tpes Rel roots Line does not intersect curve Curve lies entirel bove -is (min. grph) e.g. > 0 for ll vlues of, 5 +3 > 0 lws Curve lies entirel below -is (m. grph) e.g. < 0 for ll vlues of, < 0 lws For rel roots, it cn be distinct or equl roots b 4c 0 4

5 Polnomils & Prtil Frction Recll how long division is done. Fctor Theorem A polnomil f() is divisible b ( ) => reminder = 0 => ( ) is fctor of f(). Prtil Frctions Proper rtionl lgebric epression, () () is when degree of numertor f() < degree of denomintor, g(). Note: To write s prtil frctions, first check tht the epression is proper. m+ n A B = + ( + b)( c+ d) + b c+ d p + q+ r A B C = + + ( + b)( c+ d) + b c+ d ( c+ d) p + q+ r A B+ c = + ( + b)( + c ) + b + c 3) Modulus Function The bsolute or modulus of rel number is denoted b nd defined s: The bove is useful when solving modulus eqution. Alterntivel, ou cn squre both sides to remove the modulus sign. Refer to propert 6 below. Properties of modulus: = 3. b = b 4. = 5. n = n 6. = b implies tht = b Note: modulus is lws non-negtive. 5

6 4) Coordinte Geometr B(, ) θ A(, ). For points A(, ) nd B(, ), Length AB= ) + ( ) ( + + Mid-point of AB =, Grdient of AB= = tnθ. Line L hs grdient m. Line L hs grdient m. L is prllel to L m = km. L is perpendiculr to L m m =. 6

7 3. Ares of tringle: φ θ Are of ABC = = ( 3 3 OR Are of ABC = ( AB)( AC)sinθ (nti - clockwise direction) Eqution of stright line: = m + c, where m = grdient, c = -intercept Eqution of horizontl line: =, where is constnt. Eqution of verticl line: =, where is constnt. An -intercept is point where the grph cuts the -is. It is found b letting = 0. An -intercept is point where the grph cuts the -is. It is found b letting = 0. Sine Rule sinθ sinφ = BC AB Cosine Rule BC = AB + AC ( AB)( AC) cosθ 5) Liner Lw Re-write Non-Liner equtions into Liner equtions of form = m + c ) Non-liner equtions Liner Equtions Y-is X-is Grdient, m Y-intercept, C = m ln = m ln + ln ln ln m ln = b ln = ln b + ln ln ln b ln 7

8 6) Trigometric Function Trigonometric Rtio of Specil Angles sin = 3 cos 3 = tn 3 = Note: For ese of remembering, observe tht vlue of sin increses from to to. Alws write the vlue of the specil ngles in terms of the surd form insted of for or for Complementr Angles sin (90 - θ) = cos θ tn (90 - θ) = cot θ sec(90 - θ)= ( ) = = cosecθ cos (90 - θ) = sin θ cot (90 - θ) = tn θ cosec(90 - θ)= ( ) = = secθ Negtive Angles A positive ngle is n nti-clockwise rottion from the positive -is bout the origin. A negtive ngle is clockwise rottion from the positive -is bout the origin. For n ngle θ, cos (-θ) = sin (-θ) = tn (-θ) = cos θ - sin θ - tn θ 8

9 Smll Angle Approimtion Note: From the power series epnsions of sin, cos nd tn respectivel, when is smll nd mesured in rdins, sin cos tn.. Quotient Reltionships sinθ tn θ = cosθ cosθ cot θ = sinθ secθ = cosθ cos ecθ = sin b. Pthgors Trigonometric Identities c. R-Formul θ sin θ + cos θ = tn θ + = sec θ + cot θ = cos ec θ For > 0, b > 0 & cute ngle α, cos θ ± b sin θ R cos (θ α ) sin θ ± b cos θ R sin (θ ± α ) where R = +, tn α = The following re provided in the MF5: - Addition Formul - Double Angle Formul - Fctor Formul 9

10 d. Grphs of sin, cos nd tn functions = sin = cos = tn Amplitude = Amplitude = Amplitude undefined Period = 360º ( ) Period = 360º ( ) Period = 80º ( ) 7) Tngent nd Norml For curve = f(), d (i) Grdient of tngent = or f '( ) d d (ii) Grdient of tngent t (, b) = or f '( ). d (iii) Eqution of tngent t (, b): b= f '( )( ). (iv) Eqution of norml t (, b): b= ( ). f '( ) 8) Properties of Circle Smmetricl Properties of Circles A line through the centre of circle nd perpendiculr to chord will bisect the chord. = If OMQ = 90º then MP = MQ or vice vers Note: This propert is true for ll isosceles or equilterl tringle. Equl chords re equidistnt from the centre. If PQ = RS then OX = OY or vice vers 0

11 A tngent to circle is perpendiculr to the rdius. Tngents from n eternl point to circle re equl nd subtend equl ngles t the centre. OT bisects POQ & PTQ i.e. TP = TQ, POT = QOT & PTO = QTO Angle Properties of Circles Angle t Centre = Twice Angle t circumference POR = PQR Note: refer to three different digrms Angle in semi-circle PQR = 90º Note: PR is dimeter of circle

12 Angles in the sme segment PQR = QPR = PRS QSR Note: ngles t the circumference re subtended b the sme rc Opp. Angles in cclic qud SRQ + SPQ = 80º RSP + RQP = 80º Note: ll 4 sides must touch the circumference.

### The 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

### A 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

### Loudoun 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!

### 03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note t

A-PDF Wtermrk DEMO: Purchse from www.a-pdf.com to remove the wtermrk Add Mths Formule List: Form 4 (Updte 8/9/08) 0 Functions Asolute Vlue Function Inverse Function If f ( x ), if f ( x ) 0 f ( x) y f

### 15 - 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

### Edexcel 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

### R(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

### ( 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

### BRIEF NOTES ADDITIONAL MATHEMATICS FORM

BRIEF NOTES ADDITIONAL MATHEMATICS FORM CHAPTER : FUNCTION. : + is the object, + is the imge : + cn be written s () = +. To ind the imge or mens () = + = Imge or is. Find the object or 8 mens () = 8 wht

### Form 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

### MA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations

LESSON 0 Chpter 7.2 Trigonometric Integrls. Bsic trig integrls you should know. sin = cos + C cos = sin + C sec 2 = tn + C sec tn = sec + C csc 2 = cot + C csc cot = csc + C MA 6200 Em 2 Study Guide, Fll

### Ellipse. 1. Defini t ions. FREE Download Study Package from website: 11 of 91CONIC SECTION

FREE Downlod Stud Pckge from wesite: www.tekoclsses.com. Defini t ions Ellipse It is locus of point which moves in such w tht the rtio of its distnce from fied point nd fied line (not psses through fied

### CONIC SECTIONS. Chapter 11

CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round

### Higher 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

### P 1 (x 1, y 1 ) is given by,.

MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce

JEE Advnced Mths Assignment Onl One Correct Answer Tpe. The locus of the orthocenter of the tringle formed the lines (+P) P + P(+P) = 0, (+q) q+q(+q) = 0 nd = 0, where p q, is () hperol prol n ellipse

### 10 If 3, a, b, c, 23 are in A.S., then a + b + c = 15 Find the perimeter of the sector in the figure. A. 1:3. A. 2.25cm B. 3cm

HK MTHS Pper II P. If f ( x ) = 0 x, then f ( y ) = 6 0 y 0 + y 0 y 0 8 y 0 y If s = ind the gretest vlue of x + y if ( x, y ) is point lying in the region O (including the boundry). n [ + (n )d ], then

### Polynomials 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

### ICSE Board Class IX Mathematics Paper 4 Solution

ICSE Bord Clss IX Mthemtics Pper Solution SECTION A (0 Mrks) Q.. () Consider x y 6 5 5 x y 6 5 5 0 6 0 6 x y 6 50 8 5 6 7 6 x y 6 7 6 x y 6 x 7,y (b) Dimensions of the brick: Length (l) = 0 cm, bredth

### GEOMETRICAL PROPERTIES OF ANGLES AND CIRCLES, ANGLES PROPERTIES OF TRIANGLES, QUADRILATERALS AND POLYGONS:

GEOMETRICL PROPERTIES OF NGLES ND CIRCLES, NGLES PROPERTIES OF TRINGLES, QUDRILTERLS ND POLYGONS: 1.1 TYPES OF NGLES: CUTE NGLE RIGHT NGLE OTUSE NGLE STRIGHT NGLE REFLEX NGLE 40 0 4 0 90 0 156 0 180 0

### REVIEW SHEET FOR PRE-CALCULUS MIDTERM

. If A, nd B 8, REVIEW SHEET FOR PRE-CALCULUS MIDTERM. For the following figure, wht is the eqution of the line?, write n eqution of the line tht psses through these points.. Given the following lines,

TABLE OF CONTENTS 3 CHAPTER 1 Set Lnguge & Nottion 3 CHAPTER 2 Functions 3 CHAPTER 3 Qudrtic Functions 4 CHAPTER 4 Indices & Surds 4 CHAPTER 5 Fctors of Polynomils 4 CHAPTER 6 Simultneous Equtions 4 CHAPTER

### Lesson-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).

### MTH 4-16a Trigonometry

MTH 4-16 Trigonometry Level 4 [UNIT 5 REVISION SECTION ] I cn identify the opposite, djcent nd hypotenuse sides on right-ngled tringle. Identify the opposite, djcent nd hypotenuse in the following right-ngled

### Thomas 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

### Unit 1 Exponentials and Logarithms

HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 1 Unit 1 Eponentils nd Logrithms (2) Eponentil Functions (3) The number e (4) Logrithms (5) Specil Logrithms (7) Chnge of Bse Formul (8) Logrithmic Functions (10)

### Linear 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

### Chapter 1: Logarithmic functions and indices

Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4

### Optimization 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

### Higher Maths. Self Check Booklet. visit for a wealth of free online maths resources at all levels from S1 to S6

Higher Mths Self Check Booklet visit www.ntionl5mths.co.uk for welth of free online mths resources t ll levels from S to S6 How To Use This Booklet You could use this booklet on your own, but it my be

### A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)

A-Level Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length: - hours work (depending on prior knowledge) This trnsition tsk provides revision

### ( ) 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

### SCHEME OF WORK FOR IB MATHS STANDARD LEVEL

Snnrpsgymnsiet Lott Hydén Mthemtics, Stndrd Level Curriculum SCHEME OF WORK FOR IB MATHS STANDARD LEVEL Min resource: Mthemtics for the interntionl student, Mthemtics SL, Hese PART 1 Sequences nd Series

### TO: Next Year s AP Calculus Students

TO: Net Yer s AP Clculus Students As you probbly know, the students who tke AP Clculus AB nd pss the Advnced Plcement Test will plce out of one semester of college Clculus; those who tke AP Clculus BC

### Lecture 0. MATH REVIEW for ECE : LINEAR CIRCUIT ANALYSIS II

Lecture 0 MATH REVIEW for ECE 000 : LINEAR CIRCUIT ANALYSIS II Aung Kyi Sn Grdute Lecturer School of Electricl nd Computer Engineering Purdue University Summer 014 Lecture 0 : Mth Review Lecture 0 is intended

### First Semester Review Calculus BC

First Semester Review lculus. Wht is the coordinte of the point of inflection on the grph of Multiple hoice: No lcultor y 3 3 5 4? 5 0 0 3 5 0. The grph of piecewise-liner function f, for 4, is shown below.

### / 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

### 6.2 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.

### PARABOLA 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

### CET 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

### Drill Exercise Find the coordinates of the vertices, foci, eccentricity and the equations of the directrix of the hyperbola 4x 2 25y 2 = 100.

Drill Exercise - 1 1 Find the coordintes of the vertices, foci, eccentricit nd the equtions of the directrix of the hperol 4x 5 = 100 Find the eccentricit of the hperol whose ltus-rectum is 8 nd conjugte

### ARITHMETIC 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

### Section 13.1 Right Triangles

Section 13.1 Right Tringles Ojectives: 1. To find vlues of trigonometric functions for cute ngles. 2. To solve tringles involving right ngles. Review - - 1. SOH sin = Reciprocl csc = 2. H cos = Reciprocl

### ES.182A Topic 32 Notes Jeremy Orloff

ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In

### approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below

. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.

### , MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF

DOWNLOAD FREE FROM www.tekoclsses.com, PH.: 0 903 903 7779, 98930 5888 Some questions (Assertion Reson tpe) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs

### Mathematics 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

### Math 113 Exam 2 Practice

Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number

### + R 2 where R 1. MULTIPLE CHOICE QUESTIONS (MCQ's) (Each question carries one mark)

2. C h p t e r t G l n c e is the set of ll points in plne which re t constnt distnce from fixed point clled centre nd constnt distnce is known s rdius of circle. A tngent t ny point of circle is perpendiculr

### Chapter 8: Methods of Integration

Chpter 8: Methods of Integrtion Bsic Integrls 8. Note: We hve the following list of Bsic Integrls p p+ + c, for p sec tn + c p + ln + c sec tn sec + c e e + c tn ln sec + c ln + c sec ln sec + tn + c ln

### Chapter 6 Techniques of Integration

MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln

### THE DISCRIMINANT & ITS APPLICATIONS

THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used

### MDPT Practice Test 1 (Math Analysis)

MDPT Prctice Test (Mth Anlysis). Wht is the rdin mesure of n ngle whose degree mesure is 7? ) 5 π π 5 c) π 5 d) 5 5. In the figure to the right, AB is the dimeter of the circle with center O. If the length

### AP Calculus AB Summer Packet

AP Clculus AB Summer Pcket Nme: Welcome to AP Clculus AB! Congrtultions! You hve mde it to one of the most dvnced mth course in high school! It s quite n ccomplishment nd you should e proud of yourself

### Year 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

### Precalculus Due Tuesday/Wednesday, Sept. 12/13th Mr. Zawolo with questions.

Preclculus Due Tuesd/Wednesd, Sept. /th Emil Mr. Zwolo (isc.zwolo@psv.us) with questions. 6 Sketch the grph of f : 7! nd its inverse function f (). FUNCTIONS (Chpter ) 6 7 Show tht f : 7! hs n inverse

### Set 1 Paper 2. 1 Pearson Education Asia Limited 2017

. A. A. C. B. C 6. A 7. A 8. B 9. C. D. A. B. A. B. C 6. D 7. C 8. B 9. C. D. C. A. B. A. A 6. A 7. A 8. D 9. B. C. B. D. D. D. D 6. D 7. B 8. C 9. C. D. B. B. A. D. C Section A. A (68 ) [ ( ) n ( n 6n

### Time : 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

### ( x )( x) dx. Year 12 Extension 2 Term Question 1 (15 Marks) (a) Sketch the curve (x + 1)(y 2) = 1 2

Yer Etension Term 7 Question (5 Mrks) Mrks () Sketch the curve ( + )(y ) (b) Write the function in prt () in the form y f(). Hence, or otherwise, sketch the curve (i) y f( ) (ii) y f () (c) Evlute (i)

### 1. 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)

### KEY 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

### Level I MAML Olympiad 2001 Page 1 of 6 (A) 90 (B) 92 (C) 94 (D) 96 (E) 98 (A) 48 (B) 54 (C) 60 (D) 66 (E) 72 (A) 9 (B) 13 (C) 17 (D) 25 (E) 38

Level I MAML Olympid 00 Pge of 6. Si students in smll clss took n em on the scheduled dte. The verge of their grdes ws 75. The seventh student in the clss ws ill tht dy nd took the em lte. When her score

### IMPORTANT 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

### Summary Information and Formulae MTH109 College Algebra

Generl Formuls Summry Informtion nd Formule MTH109 College Algebr Temperture: F = 9 5 C + 32 nd C = 5 ( 9 F 32 ) F = degrees Fhrenheit C = degrees Celsius Simple Interest: I = Pr t I = Interest erned (chrged)

### Algebra & Functions (Maths ) opposite side

Instructor: Dr. R.A.G. Seel Trigonometr Algebr & Functions (Mths 0 0) 0th Prctice Assignment hpotenuse hpotenuse side opposite side sin = opposite hpotenuse tn = opposite. Find sin, cos nd tn in 9 sin

### Year 12 Trial Examination Mathematics Extension 1. Question One 12 marks (Start on a new page) Marks

THGS Mthemtics etension Tril 00 Yer Tril Emintion Mthemtics Etension Question One mrks (Strt on new pge) Mrks ) If P is the point (-, 5) nd Q is the point (, -), find the co-ordintes of the point R which

### Trigonometric Functions

Trget Publictions Pvt. Ltd. Chpter 0: Trigonometric Functions 0 Trigonometric Functions. ( ) cos cos cos cos (cos + cos ) Given, cos cos + 0 cos (cos + cos ) + ( ) 0 cos cos cos + 0 + cos + (cos cos +

### Alg. 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 ) ) (,

### Quotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m

Formuls nd Concepts MAT 099: Intermedite Algebr repring for Tests: The formuls nd concepts here m not be inclusive. You should first tke our prctice test with no notes or help to see wht mteril ou re comfortble

### 13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in

### Nat 5 USAP 3(b) This booklet contains : Questions on Topics covered in RHS USAP 3(b) Exam Type Questions Answers. Sourced from PEGASYS

Nt USAP This ooklet contins : Questions on Topics covered in RHS USAP Em Tpe Questions Answers Sourced from PEGASYS USAP EF. Reducing n lgeric epression to its simplest form / where nd re of the form (

### Stage 11 Prompt Sheet

Stge 11 rompt Sheet 11/1 Simplify surds is NOT surd ecuse it is exctly is surd ecuse the nswer is not exct surd is n irrtionl numer To simplify surds look for squre numer fctors 7 = = 11/ Mnipulte expressions

### at its center, then the measure of this angle in radians (abbreviated rad) is the length of the arc that subtends the angle.

Notes 6 ngle Mesure Definition of Rdin If circle of rdius is drwn with the vertex of n ngle Mesure: t its center, then the mesure of this ngle in rdins (revited rd) is the length of the rc tht sutends

### Precalculus Spring 2017

Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify

### PHYS 1114, Lecture 1, January 18 Contents:

PHYS 1114, Lecture 1, Jnury 18 Contents: 1 Discussed Syllus (four pges). The syllus is the most importnt document. You should purchse the ExpertTA Access Code nd the L Mnul soon! 2 Reviewed Alger nd Strted

### 2) Three noncollinear points in Plane M. [A] A, D, E [B] A, B, E [C] A, B, D [D] A, E, H [E] A, H, M [F] H, A, B

Review Use the points nd lines in the digrm to identify the following. 1) Three colliner points in Plne M. [],, H [],, [],, [],, [],, M [] H,, M 2) Three noncolliner points in Plne M. [],, [],, [],, [],,

### Unit 5. Integration techniques

18.01 EXERCISES Unit 5. Integrtion techniques 5A. Inverse trigonometric functions; Hyperbolic functions 5A-1 Evlute ) tn 1 3 b) sin 1 ( 3/) c) If θ = tn 1 5, then evlute sin θ, cos θ, cot θ, csc θ, nd

### MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK 11 WRITTEN EXAMINATION 2 SOLUTIONS SECTION 1 MULTIPLE CHOICE QUESTIONS

MASTER CLASS PROGRAM UNIT 4 SPECIALIST MATHEMATICS WEEK WRITTEN EXAMINATION SOLUTIONS FOR ERRORS AND UPDATES, PLEASE VISIT WWW.TSFX.COM.AU/MC-UPDATES SECTION MULTIPLE CHOICE QUESTIONS QUESTION QUESTION

### AQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions

Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic

### NAME: MR. WAIN FUNCTIONS

NAME: M. WAIN FUNCTIONS evision o Solving Polnomil Equtions i one term in Emples Solve: 7 7 7 0 0 7 b.9 c 7 7 7 7 ii more thn one term in Method: Get the right hnd side to equl zero = 0 Eliminte ll denomintors

### Obj: SWBAT Recall the many important types and properties of functions

Obj: SWBAT Recll the mny importnt types nd properties of functions Functions Domin nd Rnge Function Nottion Trnsformtion of Functions Combintions/Composition of Functions One-to-One nd Inverse Functions

### Chapter 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

### MATHEMATICS 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

### Mathematics 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

### Mathematics 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

### Answers to Exercises. c 2 2ab b 2 2ab a 2 c 2 a 2 b 2

Answers to Eercises CHAPTER 9 CHAPTER LESSON 9. CHAPTER 9 CHAPTER. c 9. cm. cm. b 5. cm. d 0 cm 5. s cm. c 8.5 cm 7. b cm 8.. cm 9. 0 cm 0. s.5 cm. r cm. 7 ft. 5 m.. cm 5.,, 5. 8 m 7. The re of the lrge

### ( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.

AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find

### A sequence is a list of numbers in a specific order. A series is a sum of the terms of a sequence.

Core Module Revision Sheet The C exm is hour 30 minutes long nd is in two sections. Section A (36 mrks) 8 0 short questions worth no more thn 5 mrks ech. Section B (36 mrks) 3 questions worth mrks ech.

### 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) (, )

### SOLUTION OF TRIANGLES

SOLUTION OF TIANGLES DPP by VK Sir B.TEH., IIT DELHI VK lsses, -9-40, Indr Vihr, Kot. Mob. No. 989060 . If cos A + cosb + cos = then the sides of the AB re in A.P. G.P H.P. none. If in tringle sin A :

### ES 111 Mathematical Methods in the Earth Sciences Lecture Outline 1 - Thurs 28th Sept 17 Review of trigonometry and basic calculus

ES 111 Mthemticl Methods in the Erth Sciences Lecture Outline 1 - Thurs 28th Sept 17 Review of trigonometry nd bsic clculus Trigonometry When is it useful? Everywhere! Anything involving coordinte systems

### Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...

Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting

### RAM RAJYA MORE, SIWAN. XI th, XII th, TARGET IIT-JEE (MAIN + ADVANCE) & COMPATETIVE EXAM FOR XII (PQRS) INDEFINITE INTERATION & Their Properties

M.Sc. (Mths), B.Ed, M.Phil (Mths) MATHEMATICS Mob. : 947084408 9546359990 M.Sc. (Mths), B.Ed, M.Phil (Mths) RAM RAJYA MORE, SIWAN XI th, XII th, TARGET IIT-JEE (MAIN + ADVANCE) & COMPATETIVE EXAM FOR XII

### m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r

CO-ORDINTE GEOMETR II I Qudrnt Qudrnt (-.+) (++) X X - - - 0 - III IV Qudrnt - Qudrnt (--) - (+-) Region CRTESIN CO-ORDINTE SSTEM : Retngulr Co-ordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr

### ntegration (p3) Integration by Inspection When differentiating using function of a function or the chain rule: If y = f(u), where in turn u = f(x)

ntegrtion (p) Integrtion by Inspection When differentiting using function of function or the chin rule: If y f(u), where in turn u f( y y So, to differentite u where u +, we write ( + ) nd get ( + ) (.

### Anti-derivatives/Indefinite Integrals of Basic Functions

Anti-derivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second

### k ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola.

Stndrd Eqution of Prol with vertex ( h, k ) nd directrix y = k p is ( x h) p ( y k ) = 4. Verticl xis of symmetry Stndrd Eqution of Prol with vertex ( h, k ) nd directrix x = h p is ( y k ) p( x h) = 4.

### AB Calculus Review Sheet

AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is