STRAND B: NUMBER THEORY

Size: px
Start display at page:

Download "STRAND B: NUMBER THEORY"

Transcription

1 Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet STRAND B: NUMBER THEORY B Indices nd Fctors Tet Contents Section B. Squres, Cubes, Squre Roots nd Cube Roots B. Inde Nottion B. Fctors B. Prime Fctors, HCF nd LCM B. Further Inde Nottion

2 Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet B Indices nd Fctors B. Squres, Cubes, Squre Roots nd Cube Roots When number is multiplied by itself, we sy tht the number hs been squred. For emple, squred mens 9. This is written s 9. We could lso sy tht 9 is the squre of. When number is cubed it is written down times nd multiplied. For emple cubed mens 8. This is written s 8. We could lso sy tht 8 is the cube of. Sometimes the reverse process is needed to nswer questions such s: Wht number squred gives The nswer would be. We sy tht is the squre root of, or write. Another question might be: Wht number cubed gives 8 The nswer would be. We would sy tht the cube root of 8 is. We could lso write 8. Worked Emple Find () 8 (b) (c). Use your nswers to find (d) (e) (f) () (b) (c) (d) 8 becuse 8 (e) becuse (f) becuse

3 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet Eercises. Find () (b) (c) (d) Use your nswers to find (e) (f) (g) 9 (h). Find () (b) (c) (d) 0 Use your nswers to find (e) (f) 000 (g) (h). Find () 0 (b) (c) (d) (e) 8 (f) 9 (g) (h) (i) 8 (j) 0 (k) 0 (l). Find () 00 (b) (c) 8 (d) (e) (f) 9. Use clcultor to find () (b) (c) (d) (e) (f) (g) 0 (h) Without clcultor, find (i) (j) 00 (k) 9 (l) (m) (n) 9 (o) (p). Find () + (b) (c) 0 + (d) + (e) (f) + (g) + 0 (h) + 8 Informtion On verge, humn hert bets times minute, 00 times n hour, times dy, times yer nd times for someone who lives 80 yers.

4 Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet B. Inde Nottion Inde nottion is very useful wy of writing epressions like in shorter formt. The bove could be written with inde nottion s. The smll number,, is clled the inde or power. Worked Emple Find () (b) (c) () (b) (c) 8 0 Worked Emple Find the missing number. () (b) (c) ( ) ( ) () 0 ( ) ( ) (b) (c) Note m n m+ n nd n m nm These rules pply whenever inde nottion is used.

5 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet Using these rules, 0 or So 0 Worked Emple Find () ( ) (b) ( ) ( ) ( ) ( ) ( ) ( ) () ( ) ( ) ( ) ( ) (b) Note Eercises ( ) m n m n. Write ech of the following using inde nottion. () (b) (c) (d) (e) (f) 9 9 (g) (h) (i) (j) Find the vlue of ech of the following. () (b) (c) (d) 0 (e) 0 (f) (g) (h) (i) 8 (j) (k) 0 (l)

6 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet. Fill in the missing numbers. () (b) (c) (d) (e) (f) 9 (g) (h) (i) 0 (j) (k) (m) (l) (n) 0 (o) 8 (p) (q) (r) 0 (s) (t) 8 0 (u). Fill in the missing numbers. () (b) 8 (c) (d) (e) (f) (g) (h) 8 (i). Simplify the following epressions, giving your nswer in inde nottion. () (b) (c) (d) (e) (f) (g) (h) (i) (j) 8 (k) 8 (m) (n) (l) 9 9 (o) 0 8. Fill in the missing powers. () 8 (b) (c) (d) (e) 8 (f) (g) (h) (i) 9 (j) (k) (l). Simplify the following, giving your nswers in inde form. () ( ) (b) ( ) (c) ( ) (d) ( ) (e) ( ) (f) ( ) (g) ( ) (h) ( ) (i) ( )

7 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet 8. Fill in the missing numbers. ( ) (b) ( ) (c) 0 ( ) 0 ( ) (e) ( 0 ) 0 (f) ( ) () (d) 9. Simplify ech of the following, giving your nswer in inde nottion. 0 () (b) (c) (d) (e) (f) 8 (g) (h) 8 9 (i) 0 0. Simplify ech of the following epressions. () (b) (c) 0 (d) (e) y y (f) p p (g) q q (h) (i) b b (j) (m) b 0 b (k) c c (l) y (n) y (o) 8 (p) p p p (q) 0 (r) y y y y (s) (t) (u) 8 0 (v) ( ) (w) ( ) () ( ). cn be written s. Find the vlues of p nd q in the following: () p (b) q. Epress s simply s possible: Chllenge! You open book. Two pges fce you. If the product of the two pge numbers is 9, wht re the two pge numbers

8 Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet B. Fctors A fctor of positive whole number is positive whole number tht will divide ectly into it. Worked Emple List ll the fctors of 0. The fctors of 0 re:,,,, 0, 0 These re ll numbers tht divide ectly into 0. Worked Emple Write the number s the product of two fctors in s mny wys s possible. Eercises. List the fctors of these numbers. () (b) (c) (d) (e) 8 (f) (g) 0 (h) 00 (i) (j) 0 (k) (l) 8. Write ech number below s the product of two fctors in s mny wys s possible. () 0 (b) 8 (c) (d) 9 (e) (f) (g) (h). Fill in the missing numbers. () (b) (c) (d) 0 (e) 0 (f) 88 (g) (h). Here is Bingo crd

9 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet () (b) Circle those numbers tht will divide into ectly. Cross out those numbers tht will divide into ectly () In the row of numbers bove: (i) circle ll numbers divisible by, e.g. 0 (ii) cross out ll numbers divisible by, e.g. (iii) underline ll numbers divisible by. e.g. (b) Describe the numbers which re not circled, crossed out or underlined.. A pttern of counting numbers is shown.,,,, 8, 9, 0,... () (i) Which of these numbers is squre number (ii) Which of these numbers is multiple of nine The pttern is continued. (b) (i) Wht is the net squre number (ii) Wht is the net number tht is multiple of nine B. Prime Fctors, HCF nd LCM Any positive whole number cn be written s the product of number of prime fctors. For emple, 0 Note or 80 A prime number is positive whole number with ectly two fctors; nd itself. The first few prime numbers re,,,,,... Worked Emple Write the number s product of prime numbers. Write s product of two fctors: 8 But 8 9 so 9 But 9 so 8

10 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet This epression contins only prime numbers, so This is clled the product of prime fctors. Another importnt concept is tht of the highest common fctor (HCF) of two (or more) positive integers. The HCF is the lrgest number which is fctor of both (or ll) the numbers. Worked Emple Find the HCF of 0 nd 0. Epressing both 0 nd 0 in terms of their prime fctors gives 0 0 It is esy now to see tht the highest common fctor is. Worked Emple () Write the numbers 0 nd 0 s the product of prime fctors. (b) Find the lrgest common fctor tht will divide into both 0 nd 0. () 0 0 So s product of prime fctors, 0 So s product of prime fctors, 0 0 (b) To find the lrgest common fctor tht will divide into both 0 nd 0, look t the fctors common to ech of the products of primes. The numbers tht pper in both re nd, so the lrgest number tht will divide into both 0 nd 0 is 0. So 0 is the HCF of 0 nd 0. 9

11 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet A relted concept is tht of the lowest common multiple, LCM, of two (or more) positive integers. This is the lowest number into which the two (or ll) numbers cn divide ectly. Worked Emple Find the LCM of nd 0. One wy to find the LCM is to write out multiples of ech number. For emple, multiples of re, 8,, 9, 0,, 8, 9,, 0,... multiples of 0 re 0, 0, 80, 0, 00, 0,... It is esy to see tht 0 is the LCM of nd 0. Another wy is to epress nd 0 in terms of their prime fctors: ( ) ( ) 0 Noting tht ( ) is common to both numbers, the LCM is given by 0 or 0 0. So the LCM 0 Note tht 0 nd 0 0. Eercises. Which of the following re prime numbers,,,,, 9,,, 8, 9,,,. Which numbers between 0 nd 0 re prime numbers. Write ech number below s product of prime fctors. () 0 (b) (c) 8 (d) 8 (e) 0 (f) 0 (g) 9 (h) 8 (i) 00. () Epress nd s the product of prime fctors. (b) By compring the nswers to () find the HCF of nd.. Find the highest common fctors of ech pir of numbers below. (), (b) 0, (c), 0 (d), 0 (e) 0, 80 (f) 0, (g), 0 (h), (i), 0

12 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet. () Epress ech of the following numbers s the product of prime fctors:, 99, (b) (c) By considering the products of the prime fctors, find the highest common fctor of (i) nd 99 (ii) 99 nd (iii) nd Wht is the highest common fctor of ll three numbers. Find the HCF for ech set of three numbers given below. () 0,, 0 (b) 90,, 0 (c),, (d) 00, 0, 0 (e), 8, (f) 0, 0, 0 (g),, (h) 0,, 8 (i) 008, 0, 8. Find the LCM of () nd (b) nd 0 (c) 8 nd 9 (d) 9 nd (e) 0 nd (f) 8 nd 9 9. Find the LCM of ech of the following sets of numbers. () 8,, 0 (b), 8, (c), 0, (d) 9, 8, (e) 90, 80, (f), 0, 8 B. Further Inde Nottion Indices cn lso be negtive or frctions. The rules below eplin how to use these types of indices. n n This is clled the reciprocl of. n n Worked Emple Find: () (b) (c) (d) (e) 8 (d) 9

13 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet () (b) 9 (c) (d) ( ) (e) 8 8 (f) 9 9 Worked Emple Find () (b) m m (d) 8 ( ) (e) b ( ) (c) (f) m + () (b) m m m m m (c) 8 8+ ( ) ( ) (d) 9 ( ) 9

14 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet ( ) (e) b b (f) m ( m ) b m Eercises m. Find s frctions tht do not involve indices, without using clcultor: () (b) (c) (d) (e) 9 (f) (g) (h) (i) (j) (k) (l) (m) 9 (n) (o) 8. Complete the missing numbers, without using clcultor. () (b) (c) 8 (d) (e) (f) 9 (g) (h) (i) (j) (m) (k) (n) m m (l) (o) 00 p 0 p (p) q q (q) q q (r) q q. Use clcultor to find: () 8 (b) 0 (c) (d) (e) (f) 0 (g) 8 (h) (i) (j) (k) 9 (l). Simplify the following epressions, so tht they contin no negtive indices. () (b) (c) 9 (d) (e) ( ) (f) ( )

15 B. Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet ( ) (g) (j) (h) ( ) (k) 9 ( ) ( ) (i) ( ) ( ) (l) (m) b ( ) (n) b (o) b ( ) (p) ( b ) (q) (r) m n b ( ) (s) b 0 (t) m (u) 8 b c (v) m (w) y z () 8 ( b ). () Epress 8 (b) Simplify. s frction in the form, where nd b re integers. b y (c) Find the vlue of y for which. Investigtion Find four integers,, b, c nd d such tht + b + c d.

Chapter 1: Logarithmic functions and indices

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

More information

Special Numbers, Factors and Multiples

Special Numbers, Factors and Multiples Specil s, nd Student Book - Series H- + 3 + 5 = 9 = 3 Mthletics Instnt Workooks Copyright Student Book - Series H Contents Topics Topic - Odd, even, prime nd composite numers Topic - Divisiility tests

More information

5.2 Exponent Properties Involving Quotients

5.2 Exponent Properties Involving Quotients 5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use

More information

MATHEMATICS AND STATISTICS 1.2

MATHEMATICS AND STATISTICS 1.2 MATHEMATICS AND STATISTICS. Apply lgebric procedures in solving problems Eternlly ssessed 4 credits Electronic technology, such s clcultors or computers, re not permitted in the ssessment of this stndr

More information

Adding and Subtracting Rational Expressions

Adding and Subtracting Rational Expressions 6.4 Adding nd Subtrcting Rtionl Epressions Essentil Question How cn you determine the domin of the sum or difference of two rtionl epressions? You cn dd nd subtrct rtionl epressions in much the sme wy

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils

More information

Lesson 1: Quadratic Equations

Lesson 1: Quadratic Equations Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring

More information

Before we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!

Before we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!! Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble

More information

SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER KNOWHOW STUDY AND LEARNING CENTRE SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18

More information

Equations and Inequalities

Equations and Inequalities Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in

More information

Sections 1.3, 7.1, and 9.2: Properties of Exponents and Radical Notation

Sections 1.3, 7.1, and 9.2: Properties of Exponents and Radical Notation Sections., 7., nd 9.: Properties of Eponents nd Rdicl Nottion Let p nd q be rtionl numbers. For ll rel numbers nd b for which the epressions re rel numbers, the following properties hold. i = + p q p q

More information

Consolidation Worksheet

Consolidation Worksheet Cmbridge Essentils Mthemtics Core 8 NConsolidtion Worksheet N Consolidtion Worksheet Work these out. 8 b 7 + 0 c 6 + 7 5 Use the number line to help. 2 Remember + 2 2 +2 2 2 + 2 Adding negtive number is

More information

0.1 THE REAL NUMBER LINE AND ORDER

0.1 THE REAL NUMBER LINE AND ORDER 6000_000.qd //0 :6 AM Pge 0-0- CHAPTER 0 A Preclculus Review 0. THE REAL NUMBER LINE AND ORDER Represent, clssify, nd order rel numers. Use inequlities to represent sets of rel numers. Solve inequlities.

More information

MATH FIELD DAY Contestants Insructions Team Essay. 1. Your team has forty minutes to answer this set of questions.

MATH FIELD DAY Contestants Insructions Team Essay. 1. Your team has forty minutes to answer this set of questions. MATH FIELD DAY 2012 Contestnts Insructions Tem Essy 1. Your tem hs forty minutes to nswer this set of questions. 2. All nswers must be justified with complete explntions. Your nswers should be cler, grmmticlly

More information

fractions Let s Learn to

fractions Let s Learn to 5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin

More information

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

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

More information

Sample pages. 9:04 Equations with grouping symbols

Sample pages. 9:04 Equations with grouping symbols Equtions 9 Contents I know the nswer is here somewhere! 9:01 Inverse opertions 9:0 Solving equtions Fun spot 9:0 Why did the tooth get dressed up? 9:0 Equtions with pronumerls on both sides GeoGebr ctivity

More information

3 x x x 1 3 x a a a 2 7 a Ba 1 NOW TRY EXERCISES 89 AND a 2/ Evaluate each expression.

3 x x x 1 3 x a a a 2 7 a Ba 1 NOW TRY EXERCISES 89 AND a 2/ Evaluate each expression. SECTION. Eponents nd Rdicls 7 B 7 7 7 7 7 7 7 NOW TRY EXERCISES 89 AND 9 7. EXERCISES CONCEPTS. () Using eponentil nottion, we cn write the product s. In the epression 3 4,the numer 3 is clled the, nd

More information

Calculus 2: Integration. Differentiation. Integration

Calculus 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 information

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

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

More information

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

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

More information

LCM AND HCF. Type - I. Type - III. Type - II

LCM AND HCF. Type - I. Type - III. Type - II LCM AND HCF Type - I. The HCF nd LCM of two numbers re nd 9 respectively. Then the number of such pirs () 0 () () (SSC CGL Tier-I Exm. 0 Second Sitting). The product of two numbers 08 nd their HCF. The

More information

Multiplying integers EXERCISE 2B INDIVIDUAL PATHWAYS. -6 ì 4 = -6 ì 0 = 4 ì 0 = -6 ì 3 = -5 ì -3 = 4 ì 3 = 4 ì 2 = 4 ì 1 = -5 ì -2 = -6 ì 2 = -6 ì 1 =

Multiplying integers EXERCISE 2B INDIVIDUAL PATHWAYS. -6 ì 4 = -6 ì 0 = 4 ì 0 = -6 ì 3 = -5 ì -3 = 4 ì 3 = 4 ì 2 = 4 ì 1 = -5 ì -2 = -6 ì 2 = -6 ì 1 = EXERCISE B INDIVIDUAL PATHWAYS Activity -B- Integer multipliction doc-69 Activity -B- More integer multipliction doc-698 Activity -B- Advnced integer multipliction doc-699 Multiplying integers FLUENCY

More information

UNIT 3 Indices and Standard Form Activities

UNIT 3 Indices and Standard Form Activities UNIT 3 Indices nd Stndrd Form Activities Activities 3.1 Towers 3.2 Bode's Lw 3.3 Mesuring nd Stndrd Form 3.4 Stndrd Inde Form Notes nd Solutions (1 pge) ACTIVITY 3.1 Towers How mny cubes re needed to build

More information

Chapter 6 Notes, Larson/Hostetler 3e

Chapter 6 Notes, Larson/Hostetler 3e Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn

More information

Lesson 2.4 Exercises, pages

Lesson 2.4 Exercises, pages Lesson. Exercises, pges A. Expnd nd simplify. ) + b) ( ) () 0 - ( ) () 0 c) -7 + d) (7) ( ) 7 - + 8 () ( 8). Expnd nd simplify. ) b) - 7 - + 7 7( ) ( ) ( ) 7( 7) 8 (7) P DO NOT COPY.. Multiplying nd Dividing

More information

Add and Subtract Rational Expressions. You multiplied and divided rational expressions. You will add and subtract rational expressions.

Add and Subtract Rational Expressions. You multiplied and divided rational expressions. You will add and subtract rational expressions. TEKS 8. A..A, A.0.F Add nd Subtrct Rtionl Epressions Before Now You multiplied nd divided rtionl epressions. You will dd nd subtrct rtionl epressions. Why? So you cn determine monthly cr lon pyments, s

More information

7-1: Zero and Negative Exponents

7-1: Zero and Negative Exponents 7-: Zero nd Negtive Exponents Objective: To siplify expressions involving zero nd negtive exponents Wr Up:.. ( ).. 7.. Investigting Zero nd Negtive Exponents: Coplete the tble. Write non-integers s frctions

More information

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite

More information

Chapter 1: Fundamentals

Chapter 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 information

Review Factoring Polynomials:

Review Factoring Polynomials: Chpter 4 Mth 0 Review Fctoring Polynomils:. GCF e. A) 5 5 A) 4 + 9. Difference of Squres b = ( + b)( b) e. A) 9 6 B) C) 98y. Trinomils e. A) + 5 4 B) + C) + 5 + Solving Polynomils:. A) ( 5)( ) = 0 B) 4

More information

Exponentials - Grade 10 [CAPS] *

Exponentials - Grade 10 [CAPS] * OpenStx-CNX module: m859 Exponentils - Grde 0 [CAPS] * Free High School Science Texts Project Bsed on Exponentils by Rory Adms Free High School Science Texts Project Mrk Horner Hether Willims This work

More information

Pre Regional Mathematical Olympiad, 2016 Delhi Region Set C

Pre Regional Mathematical Olympiad, 2016 Delhi Region Set C Pre Regionl Mthemticl Olympid, 06 Delhi Region Set C Mimum Mrks: 50 Importnt Note: The nswer to ech question is n integer between 0 nd 06. Ech Cndidte must write the finl nswer (in the spce provided) s,

More information

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O 1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

More information

MTH 4-16a Trigonometry

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

More information

Jim Lambers MAT 169 Fall Semester Lecture 4 Notes

Jim Lambers MAT 169 Fall Semester Lecture 4 Notes Jim Lmbers MAT 169 Fll Semester 2009-10 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of

More information

Identify graphs of linear inequalities on a number line.

Identify graphs of linear inequalities on a number line. COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line. - When grphing first-degree eqution, solve for the vrible. The grph of this solution will be single point

More information

Chapter 8.2: The Integral

Chapter 8.2: The Integral Chpter 8.: The Integrl You cn think of Clculus s doule-wide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in

More information

Each term is formed by adding a constant to the previous term. Geometric progression

Each term is formed by adding a constant to the previous term. Geometric progression Chpter 4 Mthemticl Progressions PROGRESSION AND SEQUENCE Sequence A sequence is succession of numbers ech of which is formed ccording to definite lw tht is the sme throughout the sequence. Arithmetic Progression

More information

Advanced Algebra & Trigonometry Midterm Review Packet

Advanced Algebra & Trigonometry Midterm Review Packet Nme Dte Advnced Alger & Trigonometry Midterm Review Pcket The Advnced Alger & Trigonometry midterm em will test your generl knowledge of the mteril we hve covered since the eginning of the school yer.

More information

Log1 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?

Log1 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 information

By Ken Standfield, Director Research & Development, KNOWCORP

By Ken Standfield, Director Research & Development, KNOWCORP 1 THE NORMAL DISTRIBUTION METHOD ARTICLE NO.: 10080 By Ken Stndfield, Director Reserch & Development, KNOWCORP http://www.knowcorp.com Emil: ks@knowcorp.com INTRODUCTION The following methods hve been

More information

Algebra Readiness PLACEMENT 1 Fraction Basics 2 Percent Basics 3. Algebra Basics 9. CRS Algebra 1

Algebra Readiness PLACEMENT 1 Fraction Basics 2 Percent Basics 3. Algebra Basics 9. CRS Algebra 1 Algebr Rediness PLACEMENT Frction Bsics Percent Bsics Algebr Bsics CRS Algebr CRS - Algebr Comprehensive Pre-Post Assessment CRS - Algebr Comprehensive Midterm Assessment Algebr Bsics CRS - Algebr Quik-Piks

More information

Section 7.1 Area of a Region Between Two Curves

Section 7.1 Area of a Region Between Two Curves Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region

More information

We 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.

We 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 information

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check

More information

Precalculus Spring 2017

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

More information

1. Extend QR downwards to meet the x-axis at U(6, 0). y

1. Extend QR downwards to meet the x-axis at U(6, 0). y In the digrm, two stright lines re to be drwn through so tht the lines divide the figure OPQRST into pieces of equl re Find the sum of the slopes of the lines R(6, ) S(, ) T(, 0) Determine ll liner functions

More information

Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.

Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral. Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:

More information

THE DISCRIMINANT & ITS APPLICATIONS

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

More information

Matrix Eigenvalues and Eigenvectors September 13, 2017

Matrix Eigenvalues and Eigenvectors September 13, 2017 Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues

More information

Chapter 9 Definite Integrals

Chapter 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

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

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

More information

Here we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system.

Here we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system. Section 24 Nonsingulr Liner Systems Here we study squre liner systems nd properties of their coefficient mtrices s they relte to the solution set of the liner system Let A be n n Then we know from previous

More information

The practical version

The practical version Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht

More information

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

More information

Mathematics Number: Logarithms

Mathematics Number: Logarithms plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement

More information

The graphs of Rational Functions

The graphs of Rational Functions Lecture 4 5A: The its of Rtionl Functions s x nd s x + The grphs of Rtionl Functions The grphs of rtionl functions hve severl differences compred to power functions. One of the differences is the behvior

More information

Bridging the gap: GCSE AS Level

Bridging the gap: GCSE AS Level Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions

More information

Mathematics Extension 1

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

More information

Read section 3.3, 3.4 Announcements:

Read section 3.3, 3.4 Announcements: Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f

More information

p-adic Egyptian Fractions

p-adic Egyptian Fractions p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction

More information

Chapter 5. , r = r 1 r 2 (1) µ = m 1 m 2. r, r 2 = R µ m 2. R(m 1 + m 2 ) + m 2 r = r 1. m 2. r = r 1. R + µ m 1

Chapter 5. , r = r 1 r 2 (1) µ = m 1 m 2. r, r 2 = R µ m 2. R(m 1 + m 2 ) + m 2 r = r 1. m 2. r = r 1. R + µ m 1 Tor Kjellsson Stockholm University Chpter 5 5. Strting with the following informtion: R = m r + m r m + m, r = r r we wnt to derive: µ = m m m + m r = R + µ m r, r = R µ m r 3 = µ m R + r, = µ m R r. 4

More information

Infinite Geometric Series

Infinite Geometric Series Infinite Geometric Series Finite Geometric Series ( finite SUM) Let 0 < r < 1, nd let n be positive integer. Consider the finite sum It turns out there is simple lgebric expression tht is equivlent to

More information

Chapter 11. Sequence and Series

Chapter 11. Sequence and Series Chpter 11 Sequence nd Series Lesson 11-1 Mthemticl Ptterns Sequence A sequence is n ordered list of numbers clled terms. Exmple Pge 591, #2 Describe ech pttern formed. Find the next three terms 4,8,16,32,64,...

More information

4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve

4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions

More information

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions

More information

FUNCTIONS: Grade 11. or y = ax 2 +bx + c or y = a(x- x1)(x- x2) a y

FUNCTIONS: Grade 11. or y = ax 2 +bx + c or y = a(x- x1)(x- x2) a y FUNCTIONS: Grde 11 The prbol: ( p) q or = +b + c or = (- 1)(- ) The hperbol: p q The eponentil function: b p q Importnt fetures: -intercept : Let = 0 -intercept : Let = 0 Turning points (Where pplicble)

More information

7h1 Simplifying Rational Expressions. Goals:

7h1 Simplifying Rational Expressions. Goals: h Simplifying Rtionl Epressions Gols Fctoring epressions (common fctor, & -, no fctoring qudrtics) Stting restrictions Epnding rtionl epressions Simplifying (reducin rtionl epressions (Kürzen) Adding nd

More information

Individual Events I3 a 10 I4. d 90 angle 57 d Group Events. d 220 Probability

Individual Events I3 a 10 I4. d 90 angle 57 d Group Events. d 220 Probability Answers: (98-8 HKMO Finl Events) Creted by: Mr. Frncis Hung Lst updted: 8 Jnury 08 I 800 I Individul Events I 0 I4 no. of routes 6 I5 + + b b 0 b b c *8 missing c 0 c c See the remrk 600 d d 90 ngle 57

More information

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10 University of Wshington Deprtment of Chemistry Chemistry 45 Winter Qurter Homework Assignment 4; Due t 5p.m. on // We lerned tht the Hmiltonin for the quntized hrmonic oscilltor is ˆ d κ H. You cn obtin

More information

Math 131. Numerical Integration Larson Section 4.6

Math 131. Numerical Integration Larson Section 4.6 Mth. Numericl Integrtion Lrson Section. This section looks t couple of methods for pproimting definite integrls numericlly. The gol is to get good pproimtion of the definite integrl in problems where n

More information

AT100 - Introductory Algebra. Section 2.7: Inequalities. x a. x a. x < a

AT100 - Introductory Algebra. Section 2.7: Inequalities. x a. x a. x < a Section 2.7: Inequlities In this section, we will Determine if given vlue is solution to n inequlity Solve given inequlity or compound inequlity; give the solution in intervl nottion nd the solution 2.7

More information

Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40

Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40 Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since

More information

Matrices and Determinants

Matrices and Determinants Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd Guss-Jordn elimintion to solve systems of liner

More information

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

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.

More information

A B= ( ) because from A to B is 3 right, 2 down.

A B= ( ) because from A to B is 3 right, 2 down. 8. Vectors nd vector nottion Questions re trgeted t the grdes indicted Remember: mgnitude mens size. The vector ( ) mens move left nd up. On Resource sheet 8. drw ccurtely nd lbel the following vectors.

More information

GRADE 4. Division WORKSHEETS

GRADE 4. Division WORKSHEETS GRADE Division WORKSHEETS Division division is shring nd grouping Division cn men shring or grouping. There re cndies shred mong kids. How mny re in ech shre? = 3 There re 6 pples nd go into ech bsket.

More information

Working with Powers and Exponents

Working with Powers and Exponents Working ith Poer nd Eponent Nme: September. 00 Repeted Multipliction Remember multipliction i y to rite repeted ddition. To y +++ e rite. Sometime multipliction i done over nd over nd over. To rite e rite.

More information

NAME: MR. WAIN FUNCTIONS

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

More information

AQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system

AQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex

More information

List all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.

List all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1. Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show

More information

I do slope intercept form With my shades on Martin-Gay, Developmental Mathematics

I do slope intercept form With my shades on Martin-Gay, Developmental Mathematics AAT-A Dte: 1//1 SWBAT simplify rdicls. Do Now: ACT Prep HW Requests: Pg 49 #17-45 odds Continue Vocb sheet In Clss: Complete Skills Prctice WS HW: Complete Worksheets For Wednesdy stmped pges Bring stmped

More information

x means to use x as a factor five times, or x x x x x (2 c ) means to use 2c as a factor four times, or

x means to use x as a factor five times, or x x x x x (2 c ) means to use 2c as a factor four times, or 14 DAY 1 CHAPTER FIVE Wht fscinting mthemtics is now on our gend? We will review the pst four chpters little bit ech dy becuse mthemtics builds. Ech concept is foundtion for nother ide. We will hve grph

More information

12.1 Introduction to Rational Expressions

12.1 Introduction to Rational Expressions . Introduction to Rtionl Epressions A rtionl epression is rtio of polynomils; tht is, frction tht hs polynomil s numertor nd/or denomintor. Smple rtionl epressions: 0 EVALUATING RATIONAL EXPRESSIONS To

More information

Math 113 Exam 2 Practice

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

More information

Loudoun Valley High School Calculus Summertime Fun Packet

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!

More information

Worksheet A EXPONENTIALS AND LOGARITHMS PMT. 1 Express each of the following in the form log a b = c. a 10 3 = 1000 b 3 4 = 81 c 256 = 2 8 d 7 0 = 1

Worksheet A EXPONENTIALS AND LOGARITHMS PMT. 1 Express each of the following in the form log a b = c. a 10 3 = 1000 b 3 4 = 81 c 256 = 2 8 d 7 0 = 1 C Worksheet A Epress ech of the following in the form log = c. 0 = 000 4 = 8 c 56 = 8 d 7 0 = e = f 5 = g 7 9 = 9 h 6 = 6 Epress ech of the following using inde nottion. log 5 5 = log 6 = 4 c 5 = log 0

More information

20 MATHEMATICS POLYNOMIALS

20 MATHEMATICS POLYNOMIALS 0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

More information

Geometric Sequences. Geometric Sequence a sequence whose consecutive terms have a common ratio.

Geometric Sequences. Geometric Sequence a sequence whose consecutive terms have a common ratio. Geometric Sequences Geometric Sequence sequence whose consecutive terms hve common rtio. Geometric Sequence A sequence is geometric if the rtios of consecutive terms re the sme. 2 3 4... 2 3 The number

More information

Math 1051 Diagnostic Pretest Key and Homework

Math 1051 Diagnostic Pretest Key and Homework Mth 1051 Dignostic Pretest Ke nd Hoework HW1 The dignostic test is designed to give us n ide of our level of skill in doing high school lgebr s ou begin Mth 1051. You should be ble to do these probles

More information

Find the value of x. Give answers as simplified radicals.

Find the value of x. Give answers as simplified radicals. 9.2 Dy 1 Wrm Up Find the vlue of. Give nswers s simplified rdicls. 1. 2. 3 3 3. 4. 10 Mrch 2, 2017 Geometry 9.2 Specil Right Tringles 1 Geometry 9.2 Specil Right Tringles 9.2 Essentil Question Wht is the

More information

and that at t = 0 the object is at position 5. Find the position of the object at t = 2.

and that at t = 0 the object is at position 5. Find the position of the object at t = 2. 7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we

More information

Introduction to Group Theory

Introduction to Group Theory Introduction to Group Theory Let G be n rbitrry set of elements, typiclly denoted s, b, c,, tht is, let G = {, b, c, }. A binry opertion in G is rule tht ssocites with ech ordered pir (,b) of elements

More information

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 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

More information

Section 7.1 Integration by Substitution

Section 7.1 Integration by Substitution Section 7. Integrtion by Substitution Evlute ech of the following integrls. Keep in mind tht using substitution my not work on some problems. For one of the definite integrls, it is not possible to find

More information

Section 6.3 The Fundamental Theorem, Part I

Section 6.3 The Fundamental Theorem, Part I Section 6.3 The Funmentl Theorem, Prt I (3//8) Overview: The Funmentl Theorem of Clculus shows tht ifferentition n integrtion re, in sense, inverse opertions. It is presente in two prts. We previewe Prt

More information

4.5 THE FUNDAMENTAL THEOREM OF CALCULUS

4.5 THE FUNDAMENTAL THEOREM OF CALCULUS 4.5 The Funmentl Theorem of Clculus Contemporry Clculus 4.5 THE FUNDAMENTAL THEOREM OF CALCULUS This section contins the most importnt n most use theorem of clculus, THE Funmentl Theorem of Clculus. Discovere

More information

Exploring parametric representation with the TI-84 Plus CE graphing calculator

Exploring parametric representation with the TI-84 Plus CE graphing calculator Exploring prmetric representtion with the TI-84 Plus CE grphing clcultor Richrd Prr Executive Director Rice University School Mthemtics Project rprr@rice.edu Alice Fisher Director of Director of Technology

More information

10.2 The Ellipse and the Hyperbola

10.2 The Ellipse and the Hyperbola CHAPTER 0 Conic Sections Solve. 97. Two surveors need to find the distnce cross lke. The plce reference pole t point A in the digrm. Point B is meters est nd meter north of the reference point A. Point

More information