2000 Solutions Fermat Contest (Grade 11)


 Ethelbert Ramsey
 2 years ago
 Views:
Transcription
1 anadian Mathematics ompetition n activity of The entre for Education in Mathematics and omputing, University of Waterloo, Waterloo, Ontario 000 s Fermat ontest (Grade ) for The ENTRE for EDUTION in MTHEMTIS and OMPUTING wards 000 Waterloo Mathematics Foundation
2 000 Fermat s Part :. The sum is equal to () 6 () 4 4 () 8 8 (D) 64 0 (E) = 64 = 6 NSWER: (E). If the following sequence of five arrows repeats itself continuously, what arrow will be in the 48th position?,,,, () () () (D) (E) Since this sequence repeats itself, once it has completed nine cycles it will be the same as starting at the beginning. Thus the 48th arrow will be the same as the third one. NSWER: () 3. farmer has 7 cows, 8 sheep and 6 goats. How many more goats should be bought so that half of her animals will be goats? () 8 () () (D) 9 (E) 6 If the cows and sheep were themselves goats we would have goats. This means that she would need nine extra goats.
3 000 Fermat s 3 Let the number of goats added be x. Therefore, 6 + x + x = ross multiplying gives, ( 6+ x)= + x + x = + x x = 9. s in solution, she would add 9 goats. NSWER: (D) 4. The square of 9 is divided by the cube root of. What is the remainder? () 6 () 3 () 6 (D) (E) The square of 9 is 8 and the cube root of is. When 8 is divided by, the quotient is 6 and the remainder is. Since 8 = 6 +, the remainder is. NSWER: (E). The product of, 3,, and y is equal to its sum. What is the value of y? () 3 () 0 3 () 0 9 (D) 3 0 (E) 0 3 Since ( )( 3)( )( y)= y. 30y= 0 + y 9y = 0 and y = 0. 9 NSWER: () 6. student uses a calculator to find an answer but instead of pressing the x key presses the x key by mistake. The student s answer was 9. What should the answer have been? () 43 () 8 () 79 (D) 3 (E) 66 Since x = 9, x = 8. Thus, x = 8 = 66.
4 000 Fermat s 4 Since the square root of the number entered is 9, the number must have been 8. The answer desired is 8 = 66. NSWER: (E) 7. The sum of the arithmetic series ( 300)+ ( 97)+ ( 94) is () 309 () 97 () 6 (D) 98 (E) 8 The given series is ( 300)+ ( 97)+ ( 94) ( 3) The sum of the terms from 300 to 300 is 0. The sum of all the terms is thus = 98. NSWER: (D) 8. In a school referendum, 3 of a student body voted yes and 8% voted no. If there were no spoiled ballots, what percentage of the students did not vote? () 7% () 40% () 3% (D) % (E) 88% Since 3 or 60% of the students voted yes and 8% voted no, 88% of the student body voted. Hence, 00% 88% or %, of the students did not vote. NSWER: (D) 9. The numbers 6, 4, x, 7, 9, y, 0 have a mean of 3. What is the value of x+ y? () 0 () () 3 (D) (E) 3 If the 7 numbers have a mean of 3, this implies that these numbers would have a sum of 7 3 or 9. We now can calculate x+ y since, x y + 0 = 9 or, x + y+ 6 = 9 Therefore, x+ y=3. NSWER: (E) 0. 3 If xxx ( ( + )+ )+ 3= x + x + x 6 then x is equal to () () 9 () 4 or 3 (D) or 0 (E)
5 000 Fermat s xxx ( ( + )+ )+ 3= xx ( + x+ )+ 3 3 = x + x + x Since x + x + x+ 3= x + x + x 6 x+ 3= x 6 x = 9 NSWER: () Part :. When the regular pentagon is reflected in the line PQ, and then rotated clockwise 44 about the centre of the pentagon, its position is P Q () P Q () P Q () (D) P Q (E) P Q fter the reflection, the new pentagon is Since each of the five central angles equals 7 o o, a clockwise rotation of 44 7 will place the pentagon in the shown position. o ( ) about its centre NSWER: () 6 7. If the expression 8 was evaluated, it would end with a string of consecutive zeros. How many zeros are in this string? () 0 () 8 () 6 (D) 3 (E) zero at the end of a number results from the product of and. The number of zeros at the end of a number equals the number of product pairs of and that can be formed from the prime factorization of that number Since 8 = ( ) ( ) ( )
6 000 Fermat s = = There will be ten zeros in the string at the end of the number. NSWER: () 3. Rectangle D is divided into five congruent rectangles as shown. The ratio : is () 3: () : () : (D) :3 (E) 4:3 D If we let the width of each rectangle be x units then the length of each rectangle is 3x units. (This is illustrated in the diagram.) The length,, is now 3x+ x+ x or x units and = 3 x. Thus : = x: 3x = 3 :, x 0. D 3x x x x x x 3x NSWER: (D) 4. In the regular hexagon DEF, two of the diagonals, F and D, intersect at G. The ratio of the area of quadrilateral FEDG to the area of G is F G () 3 3: () 4 : () 6 : (D) 3: (E) : E D Join E to and D to as shown. lso join E to and draw a line parallel to E through the point of intersection of E and D. Quadrilateral FEDG is now made up of five triangles each of which has the same area as G. The required ratio is :. F E G D
7 000 Fermat s 7 For convenience, assume that each side of the hexagon has a length of units. Each angle in the hexagon equals 0 o so G = ( 0 )= 60. Now label G as shown. Using the standard ratios for a triangle we have G = 3 and G =. 3 G 60 The area of G = () 3 =. Dividing the quadrilateral FGDE as illustrated, it will have an area of ( 3 )+ ()( 3)= 3. The required ratio is 3 3 : or :, as in solution. 3 F E G 3 3 D NSWER: (E). In a sequence, every term after the second term is twice the sum of the two preceding terms. The seventh term of the sequence is 8, and the ninth term is 4. What is the eleventh term of the sequence? () 60 () 304 () 8 (D) 6 (E) 64 Let the seventh through eleventh terms of the sequence be t7, t8, t9, t0, and t. Since t = t + t = 8 + t = 8 + t ( ) ( ) t8 = 4. Hence t0 = ( t8 + t9)= ( 4+ 4)= 6 and t = 4 ( + 6) = , NSWER: () 6. The digits,, 3, and are randomly arranged to form a four digit number. What is the probability that the sum of the first and last digits is even? () 4 () 3 () 6 (D) (E) 3
8 000 Fermat s 8 The numbers that can be formed from the digits,, 3, and, in ascending order, are 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, and 3. These are twelve possibilities. For the sum of two digits to be even, both must be even or both must be odd. From the above list, the numbers for which the sum of the first and last digits is even, are 3, 3, 3, and 3. These are four possibilities. Thus, the probability of getting one of these numbers is 4 or. 3 NSWER: () 7. Three circles have centres, and with radii, 4 and 6 respectively. The circles are tangent to each other as shown. Triangle has () obtuse () = 90 () = 90 (D) all angles acute (E) = Since the circles are mutually tangent, the lines joining their centres pass through the points of tangency. Thus, the sides of have lengths 6, 8 and 0 if we write in the radii as shown. Since 0 = 6 + 8, the triangle is rightangled at. NSWER: () If P = and Q = 3 3 then the value of P Q is () () () 0 (D) (E) 4 P Q = ( P+ Q) ( P Q)= [( )+( 3 3 )]( ) [ ( )] = ( )( ) = 43 = 4. o P Q = ( ) ( )
9 000 Fermat s 9 = = = o o 4000 NSWER: (E) 9. n ant walks inside a 8 cm by 0 cm rectangle. The ant s path follows straight lines which always make angles of 4 to the sides of the rectangle. The ant starts from a point X on one of the shorter sides. The first time the ant reaches the opposite side, it arrives at the midpoint. What is the distance, in centimetres, from X to the nearest corner of the rectangle? () 3 () 4 () 6 (D) 8 (E) 9 If we took a movie of the ant s path and then played it backwards, the ant would now start at the point E and would then end up at point X. Since the ant now starts at a point nine cm from the corner, the first part of his journey is from E to. This amounts to nine cm along the length of the rectangle since E is an isosceles rightangled triangle. This process continues as illustrated, until the ant reaches point. y the time the ant has reached, it has travelled or 3 cm along the length of the rectangle. To travel from to X, the ant must travel cm along the length of the rectangle which puts the ant 3 cm from the closest vertex. 9 cm E 9 cm 36 cm 8 cm 8 cm 8 cm 8 cm X 3 cm cm 9 cm 8 cm 8 cm 8 cm 36 cm 36 cm 3 cm cm NSWER: () 0. Given a+ b+ 3c+ 4d+ e= k and a= 4b= 3c= d = e find the smallest positive integer value for k so that a, b, c, d, and e are all positive integers. () 87 () () 0 (D) 0 (E) 60 From the equalities a= 4b= 3c= d = e, we conclude that e is the largest and a is the smallest of the integers. Since e is divisible by, 4, 3, and, the smallest possible value for e is 60. The corresponding values for a, b, c, and d, are,, 0, and 30, respectively. Thus, the smallest positive integer value for k is ( )+ ( )+ ( )+ ( )=.
10 000 Fermat s 0 NSWER: () Part :. Two circles of radius 0 are tangent to each other. tangent is drawn from the centre of one of the circles to the second circle. To the nearest integer, what is the area of the shaded region? () 6 () 7 () 8 (D) 9 (E) 0 Let the centres of the circles be and and the point of contact of the tangent line from to the circle with centre be. Thus, = 90 o using properties of tangents to a circle. Since the sides of rightangled triangle are in the ratio of : : 3, = 30 o and = 60 o. The area of the shaded region is equal to the area of minus the area of the two sectors of the circles. The area of the two sectors is equivalent to the area of a circle with radius 0. 4 The area of the shaded region = ( 0)( 0 3) π( 0) = 0 3 π To the nearest integer, the area is 8. 4 NSWER: (). The left most digit of an integer of length 000 digits is 3. In this integer, any two consecutive digits must be divisible by 7 or 3. The 000th digit may be either a or b. What is the value of a+ b? () 3 () 7 () 4 (D) 0 (E) 7 We start by noting that the twodigit multiples of 7 are 7, 34,, 68, and 8. Similarly we note that the twodigit multiples of 3 are 3, 46, 69, and 9. The first digit is 3 and since the only twodigit number in the two lists starting with 3 is 34, the second digit is 4. Similarly the third digit must be 6. The fourth digit, however, can be either 8 or 9. Let s consider this in two cases.
11 000 Fermat s ase If the fourth digit is 8, the number would be and would stop here since there isn t a number in the two lists starting with 7. ase If the fourth digit is 9, the number would be and the five digits 3469 would continue repeating indefinitely as long as we choose 9 to follow 6. If we consider a 000 digit number, its first 99 digits must contain 399 groups of The last groups of five digits could be either 3469 or 3468 which means that the 000th digit may be either or so that a+ b= + = 7. NSWER: () 3. circle is tangent to three sides of a rectangle having side lengths and 4 as shown. diagonal of the rectangle intersects the circle at points and. The length of is () () 4 () (D) (E) Of the many ways to solve this problem perhaps the easiest is to use a artesian coordinate system. There are two choices for the origin, the centre of the circle and the bottom left vertex of the rectangle. The first solution presented uses the centre of the circle as the origin where the axes are lines drawn parallel to the sides of the rectangle through ( 0, 0). The equation of the circle is now x + y = and the equation of the line containing the diagonal is y+ = x+ or x y=. ( ) y (0, ) (, 0) x 4 Equation of Line: y+ = ( x+ ) x y= Equation of ircle: x + y = ( ) + = Solving to find the intersection points of the line and circle gives, y+ y y + 4y= 0 y( y+ 4)= 0 y= 0 or y= If we substitute these values of y into x y= we find, 3 4 The points of intersection are, and, 0. 4 ( ) ( )
12 000 Fermat s The length of is = =. Note: The other suggested choice for the origin would involve using the equation of the line y x and the circle x y. = ( ) + ( ) = The diameter, G, is parallel to the sides of the rectangle and has its endpoint at which is the centre of the rectangle. Draw O where O is the centre of the circle and by chord properties of a circle, =. Since O DE, DE = O =α. Thus, O is similar to FDE. Since DF =, = and = 4. The length of is or 4. NSWER: () 4 4. For the system of equations x + x y + x y = and x+ xy+ xy = 3, the sum of the real y values that satisfy the equations is () 0 () () (D) (E) onsider the system of equations 4 x + x y + x y = () and x+ xy+ xy = 3 () The expression on the left side of equation () can be rewritten as, ( )( + + ) Thus, x + xy xy x xy xy Substituting from () gives, x+ xy xy Now subtracting (3) from (), xy = 0, x Substituting for x in (3) gives, ( ) ( x xy xy) ( x + xy + xy) 4 x + x y + x y = x+ xy x y ( )( )= = + = 3 or, x+ xy xy = (3) 0 y = 0 y + 0y 0 =
13 000 Fermat s 3 0y y+ 0 = 0 y y+ = 0 ( y ) ( y )= 0 y = or y = The sum of the real y values satisfying the system is. NSWER: (E). The given cube is cut into four pieces by two planes. The first plane is parallel to face D and passes through the midpoint of edge G. The second plane passes through the midpoints of edges, D, HE, and GH. Determine the ratio of the volumes of the smallest and largest of the four pieces. F D E () 3:8 () 7:4 () 7: (D) 7:7 (E) : G H T For convenience, let each edge of the cube have length. The plane PQRS through the midpoint of G and parallel N D to face D bisects the volume of the cube. The plane K through K, L, M, and N also bisects the volume of the S cube and contains the line segment QS. Hence the two P planes divide the cube into four pieces, two equal Q E F R smaller pieces and two equal larger pieces. Extend M QK and SN to meet at T. G L H We note that TN is similar to TPS. Since T N = and N =, PS = and P = it is easy TP PS to calculate to find T =. The volume of the upper smallest piece is equal to the volume of the tetradedron TQSP  volume of the tetradedron TKN. This volume is, 3 ( )( )( ) 3 ()()() 4 7 = = The volume of a largest piece is, 7 7 =. 6 6 The ratio of the volume of the smallest piece to that of the largest piece is 7:7. NSWER: (D)
Possible Stage Two Mathematics Test Topics
Possible Stage Two Mathematics Test Topics The Stage Two Mathematics Test questions are designed to be answerable by a good problemsolver with a strong mathematics background. It is based mainly on material
More informationIf A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?
Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question
More information13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant
æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the
More informationCollege of Charleston Math Meet 2008 Written Test Level 1
College of Charleston Math Meet 2008 Written Test Level 1 1. Three equal fractions, such as 3/6=7/14=29/58, use all nine digits 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly one time. Using all digits exactly one
More informationSPECIAL PRODUCTS AND FACTORS
CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 111 Factors and Factoring 112 Common Monomial Factors 113 The Square of a Monomial 114 Multiplying the Sum and the Difference of Two Terms 115 Factoring the
More informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More information121 Representations of ThreeDimensional Figures
Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 121 Representations of ThreeDimensional Figures Use isometric dot paper to sketch each prism. 1. triangular
More informationWarmUp 1. 1. What is the least common multiple of 6, 8 and 10?
WarmUp 1 1. What is the least common multiple of 6, 8 and 10? 2. A 16page booklet is made from a stack of four sheets of paper that is folded in half and then joined along the common fold. The 16 pages
More informationYOU CAN COUNT ON NUMBER LINES
Key Idea 2 Number and Numeration: Students use number sense and numeration to develop an understanding of multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationMEP Pupil Text 12. A list of numbers which form a pattern is called a sequence. In this section, straightforward sequences are continued.
MEP Pupil Text Number Patterns. Simple Number Patterns A list of numbers which form a pattern is called a sequence. In this section, straightforward sequences are continued. Worked Example Write down the
More informationGeoGebra. 10 lessons. Gerrit Stols
GeoGebra in 10 lessons Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It was developed by Markus Hohenwarter
More information2009 Chicago Area AllStar Math Team Tryouts Solutions
1. 2009 Chicago Area AllStar Math Team Tryouts Solutions If a car sells for q 1000 and the salesman earns q% = q/100, he earns 10q 2. He earns an additional 100 per car, and he sells p cars, so his total
More informationIntroduction Assignment
PRECALCULUS 11 Introduction Assignment Welcome to PREC 11! This assignment will help you review some topics from a previous math course and introduce you to some of the topics that you ll be studying
More informationFURTHER VECTORS (MEI)
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level  MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: 97 Mathematics
More information2010 Solutions. a + b. a + b 1. (a + b)2 + (b a) 2. (b2 + a 2 ) 2 (a 2 b 2 ) 2
00 Problem If a and b are nonzero real numbers such that a b, compute the value of the expression ( ) ( b a + a a + b b b a + b a ) ( + ) a b b a + b a +. b a a b Answer: 8. Solution: Let s simplify the
More informationBaltic Way 1995. Västerås (Sweden), November 12, 1995. Problems and solutions
Baltic Way 995 Västerås (Sweden), November, 995 Problems and solutions. Find all triples (x, y, z) of positive integers satisfying the system of equations { x = (y + z) x 6 = y 6 + z 6 + 3(y + z ). Solution.
More informationSuch As Statements, Kindergarten Grade 8
Such As Statements, Kindergarten Grade 8 This document contains the such as statements that were included in the review committees final recommendations for revisions to the mathematics Texas Essential
More informationThe Australian Curriculum Mathematics
The Australian Curriculum Mathematics Mathematics ACARA The Australian Curriculum Number Algebra Number place value Fractions decimals Real numbers Foundation Year Year 1 Year 2 Year 3 Year 4 Year 5 Year
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More informationGraphing and Solving Nonlinear Inequalities
APPENDIX LESSON 1 Graphing and Solving Nonlinear Inequalities New Concepts A quadratic inequality in two variables can be written in four different forms y < a + b + c y a + b + c y > a + b + c y a + b
More information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two nonzero vectors u and v,
1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It
More informationPrimes. Name Period Number Theory
Primes Name Period A Prime Number is a whole number whose only factors are 1 and itself. To find all of the prime numbers between 1 and 100, complete the following exercise: 1. Cross out 1 by Shading in
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationMath Games For Skills and Concepts
Math Games p.1 Math Games For Skills and Concepts Other material copyright: Investigations in Number, Data and Space, 1998 TERC. Connected Mathematics Project, 1998 CMP Original material 20012006, John
More information+ 4θ 4. We want to minimize this function, and we know that local minima occur when the derivative equals zero. Then consider
Math Xb Applications of Trig Derivatives 1. A woman at point A on the shore of a circular lake with radius 2 miles wants to arrive at the point C diametrically opposite A on the other side of the lake
More informationKeystone National High School Placement Exam Math Level 1. Find the seventh term in the following sequence: 2, 6, 18, 54
1. Find the seventh term in the following sequence: 2, 6, 18, 54 2. Write a numerical expression for the verbal phrase. sixteen minus twelve divided by six Answer: b) 1458 Answer: d) 16 12 6 3. Evaluate
More informationInversion. Chapter 7. 7.1 Constructing The Inverse of a Point: If P is inside the circle of inversion: (See Figure 7.1)
Chapter 7 Inversion Goal: In this chapter we define inversion, give constructions for inverses of points both inside and outside the circle of inversion, and show how inversion could be done using Geometer
More informationGEOMETRY OF THE CIRCLE
HTR GMTRY F TH IRL arly geometers in many parts of the world knew that, for all circles, the ratio of the circumference of a circle to its diameter was a constant. Today, we write d 5p, but early geometers
More informationdiscuss how to describe points, lines and planes in 3 space.
Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position
More informationGeneral Certificate of Secondary Education Additional Mathematics 9306
version 1.0 abc General Certificate of Secondary Education Additional Mathematics 9306 Pilot Specification 2008 PROBLEMSOLVING QUESTIONS Further copies of this resource are available from: The GCSE Mathematics
More information(15.) To find the distance from point A to point B across. a river, a base line AC is extablished. AC is 495 meters
(15.) To find the distance from point A to point B across a river, a base line AC is extablished. AC is 495 meters long. Angles
More informationSupport Materials for Core Content for Assessment. Mathematics
Support Materials for Core Content for Assessment Version 4.1 Mathematics August 2007 Kentucky Department of Education Introduction to Depth of Knowledge (DOK)  Based on Norman Webb s Model (Karin Hess,
More informationA Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles
A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...
More information9 Areas and Perimeters
9 Areas and Perimeters This is is our next key Geometry unit. In it we will recap some of the concepts we have met before. We will also begin to develop a more algebraic approach to finding areas and perimeters.
More informationCHAPTER 1. LINES AND PLANES IN SPACE
CHAPTER 1. LINES AND PLANES IN SPACE 1. Angles and distances between skew lines 1.1. Given cube ABCDA 1 B 1 C 1 D 1 with side a. Find the angle and the distance between lines A 1 B and AC 1. 1.2. Given
More informationStar and convex regular polyhedra by Origami.
Star and convex regular polyhedra by Origami. Build polyhedra by Origami.] Marcel Morales Alice Morales 2009 E D I T I O N M O R A L E S Polyhedron by Origami I) Table of convex regular Polyhedra... 4
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationFactoring and Applications
Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the
More informationSection 9.5: Equations of Lines and Planes
Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 35 odd, 237 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that
More informationWarmUp 3 Solutions. Peter S. Simon. October 13, 2004
WarmUp 3 Solutions Peter S. Simon October 13, 2004 Problem 1 An automobile insurance company has compiled data from a survey of 1000 16yearold drivers during the year 2003. According to the results
More informationBlue Pelican Alg II First Semester
Blue Pelican Alg II First Semester Teacher Version 1.01 Copyright 2009 by Charles E. Cook; Refugio, Tx (All rights reserved) Alg II Syllabus (First Semester) Unit 1: Solving linear equations and inequalities
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a selfadaptive test, which potentially tests students within four different levels of math including prealgebra, algebra, college algebra, and trigonometry.
More informationSOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve
SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives
More informationFACTORING OUT COMMON FACTORS
278 (6 2) Chapter 6 Factoring 6.1 FACTORING OUT COMMON FACTORS In this section Prime Factorization of Integers Greatest Common Factor Finding the Greatest Common Factor for Monomials Factoring Out the
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationAMC 10 Solutions Pamphlet TUESDAY, FEBRUARY 13, 2001 Sponsored by Mathematical Association of America University of Nebraska
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO AMERICAN MATHEMATICS COMPETITIONS nd Annual Mathematics Contest 10 AMC 10 Solutions Pamphlet TUESDAY, FEBRUARY 1, 001
More informationSECTION 16 Quadratic Equations and Applications
58 Equations and Inequalities Supply the reasons in the proofs for the theorems stated in Problems 65 and 66. 65. Theorem: The complex numbers are commutative under addition. Proof: Let a bi and c di be
More information( ) FACTORING. x In this polynomial the only variable in common to all is x.
FACTORING Factoring is similar to breaking up a number into its multiples. For example, 10=5*. The multiples are 5 and. In a polynomial it is the same way, however, the procedure is somewhat more complicated
More informationExtra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.
Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson
More informationMATH 90 CHAPTER 6 Name:.
MATH 90 CHAPTER 6 Name:. 6.1 GCF and Factoring by Groups Need To Know Definitions How to factor by GCF How to factor by groups The Greatest Common Factor Factoring means to write a number as product. a
More informationFree PreAlgebra Lesson 55! page 1
Free PreAlgebra Lesson 55! page 1 Lesson 55 Perimeter Problems with Related Variables Take your skill at word problems to a new level in this section. All the problems are the same type, so that you can
More informationDefinitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).
Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationfor the Common Core State Standards 2012
A Correlation of for the Common Core State s 2012 to the Common Core Georgia Performance s Grade 2 FORMAT FOR CORRELATION TO THE COMMON CORE GEORGIA PERFORMANCE STANDARDS (CCGPS) Subject Area: K12 Mathematics
More informationFACTORING QUADRATICS 8.1.1 and 8.1.2
FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationReview of Basic Algebraic Concepts
Section. Sets of Numbers and Interval Notation Review of Basic Algebraic Concepts. Sets of Numbers and Interval Notation. Operations on Real Numbers. Simplifying Expressions. Linear Equations in One Variable.
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
More informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationPolynomials and Factoring
7.6 Polynomials and Factoring Basic Terminology A term, or monomial, is defined to be a number, a variable, or a product of numbers and variables. A polynomial is a term or a finite sum or difference of
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationMath BINGO MOST POPULAR. Do you have the lucky card? B I N G O
MOST POPULAR Math BINGO Do you have the lucky card? Your club members will love this MATHCOUNTS reboot of a classic game. With the perfect mix of luck and skill, this is a game that can be enjoyed by students
More informationFactoring, Solving. Equations, and Problem Solving REVISED PAGES
05W4801AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring
More informationSOLUTIONS TO HANDBOOK PROBLEMS
SOLUTIONS TO HNDBOOK PROBLEMS The solutions provided here are only possible solutions. It is very likely that you or your students will come up with additional and perhaps more elegant solutions. Happy
More informationIn this section, you will develop a method to change a quadratic equation written as a sum into its product form (also called its factored form).
CHAPTER 8 In Chapter 4, you used a web to organize the connections you found between each of the different representations of lines. These connections enabled you to use any representation (such as a graph,
More informationNAKURU DISTRICT SEC. SCHOOLS TRIAL EXAM 2011. Kenya Certificate of Secondary Education MATHEMATICS PAPER2 2½ HOURS
NAME:. INDEX NO: CANDIDATE S SINGNATURE DATE:. 121 /2 MATHEMATICS ALT A. PAPER2 JULY / AUGUST 2011 2 ½ HOURS NAKURU DISTRICT SEC. SCHOOLS TRIAL EXAM 2011 INSTRUCTIONS TO CANDIDATES Kenya Certificate of
More informationVISUAL ALGEBRA FOR COLLEGE STUDENTS. Laurie J. Burton Western Oregon University
VISUAL ALGEBRA FOR COLLEGE STUDENTS Laurie J. Burton Western Oregon University VISUAL ALGEBRA FOR COLLEGE STUDENTS TABLE OF CONTENTS Welcome and Introduction 1 Chapter 1: INTEGERS AND INTEGER OPERATIONS
More informationThe Distributive Property
The Distributive Property Objectives To recognize the general patterns used to write the distributive property; and to mentally compute products using distributive strategies. www.everydaymathonline.com
More informationContent. Chapter 4 Functions 61 4.1 Basic concepts on real functions 62. Credits 11
Content Credits 11 Chapter 1 Arithmetic Refresher 13 1.1 Algebra 14 Real Numbers 14 Real Polynomials 19 1.2 Equations in one variable 21 Linear Equations 21 Quadratic Equations 22 1.3 Exercises 28 Chapter
More informationA.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it
Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply
More informationTHE COLLEGES OF OXFORD UNIVERSITY MATHEMATICS, JOINT SCHOOLS AND COMPUTER SCIENCE. Sample Solutions for Specimen Test 2
THE COLLEGES OF OXFORD UNIVERSITY MATHEMATICS, JOINT SCHOOLS AND COMPUTER SCIENCE Sample Solutions for Specimen Test. A. The vector from P (, ) to Q (8, ) is PQ =(6, 6). If the point R lies on PQ such
More informationhttp://schoolmaths.com Gerrit Stols
For more info and downloads go to: http://schoolmaths.com Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More information(Refer Slide Time: 01.26)
Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture # 27 Pigeonhole Principle In the next few lectures
More informationhow to use dual base log log slide rules
how to use dual base log log slide rules by Professor Maurice L. Hartung The University of Chicago Pickett The World s Most Accurate Slide Rules Pickett, Inc. Pickett Square Santa Barbara, California 93102
More informationReview Sheet for Test 1
Review Sheet for Test 1 Math 26100 2 6 2004 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And
More informationINDEX. SR NO NAME OF THE PRACTICALS Page No. Measuring the bearing of traverse lines, calculation of included angles and check.
INDEX SR NO NAME OF THE PRACTICALS Page No 1 Measuring the bearing of traverse lines, calculation of included angles and check. 1 2 To study the essential parts of dumpy level & reduction of levels 3 To
More informationEquations Involving Lines and Planes Standard equations for lines in space
Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity
More informationObjectives After completing this section, you should be able to:
Chapter 5 Section 1 Lesson Angle Measure Objectives After completing this section, you should be able to: Use the most common conventions to position and measure angles on the plane. Demonstrate an understanding
More informationMath 53 Worksheet Solutions Minmax and Lagrange
Math 5 Worksheet Solutions Minmax and Lagrange. Find the local maximum and minimum values as well as the saddle point(s) of the function f(x, y) = e y (y x ). Solution. First we calculate the partial
More informationJust want the standards alone? You can find the standards alone at www.corestandards.org. 7 th Grade Mathematics Unpacked Content February, 2012
7 th Grade Mathematics Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 School Year. This document is designed to help North Carolina
More information1. How much does the result of
THE QUESTIONS OF OÞMY FOR LSS 5 FIRST PRT 3. TTENTION! In this part there are questions which have the value 1,25 point from 1 to 10. 1. How much does the result of 1 1 + 0,25 2 8 lack from 1? 1 3 3 5
More information1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives
TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of
More information6706_PM10SB_C4_CO_pp192193.qxd 5/8/09 9:53 AM Page 192 192 NEL
92 NEL Chapter 4 Factoring Algebraic Epressions GOALS You will be able to Determine the greatest common factor in an algebraic epression and use it to write the epression as a product Recognize different
More informationRegents Examination in Geometry (Common Core) Sample and Comparison Items Spring 2014
Regents Examination in Geometry (Common Core) Sample and Comparison Items Spring 2014 i May 2014 777777 THE STATE EDUCATION DEPARTMENT / THE UNIVERSITY OF THE STATE OF NEW YORK / ALBANY, NY 12234 New York
More informationCatalan Numbers. Thomas A. Dowling, Department of Mathematics, Ohio State Uni versity.
7 Catalan Numbers Thomas A. Dowling, Department of Mathematics, Ohio State Uni Author: versity. Prerequisites: The prerequisites for this chapter are recursive definitions, basic counting principles,
More informationThe Clar Structure of Fullerenes
The Clar Structure of Fullerenes Liz Hartung Massachusetts College of Liberal Arts June 12, 2013 Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 1 / 25
More information83 Dot Products and Vector Projections
83 Dot Products and Vector Projections Find the dot product of u and v Then determine if u and v are orthogonal 1u =, u and v are not orthogonal 2u = 3u =, u and v are not orthogonal 6u = 11i + 7j; v
More informationFORM 3 MATHEMATICS SCHEME C TIME: 30 minutes Non Calculator Paper INSTRUCTIONS TO CANDIDATES
DIRECTORATE FOR QUALITY AND STANDARDS IN EDUCATION Department for Curriculum Management and elearning Educational Assessment Unit Annual Examinations for Secondary Schools 2011 C FORM 3 MATHEMATICS SCHEME
More informationAppendix A Designing High School Mathematics Courses Based on the Common Core Standards
Overview: The Common Core State Standards (CCSS) for Mathematics are organized by grade level in Grades K 8. At the high school level, the standards are organized by strand, showing a logical progression
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationCHAPTER 7. Think & Discuss (p. 393) m Z 55 35 180. m Z 90 180. m Z 90 QR 2 RP 2 PQ 2 QR 2 10 2 12.2 2 QR 2 100 148.84 QR 2 48.84 AB 1 6 2 3 4 2 QR 7.
HPTER 7 Think & Discuss (p. 393). The image in bo is flipped to get the image in bo. The image in bo is turned to get the image in bo D.. Sample answer: If ou look at the picture as a whole, the right
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationXV. Mathematics, Grade 10
XV. Mathematics, Grade 10 Grade 10 Mathematics Test The spring 2011 grade 10 MCAS Mathematics test was based on learning standards in the Massachusetts Mathematics Curriculum Framework (2000). The Framework
More informationDRAFT. New York State Testing Program Grade 8 Common Core Mathematics Test. Released Questions with Annotations
DRAFT New York State Testing Program Grade 8 Common Core Mathematics Test Released Questions with Annotations August 2014 Developed and published under contract with the New York State Education Department
More informationSolving Quadratic Equations by Factoring
4.7 Solving Quadratic Equations by Factoring 4.7 OBJECTIVE 1. Solve quadratic equations by factoring The factoring techniques you have learned provide us with tools for solving equations that can be written
More information