Minnesota State High School Mathematics League
|
|
- Joseph Chambers
- 5 years ago
- Views:
Transcription
1 03-4 Meet, Individual Event Question # is intended to be a quickie and is worth point. Each of the next three questions is worth points. Place your answer to each question on the line provided. You have minutes for this event. NO LULTORS are allowed on this event.. Determine exactly the value of. LM 0,4 GD 0,4. The expression 4 3 can be simplified and written as a single rational number. Determine exactly that rational number. Day # 3. The epad was originally priced at $00, but the newest model of epad is coming out, so the old one is going on sale. Each day, the price is reduced by 0%, and rounded down to the nearest dollar, as necessary. (On Day, the price is reduced to $90; on Day it is $8; on Day 3 it is $7, and so on.) What is the First day on which the old epad will cost $? 4. Express as a repeating decimal. Name: Team:
2 03-4 Meet, Individual Event SOLUTIONS NO LULTORS are allowed on this event. 70. Determine exactly the value of. LM 0,4 GD 0,4 LM 0,4 GD 0,4 7 LM, 7 GD, The expression 4 3 can be simplified and written as a single rational number. Determine exactly that rational number. Work from the inside out: Day # 7 3. The epad was originally priced at $00, but the newest model of epad is coming out, so the old one is going on sale. Each day, the price is reduced by 0%, and rounded down to the nearest dollar, as necessary. (On Day, the price is reduced to $90; on Day it is $8; on Day 3 it is $7, and so on.) What is the First day on which the old epad will cost $? The basic idea is that we re subtracting the tens place of the previous day s cost, along with an additional dollar if the ones place is greater than zero. (For example, 7 7. rounds to ) Make a table: Day # ost($) Express as a repeating decimal. 9 0., 0.09, We have repetends of lengths,, and 3, so to add them, we 999 need to extend to a common multiple: 6 decimal places: ( )
3 03-4 Meet, Individual Event Question # is intended to be a quickie and is worth point. Each of the next three questions is worth points. Place your answer to each question on the line provided. You have minutes for this event.. If 3 and 4 are the lengths of the two legs of a right triangle, determine exactly the length of the triangle s hypotenuse. m PQ s r. In Figure, uu sur P D sur, PQ is a transversal, P m PQ x, and m DQP x. Determine exactly the measure of angle PQ. x x - Q D Figure m FKH 3. In isosceles triangle FGH (), the vertex angle measures 0 and its opposite side is 0 cm long. G segment from point H meets side FG at point K so that HK 0 cm. Determine the measure of angle FKH. K 0 F 0 cm H m WVZ 4. In Figure 4, XY XZ, m YXV 30, and XV XW. Determine exactly the measure of angle WVZ. X W Y V Figure 4 Z Name: Team:
4 03-4 Meet, Individual Event SOLUTIONS or If 3 and 4 are the lengths of the two legs of a right triangle, determine exactly the length of the triangle s hypotenuse. a + b c c c c 44. m PQ 4 s r. In Figure, uu sur P D sur, PQ is a transversal, m PQ x, and m DQP x. Determine exactly the measure of angle PQ. P x PQ and DQP are same- side interior angles, so 80 they are supplementary: x + x 7x 3 x 33, and since PQ and DQP Q x - Figure D are congruent (alternate interior angles), m PQ x ( 33) 6 4. G m FKH In isosceles triangle FGH (), the vertex angle measures 0 and its opposite side is 0 cm long. segment from point H meets side FG at point K so that HK 0 cm. Determine the measure of angle FKH. K 0 θ VHFK is isosceles, so label m FKH m F θ. We are told that VFGH is isosceles, so m FHG θ also. onsidering the three angles in VFGH, 0+θ +θ 80 θ 80. F θ 0 cm H m WVZ 4. In Figure 4, XY XZ, m YXV 30, and XV XW. Determine exactly the measure of angle WVZ. Let m VXW α. Then m YXZ 30 +α, and 90 α m Z α 7 α. lso, m VWX 80 α. VWX is Y X α 30 V Figure 4 W Z exterior to VVWZ, so m VWX m WVZ + m Z 90 α m WVZ + 7 α m WVZ.
5 03-4 Meet, Individual Event Question # is intended to be a quickie and is worth point. Each of the next three questions is worth points. Place your answer to each question on the line provided. You have minutes for this event. a. In Figure, point P(a, b) is located in the second quadrant on the circumference of a circle of radius r. Express a in terms of b and r. P (a,b) r Figure sinθ. line segment drawn from the origin to a point (m, n) in the fourth quadrant makes an angle of \ with the positive x- axis. Express sin \ in terms of m and n. tan α 3. The point (^, k) lies on the graph of y cos x (), beneath the y- axis and with ^ between 4 and radians. Express tan ^ in terms of k. 4 6 (α, k) D 4. Semicircle O, pictured in Figure 4, has diameter 0. rc also has length 0. From, a perpendicular is dropped to meet segment at D. alculate the length D. O Figure 4 Name: Team:
6 a r b Minnesota State High School Mathematics League 03-4 Meet, Individual Event SOLUTIONS. In Figure, point P(a, b) is located in the second quadrant on the circumference of a circle of radius r. Express a in terms of b and r. P (a,b) b a r Drop a perpendicular from P to the x- axis as shown, creating a right triangle with legs a, b and hypotenuse r. a + b r a r b a ± r b, and since a lies on the negative x- axis, a r b. Figure sinθ n m + n. line segment drawn from the origin to a point (m, n) in the fourth quadrant makes an angle of \ with the positive x- axis. Express sin \ in terms of m and n. n m + n or m + n or equivalent See Figure. sin θ opp hyp n c, and since m + n c, we have c m + n sinθ n n m + n. m + n m + n c m θ n tan α k k Figure (m,n) 3. The point (^, k) lies on the graph of y cos x (), beneath the y- axis and with ^ between 4 and radians. Express tan ^ in terms of k. cos α k, which is negative. π <α < 3π, so sin α is also negative. sin α + cos α sinα ± cos α ± k. tan α sinα k cosα k. 4 6 (α, k) D +cos( ) or.99 Graders: if students use 360 as an π angle equivalent, the degree sign must be included. 4. Semicircle O, pictured in Figure 4, has diameter 0. rc also has length 0. From, a perpendicular is dropped to meet segment at D. alculate the length D. Since has a length equal to twice the semicircle s radius, by de`inition of a radian, angle O measures exactly, but also, radians. cos OD OD and cos O cos since OD and O are supplementary, their cosines are opposites. OD cos. D O OD cos( ) 0 O Figure 4 D
7 03-4 Meet, Individual Event D Question # is intended to be a quickie and is worth point. Each of the next three questions is worth points. Place your answer to each question on the line provided. You have minutes for this event. NO LULTORS are allowed on this event. r. Let r and r be the distinct roots of r r 0, with r < r. Determine r exactly. x + x. Let x and x be the solutions of x 0x Determine x + x exactly. f ( ) 3. Let f x be a quadratic polynomial. If it is known that f ( 0), f, and f ( ) 0, exactly. determine f Let ( X ) ( X ) ( X ) ( X + 3)+ ( X ) ( 3X )+ ( X + 3) ( X ), where,, and are all real numbers. Determine the sum + + exactly. Name: Team:
8 03-4 Meet, Individual Event D SOLUTIONS NO LULTORS are allowed on this event. r. Let r and r be the distinct roots of r r 0, with r < r. Determine r exactly. r r 0 ( r ) ( r + 4) 0 r 4 or. Since 4 <, r 4 and r. x + x 0 3. Let x and x be the solutions of x 0x Determine x + x exactly. or 7 3 Of course we could use the quadratic formula to `ind both roots, but it s more elegant to use the formulas for the sum & product of roots: + x + x x + x sum of roots x x x x x x x x product of roots 0 3. f ( ) 3 3. Let f x be a quadratic polynomial. If it is known that f ( 0), f, and f ( ) 0, exactly. determine f Let f ( x) ax + bx + c. Then f ( 0) a( 0) + b( 0)+ c c, f a + b a+ b+ a+ b, and f ( ) a( ) + b + c + c 4a+b+ c a+( a+b)+ c 0 a++ 0 a 3, b. Thus our quadratic polynomial is f ( x) 3 x + x +, and we have f ( ) 3 ( ) + ( ) Let ( X ) ( X ) ( X ) ( X + 3)+ ( X ) ( 3X )+ ( X + 3) ( X ), where,, and are all real numbers. Determine the sum + + exactly. or 6 6 Rearrange the right side of the equation, then factor out X + 3 ( X + 3) ( X )+ ( X ) + ( X )( 3X ) ( X + 3) ( 3X )+ X from two of the terms: ( 3X ) ( X +) ( 3X ). Since the coef`icient of the X term in each factor must be, we factor out of the `irst factor and 3 out of the second factor to yield: X + 3 X 3 6 X + X 3 X So 6,, ( X ).
9 03-4 Meet, Team Event Each question is worth 4 points. Team members may cooperate in any way, but at the end of 0 minutes, submit only one set of answers. Place your answer to each question on the line provided. sum. Find the sum of all positive integers n for which LM(, n) GD(n, 0). m. On isosceles V, points P and Q are on sides and respectively, distinct from,, and, so that P PQ Q. Determine exactly the measure of angle. 3. shows a semicircle with radius. If arc ª also has length, calculate the length of chord. a 4. Let f t t + at. Given that at least one of the coefficients of the degree- four f ( t ) polynomial f is zero, list all possible values of a. y k. Figure shows rays drawn from the origin at angular intervals of 0, intersecting the line x at y, y, etc. Find the smallest positive integer k for which y k y k will equal or exceed y. y 6 y y 4 y 3 y x x"" Figure b 6. Let N be a number in base b such that N b 4 b 7 b. What is the greatest base b for which N b would be written with as its left- most digit? Team:
10 03-4 Meet, Team Event SOLUTIONS (page ) sum 384. Find the sum of all positive integers n for which LM(, n) GD(n, 0). m 7 80 or 7. On isosceles V, points P and Q are on sides and respectively, distinct from,, and, so that P PQ Q. Determine exactly the measure of angle. Q m m m Figure shows a semicircle with radius. If arc m m 80-4m 3m 3m P or 0.98 or sin( 0.) or cos ª also has length, calculate the length of chord. D a, 0,, Graders: award point per correct value 4. Let f t t + at. Given that at least one of the coefficients of the degree- four f ( t ) polynomial f is zero, list all possible values of a. y y 6 k. Figure shows rays drawn from the origin at angular intervals of 0, intersecting the line x at y, y, etc. Find the smallest positive integer k for which y k y k will equal or exceed y. y y 4 y 3 y x"" x b 3 6. Let N be a number in base b such that N b 4 b 7 b. What is the greatest base b for which N b would be written with as its left- most digit?
11 03-4 Meet, Team Event SOLUTIONS (page ). For LM(, n) GD(n, 0), both quantities must be equal to n. This means n is both a multiple of and a factor of 0. Examining the prime factorization of 0 ( 3 7 ), the acceptable values of n must be of the form 3 a b 7 c, where a, b, and c are all either 0 or. This yields eight values:, 6, 0, 4, 30, 4, 70, and 0. Their sum is See Figure. egin by labeling m as m. Then in isosceles VPQ, m P m also. PQ is an exterior angle of this triangle, and so its measure is m. Using isosceles VQP, m m, and m QP 80 4m. Subtracting along straight angle P, m P 3m, and in isosceles VP, m 3m also. Now we must remember that the original triangle, V, was given as isosceles, so m Q 3m m+ m P m P m. We now have expressions for all 80 angle measures in VP : m+ 3m+ 3m 80 m Drawing segments D and creates VD, and because it is inscribed in a semicircle, it is a right triangle with right angle at. ecause arc has the same length as its circle s radius, by the de`inition of radian, m radian, and m D 0. radians. Using trigonometry, sin( 0.) sin( 0.) f ( t + at ) t ( + at ) + a ( t + at ) t 4 + at 3 + ( a )t at + + ( at + a t a) t 4 + at 3 + ( a + a )t + ( a a)t a. Since one of the coef`icients is zero, our possibilities are a 0 a 0 ; a + a 0 a + ( a ) 0 a or ; a a 0 a( a ) 0 a 0 or (only is unique), or a 0 (repeat value again). The unique values of a are, 0,,.. y k y k y + y + + y k y + y + + y k tan 0k y Therefore, we want tan 0k tan( 0( k ) ). y tan , so tan( 0( k ) ) Generate a table of possible values for k: tan 0k k tan( 0( k ) ) The `irst value that satis`ies the inequality is k. (It turns out, in fact, that tan0 tan40 tan0. an you prove why this is true?) 6. N b 4 b 7 b N ( b + 4) ( b + 7) b +b + 8. ecause we want a quantity of in the b column, we need b + 8 b (and also b + 8 < b ). Either solve the inequality or try values of b > 0 to `ind the greatest b 3.
Minnesota State High School Mathematics League
011-1 Meet 1, Individual Event Question #1 is intended to be a quickie and is worth 1 point. Each of the next three questions is worth points. Place your answer to each question on the line provided. You
More informationQUESTION BANK ON STRAIGHT LINE AND CIRCLE
QUESTION BANK ON STRAIGHT LINE AND CIRCLE Select the correct alternative : (Only one is correct) Q. If the lines x + y + = 0 ; 4x + y + 4 = 0 and x + αy + β = 0, where α + β =, are concurrent then α =,
More informationUNC Charlotte 2005 Comprehensive March 7, 2005
March 7, 2005 1. The numbers x and y satisfy 2 x = 15 and 15 y = 32. What is the value xy? (A) 3 (B) 4 (C) 5 (D) 6 (E) none of A, B, C or D Solution: C. Note that (2 x ) y = 15 y = 32 so 2 xy = 2 5 and
More informationMinnesota State High School Mathematics League
01-1 Meet, Individual Event A Question #1 is intended to be a quickie and is worth 1 point. Each of the next three questions is worth points. Place your answer to each question on the line provided. You
More informationSAGINAW VALLEY STATE UNIVERSITY SOLUTIONS OF 2013 MATH OLYMPICS LEVEL II. 1 4n + 1. n < < n n n n + 1. n < < n n 1. n 1.
SAGINAW VALLEY STATE UNIVERSITY SOLUTIONS OF 03 MATH OLYMPICS LEVEL II. The following inequalities hold for all positive integers n: n + n < 4n + < n n. What is the greatest integer which is less than
More informationChapter 13: Trigonometry Unit 1
Chapter 13: Trigonometry Unit 1 Lesson 1: Radian Measure Lesson 2: Coterminal Angles Lesson 3: Reference Angles Lesson 4: The Unit Circle Lesson 5: Trig Exact Values Lesson 6: Trig Exact Values, Radian
More informationC=2πr C=πd. Chapter 10 Circles Circles and Circumference. Circumference: the distance around the circle
10.1 Circles and Circumference Chapter 10 Circles Circle the locus or set of all points in a plane that are A equidistant from a given point, called the center When naming a circle you always name it by
More informationTheorem 1.2 (Converse of Pythagoras theorem). If the lengths of the sides of ABC satisfy a 2 + b 2 = c 2, then the triangle has a right angle at C.
hapter 1 Some asic Theorems 1.1 The ythagorean Theorem Theorem 1.1 (ythagoras). The lengths a b < c of the sides of a right triangle satisfy the relation a + b = c. roof. b a a 3 b b 4 b a b 4 1 a a 3
More informationSummer Work for students entering PreCalculus
Summer Work for students entering PreCalculus Name Directions: The following packet represent a review of topics you learned in Algebra 1, Geometry, and Algebra 2. Complete your summer packet on separate
More informationSecondary School Certificate Examination Syllabus MATHEMATICS. Class X examination in 2011 and onwards. SSC Part-II (Class X)
Secondary School Certificate Examination Syllabus MATHEMATICS Class X examination in 2011 and onwards SSC Part-II (Class X) 15. Algebraic Manipulation: 15.1.1 Find highest common factor (H.C.F) and least
More informationIntegrated Math II. IM2.1.2 Interpret given situations as functions in graphs, formulas, and words.
Standard 1: Algebra and Functions Students graph linear inequalities in two variables and quadratics. They model data with linear equations. IM2.1.1 Graph a linear inequality in two variables. IM2.1.2
More informationGlossary. Glossary 981. Hawkes Learning Systems. All rights reserved.
A Glossary Absolute value The distance a number is from 0 on a number line Acute angle An angle whose measure is between 0 and 90 Addends The numbers being added in an addition problem Addition principle
More informationPractice Test Geometry 1. Which of the following points is the greatest distance from the y-axis? A. (1,10) B. (2,7) C. (3,5) D. (4,3) E.
April 9, 01 Standards: MM1Ga, MM1G1b Practice Test Geometry 1. Which of the following points is the greatest distance from the y-axis? (1,10) B. (,7) C. (,) (,) (,1). Points P, Q, R, and S lie on a line
More informationGrade 11 Pre-Calculus Mathematics (1999) Grade 11 Pre-Calculus Mathematics (2009)
Use interval notation (A-1) Plot and describe data of quadratic form using appropriate scales (A-) Determine the following characteristics of a graph of a quadratic function: y a x p q Vertex Domain and
More informationSummer Work for students entering PreCalculus
Summer Work for students entering PreCalculus Name Directions: The following packet represent a review of topics you learned in Algebra 1, Geometry, and Algebra 2. Complete your summer packet on separate
More information2018 LEHIGH UNIVERSITY HIGH SCHOOL MATH CONTEST
08 LEHIGH UNIVERSITY HIGH SCHOOL MATH CONTEST. A right triangle has hypotenuse 9 and one leg. What is the length of the other leg?. Don is /3 of the way through his run. After running another / mile, he
More informationUNCC 2001 Comprehensive, Solutions
UNCC 2001 Comprehensive, Solutions March 5, 2001 1 Compute the sum of the roots of x 2 5x + 6 = 0 (A) (B) 7/2 (C) 4 (D) 9/2 (E) 5 (E) The sum of the roots of the quadratic ax 2 + bx + c = 0 is b/a which,
More informationPRACTICE TEST 1 Math Level IC
SOLID VOLUME OTHER REFERENCE DATA Right circular cone L = cl V = volume L = lateral area r = radius c = circumference of base h = height l = slant height Sphere S = 4 r 2 V = volume r = radius S = surface
More informationTrigonometric ratios:
0 Trigonometric ratios: The six trigonometric ratios of A are: Sine Cosine Tangent sin A = opposite leg hypotenuse adjacent leg cos A = hypotenuse tan A = opposite adjacent leg leg and their inverses:
More information4.3 TRIGONOMETRY EXTENDED: THE CIRCULAR FUNCTIONS
4.3 TRIGONOMETRY EXTENDED: THE CIRCULAR FUNCTIONS MR. FORTIER 1. Trig Functions of Any Angle We now extend the definitions of the six basic trig functions beyond triangles so that we do not have to restrict
More informationMATH II CCR MATH STANDARDS
RELATIONSHIPS BETWEEN QUANTITIES M.2HS.1 M.2HS.2 M.2HS.3 M.2HS.4 M.2HS.5 M.2HS.6 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents
More informationThings You Should Know Coming Into Calc I
Things You Should Know Coming Into Calc I Algebraic Rules, Properties, Formulas, Ideas and Processes: 1) Rules and Properties of Exponents. Let x and y be positive real numbers, let a and b represent real
More informationSM2H Unit 6 Circle Notes
Name: Period: SM2H Unit 6 Circle Notes 6.1 Circle Vocabulary, Arc and Angle Measures Circle: All points in a plane that are the same distance from a given point, called the center of the circle. Chord:
More information(D) (A) Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point ? 2 (A) 2
CIRCLE [STRAIGHT OBJECTIVE TYPE] Q. The line x y + = 0 is tangent to the circle at the point (, 5) and the centre of the circles lies on x y = 4. The radius of the circle is (A) 3 5 (B) 5 3 (C) 5 (D) 5
More informationIntroductory Chapter(s)
Course 1 6 11-12 years 1 Exponents 2 Sequences Number Theory 3 Divisibility 4 Prime Factorization 5 Greatest Common Factor Fractions and Decimals 6 Comparing and Ordering Fractions 7 Comparing and Ordering
More informationAlgebra II/Geometry Curriculum Guide Dunmore School District Dunmore, PA
Algebra II/Geometry Dunmore School District Dunmore, PA Algebra II/Geometry Prerequisite: Successful completion of Algebra 1 Part 2 K Algebra II/Geometry is intended for students who have successfully
More information(A) 20% (B) 25% (C) 30% (D) % (E) 50%
ACT 2017 Name Date 1. The population of Green Valley, the largest suburb of Happyville, is 50% of the rest of the population of Happyville. The population of Green Valley is what percent of the entire
More informationUnit Circle. Return to. Contents
Unit Circle Return to Table of Contents 32 The Unit Circle The circle x 2 + y 2 = 1, with center (0,0) and radius 1, is called the unit circle. Quadrant II: x is negative and y is positive (0,1) 1 Quadrant
More informationChapter 1. Some Basic Theorems. 1.1 The Pythagorean Theorem
hapter 1 Some asic Theorems 1.1 The ythagorean Theorem Theorem 1.1 (ythagoras). The lengths a b < c of the sides of a right triangle satisfy the relation a 2 + b 2 = c 2. roof. b a a 3 2 b 2 b 4 b a b
More informationChapter (Circle) * Circle - circle is locus of such points which are at equidistant from a fixed point in
Chapter - 10 (Circle) Key Concept * Circle - circle is locus of such points which are at equidistant from a fixed point in a plane. * Concentric circle - Circle having same centre called concentric circle.
More informationMath 5 Trigonometry Review Sheet for Chapter 5
Math 5 Trigonometry Review Sheet for Chapter 5 Key Ideas: Def: Radian measure of an angle is the ratio of arclength subtended s by that central angle to the radius of the circle: θ s= rθ r 180 = π radians.
More informationInstructions. Do not open your test until instructed to do so!
st Annual King s College Math Competition King s College welcomes you to this year s mathematics competition and to our campus. We wish you success in this competition and in your future studies. Instructions
More information9-12 Mathematics Vertical Alignment ( )
Algebra I Algebra II Geometry Pre- Calculus U1: translate between words and algebra -add and subtract real numbers -multiply and divide real numbers -evaluate containing exponents -evaluate containing
More informationCircles in Neutral Geometry
Everything we do in this set of notes is Neutral. Definitions: 10.1 - Circles in Neutral Geometry circle is the set of points in a plane which lie at a positive, fixed distance r from some fixed point.
More information02)
GRE / GMATmath,! abscissa, scalene, intercept, markup, such that, break even. abscissa. (4, 2) 4abscissa, 2ordinate absolute value acre add adjacent angles altitude ; angle () acute angle (90 ) right angle
More information2009 Math Olympics Level II Solutions
Saginaw Valley State University 009 Math Olympics Level II Solutions 1. f (x) is a degree three monic polynomial (leading coefficient is 1) such that f (0) 3, f (1) 5 and f () 11. What is f (5)? (a) 7
More informationAustralian Intermediate Mathematics Olympiad 2016
A u s t r a l i a n M at h e m at i c a l O ly m p i a d C o m m i t t e e a d e p a r t m e n t o f t h e a u s t r a l i a n m at h e m at i c s t r u s t Australian Intermediate Mathematics Olympiad
More information4 The Trigonometric Functions
Mathematics Learning Centre, University of Sydney 8 The Trigonometric Functions The definitions in the previous section apply to between 0 and, since the angles in a right angle triangle can never be greater
More informationIncoming Magnet Precalculus / Functions Summer Review Assignment
Incoming Magnet recalculus / Functions Summer Review ssignment Students, This assignment should serve as a review of the lgebra and Geometry skills necessary for success in recalculus. These skills were
More informationa. 1 b. i c. 1-2i d. i e. NOTA
Theta Individual State Convention 017 1. 15 1 16 15 i i i i i =? a. 1 b. i c. 1- i d. i e. NOTA. Mr. Lu has $.7 in pennies, nickels, dimes, quarters and half dollars. If he has an equal number of coins
More informationas a base-six number.
Round 1: Number Theory Note: a subscript indicates a number s base 1 1. Express the sum 354 6 + 43 6 as a base-six number.. The three-digit base-ten number 3A3 is added to the base-ten number 44 to give
More informationAustralian Intermediate Mathematics Olympiad 2016
Australian Intermediate Mathematics Olympiad 06 Questions. Find the smallest positive integer x such that x = 5y, where y is a positive integer. [ marks]. A 3-digit number in base 7 is also a 3-digit number
More informationUNC Charlotte 2005 Comprehensive March 7, 2005
March 7, 2005 1 The numbers x and y satisfy 2 x = 15 and 15 y = 32 What is the value xy? (A) 3 (B) 4 (C) 5 (D) 6 (E) none of A, B, C or D 2 Suppose x, y, z, and w are real numbers satisfying x/y = 4/7,
More informationCandidates are expected to have available a calculator. Only division by (x + a) or (x a) will be required.
Revision Checklist Unit C2: Core Mathematics 2 Unit description Algebra and functions; coordinate geometry in the (x, y) plane; sequences and series; trigonometry; exponentials and logarithms; differentiation;
More informationGlossary. Glossary Hawkes Learning Systems. All rights reserved.
A Glossary Absolute value The distance a number is from 0 on a number line Acute angle An angle whose measure is between 0 and 90 Acute triangle A triangle in which all three angles are acute Addends The
More informationDESK Secondary Math II
Mathematical Practices The Standards for Mathematical Practice in Secondary Mathematics I describe mathematical habits of mind that teachers should seek to develop in their students. Students become mathematically
More informationGeometry Chapter 7 7-4: SPECIAL RIGHT TRIANGLES
Geometry Chapter 7 7-4: SPECIAL RIGHT TRIANGLES Warm-Up Simplify the following. 1.) 10 30 2.) 45 5 3.) 88 8 4.) 3 27 Special Right Triangles Objective: Students will be able to use the relationships amongst
More informationCoach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers
Coach Stones Expanded Standard Pre-Calculus Algorithm Packet Page 1 Section: P.1 Algebraic Expressions, Mathematical Models and Real Numbers CLASSIFICATIONS OF NUMBERS NATURAL NUMBERS = N = {1,2,3,4,...}
More informationSince x + we get x² + 2x = 4, or simplifying it, x² = 4. Therefore, x² + = 4 2 = 2. Ans. (C)
SAT II - Math Level 2 Test #01 Solution 1. x + = 2, then x² + = Since x + = 2, by squaring both side of the equation, (A) - (B) 0 (C) 2 (D) 4 (E) -2 we get x² + 2x 1 + 1 = 4, or simplifying it, x² + 2
More informationCLASS X FORMULAE MATHS
Real numbers: Euclid s division lemma Given positive integers a and b, there exist whole numbers q and r satisfying a = bq + r, 0 r < b. Euclid s division algorithm: This is based on Euclid s division
More informationMAT1035 Analytic Geometry
MAT1035 Analytic Geometry Lecture Notes R.A. Sabri Kaan Gürbüzer Dokuz Eylül University 2016 2 Contents 1 Review of Trigonometry 5 2 Polar Coordinates 7 3 Vectors in R n 9 3.1 Located Vectors..............................................
More information7th Grade Curriculum
Unit #1 Number Systems 1.1 Number Systems 7th Grade Curriculum Distinguish between the various subsets of real numbers (Counting/natural numbers, whole numbers, integers, rational numbers, and irrational
More informationACT MATH MUST-KNOWS Pre-Algebra and Elementary Algebra: 24 questions
Pre-Algebra and Elementary Algebra: 24 questions Basic operations using whole numbers, integers, fractions, decimals and percents Natural (Counting) Numbers: 1, 2, 3 Whole Numbers: 0, 1, 2, 3 Integers:
More informationThanks for downloading this product from Time Flies!
Thanks for downloading this product from Time Flies! I hope you enjoy using this product. Follow me at my TpT store! My Store: https://www.teacherspayteachers.com/store/time-flies 2018 Time Flies. All
More informationName Score Period Date. m = 2. Find the geometric mean of the two numbers. Copy and complete the statement.
Chapter 6 Review Geometry Name Score Period Date Solve the proportion. 3 5 1. = m 1 3m 4 m = 2. 12 n = n 3 n = Find the geometric mean of the two numbers. Copy and complete the statement. 7 x 7? 3. 12
More informationWritten test, 25 problems / 90 minutes
Sponsored by: UGA Math Department and UGA Math Club Written test, 5 problems / 90 minutes October, 06 WITH SOLUTIONS Problem. Let a represent a digit from to 9. Which a gives a! aa + a = 06? Here aa indicates
More informationUnit Circle: The unit circle has radius 1 unit and is centred at the origin on the Cartesian plane. POA
The Unit Circle Unit Circle: The unit circle has radius 1 unit and is centred at the origin on the Cartesian plane THE EQUATION OF THE UNIT CIRCLE Consider any point P on the unit circle with coordinates
More informationCrash Course in Trigonometry
Crash Course in Trigonometry Dr. Don Spickler September 5, 003 Contents 1 Trigonometric Functions 1 1.1 Introduction.................................... 1 1. Right Triangle Trigonometry...........................
More informationCongratulations! You ve completed Practice Test 1! You re now ready to check your
Practice Test 1: Answers and Explanations Congratulations! You ve completed Practice Test 1! You re now ready to check your answers to see how you fared. In this chapter, I provide the answers, including
More informationDue to the detail of some problems, print the contests using a normal or high quality setting.
General Contest Guidelines: Keep the contests secure. Discussion about contest questions is not permitted prior to giving the contest. Due to the detail of some problems, print the contests using a normal
More informationCorrelation of 2012 Texas Essential Knowledge and Skills (TEKS) for Algebra I and Geometry to Moving with Math SUMS Moving with Math SUMS Algebra 1
Correlation of 2012 Texas Essential Knowledge and Skills (TEKS) for Algebra I and Geometry to Moving with Math SUMS Moving with Math SUMS Algebra 1 ALGEBRA I A.1 Mathematical process standards. The student
More informationStatistics. To find the increasing cumulative frequency, we start with the first
Statistics Relative frequency = frequency total Relative frequency in% = freq total x100 To find the increasing cumulative frequency, we start with the first frequency the same, then add the frequency
More informationThe Grade Descriptors below are used to assess work and student progress in Mathematics from Year 7 to
Jersey College for Girls Assessment criteria for KS3 and KS4 Mathematics In Mathematics, students are required to become familiar, confident and competent with a range of content and procedures described
More informationNew Jersey Center for Teaching and Learning. Progressive Mathematics Initiative
Slide 1 / 150 New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students
More informationIntegers, Fractions, Decimals and Percentages. Equations and Inequations
Integers, Fractions, Decimals and Percentages Round a whole number to a specified number of significant figures Round a decimal number to a specified number of decimal places or significant figures Perform
More information2005 Palm Harbor February Invitational Geometry Answer Key
005 Palm Harbor February Invitational Geometry Answer Key Individual 1. B. D. C. D 5. C 6. D 7. B 8. B 9. A 10. E 11. D 1. C 1. D 1. C 15. B 16. B 17. E 18. D 19. C 0. C 1. D. C. C. A 5. C 6. C 7. A 8.
More information9.7 Extension: Writing and Graphing the Equations
www.ck12.org Chapter 9. Circles 9.7 Extension: Writing and Graphing the Equations of Circles Learning Objectives Graph a circle. Find the equation of a circle in the coordinate plane. Find the radius and
More informationA List of Definitions and Theorems
Metropolitan Community College Definition 1. Two angles are called complements if the sum of their measures is 90. Two angles are called supplements if the sum of their measures is 180. Definition 2. One
More information1. Town A is 48 km from town B and 32 km from town C as shown in the diagram. A 48km
1. Town is 48 km from town and 32 km from town as shown in the diagram. 32km 48km Given that town is 56 km from town, find the size of angle (Total 4 marks) Â to the nearest degree. 2. The diagram shows
More informationPre RMO Exam Paper Solution:
Paper Solution:. How many positive integers less than 000 have the property that the sum of the digits of each such number is divisible by 7 and the number itself is divisible by 3? Sum of Digits Drivable
More informationRemember, you may not use a calculator when you take the assessment test.
Elementary Algebra problems you can use for practice. Remember, you may not use a calculator when you take the assessment test. Use these problems to help you get up to speed. Perform the indicated operation.
More informationMth 133 Trigonometry Review Problems for the Final Examination
Mth 1 Trigonometry Review Problems for the Final Examination Thomas W. Judson Stephen F. Austin State University Fall 017 Final Exam Details The final exam for MTH 1 will is comprehensive and will cover
More informationCatchup Math and the Common Core Standards. Spring 2011
Catchup Math and the Common Core Standards Spring 2011 The Catchup Math curriculum addresses nearly all the Common Core Mathematics Standards for Grades 6 8 and High School (including most of the optional
More informationMATH 32 FALL 2012 FINAL EXAM - PRACTICE EXAM SOLUTIONS
MATH 3 FALL 0 FINAL EXAM - PRACTICE EXAM SOLUTIONS () You cut a slice from a circular pizza (centered at the origin) with radius 6 along radii at angles 4 and 3 with the positive horizontal axis. (a) (3
More informationChapter 3. Radian Measure and Circular Functions. Section 3.1: Radian Measure. π 1.57, 1 is the only integer value in the
Chapter Radian Measure and Circular Functions Section.: Radian Measure. Since θ is in quadrant I, 0 < θ
More informationTEST CODE FORM TP JANUARY 2015 C A R I B B E A N E X A M I N A T I O N S C O U N C I L
TEST CODE 01234020 FORM TP 2015017 JANUARY 2015 C A R I B B E A N E X A M I N A T I O N S C O U N C I L CARIBBEAN SECONDARY EDUCATION CERTIFICATE EXAMINATION MATHEMATICS Paper 02 General Proficiency 2
More informationGive a geometric description of the set of points in space whose coordinates satisfy the given pair of equations.
1. Give a geometric description of the set of points in space whose coordinates satisfy the given pair of equations. x + y = 5, z = 4 Choose the correct description. A. The circle with center (0,0, 4)
More informationMath 5 Trigonometry Final Exam Spring 2009
Math 5 Trigonometry Final Exam Spring 009 NAME Show your work for credit. Write all responses on separate paper. There are 13 problems, all weighted equally. Your 3 lowest scoring answers problem will
More informationPrecalculus Summer Assignment 2015
Precalculus Summer Assignment 2015 The following packet contains topics and definitions that you will be required to know in order to succeed in CP Pre-calculus this year. You are advised to be familiar
More informationSection 6.2 Notes Page Trigonometric Functions; Unit Circle Approach
Section Notes Page Trigonometric Functions; Unit Circle Approach A unit circle is a circle centered at the origin with a radius of Its equation is x y = as shown in the drawing below Here the letter t
More information2015 Canadian Team Mathematics Contest
The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca 205 Canadian Team Mathematics Contest April 205 Solutions 205 University of Waterloo 205 CTMC Solutions Page 2 Individual Problems.
More informationTrigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters
Trigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters α( alpha), β ( beta), θ ( theta) as well as upper case letters A,B,
More informationThe Learning Objectives of the Compulsory Part Notes:
17 The Learning Objectives of the Compulsory Part Notes: 1. Learning units are grouped under three strands ( Number and Algebra, Measures, Shape and Space and Data Handling ) and a Further Learning Unit.
More informationIntermediate Math Circles Wednesday October Problem Set 3
The CETRE for EDUCTI in MTHEMTICS and CMPUTIG Intermediate Math Circles Wednesday ctober 24 2012 Problem Set 3.. Unless otherwise stated, any point labelled is assumed to represent the centre of the circle.
More informationFunctions and their Graphs
Chapter One Due Monday, December 12 Functions and their Graphs Functions Domain and Range Composition and Inverses Calculator Input and Output Transformations Quadratics Functions A function yields a specific
More informationCALCULUS ASSESSMENT REVIEW
CALCULUS ASSESSMENT REVIEW DEPARTMENT OF MATHEMATICS CHRISTOPHER NEWPORT UNIVERSITY 1. Introduction and Topics The purpose of these notes is to give an idea of what to expect on the Calculus Readiness
More informationArcs and Inscribed Angles of Circles
Arcs and Inscribed Angles of Circles Inscribed angles have: Vertex on the circle Sides are chords (Chords AB and BC) Angle ABC is inscribed in the circle AC is the intercepted arc because it is created
More informationMASSACHUSETTS ASSOCIATION OF MATHEMATICS LEAGUES STATE PLAYOFFS Arithmetic and Number Theory 1.
STTE PLYOFFS 004 Round 1 rithmetic and Number Theory 1.. 3. 1. How many integers have a reciprocal that is greater than 1 and less than 1 50. 1 π?. Let 9 b,10 b, and 11 b be numbers in base b. In what
More informationOBJECTIVE TEST. Answer all questions C. N3, D. N3, Simplify Express the square root of in 4
. In a particular year, the exchange rate of Naira (N) varies directly with the Dollar ($). If N is equivalent to $8, find the Naira equivalent of $6. A. N8976 B. N049 C. N40. D. N.7. If log = x, log =
More informationGRE Quantitative Reasoning Practice Questions
GRE Quantitative Reasoning Practice Questions y O x 7. The figure above shows the graph of the function f in the xy-plane. What is the value of f (f( ))? A B C 0 D E Explanation Note that to find f (f(
More informationUnit 4-Review. Part 1- Triangle Theorems and Rules
Unit 4-Review - Triangle Theorems and Rules Name of Theorem or relationship In words/ Symbols Diagrams/ Hints/ Techniques 1. Side angle relationship 2. Triangle inequality Theorem 3. Pythagorean Theorem
More informationScope and Sequence: National Curriculum Mathematics from Haese Mathematics (7 10A)
Scope and Sequence: National Curriculum Mathematics from Haese Mathematics (7 10A) Updated 06/05/16 http://www.haesemathematics.com.au/ Note: Exercises in red text indicate material in the 10A textbook
More informationPre-Algebra (7) B Mathematics
Course Overview Students will develop skills in using variables, evaluating algebraic expressions by the use of the order of operations, solving equations and inequalities, graphing linear equations, functions
More informationMu Alpha Theta National Convention 2013
Practice Round Alpha School Bowl P1. What is the common difference of the arithmetic sequence 10, 23,? P2. Find the sum of the digits of the base ten representation of 2 15. P3. Find the smaller value
More informationCERT Grade 11 Mathematics Test 2 60 Minutes 60 Questions
DIRECTIONS: Solve each problem, choose the correct answer, and then fill in the corresponding oval on your answer sheet. Do not linger over problems that take too much time. Solve as many as you can; then
More informationChapter 6. Worked-Out Solutions AB 3.61 AC 5.10 BC = 5
27. onstruct a line ( DF ) with midpoint P parallel to and twice the length of QR. onstruct a line ( EF ) with midpoint R parallel to and twice the length of QP. onstruct a line ( DE ) with midpoint Q
More informationAnswer Explanations for: ACT June 2012, Form 70C
Answer Explanations for: ACT June 2012, Form 70C Mathematics 1) C) A mean is a regular average and can be found using the following formula: (average of set) = (sum of items in set)/(number of items in
More informationMath 9 Unit 8: Circle Geometry Pre-Exam Practice
Math 9 Unit 8: Circle Geometry Pre-Exam Practice Name: 1. A Ruppell s Griffon Vulture holds the record for the bird with the highest documented flight altitude. It was spotted at a height of about 11 km
More informationAll E Maths Formulas for O levels E Maths by Ethan Wu
All E Maths Formulas for O levels E Maths by Ethan Wu Chapter 1: Indices a 5 = a x a x a x a x a a m a n = a m + n a m a n = a m n (a m ) n = a m n (ab) n = a n b n ( a b )n = an b n a 0 = 1 a -n = 1 a
More informationAppendix C: Event Topics per Meet
Appendix C: Event Topics per Meet Meet 1 1A Pre-algebra Topics Fractions to add and express as the quotient of two relatively prime integers Complex fractions and continued fractions Decimals, repeating
More information