The Cauchy-Schwarz inequality
|
|
- Alexander Flynn
- 6 years ago
- Views:
Transcription
1 The Cauchy-Schwarz inequality Finbarr Holland, April, Introduction The inequality known as the Cauchy-Schwarz inequality, CS for short, is probably the most useful of all inequalities, and arises in most areas of Mathematics. It comes in various shapes and sizes, and can involve finite or infinitely many variables, and be discrete or continuous, and a mixture of all of these kinds. Here we discuss the simplest kind, which involves two vectors in the same finite-dimensional vector space. The statement Let a = (a 1, a,..., a n ), b = (b 1, b,..., b n ) be two vectors with real coordinates, i.e., a, b R n. Their inner-product is the expression < a, b >= a 1 b 1 + a b + + a n b n = a i b i, and their lengths are given by a = < a, a > = n a i, b = < b, b > = n b i. A concise version of CS reads as follows:- Theorem 1. For all vectors a, b R n, < a, b > a b, with equality iff there are real numbers λ, µ, not both equal to zero, such that λa = µb. 1
2 There are many proofs of this result. We outline two. First Proof. We present one that relies on three basic facts. One is that the square of every real number is nonnegative. Another is that the sum of nonnegative numbers is nonnegative. The third is that a real nonnegative quadratic polynomial has either no real root or two coincident real roots, and the condition that one of these options occurs is that the discriminant of the quadratic be nonpositive. First of all, note that if a b = 0, then one of a, b is the zero vector, in which case their inner-product is also zero, and the inequality becomes an equality. So, it s enough to establish the result when a b > 0. To do so, let t be any real number, and note that then 0 (ta i b i ) p(t). Now, for all real t, (ta i b i ) = (t a i ta i b i + b i ) = t a i t a i b i + b i ) = t a t < a, b > + b. Thus, p is the quadratic polynomial whose values are nonnegative: p(t) = t a t < a, b > + b 0, t R. This means that, since its graph lies in the upper-half plane, either p has no real root or precisely one. In other words, its discriminant ( < a, b >) 4 a b is nonpositive. It follows that < a, b > = < a, b > a b = a b = a b. The inequality is strict unless the discriminant of p is zero. If this occurs, then p has only one real root, viz., < a, b > / a, in which case 0 = ( < a, b > a a i b i ), which forces < a, b > a i a b i = 0, i = 1,,..., n, i.e., λa = µb, where λ =< a, b >, µ = a. Second Proof. This relies on the observation that, if t > 0, ab ta + b, a, b R, (1) t
3 with equality iff ta = b. As before, we may assume that a b > 0. Applying (1), it follows that a i b i ta i + b i, i = 1,,..., n, t whence, adding these we infer that, for all t > 0, But, for all t > 0, < a, b > t a + b. () t a b t a + b, t with equality iff t = b / a. Since the left-hand side of () is independent of t it follows that < a, b > a b, < a, b > a b. Replacing a in this by its additive inverse a = ( a 1, a,..., a n ), and noting that a = a we deduce that so that < a, b >=< a, b > a b = a b, a b < a, b > a b, < a, b > a b. This establishes the inequality. If the equality sign prevails, then either < a, b >= a b or < a, b >= a b. If the former happens, there is equality in () when t a = b, and so in each of the preceding inequalities for this same t. In other words, only when a a i = b b i, i = 1,,..., n. Example 1. Suppose a, b are real numbers. Then, for all real x, a cos x + b sin x a + b, with equality iff a sin x = b cos x. Solution. Apply the CS inequality with (a 1, a ) = (a, b), (b 1, b ) = (cos x, sin x). Then a cos x + b sin x a + b cos x + sin x = a + b, since cos x + sin x = 1, x R.
4 Equality can occur iff there are constants λ, µ, not both equal to zero, such that λa = µ cos x, λb = µ sin x, i.e., a sin x = b cos x, i.e., tan x = b, if a 0. a Example. Suppose a, b, c are real numbers. Then, for all real x, y, a cos x cos y + b sin x cos y + c sin y a + b + c. Solution. Apply the CS inequality with Then (a 1, a, a ) = (a, b, c), (b 1, b, b ) = (cos x cos y, sin x cos y, sin y). a cos x cos y + b sin x cos y + c sin y a + b + c cos x cos y + sin x cos y + sin y = a + b + c (cos x + sin x) cos y + sin y = a + b + c (cos y + sin y = a + b + c, with equality, for some pair of real numbers x, y, iff there are constants λ, µ, not both equal to zero, such that λa = µ cos x cos y, λb = µ sin x cos y, λc = µ sin y. Exercise 1. Let r > 0. Show that, for all real x, y, the point (r cos x cos y, r sin x cos y, r sin y) lies on the sphere centred at (0, 0, 0) whose radius is r. Example. Let a, b, c, d be real numbers with a + b + c > 0. Determine the distance from the origin to the plane Answer: the distance is equal to M = {(x, y, z) R : ax + by + cz = d}. d a + b + c. For, if P = (x, y, z) M, then, by two applications of Pythagoras theorem, its distance from the origin is equal to x + y + z. But ax + by + cz = d, 4
5 and, by the CS inequality, d = ax + by + cz a + b + c x + y + z. Hence x + y + z d a + b + c. This means that no point in M lies inside any sphere centred at (0, 0, 0) whose radius is less than d / a + b + c. In addition, there is equality in the CS inequality, for some point P M, iff x + y + z = d, and ax + by + cz = d. a + b + c In other words, iff there are constants λ, µ, not both of which are zero, such that λa = µx, λb = µy, λc = µz, and ax + by + cz = d. In geometrical language, iff P is a point of intersection of the sphere centred at (0, 0, 0) whose radius is equal to d / a + b + c and the plane M, equivalently, iff P is the point of tangency of the plane with the sphere. Corollary 1. Let a, b, c, d be real numbers with a +b +c > 0. Let P 0 = (x 0, y 0, z 0 ) be any point in R. Show that the distance from P 0 to the plane M = {(x, y, z) R : ax + by + cz = d} is given by ax 0 + by 0 + cz 0 d a + b + c. Remark. This is the analogue in -space, R, of a result in the plane, R, with which you are familiar from school. Example 4. Suppose, in the usual notation, that a, b, c are the lengths of the sides of a triangle ABC. Let m a, m b, m c denote the lengths of the medians from the vertices A, B, C, respectively, to the opposite sides. Then am a + bm b + cm c with equality iff ABC is equilateral. (a + b + c ),.Solution. We begin by establishing a formula for the length of the median from A. Let D be the mid-point of the side BC. Since BD = a = DC, by two applications of the Cosine Rule, m a + ( a ) c am a = cos ADB = cos ADC = m a + ( a ) b am a. 5
6 Hence Similarly, by cyclicity, m a = b + c a. and so, in particular, Hence, by the CS inequality, m b = c + a b m c = a + b c, am a + bm b + cm c m a + m b + m c = 4 (a + b + c ). = ma + m b + c m a + b + c (a + b + c ), with equality iff there are constants λ, µ, not both of which are zero, such that λm a = µa, λm b = µb, λm c = µc. But neither λ nor µ can be zero. Hence there is equality iff am a + bm b + cm c = and there is a constant ν > 0 such that (a + b + c ), m a = νa, m b = νb, m c = νc. It follows that ν = /, whence m a = a, m a = 4 a, b + c a = a 4, i.e., a = b + c. Similarly, b + c + a and c = a + b. From these equations it follows that a = b = c, as desired. Exercise. In the same notation, show that m a m b = 9 a b, m 4 a = 9 a Exercise. In the same notation, show that a, b, c and m a, m b, m c are oppositely ordered Exercise 4. In the same notation, show that m a, m b, m c are the lengths of the sides of a triangle, whose area is equal to 4. 6
7 Exercise 5. In the same notation, show that with equality iff a = b = c. am a + bm b + cm c 1 (a + b + c)(m a + m b + m c ), Example 5. Let P be an arbitrary point within a triangle ABC whose area is. Denote the distances from P to the sides BC, CA and AB by x, y, z, respectively. Then x + y + z 4 a + b + c, with equality iff P is such that x a = y b = z c = a + b + c. Solution. The triangle ABC is the union of the triangles BP C, CP A, AP B, which are disjoint if we ignore common sides, which have zero area. Hence the area of ABC is the sum of the areas of BP C, CP A, AP B. Therefore i.e., so that with equality iff and for some ν. This happens iff = 1 xa + 1 by + 1 zc, = ax + by + cz a + b + c x + y + z, x + y + z x + y + z = 4 a + b + c, 4 a + b + c, x = νa, y = νb, z = νc ν = a + b + c. Hence the point P which furnishes the minimum of x + y + z is such that the lengths of the perpendiculars from it to the sides of the triangle are proportional to the lengths of the sides: x a = y b = z c = a + b + c. This point is known as the Lemoine point of the triangle. 7
8 Definition 1. In the same notation: Let x be the distance from the Lemoine point to BC. The angle ω such that is called the Brocard angle of ABC. Exercise 6. Prove that tan ω = x a, cot ω = cot A + cot B + cot C 8
Lines, parabolas, distances and inequalities an enrichment class
Lines, parabolas, distances and inequalities an enrichment class Finbarr Holland 1. Lines in the plane A line is a particular kind of subset of the plane R 2 = R R, and can be described as the set of ordered
More informationComplex numbers in the realm of Euclidean Geometry
Complex numbers in the realm of Euclidean Geometry Finbarr Holland February 7, 014 1 Introduction Before discussing the complex forms of lines and circles, we recall some familiar facts about complex numbers.
More informationTrigonometrical identities and inequalities
Trigonometrical identities and inequalities Finbarr Holland January 1, 010 1 A review of the trigonometrical functions These are sin, cos, & tan. These are discussed in the Maynooth Olympiad Manual, which
More informationAdditional Mathematics Lines and circles
Additional Mathematics Lines and circles Topic assessment 1 The points A and B have coordinates ( ) and (4 respectively. Calculate (i) The gradient of the line AB [1] The length of the line AB [] (iii)
More information1 / 23
CBSE-XII-07 EXAMINATION CBSE-X-009 EXAMINATION MATHEMATICS Series: HRL Paper & Solution Code: 0/ Time: Hrs. Max. Marks: 80 General Instuctions : (i) All questions are compulsory. (ii) The question paper
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 informationBaltic Way 2008 Gdańsk, November 8, 2008
Baltic Way 008 Gdańsk, November 8, 008 Problems and solutions Problem 1. Determine all polynomials p(x) with real coefficients such that p((x + 1) ) = (p(x) + 1) and p(0) = 0. Answer: p(x) = x. Solution:
More information1 Sets of real numbers
1 Sets of real numbers Outline Sets of numbers, operations, functions Sets of natural, integer, rational and real numbers Operations with real numbers and their properties Representations of real numbers
More informationObjective Mathematics
Multiple choice questions with ONE correct answer : ( Questions No. 1-5 ) 1. If the equation x n = (x + ) is having exactly three distinct real solutions, then exhaustive set of values of 'n' is given
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 informationCircles, Mixed Exercise 6
Circles, Mixed Exercise 6 a QR is the diameter of the circle so the centre, C, is the midpoint of QR ( 5) 0 Midpoint = +, + = (, 6) C(, 6) b Radius = of diameter = of QR = of ( x x ) + ( y y ) = of ( 5
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 informationCO-ORDINATE GEOMETRY. 1. Find the points on the y axis whose distances from the points (6, 7) and (4,-3) are in the. ratio 1:2.
UNIT- CO-ORDINATE GEOMETRY Mathematics is the tool specially suited for dealing with abstract concepts of any ind and there is no limit to its power in this field.. Find the points on the y axis whose
More information(c) n (d) n 2. (a) (b) (c) (d) (a) Null set (b) {P} (c) {P, Q, R} (d) {Q, R} (a) 2k (b) 7 (c) 2 (d) K (a) 1 (b) 3 (c) 3xyz (d) 27xyz
318 NDA Mathematics Practice Set 1. (1001)2 (101)2 (110)2 (100)2 2. z 1/z 2z z/2 3. The multiplication of the number (10101)2 by (1101)2 yields which one of the following? (100011001)2 (100010001)2 (110010011)2
More information( 1 ) Show that P ( a, b + c ), Q ( b, c + a ) and R ( c, a + b ) are collinear.
Problems 01 - POINT Page 1 ( 1 ) Show that P ( a, b + c ), Q ( b, c + a ) and R ( c, a + b ) are collinear. ( ) Prove that the two lines joining the mid-points of the pairs of opposite sides and the line
More informationYear 11 Math Homework
Yimin Math Centre Year 11 Math Homework Student Name: Grade: Date: Score: Table of contents 8 Year 11 Topic 8 Trigonometry Part 5 1 8.1 The Sine Rule and the Area Formula........................... 1 8.1.1
More informationTriangles. Example: In the given figure, S and T are points on PQ and PR respectively of PQR such that ST QR. Determine the length of PR.
Triangles Two geometric figures having the same shape and size are said to be congruent figures. Two geometric figures having the same shape, but not necessarily the same size, are called similar figures.
More informationDot Products. K. Behrend. April 3, Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem.
Dot Products K. Behrend April 3, 008 Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem. Contents The dot product 3. Length of a vector........................
More informationy mx 25m 25 4 circle. Then the perpendicular distance of tangent from the centre (0, 0) is the radius. Since tangent
Mathematics. The sides AB, BC and CA of ABC have, 4 and 5 interior points respectively on them as shown in the figure. The number of triangles that can be formed using these interior points is () 80 ()
More informationLinear Algebra I. Ronald van Luijk, 2015
Linear Algebra I Ronald van Luijk, 2015 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents Dependencies among sections 3 Chapter 1. Euclidean space: lines and hyperplanes 5 1.1. Definition
More informationMaharashtra Board Class X Mathematics - Geometry Board Paper 2014 Solution. Time: 2 hours Total Marks: 40
Maharashtra Board Class X Mathematics - Geometry Board Paper 04 Solution Time: hours Total Marks: 40 Note: - () All questions are compulsory. () Use of calculator is not allowed.. i. Ratio of the areas
More informationChapter 1. Measure Spaces. 1.1 Algebras and σ algebras of sets Notation and preliminaries
Chapter 1 Measure Spaces 1.1 Algebras and σ algebras of sets 1.1.1 Notation and preliminaries We shall denote by X a nonempty set, by P(X) the set of all parts (i.e., subsets) of X, and by the empty set.
More informationCBSE QUESTION PAPER CLASS-X MATHS
CBSE QUESTION PAPER CLASS-X MATHS SECTION - A Question 1:If sin α = 1 2, then the value of 4 cos3 α 3 cos α is (a)0 (b)1 (c) 1 (d)2 Question 2: If cos 2θ = sin(θ 12 ), where2θ and (θ 12 ) are both acute
More informationReview exercise 2. 1 The equation of the line is: = 5 a The gradient of l1 is 3. y y x x. So the gradient of l2 is. The equation of line l2 is: y =
Review exercise The equation of the line is: y y x x y y x x y 8 x+ 6 8 + y 8 x+ 6 y x x + y 0 y ( ) ( x 9) y+ ( x 9) y+ x 9 x y 0 a, b, c Using points A and B: y y x x y y x x y x 0 k 0 y x k ky k x a
More informationMockTime.com. NDA Mathematics Practice Set 1.
346 NDA Mathematics Practice Set 1. Let A = { 1, 2, 5, 8}, B = {0, 1, 3, 6, 7} and R be the relation is one less than from A to B, then how many elements will R contain? 2 3 5 9 7. 1 only 2 only 1 and
More informationNEW YORK CITY INTERSCHOLASTIC MATHEMATICS LEAGUE Senior A Division CONTEST NUMBER 1
Senior A Division CONTEST NUMBER 1 PART I FALL 2011 CONTEST 1 TIME: 10 MINUTES F11A1 Larry selects a 13-digit number while David selects a 10-digit number. Let be the number of digits in the product of
More informationa. Define a function called an inner product on pairs of points x = (x 1, x 2,..., x n ) and y = (y 1, y 2,..., y n ) in R n by
Real Analysis Homework 1 Solutions 1. Show that R n with the usual euclidean distance is a metric space. Items a-c will guide you through the proof. a. Define a function called an inner product on pairs
More informationMATH Linear Algebra
MATH 4 - Linear Algebra One of the key driving forces for the development of linear algebra was the profound idea (going back to 7th century and the work of Pierre Fermat and Rene Descartes) that geometry
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 information2008 Euclid Contest. Solutions. Canadian Mathematics Competition. Tuesday, April 15, c 2008 Centre for Education in Mathematics and Computing
Canadian Mathematics Competition An activity of the Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario 008 Euclid Contest Tuesday, April 5, 008 Solutions c 008
More informationTrigonometry. Sin θ Cos θ Tan θ Cot θ Sec θ Cosec θ. Sin = = cos = = tan = = cosec = sec = 1. cot = sin. cos. tan
Trigonometry Trigonometry is one of the most interesting chapters of Quantitative Aptitude section. Basically, it is a part of SSC and other bank exams syllabus. We will tell you the easy method to learn
More informationGeometry in the Complex Plane
Geometry in the Complex Plane Hongyi Chen UNC Awards Banquet 016 All Geometry is Algebra Many geometry problems can be solved using a purely algebraic approach - by placing the geometric diagram on a coordinate
More informationYear 12 into 13 Maths Bridging Tasks
Year 1 into 13 Maths Bridging Tasks Topics covered: Surds Indices Curve sketching Linear equations Quadratics o Factorising o Completing the square Differentiation Factor theorem Circle equations Trigonometry
More informationMATHEMATICS. (Two hours and a half) Answers to this Paper must be written on the paper provided separately.
CLASS IX MATHEMATICS (Two hours and a half) Answers to this Paper must be written on the paper provided separately. You will not be allowed to write during the first 15 minutes. This time is to be spent
More informationDISCUSSION CLASS OF DAX IS ON 22ND MARCH, TIME : 9-12 BRING ALL YOUR DOUBTS [STRAIGHT OBJECTIVE TYPE]
DISCUSSION CLASS OF DAX IS ON ND MARCH, TIME : 9- BRING ALL YOUR DOUBTS [STRAIGHT OBJECTIVE TYPE] Q. Let y = cos x (cos x cos x). Then y is (A) 0 only when x 0 (B) 0 for all real x (C) 0 for all real x
More informationVectors, dot product, and cross product
MTH 201 Multivariable calculus and differential equations Practice problems Vectors, dot product, and cross product 1. Find the component form and length of vector P Q with the following initial point
More informationIn Z: x + 3 = 2 3x = 2 x = 1 No solution In Q: 3x = 2 x 2 = 2. x = 2 No solution. In R: x 2 = 2 x = 0 x = ± 2 No solution Z Q.
THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS MATH 1141 HIGHER MATHEMATICS 1A ALGEBRA. Section 1: - Complex Numbers. 1. The Number Systems. Let us begin by trying to solve various
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 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 informationHigher Geometry Problems
Higher Geometry Problems (1) Look up Eucidean Geometry on Wikipedia, and write down the English translation given of each of the first four postulates of Euclid. Rewrite each postulate as a clear statement
More informationThe Advantage Testing Foundation Solutions
The Advantage Testing Foundation 2016 Problem 1 Let T be a triangle with side lengths 3, 4, and 5. If P is a point in or on T, what is the greatest possible sum of the distances from P to each of the three
More informationGrade XI Mathematics
Grade XI Mathematics Exam Preparation Booklet Chapter Wise - Important Questions and Solutions #GrowWithGreen Questions Sets Q1. For two disjoint sets A and B, if n [P ( A B )] = 32 and n [P ( A B )] =
More informationVectors - Applications to Problem Solving
BERKELEY MATH CIRCLE 00-003 Vectors - Applications to Problem Solving Zvezdelina Stankova Mills College& UC Berkeley 1. Well-known Facts (1) Let A 1 and B 1 be the midpoints of the sides BC and AC of ABC.
More informationMathematics, Algebra, and Geometry
Mathematics, Algebra, and Geometry by Satya http://www.thesatya.com/ Contents 1 Algebra 1 1.1 Logarithms............................................ 1. Complex numbers........................................
More informationAnnalee Gomm Math 714: Assignment #2
Annalee Gomm Math 714: Assignment #2 3.32. Verify that if A M, λ(a = 0, and B A, then B M and λ(b = 0. Suppose that A M with λ(a = 0, and let B be any subset of A. By the nonnegativity and monotonicity
More information11 th Philippine Mathematical Olympiad Questions, Answers, and Hints
view.php3 (JPEG Image, 840x888 pixels) - Scaled (71%) https://mail.ateneo.net/horde/imp/view.php3?mailbox=inbox&inde... 1 of 1 11/5/2008 5:02 PM 11 th Philippine Mathematical Olympiad Questions, Answers,
More informationMathematics CLASS : X. Time: 3hrs Max. Marks: 90. 2) If a, 2 are three consecutive terms of an A.P., then the value of a.
1 SAMPLE PAPER 4 (SAII) MR AMIT. KV NANGALBHUR Mathematics CLASS : X Time: 3hrs Max. Marks: 90 General Instruction:- 1. All questions are Compulsory. The question paper consists of 34 questions divided
More informationMT182 Matrix Algebra: Sheet 1
MT82 Matrix Algebra: Sheet Attempt at least questions to 5. Please staple your answers together and put your name and student number on the top sheet. Do not return the problem sheet. The lecturer will
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Second Online Version, December 1998 Comments to the author at krm@maths.uq.edu.au Contents 1 LINEAR EQUATIONS
More informationPart IA. Vectors and Matrices. Year
Part IA Vectors and Matrices Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2018 Paper 1, Section I 1C Vectors and Matrices For z, w C define the principal value of z w. State de Moivre s
More informationUNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education
www.xtremepapers.com UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education *7560400886* ADDITIONAL MATHEMATICS 0606/22 Paper 2 May/June 2011 2 hours
More informationInternational General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL EXAMINATIONS PAPER 2 MAY/JUNE SESSION 2002
International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL EXAMINATIONS ADDITIONAL MATHEMATICS 0606/2 PAPER 2 MAY/JUNE SESSION 2002 2 hours Additional materials: Answer paper Electronic
More information10. Show that the conclusion of the. 11. Prove the above Theorem. [Th 6.4.7, p 148] 4. Prove the above Theorem. [Th 6.5.3, p152]
foot of the altitude of ABM from M and let A M 1 B. Prove that then MA > MB if and only if M 1 A > M 1 B. 8. If M is the midpoint of BC then AM is called a median of ABC. Consider ABC such that AB < AC.
More informationIYGB. Special Paper U. Time: 3 hours 30 minutes. Created by T. Madas. Created by T. Madas
IYGB Special Paper U Time: 3 hours 30 minutes Candidates may NOT use any calculator Information for Candidates This practice paper follows the Advanced Level Mathematics Core Syllabus Booklets of Mathematical
More informationMathematics. Exercise 6.4. (Chapter 6) (Triangles) (Class X) Question 1: Let and their areas be, respectively, 64 cm 2 and 121 cm 2.
() Exercise 6.4 Question 1: Let and their areas be, respectively, 64 cm 2 and 121 cm 2. If EF = 15.4 cm, find BC. Answer 1: 1 () Question 2: Diagonals of a trapezium ABCD with AB DC intersect each other
More informationSec 4 Maths. SET A PAPER 2 Question
S4 Maths Set A Paper Question Sec 4 Maths Exam papers with worked solutions SET A PAPER Question Compiled by THE MATHS CAFE 1 P a g e Answer all the questions S4 Maths Set A Paper Question Write in dark
More information8 Systems of Linear Equations
8 Systems of Linear Equations 8.1 Systems of linear equations in two variables To solve a system of linear equations of the form { a1 x + b 1 y = c 1 x + y = c 2 means to find all its solutions (all pairs
More informationCurriculum Map for Mathematics HL (DP1)
Curriculum Map for Mathematics HL (DP1) Unit Title (Time frame) Sequences and Series (8 teaching hours or 2 weeks) Permutations & Combinations (4 teaching hours or 1 week) Standards IB Objectives Knowledge/Content
More information0114ge. Geometry Regents Exam 0114
0114ge 1 The midpoint of AB is M(4, 2). If the coordinates of A are (6, 4), what are the coordinates of B? 1) (1, 3) 2) (2, 8) 3) (5, 1) 4) (14, 0) 2 Which diagram shows the construction of a 45 angle?
More informationRMT 2014 Geometry Test Solutions February 15, 2014
RMT 014 Geometry Test Solutions February 15, 014 1. The coordinates of three vertices of a parallelogram are A(1, 1), B(, 4), and C( 5, 1). Compute the area of the parallelogram. Answer: 18 Solution: Note
More informationMATH 241 FALL 2009 HOMEWORK 3 SOLUTIONS
MATH 41 FALL 009 HOMEWORK 3 SOLUTIONS H3P1 (i) We have the points A : (0, 0), B : (3, 0), and C : (x, y) We now from the distance formula that AC/BC = if and only if x + y (3 x) + y = which is equivalent
More informationInternational Mathematical Olympiad. Preliminary Selection Contest 2017 Hong Kong. Outline of Solutions 5. 3*
International Mathematical Olympiad Preliminary Selection Contest Hong Kong Outline of Solutions Answers: 06 0000 * 6 97 7 6 8 7007 9 6 0 6 8 77 66 7 7 0 6 7 7 6 8 9 8 0 0 8 *See the remar after the solution
More informationRecognise the Equation of a Circle. Solve Problems about Circles Centred at O. Co-Ordinate Geometry of the Circle - Outcomes
1 Co-Ordinate Geometry of the Circle - Outcomes Recognise the equation of a circle. Solve problems about circles centred at the origin. Solve problems about circles not centred at the origin. Determine
More information2012 Mu Alpha Theta National Convention Theta Geometry Solutions ANSWERS (1) DCDCB (6) CDDAE (11) BDABC (16) DCBBA (21) AADBD (26) BCDCD SOLUTIONS
01 Mu Alpha Theta National Convention Theta Geometry Solutions ANSWERS (1) DCDCB (6) CDDAE (11) BDABC (16) DCBBA (1) AADBD (6) BCDCD SOLUTIONS 1. Noting that x = ( x + )( x ), we have circle is π( x +
More informationCambridge International Examinations Cambridge Ordinary Level
Cambridge International Examinations Cambridge Ordinary Level * 2 2 1 1 9 6 0 6 4 7 * ADDITIONAL MATHEMATICS 4037/22 Paper 2 May/June 2016 2 hours Candidates answer on the Question Paper. No Additional
More information1 Line n intersects lines l and m, forming the angles shown in the diagram below. 4 In the diagram below, MATH is a rhombus with diagonals AH and MT.
1 Line n intersects lines l and m, forming the angles shown in the diagram below. 4 In the diagram below, MATH is a rhombus with diagonals AH and MT. Which value of x would prove l m? 1) 2.5 2) 4.5 3)
More informationChapter 13: Vectors and the Geometry of Space
Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic
More informationChapter 13: Vectors and the Geometry of Space
Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic
More informationKENDRIYA VIDYALAYA GACHIBOWLI, GPRA CAMPUS, HYD 32
KENDRIYA VIDYALAYA GACHIBOWLI, GPRA CAMPUS, HYD 32 SAMPLE PAPER 02 (2018-19) SUBJECT: MATHEMATICS(041) BLUE PRINT : CLASS X Unit Chapter VSA (1 mark) SA I (2 marks) SA II (3 marks) LA (4 marks) Total Unit
More informationThis pre-publication material is for review purposes only. Any typographical or technical errors will be corrected prior to publication.
This pre-publication material is for review purposes only. Any typographical or technical errors will be corrected prior to publication. Copyright Pearson Canada Inc. All rights reserved. Copyright Pearson
More informationCalgary Math Circles: Triangles, Concurrency and Quadrilaterals 1
Calgary Math Circles: Triangles, Concurrency and Quadrilaterals 1 1 Triangles: Basics This section will cover all the basic properties you need to know about triangles and the important points of a triangle.
More informationINMO-2001 Problems and Solutions
INMO-2001 Problems and Solutions 1. Let ABC be a triangle in which no angle is 90. For any point P in the plane of the triangle, let A 1,B 1,C 1 denote the reflections of P in the sides BC,CA,AB respectively.
More informationThe CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca Euclid Contest. Wednesday, April 15, 2015
The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca 015 Euclid Contest Wednesday, April 15, 015 (in North America and South America) Thursday, April 16, 015 (outside of North America
More informationPart (1) Second : Trigonometry. Tan
Part (1) Second : Trigonometry (1) Complete the following table : The angle Ratio 42 12 \ Sin 0.3214 Cas 0.5321 Tan 2.0625 (2) Complete the following : 1) 46 36 \ 24 \\ =. In degrees. 2) 44.125 = in degrees,
More informationHILBERT SPACES AND THE RADON-NIKODYM THEOREM. where the bar in the first equation denotes complex conjugation. In either case, for any x V define
HILBERT SPACES AND THE RADON-NIKODYM THEOREM STEVEN P. LALLEY 1. DEFINITIONS Definition 1. A real inner product space is a real vector space V together with a symmetric, bilinear, positive-definite mapping,
More informationHigher Mathematics Skills Checklist
Higher Mathematics Skills Checklist 1.1 The Straight Line (APP) I know how to find the distance between 2 points using the Distance Formula or Pythagoras I know how to find gradient from 2 points, angle
More informationSolutions th AMC 12 A (E) Since $20 is 2000 cents, she pays (0.0145)(2000) = 29 cents per hour in local taxes.
Solutions 2004 55 th AMC 12 A 2 1. (E) Since $20 is 2000 cents, she pays (0.0145)(2000) = 29 cents per hour in local taxes. 2. (C) The 8 unanswered problems are worth (2.5)(8) = 20 points, so Charlyn must
More informationPaper: 03 Class-X-Math: Summative Assessment - I
1 P a g e Paper: 03 Class-X-Math: Summative Assessment - I Total marks of the paper: 90 Total time of the paper: 3.5 hrs Questions: 1] Triangle ABC is similar to triangle DEF and their areas are 64 cm
More informationQUANTUM MEASURE THEORY. Stanley Gudder. Department of Mathematics. University of Denver. Denver Colorado
QUANTUM MEASURE THEORY Stanley Gudder Department of Mathematics University of Denver Denver Colorado 828 sgudder@math.du.edu 1. Introduction A measurable space is a pair (X, A) where X is a nonempty set
More informationSOLUTIONS TO ADDITIONAL EXERCISES FOR II.1 AND II.2
SOLUTIONS TO ADDITIONAL EXERCISES FOR II.1 AND II.2 Here are the solutions to the additional exercises in betsepexercises.pdf. B1. Let y and z be distinct points of L; we claim that x, y and z are not
More informationNINETEENTH IRISH MATHEMATICAL OLYMPIAD. Saturday, 6 May a.m. 1 p.m. First Paper. 1. Are there integers x, y, and z which satisfy the equation
NINETEENTH IRISH MATHEMATICAL OLYMPIAD Saturday, 6 May 2006 10 a.m. 1 p.m. First Paper 1. Are there integers x, y, and z which satisfy the equation when (a) n = 2006 (b) n = 2007? z 2 = (x 2 + 1)(y 2 1)
More informationDO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO SO
DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO SO T.B.C. : P-AQNA-L-ZNGU Serial No.- TEST BOOKLET MATHEMATICS Test Booklet Series Time Allowed : Two Hours and Thirty Minutes Maximum Marks : 00
More informationSolutions to the February problems.
Solutions to the February problems. 348. (b) Suppose that f(x) is a real-valued function defined for real values of x. Suppose that both f(x) 3x and f(x) x 3 are increasing functions. Must f(x) x x also
More informationHigher Geometry Problems
Higher Geometry Problems (1 Look up Eucidean Geometry on Wikipedia, and write down the English translation given of each of the first four postulates of Euclid. Rewrite each postulate as a clear statement
More informationRecall the convention that, for us, all vectors are column vectors.
Some linear algebra Recall the convention that, for us, all vectors are column vectors. 1. Symmetric matrices Let A be a real matrix. Recall that a complex number λ is an eigenvalue of A if there exists
More information1. Suppose that a, b, c and d are four different integers. Explain why. (a b)(a c)(a d)(b c)(b d)(c d) a 2 + ab b = 2018.
New Zealand Mathematical Olympiad Committee Camp Selection Problems 2018 Solutions Due: 28th September 2018 1. Suppose that a, b, c and d are four different integers. Explain why must be a multiple of
More informationDefinition: A vector is a directed line segment which represents a displacement from one point P to another point Q.
THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS MATH Algebra Section : - Introduction to Vectors. You may have already met the notion of a vector in physics. There you will have
More informationGCSE LINKED PAIR PILOT 4363/02 METHODS IN MATHEMATICS UNIT 1: Methods (Non-Calculator) HIGHER TIER
Surname Centre Number Candidate Number Other Names 0 GCSE LINKED PAIR PILOT 4363/02 METHODS IN MATHEMATICS UNIT 1: Methods (Non-Calculator) HIGHER TIER A.M. THURSDAY, 26 May 2016 2 hours S16-4363-02 For
More informationTopic Learning Outcomes Suggested Teaching Activities Resources On-Line Resources
UNIT 3 Trigonometry and Vectors (P1) Recommended Prior Knowledge. Students will need an understanding and proficiency in the algebraic techniques from either O Level Mathematics or IGCSE Mathematics. Context.
More informationU e = E (U\E) e E e + U\E e. (1.6)
12 1 Lebesgue Measure 1.2 Lebesgue Measure In Section 1.1 we defined the exterior Lebesgue measure of every subset of R d. Unfortunately, a major disadvantage of exterior measure is that it does not satisfy
More informationCBSE Class X Mathematics Board Paper 2019 All India Set 3 Time: 3 hours Total Marks: 80
CBSE Class X Mathematics Board Paper 2019 All India Set 3 Time: 3 hours Total Marks: 80 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided
More information36th United States of America Mathematical Olympiad
36th United States of America Mathematical Olympiad 1. Let n be a positive integer. Define a sequence by setting a 1 = n and, for each k > 1, letting a k be the unique integer in the range 0 a k k 1 for
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 informationHigh School Math Contest
High School Math Contest University of South Carolina February th, 017 Problem 1. If (x y) = 11 and (x + y) = 169, what is xy? (a) 11 (b) 1 (c) 1 (d) (e) 8 Solution: Note that xy = (x + y) (x y) = 169
More informationObjective Mathematics
. In BC, if angles, B, C are in geometric seq- uence with common ratio, then is : b c a (a) (c) 0 (d) 6. If the angles of a triangle are in the ratio 4 : :, then the ratio of the longest side to the perimeter
More informationQUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola)
QUESTION BANK ON CONIC SECTION (Parabola, Ellipse & Hyperbola) Question bank on Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct) Q. Two mutually perpendicular tangents
More information'R'nze Allowed : 3 to 3% Hours] LMaximum Marks : 80
MODEL TEST PAPER 6 FIRST TERM (SA-I) MATHEMATICS (With ~nszuers) CLASS X 'R'nze Allowed : 3 to 3% Hours] LMaximum Marks : 80 General Instructions : (i) All questions are compulsory. (ii) The question paper
More informationSOLVED SUBJECTIVE EXAMPLES
Example 1 : SOLVED SUBJECTIVE EXAMPLES Find the locus of the points of intersection of the tangents to the circle x = r cos, y = r sin at points whose parametric angles differ by /3. All such points P
More information( y) ( ) ( ) ( ) ( ) ( ) Trigonometric ratios, Mixed Exercise 9. 2 b. Using the sine rule. a Using area of ABC = sin x sin80. So 10 = 24sinθ.
Trigonometric ratios, Mixed Exercise 9 b a Using area of ABC acsin B 0cm 6 8 sinθ cm So 0 4sinθ So sinθ 0 4 θ 4.6 or 3 s.f. (.) As θ is obtuse, ABC 3 s.f b Using the cosine rule b a + c ac cos B AC 8 +
More informationEdexcel New GCE A Level Maths workbook Circle.
Edexcel New GCE A Level Maths workbook Circle. Edited by: K V Kumaran kumarmaths.weebly.com 1 Finding the Midpoint of a Line To work out the midpoint of line we need to find the halfway point Midpoint
More information