There are 5 possible pairs: (0,0), (0,9), (9,0), (1,8), (8,1)
The zero can be in any of the 5 positions.
The other two digits in the pair can be arranged in 2 ways.
Hence total numbers = 5 × 2 × 1000 = 10,000
Jawapan: 10,000
SOALAN A4
BM Diberi ABCD satu segiempat sama sisi dengan sisi AB = 4 cm. E dan F
dua titik di atas AB dan BC masing-masing dengan AE = 2 cm dan BF = 3
cm
The composite function is gf(x) = 2x - 2. We are given f(x) = 2 - x. To find g, let f(x) = u in gf(x). Then u = 2 - x. Substitute u = 2 - x in gf(x) = 2u - 2. This gives g(x) = 2 - 2x. Therefore, fg(x) = f(2 - 2x) = (2 - (2 - 2x)) = 2x - 2.
Dokumen tersebut membahas tentang pentaksiran pendidikan moral untuk murid tahun 1 mengenai unit kebajikan. Dokumen tersebut meminta murid untuk mewarnai gambar yang menunjukkan kegembiraan keluarga dengan bantuan serta menceritakan sifat-sifat kebajikan yang dimiliki.
Mathematics Form 1-Chapter 3 Squares, Square Roots, Cubes and Cube Roots KBSM...KelvinSmart2
1. Squares are the product of a number multiplied by itself. Squares of fractions, decimals, and negative numbers can be calculated using this property.
2. Perfect squares are numbers that are the result of squaring another number. Prime factorization can help determine if a number is a perfect square.
3. The document provides examples of calculating squares and identifying perfect squares, and includes practice exercises for working with squares.
Hobi saya ialah membaca buku. Saya dapat banyak manfaat dari membaca pada waktu lapang. Membaca menambah pengetahuan saya dan saya sering meminjam buku dari perpustakaan untuk belajar. Membaca juga dapat memperbaiki bahasa saya dengan mempelajari istilah baru. Hobi membaca sangat membantu dalam pelajaran saya dan saya dapat mengulang kaji dengan mudah. Saya akan terus mengamalkan
This document provides an overview of solving simultaneous equations between linear and quadratic equations with two unknowns. It includes 4 examples of solving simultaneous equations involving a linear equation equal to a quadratic equation. The examples show the steps of substituting one equation into the other and solving. The document also includes a chapter review with 6 practice problems involving simultaneous equations.
This document provides examples of solving equations, expanding and factorizing expressions, solving simultaneous equations, working with indices and logarithms. It includes over 100 problems across these topics for students to practice. The problems range in complexity from basic single-step equations to multi-part logarithmic expressions and systems of simultaneous equations.
The composite function is gf(x) = 2x - 2. We are given f(x) = 2 - x. To find g, let f(x) = u in gf(x). Then u = 2 - x. Substitute u = 2 - x in gf(x) = 2u - 2. This gives g(x) = 2 - 2x. Therefore, fg(x) = f(2 - 2x) = (2 - (2 - 2x)) = 2x - 2.
Dokumen tersebut membahas tentang pentaksiran pendidikan moral untuk murid tahun 1 mengenai unit kebajikan. Dokumen tersebut meminta murid untuk mewarnai gambar yang menunjukkan kegembiraan keluarga dengan bantuan serta menceritakan sifat-sifat kebajikan yang dimiliki.
Mathematics Form 1-Chapter 3 Squares, Square Roots, Cubes and Cube Roots KBSM...KelvinSmart2
1. Squares are the product of a number multiplied by itself. Squares of fractions, decimals, and negative numbers can be calculated using this property.
2. Perfect squares are numbers that are the result of squaring another number. Prime factorization can help determine if a number is a perfect square.
3. The document provides examples of calculating squares and identifying perfect squares, and includes practice exercises for working with squares.
Hobi saya ialah membaca buku. Saya dapat banyak manfaat dari membaca pada waktu lapang. Membaca menambah pengetahuan saya dan saya sering meminjam buku dari perpustakaan untuk belajar. Membaca juga dapat memperbaiki bahasa saya dengan mempelajari istilah baru. Hobi membaca sangat membantu dalam pelajaran saya dan saya dapat mengulang kaji dengan mudah. Saya akan terus mengamalkan
This document provides an overview of solving simultaneous equations between linear and quadratic equations with two unknowns. It includes 4 examples of solving simultaneous equations involving a linear equation equal to a quadratic equation. The examples show the steps of substituting one equation into the other and solving. The document also includes a chapter review with 6 practice problems involving simultaneous equations.
This document provides examples of solving equations, expanding and factorizing expressions, solving simultaneous equations, working with indices and logarithms. It includes over 100 problems across these topics for students to practice. The problems range in complexity from basic single-step equations to multi-part logarithmic expressions and systems of simultaneous equations.
PMR Form 3 Mathematics Algebraic FractionsSook Yen Wong
The document provides instructions for expanding and factorizing algebraic expressions involving single and double brackets. It explains how to expand brackets by distributing terms inside brackets to each term outside. For factorizing, it describes finding common factors and grouping terms. It also covers techniques for factorizing quadratic expressions, difference of squares, and grouping. Further sections cover simplifying algebraic fractions through factorizing numerators and denominators and combining like terms.
Mathematics Form 1-Chapter 6-7 Linear Equalities Linear Inequalities KBSM of ...KelvinSmart2
This document contains notes for a mathematics chapter covering linear equations and inequalities. It introduces key topics like conversions between units of length, mass, time, and money. It also covers solving linear equations in two variables, simultaneous linear equations using substitution and elimination methods, and solving inequalities in one and two variables. Examples of each type of problem are provided.
Koleksi karangan murid galus tanpa warnasitinourcheah
Karangan ini menceritakan tentang seorang gadis bernama Siti yang merancang untuk berlibur ke kampung selama cuti sekolah. Dia akan pergi ke kampung bersama dua orang rakannya menggunakan bas dan teksi. Siti juga memberitahu datuknya agar tidak risau tentang keselamatannya dan mereka akan bersiar-siar serta memancing di sungai.
This document contains instructions and questions for a Malaysian UPSR exam consisting of three sections (A, B, and C) testing Bahasa Melayu writing skills. Section A contains a picture prompt asking students to write five complete sentences. Section B asks students to choose and write an essay of at least 80 words on one of three topics. Section C asks students to identify five moral values found in a passage about a boy named Ramli and his responsibilities caring for his elderly grandmother. Students are given suggested time limits for each section and the exam is out of 60 total marks.
This document provides notes on additional mathematics for Form 4 students. It includes definitions and examples of functions, inverse functions, quadratic equations, and logarithms. Some key points summarized:
1. A function f maps objects to images. To find the inverse function f-1, change f(x) to y and solve for x in terms of y.
2. To find the roots of a quadratic equation, one can use factorisation, the quadratic formula, or complete the square. The nature of the roots depends on the sign of b2 - 4ac.
3. To solve a system of equations involving one linear and one non-linear equation, one can substitute one equation into the other and solve
This document discusses quadratic functions and their graphs. It begins by defining the general form of a quadratic function as f(x) = ax2 + bx + c, where a ≠ 0. It then explains how to identify the shape of a quadratic graph based on the sign of a, whether it is positive or negative. Examples are provided to show how to sketch graphs, find maximum and minimum values, axes of symmetry, and zeros. The document also covers using the discriminant to determine the number and type of roots, and completing the square to find the vertex of a quadratic function.
Teknik Menjawab Kertas 1 Matematik TambahanZefry Hanif
The document provides information about the format, topics, and analysis of past year mathematics SPM 3472/1 papers from 2008 to 2011. It discusses the number of questions and marks for each paper. It also contains tables analyzing the topics that have appeared each year, including functions, quadratic equations, indices, and other topics. The document advises students to maximize the use of calculators, including the "SOLVE" and "CALC" functions. It provides examples of using these functions to solve equations.
This document contains notes on additional mathematics including topics on progression, linear laws, integration, and vectors. Some key points:
- It discusses arithmetic and geometric progressions, defining the terms and formulas for finding terms and sums. Examples are worked through finding terms, sums, and differences between sums.
- Linear laws are explained including lines of best fit, converting between linear and non-linear forms using logarithms, and working through examples of finding equations from graphs.
- Integration techniques are outlined including formulas for integrals of powers, areas under and between curves, volumes of revolution, and the basic rules of integration. Worked examples find areas and volumes.
- Vectors are introduced including addition using the triangle
This document contains a 20 question mathematics test for Form 2 students. It provides the questions along with space for students to show their work. The test covers topics like operations with integers, decimals, exponents, square roots, simplifying algebraic expressions, solving linear equations, and finding the perimeter of a triangle given side lengths. Students are instructed to answer all questions and show their calculations in the spaces provided. The test is timed for 1 hour and 45 minutes.
Mathematics Form 1-Chapter 5-6 Algebraic Expression Linear Equations KBSM of ...KelvinSmart2
This document summarizes a math chapter about algebraic expressions and linear equations. It covers topics like algebraic terms with multiple unknowns, multiplication and division of terms, and solving linear equations. It provides examples and exercises for students to practice the concepts. Key points introduced are the definitions of unknowns, coefficients, like and unlike terms, and how to perform operations and solve equations involving algebraic expressions.
This document contains a 10 question mathematics exam with multiple parts to each question. The exam covers topics such as number lines, percentages, geometry, algebra, graphs, and trigonometry. It provides diagrams, tables and questions for students to solve problems and show their work. The exam is designed to test students' understanding of essential mathematics concepts.
Pentaksiran Tingkatan 1 Bahasa Inggeris (Answer)Miz Malinz
The document provides a reading comprehension passage and questions about a trip to Sarawak. There are 10 grammatical errors in the passage that need to be corrected. The questions then ask the reader to summarize details about the trip based on the information provided and correct any errors in the passage. Further questions test understanding of vocabulary used and ability to infer implicit details.
The document contains 25 multiple choice mathematics questions covering topics such as place value, rounding numbers, addition, subtraction, multiplication, division, and word problems involving quantities such as newspaper deliveries and milk production. The questions test skills such as partitioning numbers by place value, performing calculations with large numbers, rounding to various places, and solving multi-step word problems.
PMR Form 3 Mathematics Algebraic FractionsSook Yen Wong
The document provides instructions for expanding and factorizing algebraic expressions involving single and double brackets. It explains how to expand brackets by distributing terms inside brackets to each term outside. For factorizing, it describes finding common factors and grouping terms. It also covers techniques for factorizing quadratic expressions, difference of squares, and grouping. Further sections cover simplifying algebraic fractions through factorizing numerators and denominators and combining like terms.
Mathematics Form 1-Chapter 6-7 Linear Equalities Linear Inequalities KBSM of ...KelvinSmart2
This document contains notes for a mathematics chapter covering linear equations and inequalities. It introduces key topics like conversions between units of length, mass, time, and money. It also covers solving linear equations in two variables, simultaneous linear equations using substitution and elimination methods, and solving inequalities in one and two variables. Examples of each type of problem are provided.
Koleksi karangan murid galus tanpa warnasitinourcheah
Karangan ini menceritakan tentang seorang gadis bernama Siti yang merancang untuk berlibur ke kampung selama cuti sekolah. Dia akan pergi ke kampung bersama dua orang rakannya menggunakan bas dan teksi. Siti juga memberitahu datuknya agar tidak risau tentang keselamatannya dan mereka akan bersiar-siar serta memancing di sungai.
This document contains instructions and questions for a Malaysian UPSR exam consisting of three sections (A, B, and C) testing Bahasa Melayu writing skills. Section A contains a picture prompt asking students to write five complete sentences. Section B asks students to choose and write an essay of at least 80 words on one of three topics. Section C asks students to identify five moral values found in a passage about a boy named Ramli and his responsibilities caring for his elderly grandmother. Students are given suggested time limits for each section and the exam is out of 60 total marks.
This document provides notes on additional mathematics for Form 4 students. It includes definitions and examples of functions, inverse functions, quadratic equations, and logarithms. Some key points summarized:
1. A function f maps objects to images. To find the inverse function f-1, change f(x) to y and solve for x in terms of y.
2. To find the roots of a quadratic equation, one can use factorisation, the quadratic formula, or complete the square. The nature of the roots depends on the sign of b2 - 4ac.
3. To solve a system of equations involving one linear and one non-linear equation, one can substitute one equation into the other and solve
This document discusses quadratic functions and their graphs. It begins by defining the general form of a quadratic function as f(x) = ax2 + bx + c, where a ≠ 0. It then explains how to identify the shape of a quadratic graph based on the sign of a, whether it is positive or negative. Examples are provided to show how to sketch graphs, find maximum and minimum values, axes of symmetry, and zeros. The document also covers using the discriminant to determine the number and type of roots, and completing the square to find the vertex of a quadratic function.
Teknik Menjawab Kertas 1 Matematik TambahanZefry Hanif
The document provides information about the format, topics, and analysis of past year mathematics SPM 3472/1 papers from 2008 to 2011. It discusses the number of questions and marks for each paper. It also contains tables analyzing the topics that have appeared each year, including functions, quadratic equations, indices, and other topics. The document advises students to maximize the use of calculators, including the "SOLVE" and "CALC" functions. It provides examples of using these functions to solve equations.
This document contains notes on additional mathematics including topics on progression, linear laws, integration, and vectors. Some key points:
- It discusses arithmetic and geometric progressions, defining the terms and formulas for finding terms and sums. Examples are worked through finding terms, sums, and differences between sums.
- Linear laws are explained including lines of best fit, converting between linear and non-linear forms using logarithms, and working through examples of finding equations from graphs.
- Integration techniques are outlined including formulas for integrals of powers, areas under and between curves, volumes of revolution, and the basic rules of integration. Worked examples find areas and volumes.
- Vectors are introduced including addition using the triangle
This document contains a 20 question mathematics test for Form 2 students. It provides the questions along with space for students to show their work. The test covers topics like operations with integers, decimals, exponents, square roots, simplifying algebraic expressions, solving linear equations, and finding the perimeter of a triangle given side lengths. Students are instructed to answer all questions and show their calculations in the spaces provided. The test is timed for 1 hour and 45 minutes.
Mathematics Form 1-Chapter 5-6 Algebraic Expression Linear Equations KBSM of ...KelvinSmart2
This document summarizes a math chapter about algebraic expressions and linear equations. It covers topics like algebraic terms with multiple unknowns, multiplication and division of terms, and solving linear equations. It provides examples and exercises for students to practice the concepts. Key points introduced are the definitions of unknowns, coefficients, like and unlike terms, and how to perform operations and solve equations involving algebraic expressions.
This document contains a 10 question mathematics exam with multiple parts to each question. The exam covers topics such as number lines, percentages, geometry, algebra, graphs, and trigonometry. It provides diagrams, tables and questions for students to solve problems and show their work. The exam is designed to test students' understanding of essential mathematics concepts.
Pentaksiran Tingkatan 1 Bahasa Inggeris (Answer)Miz Malinz
The document provides a reading comprehension passage and questions about a trip to Sarawak. There are 10 grammatical errors in the passage that need to be corrected. The questions then ask the reader to summarize details about the trip based on the information provided and correct any errors in the passage. Further questions test understanding of vocabulary used and ability to infer implicit details.
The document contains 25 multiple choice mathematics questions covering topics such as place value, rounding numbers, addition, subtraction, multiplication, division, and word problems involving quantities such as newspaper deliveries and milk production. The questions test skills such as partitioning numbers by place value, performing calculations with large numbers, rounding to various places, and solving multi-step word problems.
1. The document provides solutions to 7 miscellaneous physics problems involving kinematics, dynamics, and rotational motion. The problems involve calculating quantities like angular speeds, forces, and times using principles like conservation of energy, angular momentum, and equations of motion. Complex algebraic and calculus solutions are shown.
2. Key steps involve setting up and solving equations derived from applying relevant physics principles to diagrams of the systems. Calculus techniques like differentiation and numerical integration are used.
3. Emphasis is placed on being able to instantly solve equations of the form f(x)=0 that arise, with references provided to resources on techniques like Newton-Raphson iteration.
Kemahiran berfikir aras tinggi dalam pentaksiran matematikCik Niz
Kemahiran Berfikir Aras Tinggi (KBAT) dalam Pentaksiran Matematik merupakan salah satu elemen penting dalam Reformasi Pendidikan Malaysia. KBAT merujuk kepada tahap pemikiran yang lebih tinggi seperti mengaplikasi, menganalisa, menilai dan mencipta. KBAT penting untuk menghasilkan modal insan yang berfikir kritis dan kreatif untuk memenuhi cabaran abad ke-21. Guru perlu memperkenalkan so
This document contains an individual assignment for an Algebra course at Universiti Tun Hussein Onn Malaysia. The assignment includes two questions. Question 1 has parts (a) and (b) that involve solving equations using completing the squares and the quadratic formula. Question 2 asks the student to show that -1 is a root of the given equation and then factorize it completely. It also asks the student to calculate the composition of some given functions.
Ringkasan: Laporan ini menyoroti lawatan sambil belajar 42 orang pelajar dan guru Sekolah Kebangsaan Jongok Batu ke Pulau Langkawi selama 3 hari 2 malam. Mereka melawati tempat-tempat menarik seperti Underwater World, Kilang Gamat, Langkawi Cable Car, Kota Mahsuri dan Pekan Kuah. Lawatan ini bertujuan untuk menambah pengalaman pelajar, menghubungkaitkan dengan pembelajaran di kelas dan menanam rasa cinta terhad
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1. This document contains a 24-question multiple choice exam covering topics in quadratic equations, simultaneous equations, and graphing.
2. Questions involve solving equations, finding values that satisfy equations, identifying graphs that match equations, and determining properties of quadratic functions from their equations.
3. The document tests understanding of key concepts in the unit on quadratic equations through multiple choice problems.
The document contains 6 problems related to algebra and numbers along with their solutions. Problem 1 involves a number guessing game between two players and determining the minimum number of rounds needed. Problem 2 examines properties of a polynomial where the polynomial equals certain values for distinct integer inputs. Problem 3 finds all integer solutions to a system of equations involving cubes of variables. Problem 4 determines the value of a polynomial of degree 8 at a particular input, given its values at other integers. Problems 5 and 6 involve finding the smallest integer greater than an expression and the minimum possible value of a product of variables, respectively, given an equation relating the variables.
This document discusses exponential and logarithmic functions and equations. It begins by defining exponential equations as equations where the variable appears as an exponent of a constant or variable base. It provides a method for solving exponential equations by reducing both sides of the equation to the same base and then equating the exponents. The document then provides 13 worked examples of solving various exponential equations. It explores properties of exponential functions like the multiplication of powers with the same base and solving systems of exponential equations. The goal is to demonstrate methods for solving different types of exponential equations.
The document contains solutions to 4 geometry problems:
1) It is proven that if points D, E, F divide sides of triangle ABC proportionally, then ABC is equilateral.
2) It is shown that all but a finite number of natural numbers can be written as the sum of 3 numbers where one divides the next.
3) It is proven that if two polynomials P and Q have a common rational root r, then r must be an integer.
4) It is shown that given any 5 vertices of a regular 9-sided polygon, 4 can be chosen to form a trapezium.
I am Marvin Jones, a Number Theory Homework Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from Columbia University, and have been assisting students with their homework for the past six years. I specialize in number theory assignments.
For any number theory assignment solution or homework help, visit mathsassignmenthelp.com, email info@mathsassignmenthelp.com, or call +1 678 648 4277. This sample assignment solution is a prove of our work.
The document discusses different methods for solving quadratic equations. It explains that quadratic equations arise in various situations and fields of mathematics. Several methods are covered, including solving by square root property, factorization, completing the square, and using the quadratic formula. The quadratic formula provides the solutions to a quadratic equation in the form of ax2 + bx + c = 0 and depends on the discriminant to determine the number and type of solutions.
This document contains solutions to 5 calculus assignment problems:
1) Finding values of x such that an inequality is satisfied, with the solution being 3 < x < 4.
2) Proving that the square of any odd integer is odd.
3) Proving there is no rational number whose square is 2.
4) Determining the value of x where a quadratic equation has a minimum, which is x = 1.
5) Proving an equation involving sums and powers using mathematical induction, for all natural numbers n and where r ≠ 1.
MIT Math Syllabus 10-3 Lesson 7: Quadratic equationsLawrence De Vera
This document discusses different methods for solving quadratic equations:
1) Factoring - Setting each factor of the factored quadratic equation equal to zero and solving.
2) Taking square roots - Taking the square root of both sides to isolate the variable.
3) Completing the square - Adding terms to complete the quadratic into a perfect square trinomial form.
4) Quadratic formula - A general formula for solving any quadratic equation using the coefficients.
The discriminant (b^2 - 4ac) determines the nature of the solutions, with positive discriminant yielding two real solutions and negative or zero discriminant yielding non-real or repeated solutions.
The document presents solutions to 6 problems:
1) Finding the maximum area of a quadrilateral inscribed in a circle.
2) Proving that 30 divides the difference of two prime number sets with differences of 8.
3) Showing polynomials with integer coefficients satisfy a recursive relation.
4) Determining the number of points dividing a triangle into equal area triangles.
5) Proving an acute triangle is equilateral given conditions on angle bisectors and altitudes.
6) Characterizing a function on integers satisfying two properties.
Triangle ABC is given, with altitudes CD and BE from vertices C and B to opposite sides AB and AC respectively being equal. It is proved that triangle ABC must be isosceles by showing that triangles CBD and BCE are congruent by the right angle-hypotenuse (RHS) criterion, implying corresponding angles are equal, and then using corresponding parts of congruent triangles to show sides AB and AC are equal, making triangle ABC isosceles.
The document provides solutions to problems from an IIT-JEE 2004 mathematics exam. Problem 1 asks the student to find the center and radius of a circle defined by a complex number relation. The solution shows that the center is the midpoint of points dividing the join of the constants in the ratio k:1, and gives the radius. Problem 2 asks the student to prove an inequality relating dot products of four vectors satisfying certain conditions. The solution shows that the vectors must be parallel or antiparallel.
Peperiksaan pertengahan tahun t4 2012 (2)normalamahadi
This document contains 12 mathematics questions testing skills such as solving simultaneous linear equations, quadratic equations, calculating areas and perimeters of shapes, set theory, and logical reasoning. The questions cover topics like functions, sequences, proportions, geometry, and Venn diagrams. Students are required to show their work and provide answers for full marks.
The document contains a math exam with 5 questions testing students on solving equations, inequalities, and geometry proofs. Question 1 has students solve several equations. Question 2 has students solve inequalities and represent the solution sets on a number line. Question 3 asks students to find the current age of a person given information about the ages of the person and their mother now and in the future. Questions 4 and 5 involve geometry problems, with Question 5 containing multiple parts requiring proofs of properties of triangles.
- The document is the solutions leaflet for the UK Intermediate Mathematical Challenge with 25 math problems and their brief solutions.
- It provides alternative solutions for students to compare with their own work and encourages students to submit additional solutions.
- The UKMT (United Kingdom Mathematics Trust) organizes the challenge to promote mathematical problem solving among students.
The document discusses algebra and its basic concepts. It begins by introducing algebra and defining it as the aspect of mathematics involving the use of numbers and letters. It then provides examples of solving algebraic puzzles using letters to represent unknown numbers. The document goes on to define key algebraic terms like constants, variables, coefficients and terms. It also outlines fundamental algebraic rules like commutativity, associativity and distributivity. It discusses how to collect like terms and factorize expressions. Finally, it covers topics like expanding and simplifying brackets, fractions and factorizing expressions.
The document provides information about an online math class, including:
- A prayer asking God for guidance and wisdom as students wait to be taught.
- Reminders for online class such as turning on cameras, being on time, muting/unmuting audio.
- The weekly task of answering pretest questions from the module.
- An overview of the math topic for the first week of quarter 1 - quadratic equations. It discusses methods for solving quadratic equations such as extracting square roots, factoring, completing the square, and using the quadratic formula.
The document provides steps and examples for solving various types of word problems in algebra, including number, mixture, rate/time/distance, work, coin, and geometric problems. It also covers solving quadratic equations using methods like the square root property, completing the square, quadratic formula, factoring, and using the discriminant. Finally, it discusses linear inequalities, including properties related to addition, multiplication, division, and subtraction of inequalities.
Okay, let's think through this with the new information:
* The equation modeling the height is: h = -16t^2 + vt + c
* The initial height (c) is still 2 feet
* The initial velocity (v) is now 20 feet/second
* The target height (h) is still 20 feet
So the equation is:
20 = -16t^2 + 20t + 2
0 = -16t^2 + 20t + 18 (subtract 20 from both sides)
Evaluating the discriminant:
(20)^2 - 4(-16)(-18) = 400 - 288 = 112
Since the discriminant is positive
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The document provides information about sets and operations on sets such as union, intersection, complement, difference, properties of these operations, counting theorems for finite sets, and the number of elements in power sets. It defines key terms like union, intersection, complement, difference of sets. It lists properties of union, intersection, and complement. It presents counting theorems for finite sets involving union, intersection. It states that the number of elements in the power set of a set with n elements is 2n and the number of proper subsets is 2n-2.
This document contains a 2010 Additional Mathematics exam paper from the Sijil Pelajaran Malaysia (SPM). It consists of 25 multiple choice and short answer questions covering topics like:
- Relations and functions
- Quadratic equations
- Geometric and arithmetic progressions
- Trigonometry
- Probability and statistics
The questions require students to apply concepts like domain and range, inverse functions, maximum/minimum values, and normal distributions to solve problems involving graphs, equations, and word problems.
Dokumen ini membahas tentang kebarangkalian mudah untuk menentukan peluang atau kemungkinan terjadinya suatu peristiwa berdasarkan data dan eksperimen. Dibahas pula tentang penggunaan istilah untuk menggambarkan tingkat kemungkinan seperti mustahil, kecil kemungkinan, sama kemungkinan, besar kemungkinan atau pasti. Juga dibahas cara menghitung kebarangkalian eksperimen dan teori serta perband
This document provides an overview of functions and relations. It begins by defining the learning objectives and outcomes for understanding functions. It then discusses representing relations using arrow diagrams, ordered pairs, and graphs. It introduces the concepts of domain, codomain, object, image, and range for relations. Different types of relations like one-to-one, many-to-one, one-to-many, and many-to-many are classified. Functions are introduced as a special type of relation where each element in the domain maps to only one element in the codomain. Notation for expressing functions is explained along with determining the domain, object, image, and range of functions. Examples are provided to illustrate these concepts.
[Ringkasan]
Dokumen tersebut memberikan perbandingan pencapaian pelajar dalam peperiksaan SPM dan PMR di SM Sains Muzaffar Syah, Melaka dari tahun 2000 hingga 2011. Ia menyediakan statistik mengenai Gred Purata Keseluruhan (GPS), peratus pelajar mendapat gred 'A', dan bilangan pelajar mendapat Straight 'A'. Dokumen tersebut juga memaparkan sasaran pencapaian untuk SPM dan PMR 2012, serta rancangan dan aktiviti untuk
This document contains graphs and charts comparing grade point averages (GPA) and percentages of students who scored different grades for several subjects between the Technical and Vocational stream (TOV), Technical stream (PPT), and general SPM examinations from 2011-2012. The subjects analyzed include Principles of Accounting, Additional Mathematics, and Choice 1 & 2. Overall, TOV students had higher GPAs than PPT or SPM students, while PPT students showed more varied performance levels between grades compared to SPM students where most scored a grade A.
This document contains graphs and charts comparing grade point averages (GPA) and percentages of students who scored different grades for several subjects between the Technical and Vocational stream (TOV), Technical stream (PPT), and regular academic stream (PMR and SPM) in 2011 and 2012. The subjects analyzed include Principles of Accounting, Additional Mathematics, and Choice 1 and Choice 2. Overall, the TOV and PPT students had lower GPAs and higher percentages scoring lower grades compared to the PMR and SPM students.
Taklimat ini memberikan ringkasan prestasi pelajar SM Sains Muzaffar Syah Melaka pada tahun 2011 dan 2012. Ia menyoroti pencapaian GPS dan peratusan pelajar mendapat gred A untuk tingkatan 3 dan 5. Sasaran prestasi untuk tahun 2012 juga ditetapkan. Taklimat ini juga memperkenalkan anugerah 'Principal's List' untuk mengenalpasti pelajar cemerlang.
Dokumen tersebut memberikan arahan dan motivasi kepada guru dan murid di sekolah Mozac untuk mencapai kecemerlangan dalam bidang akademik, kokurikulum dan sahsiah. Ia menekankan konsep-konsep seperti kebersihan, disiplin, kepimpinan, dan komitmen untuk mencapai matlamat sekolah menjadi sekolah terunggul. Dokumen ini juga menyarankan pelbagai strategi dan inisiatif untuk memastikan kejayaan mur
Generasi muda kini memainkan peranan penting dalam masyarakat global. Lebih daripada separuh penduduk dunia berusia di bawah 30 tahun, dan penggunaan media sosial semakin meningkat di kalangan mereka. Walaupun China mempunyai populasi terbesar, negara itu tidak sepenuhnya dipengaruhi oleh Google dan Facebook kerana platform media sosial tempatan seperti Baidu, QQ dan Renren lebih popular.
The document discusses the role of excellent teachers in the era of globalization, challenges, and strategies to address changes. It notes that globalization has made the world smaller through improved communication technology. While technology has accelerated global flows, it is driven by few multinational corporations and may harm the environment. The document also discusses challenges Malaysian teachers face in preparing students for 21st century skills and the need to develop teachers' instructional leadership skills.
1. SULIT 2 OMK 2007 BONGSU
SKEMA PENYELESAIAN
BAHAGIAN A
(12 Markah)
SOALAN A1
BM Seorang lelaki memandu pada kelajuan 90 km/j. Apabila menyedari dia
telah terlewat, dia meningkatkan kelajuannya kepada 110 km/j dan
melengkapkan perjalanannya sejauh 395 km dalam masa 4 jam. Berapa
lamakah dia memandu pada kelajuan 90 km/j?
BI A man was driving at 90km/h. Realizing that he was late, he increased his
speed to 110km/h and completed his journey of 395 km in 4 hrs. For how
long did he drive at 90 km/h?
PENYELESAIAN SOALAN A1
Let the time he drives at 90 km/hr = t and the time he drives at 110 km/h = 4 – t
So,
90t + 110(4 – t) = 395
20t = 45
t = 2.25 jam atau 2 jam 15 minit atau 135 minit
t = 2.25 jam atau
Jawapan: 2 jam 15 minit atau
135 minit
SOALAN A2
BM Misalkan ABCD satu segiempat selari dengan E satu titik di atas garis AB
dengan keadaan 3BE = 2DC. Garis CE dan garis BD bersilang di titik Q.
Jika luas ∆DQC ialah 36, cari luas ∆BQE.
BI Let ABCD be a parallelogram where E is a point on AB such that 3BE =
2DC. The lines CE and BD intersect at the point Q. If the area of ∆DQC is
36, find the area of ∆BQE.
PENYELESAIAN SOALAN A2
A E B
Q
D C
2. SULIT 3 OMK 2007 BONGSU
∆DQC and ∆BQE are similar
Q
B h E
h’
D C
BE 2 h 2
= ⇒ =
DC 3 h' 3
1
Area of ∆DQC = DC × h = 36 ⇒ DC × h = 72
2
1 1 2 2 2 2
Area of ∆BQE = BE × h = × DC × h' = DC × h' = × 72 = 16
2 2 3 3 9 9
OR
Since ∆DQC and ∆BQE are similar, and since 3BE = 2DC,
All corresponding sides the same ratio
2
BE QE BQ h 2 ∆BQE ⎛ 2 ⎞ 4
= = = = ∴ =⎜ ⎟ =
DE QC DQ h' 3 ∆DQC ⎝ 3 ⎠ 9
Thus area of ∆BQE = 4
9
area ∆DQC = 16
Jawapan: 16
SOALAN A3
BM Selesaikan 200520062007 2 − 200520062006 × 200520062008 .
BI Solve 200520062007 2 − 200520062006 × 200520062008 .
PENYELESAIAN SOALAN A3
Jika t = 200520062007 maka
200520062007 2 − 200520062006 × 200520062008 =
t 2 − (t − 1)(t + 1) = t 2 − ( t 2 − 1 ) = 1
Jawapan: 1
3. SULIT 4 OMK 2007 BONGSU
SOALAN A4
BM 1 1
Misalkan 0 < x < 1 . Jika A= x , B= x 2 , C= dan D= 2 susunkan
x x
daripada nilai yang terkecil kepada yang terbesar.
BI 1 1
Let 0 < x < 1 . If A= x , B= x 2 , C= and D= 2 arrange them from the
x x
smallest to the largest value.
PENYELESAIAN SOALAN A4
0 < x < 1 ⇒ x2 < x
1
0 < x <1⇒ >1
x
1 1
x2 < x ⇒ 2 >
x x
1 1
∴ 0 < x2 < x < 1 < < 2
x x
Therefore the arrangements are: BACD
Jawapan: BACD
SOALAN A5
BM 1 1
Cari integer x , y dan z yang memenuhi xy + = yz + dengan x ≠ z .
z x
BI 1 1
Find integers x , y and z satisfying xy + = yz + where x ≠ z .
z x
PENYELESAIAN SOALAN A5
1 1 1 1 z−x
xy + = yz + ⇒ ( x − z ) y = − = ⇒ xyz = −1.
z x x z xz
Since x , y , z are integers and x ≠ z , the only possibilities are
( x , y , z ) = ( 1,1, − 1 ) or ( x , y , z ) = ( −1,1,1 ) .
( x , y , z ) = ( 1,1, − 1 )
Jawapan: atau
( x , y , z ) = ( −1,1,1 )
4. SULIT 5 OMK 2007 BONGSU
SOALAN A6
BM 20 orang menggali sebuah kolam ikan selama 12 hari jika mereka bekerja
selama 6 jam sehari. Berapakah bilangan hari yang diperlukan untuk
menggali kolam yang sama jika 4 orang bekerja selama 5 jam sehari?
BI 20 persons dig a fish pond in 12 days if they work 6 hours per day. How
many days is required to dig the same pond if 4 persons work for 5 hours
per day?
PENYELESAIAN SOALAN A6
20 × 6 × 12 is for one work done
Let x be the number of days required by 4 persons working for 5 hours per day
20 × 6 ×12
So, x = = 72 days
4×5
Jawapan: 72 hari
5. SULIT 6 OMK 2007 BONGSU
BAHAGIAN B
(18 Markah)
SOALAN B1
BM Dengan menggunakan angka 1, 2, 3, 6, 7, 9, 0 sahaja, tentukan angka mana
yang boleh dipadankan dengan huruf di bawah supaya hasil tambahnya
adalah betul.
P A K
+ M A K
P I U T
BI Using only the numbers 1, 2, 3, 6, 7, 9, 0, find which number goes with
each letter in the addition problem below to make it correct.
P A K
+ M A K
P I U T
PENYELESAIAN SOALAN B1
Hasiltambah tiga digit ≤ 2000 jadi P=1, M=9 I=0.
U dan T tidak boleh 4
K tidak boleh 7 atau 2.
Mak ayang tiggal ialah 3, 6 .
Kalau K=3 maka T=6, dan A=2
Jika K=6 maka A=3, dan T=2
Note: Kaedah cuba jaya tak diterima.
ATAU
K + K = 2K = T ⇒ T must be an even number
T = mod 2, 6, 0, but T ≠ mod 0 because then K = T
∴ T = mod 2, 6 ⇒ K = 1, 3, 6 1
Likewise, 2A = U
If T < 10, then U is even
If T > 10, then U is odd 1
In both cases, A = 1, 3, 6
Also P + M = I = 10(P) + I < 20
So P< 2, that is, P = 1 1
Test for all possibilities:
K, A ≠ 1, because P = 1
If K = 3, T =6, then A = 3 or 6 which is not possible
∴ K = 6, T = 12 mod 10 = 2
A =3, U = 3 + 3 + 1 = 7 because T > 10
6. SULIT 7 OMK 2007 BONGSU
Since P = 1, M= 9 and I = 10 mod 10 = 0
∴ PAK 1 3 6
+ MAK + 9 3 6
2
PIUT 1 0 7 2
Note: Kaedah cuba jaya tak diterima.
SOALAN B2
BM Cari jumlah sudut-sudut a+b+c+d+e+f+g+h dalam rajah berikut.
a b
300
c d
e
f
g h
BI Find the sum of angles a+b+c+d+e+f+g+h in the following diagram.
a b
300
c d
e
f
g h
7. SULIT 8 OMK 2007 BONGSU
PENYELESAIAN SOALAN B2
a b
300
E
c A B d
C
e D
f
g h
It is clear that A+B+E = π
C+D+E = π
B+a+c = π
A+b+d = π 2
C+e+g = π
D+f+h = π
Hence {a+b+c+d+e+f+g+h} + {A+B+C+D} = 4 π
{a+b+c+d+e+f+g+h} + 2 π -2E = 4 π . 4
Since E = 30 o , thus {a+b+c+d+e+f+g+h} = 420 o
8. SULIT 9 OMK 2007 BONGSU
SOALAN B3
⎛ 1 ⎞
Diberi 8⎜ y 2 + ⎟ − 56⎛ y + 1 ⎞ + 112 = 0 cari semua nilai y 2 + 1 .
⎜ ⎟
BM
⎜ ⎜ y⎟
⎝ y2 ⎟
⎠ ⎝ ⎠ y2
⎛ 1 ⎞
Given 8⎜ y 2 + ⎟ − 56⎛ y + 1 ⎞ + 112 = 0 find all the values of y 2 + 1 .
⎜ ⎟
BI ⎜ ⎜ y⎟
⎝ y2 ⎟
⎠ ⎝ ⎠ y2
PENYELESAIAN SOALAN B3
⎛ 1 ⎞
8⎜ y 2 + ⎟ − 56⎛ y + 1 ⎞ + 112 = 0
⎜ ⎟
⎜ 2⎟ ⎜ y⎟
⎝ y ⎠ ⎝ ⎠
1 1 3
Let u = y + ⇒ u 2 − 2 = y 2 + .
y y2
8( u 2 − 2 ) − 56u + 112 = 0
8u 2 − 56u + 96 = 0
u 2 − 7u + 12 = 0
( u − 3 )( u − 4 ) = 0 1
u = 3, 4
1
y2 + = 16 − 2 atau 9 − 2
y2 2
= 14 7
9. SULIT 2 OMK 2007 MUDA
SKEMA PENYELESAIAN
BAHAGIAN A
(12 Markah)
SOALAN A1
BM Diberi x + y = 2 dan 2 xy − z 2 = 1 . Dapatkan penyelesaian integer untuk
persamaan-persamaan ini.
BI Given x + y = 2 and 2 xy − z 2 = 1 . Find the integer solutions of the
equations.
PENYELESAIAN SOALAN A1
x+y = 2 and 2 xy − z 2 = 1 leads to 2 ( x − 1) 2 + z 2 = 1 , hence integer solutions
are (1,1,1),(1,1,-1)
Jawapan: (1,1,1),(1,1,-1)
SOALAN A2
BM Dalam suatu kejohanan sukan yang berlangsung selama 4 hari, terdapat n
pingat untuk dimenangi. Pada hari pertama, 1/5 daripada n pingat
dimenangi. Pada hari kedua, 2/5 daripada baki pingat pada hari pertama
dimenangi. Pada hari ketiga, 3/5 daripada baki pingat pada hari kedua
dimenangi. Pada hari keempat, 24 pingat dimenangi. Berapakah jumlah
pingat kesemuanya?
BI In a sport’s tournament lasting for 4 days, there are n medals to be won.
On the first day, 1/5 of the n medals are won. On the second day, 2/5 of the
remainder from the first day are won. On the third day, 3/5 of the
remainder from the second day are won. On the final day, 24 medals are
won. What was the total number of medals?
PENYELESAIAN SOALAN A2
Let Ti be the number of medals won on the i th day, i = 1,2,3,4.
Then
1
T1 = n ,
5
2 8n
T2 = ( n − T1 ) = ,
5 25
T3 = (n − (T 1+T2 )) =
3 36n
.
5 125
T4 = 24.
10. SULIT 3 OMK 2007 MUDA
T1 + T2 + T3 + T4 = n
101n
+ 24 = n
125
24n
= 24
125
∴ n = 125
Jawapan: n=125
SOALAN A3
BM Misalkan 2xyz7 suatu nombor lima angka sedemikian hingga hasil darab
angka-angka tersebut ialah sifar dan hasil tambah angka-angka tersebut
pula boleh dibahagikan dengan 9. Cari bilangan nombor-nombor tersebut.
BI Let 2xyz7 be a five-digit number such that the product of the digits is zero
and the sum of the digits is divisible by 9. Find how many such numbers.
PENYELESAIAN SOALAN A3
One of the digits must be zero. The sum of the other two digits must be divisible by 9.
Possible pairs are : (0,0),(0,9),(9,9),(1,8),(2,7),(3,6),(4,5).
The total number of such numbers with the given pair :
(0,0) 1, (0,9) 3, (9,9) 3, (1,8) 6, (2,7) 6, (3,6) 6, (4,5) 6.
There are 31 such numbers
Jawapan: 31
SOALAN A4
BM Misal ABCD sebagai suatu segiempat tepat. Garis DP memotong pepenjuru
AC pada Q dan membahagikannya pada nisbah 1:4. Jika luas segitiga APQ
satu unit persegi, tentukan luas segiempat tersebut.
BI Let ABCD be a rectangle. The line DP intersects the diagonal AC at Q and
divides it in the ratio of 1:4. If the area of triangle APQ is one unit square,
determine the area of the rectangle.
11. SULIT 4 OMK 2007 MUDA
PENYELESAIAN SOALAN A4
A P B
Q
D C
Biar tinggi segitiga APQ ialah x, dan tinggi segitiga AQP ialah y. Segitiga APQ dan
segitiga CDQ adalah sebentuk, maka
AP: DC = AQ:QC = x:y = 1: 4. Luas segiempat DC (x+y) = 4AP (x + 4x) = 20 AP.x
= 40 (Luas segitiga APQ) = 40
Jawapan: 40
SOALAN A5
BM Cari integer terkecil yang memenuhi syarat apabila dibahagi dengan 2
meninggalkan baki 1, apabila dibahagi dengan 3 meninggalkan baki 2,
apabila dibahagi dengan 4 meninggalkan baki 3 dan apabila dibahagi
dengan 5 meninggalkan baki 4.
BI Find the smallest integer such that if divided by 2 leaves a remainder of 1, if
divided by 3 leaves a remainder of 2, if divided by 4 leaves a remainder of
3, and if divided by 5 leaves a remainder of 4.
PENYELESAIAN SOALAN A5
Let N be the integer. Then
N = 2q1 + 1
= 3q2 + 2
= 4q3 + 3
= 5q4 + 4
Observe that
N + 1 = 2q1 + 2 = 3q2 + 3 = 4q3 + 4 = 5q4 + 5
= 2(q1 + 1) = 3(q2 + 1) = 4(q3 + 1) = 5(q4 + 1)
∴ 2|N + 1, 3|N+1, 4|N + 1 dan 5|N+1
⇒ N + 1 = LCM (2, 3, 4, 5) = 60
∴ N = 59
Jawapan: 59
12. SULIT 5 OMK 2007 MUDA
SOALAN A6
BM Andaikan f suatu fungsi ditakrif pada integer sedemikian
f(2n) = -2f(n), f(2n+1)= f(n) -1, dan f(0) = 2.
Cari nilai f(2007).
BI Let f be a function defined on integers such that
f(2n) = - 2f(n), f(2n+1) = f(n)-1, and f(0) = 2.
Find the value of f(2007).
PENYELESAIAN SOALAN A6
f(2007) = f(1003) -1 = f(501 )-2 = f(250 ) -3 = -2f(125)-3
= -2f(62)-1 = 4f(31) - 2 =4 f(15) - 6 = 4f(7) - 10 =4 f(3)-14
= 4f(1) -18 = -14
Jawapan: -14
13. SULIT 6 OMK 2007 MUDA
BAHAGIAN B
(18 Markah)
SOALAN B1
⎛ 1 ⎞
Diberi 8⎜ y 2 + ⎟ − 56⎛ y + 1 ⎞ + 112 = 0 cari semua nilai y 2 + 1 .
⎜ ⎟
BM ⎜ ⎜ y⎟
⎝ y2 ⎟
⎠ ⎝ ⎠ y2
⎛ 1 ⎞
Given 8⎜ y 2 + ⎟ − 56⎛ y + 1 ⎞ + 112 = 0 find all the values of y 2 + 1 .
⎜ ⎟
BI ⎜ ⎜ y⎟
⎝ y2 ⎟
⎠ ⎝ ⎠ y2
PENYELESAIAN SOALAN B1
⎛ 1 ⎞
8⎜ y 2 + ⎟ − 56⎛ y + 1 ⎞ + 112 = 0
⎜ ⎟
⎜ ⎜ y⎟
⎝ y2 ⎟
⎠ ⎝ ⎠
1 1 3
Let u = y + ⇒ u 2 − 2 = y 2 + .
y y2
8( u 2 − 2 ) − 56u + 112 = 0
8u 2 − 56u + 96 = 0
u 2 − 7u + 12 = 0
( u − 3 )( u − 4 ) = 0 1
u = 3, 4
1
y2 + = 16 − 2 atau 9 − 2
y2 2
= 14 7
SOALAN B2
BM Satu sisi sebuah segitiga berukuran 4 cm. Dua sisi yang lain berukuran
dalam nisbah 1:3. Cari luas yang terbesar untuk segitiga ini.
BI One side of a triangle is 4cm. The other two sides are in the ratio 1:3. What
is the largest area of the triangle?
PENYELESAIAN SOALAN B2
With Heron’s formula the area of the triangle with sides a, b , c is
A= s ( s − a )( s − b )( s − c ) 2
14. SULIT 7 OMK 2007 MUDA
Where s = a + b + c
2
Let a = 4, b = x, then c = 3x so s = 2 + 2x, hence 2
A = ( 2 + 2 x )( 2 x − 2 )( 2 + x )( 2 − x ) = 2 ( x 2 − 1 )( 4 − x 2 )
A is maximum when x 2 = 5 , largest area = 3. 2
2
SOALAN B3
BM Biar f , g dua fungsi yang tertakrif atas [0 , 2c] dengan c > 0 . Tunjukkan
bahawa wujud x , y ∈ [0, 2c] supaya
xy − f ( x ) + g ( y ) ≥ c 2 .
BI Let f , g be two functions defined on [0 , 2c] where c > 0 . Show that there
exists x , y ∈ [0, 2c] such that
xy − f ( x ) + g ( y ) ≥ c2 .
PENYELESAIAN SOALAN B3
Let h( x , y ) = xy − f ( x ) + g( y ) . Suppose that h( x , y ) < c 2 for all 0 ≤ x , y ≤ 2c .
Then 2
h( x1 , y1 ) + h( x2 , y 2 ) + h( x3 , y3 ) + h( x4 , y 4 ) < 4c 2
for all 0 ≤ xi , yi ≤ 2c ( i = 1, 2, 3, 4 ).
However, by the triangle inequality, we have
h( 0 ,0 ) + h( 0 ,2c ) + h( 2c ,0 ) + h( 2c ,2c )
2
≥ h( 0 ,0 ) − h( 0 ,2c ) − h( 2c ,0 ) + h( 2c ,2c )
= 4c 2
1
which is a contradiction.
Hence there exists x , y ∈ [ 0 ,2c ] such that
1
xy − f ( x ) + g ( y ) ≥ c 2 .
Note: Jika jawapan shj tanpa jalan kerja beri 2 markah shj.
15. SULIT 2 OMK 2007 SULONG
SKEMA PENYELESAIAN
BAHAGIAN A
(12 Markah)
SOALAN A1
BM Misalkan a = 6 ,..., a = 6 a n −1 . Cari baki apabila a
1 n 100 dibahagi dengan 11.
BI Let a1 = 6 ,…, a n = 6 a n −1 . Find the remainder when a100 is divided by 11.
PENYELESAIAN SOALAN A1
Fermat’s Little Theorem states that if p is prime and p is not divisible by a, then
a p −1 ≡ 1 mod p . Since 11 is prime and not divisible by 6, then 610 ≡ 1 mod 11 and
6 n ≡ 6 mod 10 for all positive n ie 6 n = 6 + 10 t for some t. Thus
( )
a100 ≡ 6 a99 ≡ 6 6 +10 t ≡ 6 6 610
t
≡ 6 mod 11
Hence the remainder is 6.
Jawapan: 5
SOALAN A2
BM 1
Misalkan −1 < y < 0 < x < 1 . Jika A = x 2 y , B = , C = y 2 x dan
x2 y
1
D= , susunkan daripada nilai yang terkecil kepada yang terbesar.
xy 2
BI 1 1
Let −1 < y < 0 < x < 1 . If A= x 2 y , B= 2
, C= y 2 x and D= 2 ,
x y xy
arrange them from the smallest to the greatest value
PENYELESAIAN SOALAN A2
−1 < y < 0 < x < 1 ⇒ − 1 < x 2 y < 0
1
−1 < x 2 y < 0 ⇒ < −1
x2 y
−1 < y < 0 < x < 1 ⇒ 0 < xy 2 < 1
1
0 < xy 2 < 1 ⇒ >1
xy 2
1 1
∴ 2
< −1 < x 2 y < 0 < xy 2 < 1 < 2
x y xy
Therefore the arrangement are: BACD
Jawapan: BACD
16. SULIT 3 OMK 2007 SULONG
SOALAN A3
BM Satu sisi sebuah segitiga berukuran 4 cm. Dua sisi yang lain berukuran
dalam nisbah 1:3. Cari luas yang terbesar untuk segitiga ini.
BI One side of a triangle is 4 cm. The other two sides are in the ratio 1:3.
Find the largest area of the triangle.
PENYELESAIAN SOALAN A3
With Heron’s formula the area of the triangle with sides a, b , c is
A = s ( s − a )( s − b )( s − c )
Where s = a + b + c
2
Let a = 4, b = x, then c = 3x so s = 2 + 2x, hence
A = ( 2 + 2 x )( 2 x − 2 )( 2 + x )( 2 − x ) = 2 ( x 2 − 1 )( 4 − x 2 )
A is maximum when x 2 = 5 , largest area = 3.
2
Jawapan: 3
SOALAN A4
BM Permudahkan log24·log46 log68 ... log2n(2n+2).
BI Simplify log24·log46 log68 ... log2n (2n+2)
PENYELESAIAN SOALAN A4
log 2 6 log 2 8 log 2 2n log 2 (2n + 2)
log 2 4 ⋅ log 4 6 ⋅ log 6 8 ⋅ ... ⋅ log 2 n (2n + 2) = log 2 4 ⋅ ⋅ ⋅ ... ⋅ ⋅
log 2 4 log 2 6 log 2 (2n − 2) log 2 2n
= log 2 ( 2n + 2 ) atau log 2 2( n + 1 ) atau 1+ log 2 (n + 1)
log 2 ( 2n + 2 ) atau
Jawapan: log 2 2( n + 1 ) atau
1+ log 2 (n + 1)
17. SULIT 4 OMK 2007 SULONG
SOALAN A5
BM Tuliskan 580 sebagai hasil tambah dua nombor kuasa dua.
BI Write 580 as a sum of two squares.
PENYELESAIAN SOALAN A5
By prime power the composition, we have
580 = 2 2 .5.29
= 2 2 .( 2 2 + 1 ).( 5 2 + 2 2 )
= 2 2 (( 2.5 + 1.2 ) 2 + ( 2.2 − 1.5 )2 )
= 2 2 ( 12 2 + 12 )
= 24 2 + 2 2
Jawapan: 24 2 + 2 2
SOALAN A6
BM Misalkan f suatu fungsi tertakrif pada set integer bukan negatif yang
memenuhi
f(2n+1) = f(n), f(2n) = 1 – f(n).
Cari f(2007).
BI Let f be a function defined on non-negative integers satisfying the following
conditions
f(2n+1) = f(n), f(2n) = 1 – f(n).
Find f(2007).
PENYELESAIAN SOALAN A6
f(0) = ½; f(1) = f(0) = ½; and f(2) = 1 – f(1) = ½;
By induction f(n) = ½; for any n
∴f(2007) = ½
Jawapan: 1
2
18. SULIT 5 OMK 2007 SULONG
BAHAGIAN B
(18 Markah)
SOALAN B1
BM Biar f , g dua fungsi yang tertakrif atas [0 , 2c] dengan c > 0 . Tunjukkan
bahawa wujud x , y ∈ [0, 2c] supaya
xy − f ( x ) + g ( y ) ≥ c 2 .
BI Let f , g be two functions defined on [0 , 2c] where c > 0 . Show that there
exists x , y ∈ [0, 2c] such that
xy − f ( x ) + g ( y ) ≥ c2 .
PENYELESAIAN SOALAN B1
Let h( x , y ) = xy − f ( x ) + g( y ) . Suppose that h( x , y ) < c 2 for all 0 ≤ x , y ≤ 2c .
Then 2
h( x1 , y1 ) + h( x2 , y 2 ) + h( x3 , y3 ) + h( x4 , y 4 ) < 4c 2
for all 0 ≤ xi , yi ≤ 2c ( i = 1, 2, 3, 4 ).
However, by the triangle inequality, we have
h( 0 ,0 ) + h( 0 ,2c ) + h( 2c ,0 ) + h( 2c ,2c )
2
≥ h( 0 ,0 ) − h( 0 ,2c ) − h( 2c ,0 ) + h( 2c ,2c )
= 4c 2
1
which is a contradiction.
Hence there exists x , y ∈ [ 0 ,2c ] such that
1
xy − f ( x ) + g ( y ) ≥ c 2 .
Note: Jika jawapan shj tanpa jalan kerja beri 2 markah shj.
19. SULIT 6 OMK 2007 SULONG
SOALAN B2
BM Dua bulatan masing-masing berjejari 1 dan 2 bersentuhan sesama sendiri
secara luaran. Suatu bulatan lain dilukis bersentuhan dengan dua bulatan
ini dengan pusat-pusat bulatan membentuk suatu segitiga bersudut tepat.
Cari jejari bulatan ketiga.
BI Two circles of radius 1 and 2 respectively are tangential to one another
externally. Another circle is drawn tangential to both circles such that their
centres form a right angle triangle. Find the radius of the third circle.
PENYELESAIAN SOALAN B2
Bulatan ketiga boleh bersentuh secara luaran atau kedua-dua bulatan yang diberi
terterap dalam bulatan ketiga.
1
Two possiblities:
If drawn externally:
let the radius of third circle be r we have the sides of triangle r+1, r+2, and 3 1
we will have two possibilities of right angle,
Then first possiblitiy
( r + 2 ) 2 = 3 2 + ( r + 1) 2
Solving for r we get r = 3 2
Next possibility , ( r + 2 ) 2 + ( r + 1 ) 2 = 3 2
20. SULIT 7 OMK 2007 SULONG
Solve for r we get 17 − 3 2
r =
2
If drawn enclosing the two circles, sides of triangle are
r-1, r-2 and 3 1
Two possiblities ( r − 1 ) 2 = 3 2 + ( r − 2 ) 2 and ( r − 2 ) 2 + ( r − 1 ) 2 = 3 2 2
We get r = 6 and
17 + 3
r = 2
2
SOALAN B3
BM Tentukan nilai maksimum bagi m 2 + n 2 untuk m , n ∈ { ,2 ,3,… ,2007} dan
1
( n 2 − mn − m 2 ) 2 = 1
BI Determine the maximum value of m 2 + n 2 where m , n ∈ { ,2 ,3,… ,2007}
1
and ( n 2 − mn − m 2 ) 2 = 1
PENYELESAIAN SOALAN B3
Let the pair be ( m , n ) satisfying both conditions. 2
If m = 1, then ( 1,1 ) and ( 2,1 ) are the only possibilities. Suppose that (n1 , n 2 ) is one
2
of the possible solutions with n2 > 1 . As n1( n1 −n 2 ) = n2 ± 1 > 0 then we must have
n1 > n 2 .
Now let n3 = n1 − n2 . Then
(2 2 )2 (
2 2 )2
1 = n1 − n1n 2 − n 2 = n2 − n2 n3 − n3 making (n 2 , n3 ) as one of possible
solutions too with n3 > 1 . In the same way we conclude n2 > n3 . The same goes to
(n3 , n4 ) such that n4 = n2 − n3 . Hence n1 > n2 > n3 > … and must terminate ie
when nk = 1 for some k. Since (nk −1 ,1) is one of the possibilities, thus we must have
nk −1 . It looks like the sequence goes 1,2,3,5,8, …, 987, 1597 (<2007), a truncated
Fibonacci sequence.
3
It is clear that the largest possible pair is (1597, 987) 1