Apply Mathematical Techniques Practice for Test 2 (2013)(1)
数学技术用于英语学习的英文作文
数学技术用于英语学习的英文作文Mathematical techniques applied to English learningIntroductionIn today's interconnected world, proficiency in the English language has become increasingly important for individuals seeking to advance their careers or pursue higher education opportunities. However, for many learners, mastering the intricacies of English grammar, vocabulary, and pronunciation can be a daunting task. Fortunately, advancements in technology and pedagogy have led to the development of innovative approaches to language learning, including the use of mathematical techniques to enhance comprehension and retention of English language concepts.Mathematics and language learning might seem like unrelated disciplines, but they share many common characteristics. Both require analytical thinking, problem-solving skills, and the ability to recognize patterns and structures. By leveraging mathematical concepts and tools, educators can help students understand and apply complex language rules more effectively. In this essay, we will explore how mathematical techniques can be used to enhance English learning outcomes.Vocabulary acquisitionOne of the most challenging aspects of learning a new language is expanding one's vocabulary. To help learners overcome this obstacle, educators can employ mathematical techniques such as word frequency analysis and spaced repetition. Word frequency analysis involves identifying the most commonly used words in a language and prioritizing these words for learning. By focusing on high-frequency words first, students can build a strong foundation of vocabulary that will be beneficial in real-world communication.Spaced repetition is another mathematical technique that can aid in vocabulary acquisition. This method involves reviewing words at increasing intervals over time to reinforce memory retention. By spacing out practice sessions and strategically timing reviews, learners can optimize their recall of new words and prevent forgetting. Research has shown that spaced repetition can significantly improve long-term retention compared to traditional study methods.Grammar and syntaxGrammar and syntax are fundamental components of language learning, but they can be intimidating for many students. Mathematical concepts such as logic and algorithmscan help learners make sense of complex grammar rules and sentence structures. For example, educators can use logic puzzles and diagramming techniques to demonstrate the relationships between words in a sentence. By breaking down sentences into logical components and identifying the roles of different parts of speech, students can improve their understanding of grammar concepts.Additionally, algorithms can be used to generate practice exercises that target specific grammar rules and patterns. Adaptive learning platforms can analyze students' performance data and provide personalized feedback and remediation to address areas of weakness. By incorporating algorithms into language learning activities, educators can tailor instruction to meet the individual needs of each student and facilitate mastery of grammar and syntax.Pronunciation and intonationDeveloping accurate pronunciation and intonation is crucial for effective communication in a second language. Mathematical techniques such as spectrogram analysis and speech recognition technology can help learners refine their pronunciation skills and improve their spoken English proficiency. Spectrogram analysis involves visualizing speech sounds on a graph to identifypatterns and differences in vocal production. By analyzing spectrograms, students can pinpoint areas of pronunciation that require improvement and practice specific phonetic sounds more effectively.Speech recognition technology can also provide valuable feedback on pronunciation accuracy. Language learning apps and software programs use algorithms to evaluate spoken input and assess pronunciation quality based on predefined criteria. By incorporating speech recognition technology into language learning activities, students can receive instant feedback on their speaking skills and track their progress over time. This real-time feedback can motivate learners to practice more consistently and enhance their overall oral communication abilities.ConclusionIn conclusion, mathematical techniques can be valuable tools for enhancing English language learning outcomes. By leveraging word frequency analysis, spaced repetition, logic, algorithms, spectrogram analysis, and speech recognition technology, educators can support students in building vocabulary, mastering grammar and syntax, and improving pronunciation and intonation. These innovative approaches to language learning can foster deeper understanding, retention,and application of English language concepts, ultimately empowering learners to communicate more effectively in an increasingly globalized world. As technology continues to advance, the integration of mathematical techniques into language education will play a key role in helping students achieve proficiency in the English language and beyond.。
解决数学难题的英语
解决数学难题的英语English:When faced with challenging mathematical problems, it's crucial to employ a systematic approach. Start by carefully reading and understanding the problem statement, identifying key information, and defining any variables or unknowns. Next, brainstorm potential strategies or methods that could be applied to solve the problem. This might involve breaking the problem down into smaller, more manageable steps, or applying relevant mathematical concepts and techniques. As you work through the problem, be sure to check your progress regularly and verify each step for accuracy. Don't hesitate to consult textbooks, online resources, or ask for help from peers or teachers if you encounter difficulties. Additionally, consider exploring similar problems to gain insights into different solution approaches. Persistence, patience, and a willingness to learn from mistakes are essential qualities for overcoming mathematical challenges.Translated content:面对具有挑战性的数学问题时，采用系统化的方法至关重要。
学好数学的方法 英语作文
学好数学的方法英语作文Mastering the Art of Mathematics: A Comprehensive GuideMathematics is a subject that has long been regarded as a fundamental pillar of education, providing the foundation for understanding the world around us. However, for many students, the prospect of learning and excelling in mathematics can be daunting. In this essay, we will explore the various strategies and techniques that can help students effectively master the art of mathematics.Firstly, it is essential to develop a strong conceptual understanding of mathematical principles. Too often, students focus solely on memorizing formulas and algorithms without truly comprehending the underlying concepts. By taking the time to understand the why behind the what, students can build a solid foundation that will serve them well in more advanced mathematical coursework. This can be achieved through active engagement with the material, asking questions, and seeking out additional resources to supplement classroom instruction.Another crucial aspect of mastering mathematics is consistent practice. Mathematics is a subject that requires repetition andreinforcement to truly solidify the knowledge. Students should make a habit of regularly practicing problems, both those assigned by their teachers and additional ones found in textbooks or online resources. This consistent practice not only helps to strengthen problem-solving skills but also builds confidence and fluency in mathematical reasoning.Effective time management and organization are also key factors in successful mathematical learning. Students should create a dedicated study schedule, allocating specific time slots for reviewing course material, working on practice problems, and seeking help when needed. Additionally, maintaining organized notes and resources can make it easier to reference and review important concepts when preparing for exams or tackling challenging problems.One often overlooked strategy for mastering mathematics is the importance of seeking help and support when needed. Many students may feel hesitant to ask for assistance, fearing that it might be perceived as a sign of weakness. However, the reality is that seeking help is a mark of strength and a crucial step in the learning process. Whether it's through office hours with the teacher, peer study groups, or online tutoring services, students should not be afraid to reach out and get the support they need to overcome any obstacles they may encounter.Furthermore, developing a growth mindset is essential for success in mathematics. Too often, students view their mathematical abilities as fixed, believing that they either have a "knack" for the subject or they don't. However, research has shown that with the right mindset and dedication, anyone can improve their mathematical skills. By embracing a growth mindset and believing in their ability to learn and improve, students can overcome challenges and persist through difficulties, ultimately achieving greater success in the subject.In addition to the strategies mentioned above, there are several other techniques that can aid in the mastery of mathematics. For instance, visualization and the use of visual aids, such as diagrams and graphs, can help students better understand and retain mathematical concepts. Additionally, breaking down complex problems into smaller, more manageable steps can make the problem-solving process more accessible and less overwhelming.It is also important to recognize that different students may respond better to different learning styles and approaches. Some may thrive in a more structured, step-by-step approach, while others may benefit from a more exploratory, hands-on style of learning. By being adaptable and willing to experiment with various learning strategies, students can find the approach that works best for them and maximize their mathematical potential.Ultimately, mastering the art of mathematics requires a multifaceted approach that combines conceptual understanding, consistent practice, effective time management, and a growth mindset. By embracing these strategies and techniques, students can not only excel in their mathematical studies but also develop critical thinking and problem-solving skills that will serve them well in all aspects of their academic and professional lives. With dedication, perseverance, and a willingness to seek help when needed, students can unlock the endless possibilities that mathematics has to offer.。
学数学的作文英语
Learning mathematics is an essential part of education that not only sharpens analytical skills but also fosters logical thinking and problemsolving abilities.Heres a detailed essay on the importance of studying mathematics and how it can be approached effectively.Title:The Significance of Mathematics in EducationIntroduction:Mathematics is often referred to as the language of the universe.It is a subject that transcends cultural and linguistic barriers,providing a universal framework for understanding the world around us.From counting to complex calculations,math is an integral part of our daily lives.Importance of Mathematics:1.Foundation for Other Subjects:Mathematics is a fundamental building block for other sciences such as physics,chemistry,and engineering.It provides the necessary tools to analyze and interpret data,making it indispensable for scientific inquiry.2.Critical Thinking:Studying math helps develop critical thinking skills.It encourages students to approach problems systematically and logically,breaking them down into manageable parts.3.ProblemSolving Abilities:Mathematical problems often have multiple solutions, which teaches students to think creatively and find the most efficient method to solve a problem.4.RealWorld Applications:Mathematics is not confined to the classroom.It is used in everyday life,from managing finances to understanding statistical data in various fields such as economics,medicine,and social sciences.Approaches to Learning Mathematics:1.Understanding Concepts:Rather than memorizing formulas,it is crucial to understand the underlying concepts.This approach helps in applying mathematical principles to new and unfamiliar problems.2.Practice:Mathematics is a subject that requires consistent practice.Regular exercises and problemsolving can significantly improve ones ability to tackle complex problems.e of Technology:Utilizing technology such as calculators,computer software,and online resources can enhance the learning experience.They can provide visual representations of mathematical concepts and offer interactive ways to solve problems.4.Group Work:Collaborative learning can be beneficial in mathematics.Discussing problems with peers can lead to new insights and a deeper understanding of the subject.Challenges in Learning Mathematics:1.Perception of Difficulty:Many students find mathematics challenging and may develop a fear of the subject.Overcoming this fear is essential for embracing the subject fully.ck of RealWorld Connection:Sometimes,the abstract nature of mathematics can make it difficult for students to see its relevance.Connecting mathematical concepts to realworld applications can help bridge this gap.Conclusion:In conclusion,mathematics is a subject of immense importance that offers a wealth of benefits to students.It not only equips them with analytical and problemsolving skills but also prepares them for various professional fields.By adopting effective learning strategies and recognizing the relevance of mathematics in everyday life,students can overcome challenges and excel in this fascinating subject.。
如何对待化学难题英语作文
如何对待化学难题英语作文When facing challenging chemistry problems, it's essential to adopt a systematic approach and utilize effective problem-solving strategies. Here's how you can tackle them:1. Understand the Problem: Begin by carefully reading and understanding the problem statement. Identify the key concepts, equations, and information provided.2. Break Down the Problem: Break down the problem into smaller, more manageable parts. If necessary, draw diagrams or tables to organize the given data.3. Review Relevant Concepts: Review the relevant concepts and theories related to the problem. Ensure you understand the fundamental principles involved.4. Identify Knowns and Unknowns: Clearly identify what is given (knowns) and what needs to be determined(unknowns). This will help you decide which equations or methods to apply.5. Choose the Right Approach: Select the appropriate approach or method to solve the problem. This might involve using specific equations, applying mathematical techniques, or employing analytical methods.6. Apply Formulas and Equations: Use the appropriate formulas and equations to calculate the desired quantities. Make sure to use units consistently throughout your calculations.7. Check for Errors: After obtaining your solution, double-check your calculations for accuracy. Verify that your answer is reasonable and consistent with the given information.8. Seek Help if Needed: If you're stuck or uncertain about certain concepts, don't hesitate to seek help from your teacher, classmates, or online resources. Sometimes, discussing the problem with others can provide valuableinsights.9. Practice Regularly: Practice solving similar problems regularly to strengthen your understanding of key concepts and improve your problem-solving skills. The more you practice, the more confident you'll become in tackling challenging chemistry problems.10. Reflect on Your Process: After solving the problem, take some time to reflect on your problem-solving process. Identify what strategies worked well for you and areas where you can improve. Learning from your experiences will help you become a more proficient problem solver in the future.By following these steps and maintaining a positive attitude towards challenging chemistry problems, you'll be better equipped to tackle them effectively and achieve success in your studies. Remember, perseverance andpractice are key to mastering any subject, including chemistry.。
怎样学好数学的英语作文
怎样学好数学的英语作文Learning mathematics can be a challenging and rewarding experience. It is a subject that requires a unique set of skills and a deep understanding of concepts. However, with the right approach and dedication, anyone can excel in mathematics. In this essay, we will explore several strategies and techniques that can help students learn mathematics effectively.Firstly, it is essential to develop a positive attitude towards mathematics. Many students have a preconceived notion that mathematics is difficult or intimidating. This mindset can hinder their progress and make the learning process more challenging. It is crucial to understand that mathematics is a logical and structured subject, and with practice and persistence, it can become more accessible.One of the key strategies for learning mathematics well is to build a strong foundation. This means mastering the basic concepts and skills before moving on to more advanced topics. It is important to ensure that you have a solid understanding of the fundamentals,such as arithmetic operations, fractions, and algebraic expressions, before attempting to tackle more complex problems.Another important aspect of learning mathematics is active engagement. Passive learning, such as simply listening to lectures or reading textbooks, is often not enough. Students should actively participate in the learning process by asking questions, solving practice problems, and engaging in discussions with their peers and teachers.Effective time management is also crucial for learning mathematics. It is important to set aside dedicated time for studying and practicing mathematics regularly. Consistent practice is essential for developing the necessary skills and understanding. Additionally, it is important to avoid procrastination and to stay organized with your studies.Utilizing various learning resources can also be beneficial for learning mathematics. This includes textbooks, online tutorials, educational videos, and practice problem sets. Exploring different resources can help students understand concepts from multiple perspectives and reinforce their learning.Furthermore, it is important to seek help when needed. Mathematics can be a challenging subject, and it is perfectly normal to strugglewith certain concepts. Students should not hesitate to ask for help from their teachers, tutors, or peers. Collaborative learning, where students work together to solve problems, can be particularly effective.In addition to the above strategies, it is important to develop a deep understanding of mathematical concepts rather than simply memorizing formulas or algorithms. This means understanding the underlying logic and reasoning behind the mathematics, rather than just focusing on the mechanical aspects of problem-solving.Another important aspect of learning mathematics is developing problem-solving skills. Mathematics often involves solving complex problems, and the ability to break down a problem, identify the relevant information, and develop a logical solution is crucial. Students should practice a variety of problem-solving techniques, such as breaking down the problem into smaller steps, using visual representations, and exploring multiple approaches.Lastly, it is important to stay motivated and persistent in the learning process. Learning mathematics can be challenging, and it is easy to become discouraged. However, it is important to remember that with consistent effort and dedication, progress can be made. Students should celebrate their successes, no matter how small, and use setbacks as opportunities to learn and grow.In conclusion, learning mathematics effectively requires a combination of strategies and techniques. Building a strong foundation, actively engaging in the learning process, managing time effectively, utilizing various resources, seeking help when needed, developing a deep understanding of concepts, and staying motivated and persistent are all key elements for success in mathematics. By implementing these strategies, students can not only improve their mathematical skills but also develop critical thinking and problem-solving abilities that will serve them well in their academic and professional pursuits.。
解决数学问题英文作文
In the realm of mathematics, solving intricate problems often necessitates more than mere application of formulas or algorithms. It requires an astute understanding of underlying principles, a creative perspective, and the ability to analyze problems from multiple angles. This essay will delve into a hypothetical complex mathematical problem and outline a multi-faceted approach to its resolution, highlighting the importance of analytical reasoning, strategic planning, and innovative thinking.Suppose we are faced with a challenging combinatorial optimization problem – the Traveling Salesman Problem (TSP). The TSP involves finding the shortest possible route that visits every city on a list exactly once and returns to the starting point. Despite its deceptively simple description, this problem is NP-hard, which means there's no known efficient algorithm for solving it in all cases. However, we can explore several strategies to find near-optimal solutions.Firstly, **Mathematical Modeling**: The initial step is to model the problem mathematically. We would represent cities as nodes and the distances between them as edges in a graph. By doing so, we convert the real-world scenario into a mathematical construct that can be analyzed systematically. This phase underscores the significance of abstraction and formalization in mathematics - transforming a complex problem into one that can be tackled using established mathematical tools.Secondly, **Algorithmic Approach**: Implementing exact algorithms like the Held-Karp algorithm or approximation algorithms such as the nearest neighbor or the 2-approximation algorithm by Christofides can help find feasible solutions. Although these may not guarantee the absolute optimum, they provide a benchmark against which other solutions can be measured. Here, computational complexity theory comes into play, guiding our decision on which algorithm to use based on the size and characteristics of the dataset.Thirdly, **Heuristic Methods**: When dealing with large-scale TSPs, heuristic methods like simulated annealing or genetic algorithms can offerpractical solutions. These techniques mimic natural processes to explore the solution space, gradually improving upon solutions over time. They allow us to escape local optima and potentially discover globally better solutions, thereby demonstrating the value of simulation and evolutionary computation in problem-solving.Fourthly, **Optimization Techniques**: Leveraging linear programming or dynamic programming could also shed light on the optimal path. For instance, using the cutting-plane method to iteratively refine the solution space can lead to increasingly accurate approximations of the optimal tour. This highlights the importance of advanced optimization techniques in addressing complex mathematical puzzles.Fifthly, **Parallel and Distributed Computing**: Given the computational intensity of some mathematical problems, distributing the workload across multiple processors or machines can expedite the search for solutions. Cloud computing and parallel algorithms can significantly reduce the time needed to solve large instances of TSP.Lastly, **Continuous Learning and Improvement**: Each solved instance provides learning opportunities. Analyzing why certain solutions were suboptimal can inform future approaches. This iterative process of analysis and refinement reflects the continuous improvement ethos at the heart of mathematical problem-solving.In conclusion, tackling a complex mathematical problem like the Traveling Salesman Problem involves a multi-dimensional strategy that includes mathematical modeling, selecting appropriate algorithms, applying heuristic methods, utilizing optimization techniques, leveraging parallel computing, and continuously refining methodologies based on feedback. Such a comprehensive approach embodies the essence of mathematical thinking – rigorous, adaptable, and relentlessly curious. It underscores that solving math problems transcends mere calculation; it’s about weaving together diverse strands of knowledge to illuminate paths through the labyrinth of numbers and logic.Word Count: 693 words(For a full 1208-word essay, this introduction can be expanded with more detailed explanations of each strategy, case studies, or examples showcasing their implementation. Also, the conclusion can be extended to discuss broader implications of the multi-faceted approach to problem-solving in various fields beyond mathematics.)。
数学是我学过最难的科目英语作文
数学是我学过最难的科目英语作文全文共3篇示例，供读者参考篇1Mathematics has always been considered one of the most challenging subjects to study. Many students struggle with its complex concepts, intricate formulas, and abstract theories. Personally, I can attest to the fact that mathematics is indeed the most difficult subject I have ever studied.One of the reasons why I find mathematics so challenging is the level of precision and accuracy required. In subjects like English or History, there is more room for interpretation and creativity. However, in mathematics, there is only one correct answer, and the process to arrive at that answer must be flawless.A small mistake in calculation can lead to a completely different result, making it crucial to pay attention to every detail.Furthermore, mathematics requires a high level of critical thinking and problem-solving skills. It is not enough to simply memorize formulas and procedures; one must understand the underlying concepts and be able to apply them to a variety of different problems. This can be especially challenging whenfaced with complex problems that involve multiple steps and require the use of different mathematical techniques.In addition, mathematics often builds upon itself, with later concepts and topics depending on a solid understanding of earlier ones. This means that if a student falls behind or struggles to grasp a certain concept, it can be difficult to catch up and keep pace with the rest of the class. This cumulative nature of mathematics can make it a particularly challenging subject to study.Despite its difficulty, I believe that mathematics is a valuable subject to study. It teaches important skills such as logical reasoning, problem-solving, and critical thinking, which are essential not only in academics but also in many aspects of life. While it may be challenging, the sense of achievement and satisfaction that comes from solving a difficult mathematical problem is unparalleled.In conclusion, mathematics is without a doubt the most difficult subject I have ever studied. Its precision, complexity, and cumulative nature make it a formidable academic discipline. However, the skills and knowledge gained from studying mathematics are invaluable, making the challenge well worth it in the end.篇2Mathematics has always been considered one of the most challenging subjects for students to learn. It requires logic, critical thinking, problem-solving skills, and a solid understanding of various concepts and formulas. Personally, I have found mathematics to be the most difficult subject I have ever studied.One of the main reasons why I find mathematics so challenging is its abstract nature. Unlike other subjects like history or literature, where concepts are more concrete and tangible, mathematics deals with abstract ideas like numbers, variables, and equations. This abstract nature can make it difficult for students to grasp and apply mathematical concepts effectively.Another reason why I struggle with mathematics is the complexity of the problems and equations involved. Mathematics is a subject that builds upon itself, with each new concept and formula based on the ones that came before. This means that if a student does not fully understand a fundamental concept, they may have trouble with more advanced topics. As someone who sometimes struggles to grasp complex concepts quickly, this can be a major challenge for me.Additionally, mathematics requires a high level of precision and accuracy. A small mistake in a calculation or a misplaced decimal point can completely change the result of a problem. This means that students need to be extremely careful and focused when working through mathematical problems, which can be mentally taxing and frustrating at times.Despite these challenges, I have come to appreciate the importance of mathematics in various aspects of life. From calculating finances to understanding scientific theories, mathematics plays a crucial role in many fields. I have learned to approach mathematics with patience, perseverance, and a willingness to ask for help when needed.In conclusion, mathematics is undoubtedly the most difficult subject I have ever studied. Its abstract nature, complexity, and need for precision make it a challenging but rewarding discipline to master. While I may continue to struggle with mathematics at times, I am determined to improve my skills and gain a deeper understanding of this essential subject.篇3Title: Mathematics - The Most Challenging Subject I Have Ever StudiedIntroduction:Mathematics has always been regarded as one of the most difficult subjects for students to master. Many students struggle with the complex concepts and abstract reasoning required in mathematics. Personally, I have found mathematics to be the most challenging subject I have ever studied. In this essay, I will explore the reasons why I find mathematics challenging and discuss some strategies I have used to improve my understanding of the subject.Challenges in Mathematics:One of the main reasons why I find mathematics difficult is the abstract nature of the subject. Unlike other subjects which deal with concrete facts and information, mathematics involves dealing with abstract concepts such as numbers, variables, and equations. This abstraction can make it difficult to understand the underlying principles and apply them to solve problems.Another challenge I face in mathematics is the need for logical reasoning and critical thinking. Mathematics requires students to think in a systematic and logical manner, making connections between different concepts and applying them to solve problems. This can be a daunting task for many students,including myself, who may struggle with analytical thinking and problem-solving skills.Furthermore, the vast amount of content covered in mathematics can be overwhelming. From algebra and geometry to calculus and statistics, there is a wide range of topics that students need to grasp in order to succeed in mathematics. This breadth of content can make it difficult to focus on specific areas and deepen our understanding of the subject.Strategies for Improvement:Despite the challenges I face in mathematics, I have developed some strategies to help improve my understanding of the subject. One strategy I use is to break down complex problems into smaller, more manageable parts. By tackling one step at a time, I can gradually build up my understanding of the problem and find a solution.I also find it helpful to practice regularly and seek help from teachers or peers when I encounter difficulties. Mathematics is a subject that requires practice and repetition in order to improve. By working through problems and seeking feedback from others, I can identify areas where I need to improve and focus on developing my skills in those areas.In addition, I try to make connections between different concepts in mathematics to deepen my understanding of the subject. By relating new topics to familiar ones and exploring the relationships between different ideas, I can enhance my knowledge and improve my ability to apply mathematical principles to solve problems.Conclusion:In conclusion, mathematics is undeniably one of the most challenging subjects I have ever studied. The abstract nature of the subject, the need for logical reasoning and critical thinking, and the vast amount of content covered in mathematics all contribute to the difficulties I face in the subject. However, by using effective strategies such as breaking down problems, practicing regularly, seeking help from others, and making connections between different concepts, I have been able to improve my understanding of mathematics. While I may still find mathematics challenging, I am confident that with continued effort and determination, I can overcome these challenges and succeed in the subject.。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
SUBJECT NAME: APPLY MATHEMATICAL TECHNIQUES IN
MANUFACTURING ENGINEERING AND RELATED
ENVIRONMENTS
TEST DETAILS: PRACTICE for TEST 2 (Version 4)
TIME ALLOWED: ONE HOUR AND THIRTY MINUTES
ANSWER: ALL QUESTIONS
Candidates are to write their answers in the spaces provided in the examination paper. Additional blank sheets are provided at the end of the paper for use if more space is required for the answers.
All working must be shown. Merely passing on information from your (graphics) calculator is not sufficient
Candidates are to fill out the following (Print in block letters);
SURNAME: ………………………………………………………………………
GIVEN NAMES:…………………………………………………………………STUDENT I.D. NUMBER: ………………………………………………………DATE: ………………………………………………………………………
Aids permitted (candidate to supply):
Writing materials
Calculator except any of the following:
Casio CFX-9970, Casio Algebra FX 2.0, HP-40G, HP 49G, TI-89
Learning Outcome 3: Geometry
1. Make a sketch to illustrate the following
a)an acute angle
b)supplementary angles
c)vertically opposite angles
d) a reflex angle
2. Determine the value of x in each of the following diagrams.
a)
b)
c)
142 ︒ x ︒
3. Determine the value of the pronumeral(s) in each of the following diagrams.
a)
b)
c)
d)
4. Give the name which best describes the given information.
a) A triangle with all three sides equal in length
b) A polygon with six sides
c) A quadrilateral with one pair of sides parallel
d) A triangle with one angle greater than 90
5. a) What is the complement of 23o15’35” ?
b) The angles in a triangle are in the ratio 1: 2: 4. What, correct to the nearest minute,
is the size of the smallest angle?
6. a) Change 25o to radians correct to 4 decimal places.
b) Change 1.25 radians to degrees, minutes and seconds.
c) Use the formula s = r to calculate the length of an arc in a sector with central
angle 70o and radius 5.2 cm. Answer to the nearest mm.
7. Use the Theorem of Pythagoras to find the distance between the points (2, 4) and
(-1, 7). Answer correctly to 2 decimal places
(Hint: Draw the points on a Cartesian Plane)
8. A triangle has sides 2 m, 3 m and 4 m. Calculate
a)the perimeter
b) the area
9. For a semi-circle with diameter of 10.3 mm, calculate, correct to two decimal places
the
a)area
b)perimeter
10. A can is in the shape of a cylinder. It has a base area of 32.95 cm2 and a volume of
376 mL. Find, correct to the nearest mm, the height of the can.
Learning Outcome 4: Trigonometry
11. Use your calculator to obtain following ratios correct to 4 decimal places.
a) sin 83︒ b) tan 45.6︒
c) cos 35︒33’12” d) sec 42︒50’01”
π
e) cos 2 f) cot
4
12. Prove that sin2θ + cos2θ = 1.
13. Find, correct to two decimal places, the length of the side marked by x
12︒
x
18 m
14. How far must I walk on a bearing of 65︒ so that I will end 4.5 km to the east of my
starting point?Answer to the nearest metre.
15. An escalator inclined at an angle of 22.6︒ moves at a constant speed of 0.21 m/s. A
person steps on the escalator and reaches the top in 62 seconds. Calculate (correct to 3 significant figures) the vertical height that the person been lifted?
16. In triangle XYZ, 32,62,3.
=︒=︒= Calculate XZ correct to the
X Y YZ cm
nearest mm.
17. Two ships sail from port at the same time. One travels at 22 km/hr on a bearing of
325︒ and the other travels on a bearing of 270︒ at 18 km/hr. Find, correct to two decimal places, their distance apart after 4 hours.
18. Find the size of the largest angle in a triangle which has side lengths a = 2 cm,
b = 3 cm and
c = 4 cm. Give your answers in degrees, minutes an
d seconds.
END OF PAPER。