The document contains a solution to an aptitude test with 14 multiple choice questions covering topics like ratios, probability, geometry, linear equations, and mechanics. For each question, the correct answer is provided along with a brief explanation of the solution. The questions assess logical reasoning and problem-solving abilities.
The document provides solutions to 10 multiple choice questions related to aptitude. It also provides detailed working for questions 2, 5, and 7. Some key details:
- Question 2 asks about the area enclosed between two straight lines passing through the origin and calculates it to be 0.5.
- Question 5 involves calculating the relation between the areas of three squares based on a diagram with lines tangential to a circle. The relation is that the area of one square is the sum of the other two.
- Question 7 calculates what value of x makes the ratio of investments equal to the ratio of profits received, finding x to be 3000.
This document contains a 60 question mathematical reasoning practice test with multiple choice answers for each question. The test covers various math topics including arithmetic, algebra, geometry, percentages and ratios. Each question is one sentence long and has 5 possible multiple choice answers. The test is meant to help students prepare for the real mathematical reasoning entrance exam by showing them the format and style of questions that will be asked.
1. This document is a 40 question mathematics exam for 5th grade students in Sekolah Kebangsaan Sebobok, Bau.
2. The exam provides instructions for students, informing them to answer all questions by blackening the correct answer on the answer sheet.
3. It consists of questions involving operations with whole numbers, decimals, fractions and word problems.
The document provides examples of advanced quantitative questions from the GMAT exam covering topics such as fractions and ratios, percentages, integers, powers, roots, algebra, and word problems. Each section presents 3 sample problems with multiple choice answers and explanations of the correct answers. The questions require setting up and solving equations, factoring expressions, evaluating expressions, approximations, properties of integers and exponents, and interpreting word problems algebraically.
This document contains a mathematics assignment with 7 questions testing skills in rounding numbers, expressing values in standard form, solving equations, and factorizing and solving quadratic equations. Students are instructed to show their work and express answers to 3 significant figures over the course of solving 35 problems within 1 week.
The document provides information about the quantitative section of the Graduate Management Admission Test (GMAT). It details that the section contains 37 questions to be completed in 75 minutes, including both problem solving and data sufficiency questions. Problem solving questions require solving the problem and choosing the best answer, while data sufficiency questions involve determining whether one, both, or neither of the given statement(s) are sufficient to answer the question. The document provides examples of several questions and explains the relevant terms and concepts.
This document contains a math skills assessment with 40 multiple choice questions testing various math concepts including:
- Addition
- Subtraction
- Multiplication
- Division
- Word problems
- Place value
- Telling time
- Fractions
- Decimals
- Money amounts
The questions require choosing the correct answer from 4 options to complete number sentences, perform calculations, or solve word problems on a range of math topics for different grade levels.
The document provides examples of non-verbal reasoning questions and their solutions. It includes number series, letter series, logical reasoning, and mathdoku puzzles. For the number series questions, the correct answer is choosing the option that continues the same pattern to fill in the missing term. The letter and logic series involve analyzing the relationship between letters or numbers to determine the missing element. The mathdoku puzzles require logically placing the numbers 1 to 5 in the grid so that each row and column uses each number, and the sums or products of the bold outlined groups equal the given hints.
The document provides solutions to 10 multiple choice questions related to aptitude. It also provides detailed working for questions 2, 5, and 7. Some key details:
- Question 2 asks about the area enclosed between two straight lines passing through the origin and calculates it to be 0.5.
- Question 5 involves calculating the relation between the areas of three squares based on a diagram with lines tangential to a circle. The relation is that the area of one square is the sum of the other two.
- Question 7 calculates what value of x makes the ratio of investments equal to the ratio of profits received, finding x to be 3000.
This document contains a 60 question mathematical reasoning practice test with multiple choice answers for each question. The test covers various math topics including arithmetic, algebra, geometry, percentages and ratios. Each question is one sentence long and has 5 possible multiple choice answers. The test is meant to help students prepare for the real mathematical reasoning entrance exam by showing them the format and style of questions that will be asked.
1. This document is a 40 question mathematics exam for 5th grade students in Sekolah Kebangsaan Sebobok, Bau.
2. The exam provides instructions for students, informing them to answer all questions by blackening the correct answer on the answer sheet.
3. It consists of questions involving operations with whole numbers, decimals, fractions and word problems.
The document provides examples of advanced quantitative questions from the GMAT exam covering topics such as fractions and ratios, percentages, integers, powers, roots, algebra, and word problems. Each section presents 3 sample problems with multiple choice answers and explanations of the correct answers. The questions require setting up and solving equations, factoring expressions, evaluating expressions, approximations, properties of integers and exponents, and interpreting word problems algebraically.
This document contains a mathematics assignment with 7 questions testing skills in rounding numbers, expressing values in standard form, solving equations, and factorizing and solving quadratic equations. Students are instructed to show their work and express answers to 3 significant figures over the course of solving 35 problems within 1 week.
The document provides information about the quantitative section of the Graduate Management Admission Test (GMAT). It details that the section contains 37 questions to be completed in 75 minutes, including both problem solving and data sufficiency questions. Problem solving questions require solving the problem and choosing the best answer, while data sufficiency questions involve determining whether one, both, or neither of the given statement(s) are sufficient to answer the question. The document provides examples of several questions and explains the relevant terms and concepts.
This document contains a math skills assessment with 40 multiple choice questions testing various math concepts including:
- Addition
- Subtraction
- Multiplication
- Division
- Word problems
- Place value
- Telling time
- Fractions
- Decimals
- Money amounts
The questions require choosing the correct answer from 4 options to complete number sentences, perform calculations, or solve word problems on a range of math topics for different grade levels.
The document provides examples of non-verbal reasoning questions and their solutions. It includes number series, letter series, logical reasoning, and mathdoku puzzles. For the number series questions, the correct answer is choosing the option that continues the same pattern to fill in the missing term. The letter and logic series involve analyzing the relationship between letters or numbers to determine the missing element. The mathdoku puzzles require logically placing the numbers 1 to 5 in the grid so that each row and column uses each number, and the sums or products of the bold outlined groups equal the given hints.
The document discusses irrational inequalities of one variable. It explains that an irrational inequality contains a variable under the radical sign. It provides examples of different forms of irrational inequalities and the steps to determine their solution sets, which involve squaring both sides and finding the intersections of the resulting conditions. The key points are that the radical terms must be greater than or equal to zero and the inequality sign must be preserved after squaring.
This document provides instructions and information for candidates taking a preliminary mathematics examination. It consists of 9 printed pages containing 10 questions. Candidates are instructed to show all working and calculations, and to write their answers on the provided writing papers. Calculators should be used where appropriate. Formulas for topics like compound interest, mensuration, trigonometry, and statistics are also provided. The total marks for the paper are 100.
Mathematical Operations Reasoning QuestionsSandip Kar
The document contains 12 math word problems where the symbols like +, -, x, / are used with different meanings than their usual meanings. For each problem, the correct meaning of the symbols is determined and the problem is solved to find the right answer. The document tests the ability to solve math problems where the symbols are used in an unconventional way by providing the correct symbolic interpretations.
This document is a marking scheme for an Additional Mathematics paper consisting of 25 questions. It provides the solutions, working and marks allocated for each question. The marking scheme is broken down question by question with the key steps shown. Marks are awarded for method (M), answer (A) and working (W). The highest total mark for a question is indicated. The marking scheme is 6 pages for paper 1 and 10 pages for paper 2, guiding examiners on how to consistently and fairly award marks for students' responses.
This document is a 35 question mathematics assessment in Malay on algebraic expressions and formulas for Form 3 students. It covers topics like algebraic terms, factors, equations, variables, and formulas. Students are instructed to fill out an answer sheet indicating their answers for multiple choice and short answer questions. The assessment includes questions testing comprehension of algebraic concepts in English and applying formulas to solve problems.
This document introduces different methods for solving systems of linear equations: graphing, substitution, and elimination. It provides examples of solving systems by graphing them on a coordinate plane and finding the point of intersection. It also gives examples of converting equations between standard and slope-intercept form and using substitution and elimination to solve systems algebraically.
Ms Wong bought three presents for her sister. The total cost was RM 71.64. The number of sweets in each of the 36 packets if 1,944 sweets are repacked is 54. Zali traveled a total distance of 2.09 km.
This document contains solutions to mathematics problems from the Anglican High School 2009 Preliminary Examinations. Some key points summarized:
1) Problems cover topics like algebra, trigonometry, geometry, calculus and statistics. Full worked solutions are provided for each question.
2) One question involves calculating interest earned over multiple years, finding a total profit, and determining original cost and selling prices.
3) Another question calculates areas and volumes for geometric shapes like triangles, sectors, pyramids and hemispheres using trigonometric and algebraic formulas.
4) Statistics questions calculate probabilities, means, standard deviations and compare data sets.
The document provides the answers and working for an algebra assessment. It includes 6 multiple choice questions worth 1-3 marks each on simplifying expressions, expanding expressions, solving equations, solving inequalities, and solving word problems. It also includes a question worth 4 marks requiring trial and error to solve an equation between 2 and 3 to 1 decimal place. The total marks for the assessment are 35. The student's score is shown as M1-M6 for each question, with a total score of 35/35.
This document contains 42 multi-part quantitative questions along with their answer choices. The questions cover a variety of topics including geometry, algebra, probability, sequences, ratios and proportions. They range in difficulty from straightforward calculations to more complex problems requiring multiple steps.
This document contains practice questions for students in Year 4 focusing on whole numbers up to 100,000. It includes over 221 carefully selected multiple choice questions suitable for students to practice during the school holidays in topics such as whole numbers, addition within 100,000, subtraction within 100,000, multiplication up to 100,000, and division up to 100,000. The questions are at an exam standard and of high quality.
Questions on Verbal & Non Verbal ReasoningLearnPick
This document contains 7 slides with math problems and their solutions. The problems involve finding missing numbers in sequences, determining patterns in matrices, and performing calculations. For example, one problem finds the missing number in the sequence 72, 15, 31, ?, 127 by noting each term is the previous multiplied by 2 and added 1. The solutions are provided and range from simple arithmetic to more complex algebraic steps.
The document provides a review for an Algebra II final exam covering topics such as linear equations and inequalities, rational expressions, complex numbers, matrices, and polynomial functions. The review contains 50 practice problems across 7 sections testing different algebra concepts and skills needed to solve various types of problems.
What is the distance between the points B and C Experience Tradition/tutorial...pinck3124
FOR MORE CLASSES VISIT
www.tutorialoutlet.com
This paper is part of an examination of the College counting towards the award of a degree.
Examinations are governed by the College Regulations under the authority of the Academic Board.
The document contains a math exam for 6th grade with 5 questions. Question 1 has 4 math calculation problems worth 3 points total. Question 2 has 4 algebra word problems to solve for x worth 2.5 points total. Question 3 is about calculating the number of students of different achievement levels in a class of 30 students worth 1.5 points. Question 4 has 3 geometry problems about angles and lines on a plane worth 2 points total. Question 5 is a word problem about calculating the price of an item after a 20% discount worth 1 point. The document also includes an answer key showing the step-by-step work for each problem.
This document contains examples of operations with powers and radicals. It includes:
- Notes on the basic operations and properties of powers and radicals.
- 23 example problems working through simplifying expressions with powers and radicals. The examples show applying order of operations, properties of exponents, and simplifying radicals.
- The goal is to practice simplifying complex expressions involving multiple operations with powers and radicals down to their simplest form through step-by-step working.
This document contains 52 probability and combinatorics word problems with multiple choice answers. The problems cover a wide range of topics including counting arrangements, combinations, probability calculations involving drawing balls from jars or flipping coins, and more. The correct answers to each problem are provided as multiple choice options.
Strategic Intervention Material (SIM) was provided for Grade 10 students to enhance learning and to motivate and stir up the attention and interest of the students until they master the topic. This material depicts the entire definition of learning since it concludes a systematic development of students’ comprehension on a distinct lesson in Mathematics 10.
The document contains a 40-question multiple choice quiz about mathematics and word problems. The questions cover topics like arithmetic operations, fractions, time, money, measurement, and word problems. For each question there are 4 possible answer choices labelled A, B, C, or D.
Ncert solutions for class 7 maths chapter 1 integers exercise 1.4iprepkumar
The document provides solutions to 7 questions from an NCERT Class 7 math exercise on integers. Some key details:
- Question 1 involves evaluating integer expressions by dividing integers.
- Question 2 shows that a ÷ (b + c) is not always equal to (a ÷ b) + (a ÷ c) through counter examples.
- Question 3 fills in missing values to make integer division statements true.
- Question 4 gives examples of integer pairs where the first number divided by the second equals -3.
- Question 5 calculates the time when temperature reaches a given value, given an initial temperature and rate of change.
- Questions 6 and 7 involve word problems on test scoring and elevator descent time
The document provides solutions to 10 questions from an aptitude test. The solutions cover topics like roots of equations, logical reasoning, palindromes, functional dependencies in databases, probability, and functions. For each question, the full question text is included along with an explanation of the answer. Technical topics covered in some solutions include cache replacement policies, relational data models, and formal language definitions.
This document contains a sample GATE paper with questions from various subjects like mathematics, physics, chemistry and general aptitude. The questions include multiple choice, numerical answer type and explanation type questions. Some questions test concepts like differential equations, complex numbers, Laplace transforms, electric circuits etc. The document also contains information about an online portal for GATE preparation that has trained over 1 lakh students across India.
The document discusses irrational inequalities of one variable. It explains that an irrational inequality contains a variable under the radical sign. It provides examples of different forms of irrational inequalities and the steps to determine their solution sets, which involve squaring both sides and finding the intersections of the resulting conditions. The key points are that the radical terms must be greater than or equal to zero and the inequality sign must be preserved after squaring.
This document provides instructions and information for candidates taking a preliminary mathematics examination. It consists of 9 printed pages containing 10 questions. Candidates are instructed to show all working and calculations, and to write their answers on the provided writing papers. Calculators should be used where appropriate. Formulas for topics like compound interest, mensuration, trigonometry, and statistics are also provided. The total marks for the paper are 100.
Mathematical Operations Reasoning QuestionsSandip Kar
The document contains 12 math word problems where the symbols like +, -, x, / are used with different meanings than their usual meanings. For each problem, the correct meaning of the symbols is determined and the problem is solved to find the right answer. The document tests the ability to solve math problems where the symbols are used in an unconventional way by providing the correct symbolic interpretations.
This document is a marking scheme for an Additional Mathematics paper consisting of 25 questions. It provides the solutions, working and marks allocated for each question. The marking scheme is broken down question by question with the key steps shown. Marks are awarded for method (M), answer (A) and working (W). The highest total mark for a question is indicated. The marking scheme is 6 pages for paper 1 and 10 pages for paper 2, guiding examiners on how to consistently and fairly award marks for students' responses.
This document is a 35 question mathematics assessment in Malay on algebraic expressions and formulas for Form 3 students. It covers topics like algebraic terms, factors, equations, variables, and formulas. Students are instructed to fill out an answer sheet indicating their answers for multiple choice and short answer questions. The assessment includes questions testing comprehension of algebraic concepts in English and applying formulas to solve problems.
This document introduces different methods for solving systems of linear equations: graphing, substitution, and elimination. It provides examples of solving systems by graphing them on a coordinate plane and finding the point of intersection. It also gives examples of converting equations between standard and slope-intercept form and using substitution and elimination to solve systems algebraically.
Ms Wong bought three presents for her sister. The total cost was RM 71.64. The number of sweets in each of the 36 packets if 1,944 sweets are repacked is 54. Zali traveled a total distance of 2.09 km.
This document contains solutions to mathematics problems from the Anglican High School 2009 Preliminary Examinations. Some key points summarized:
1) Problems cover topics like algebra, trigonometry, geometry, calculus and statistics. Full worked solutions are provided for each question.
2) One question involves calculating interest earned over multiple years, finding a total profit, and determining original cost and selling prices.
3) Another question calculates areas and volumes for geometric shapes like triangles, sectors, pyramids and hemispheres using trigonometric and algebraic formulas.
4) Statistics questions calculate probabilities, means, standard deviations and compare data sets.
The document provides the answers and working for an algebra assessment. It includes 6 multiple choice questions worth 1-3 marks each on simplifying expressions, expanding expressions, solving equations, solving inequalities, and solving word problems. It also includes a question worth 4 marks requiring trial and error to solve an equation between 2 and 3 to 1 decimal place. The total marks for the assessment are 35. The student's score is shown as M1-M6 for each question, with a total score of 35/35.
This document contains 42 multi-part quantitative questions along with their answer choices. The questions cover a variety of topics including geometry, algebra, probability, sequences, ratios and proportions. They range in difficulty from straightforward calculations to more complex problems requiring multiple steps.
This document contains practice questions for students in Year 4 focusing on whole numbers up to 100,000. It includes over 221 carefully selected multiple choice questions suitable for students to practice during the school holidays in topics such as whole numbers, addition within 100,000, subtraction within 100,000, multiplication up to 100,000, and division up to 100,000. The questions are at an exam standard and of high quality.
Questions on Verbal & Non Verbal ReasoningLearnPick
This document contains 7 slides with math problems and their solutions. The problems involve finding missing numbers in sequences, determining patterns in matrices, and performing calculations. For example, one problem finds the missing number in the sequence 72, 15, 31, ?, 127 by noting each term is the previous multiplied by 2 and added 1. The solutions are provided and range from simple arithmetic to more complex algebraic steps.
The document provides a review for an Algebra II final exam covering topics such as linear equations and inequalities, rational expressions, complex numbers, matrices, and polynomial functions. The review contains 50 practice problems across 7 sections testing different algebra concepts and skills needed to solve various types of problems.
What is the distance between the points B and C Experience Tradition/tutorial...pinck3124
FOR MORE CLASSES VISIT
www.tutorialoutlet.com
This paper is part of an examination of the College counting towards the award of a degree.
Examinations are governed by the College Regulations under the authority of the Academic Board.
The document contains a math exam for 6th grade with 5 questions. Question 1 has 4 math calculation problems worth 3 points total. Question 2 has 4 algebra word problems to solve for x worth 2.5 points total. Question 3 is about calculating the number of students of different achievement levels in a class of 30 students worth 1.5 points. Question 4 has 3 geometry problems about angles and lines on a plane worth 2 points total. Question 5 is a word problem about calculating the price of an item after a 20% discount worth 1 point. The document also includes an answer key showing the step-by-step work for each problem.
This document contains examples of operations with powers and radicals. It includes:
- Notes on the basic operations and properties of powers and radicals.
- 23 example problems working through simplifying expressions with powers and radicals. The examples show applying order of operations, properties of exponents, and simplifying radicals.
- The goal is to practice simplifying complex expressions involving multiple operations with powers and radicals down to their simplest form through step-by-step working.
This document contains 52 probability and combinatorics word problems with multiple choice answers. The problems cover a wide range of topics including counting arrangements, combinations, probability calculations involving drawing balls from jars or flipping coins, and more. The correct answers to each problem are provided as multiple choice options.
Strategic Intervention Material (SIM) was provided for Grade 10 students to enhance learning and to motivate and stir up the attention and interest of the students until they master the topic. This material depicts the entire definition of learning since it concludes a systematic development of students’ comprehension on a distinct lesson in Mathematics 10.
The document contains a 40-question multiple choice quiz about mathematics and word problems. The questions cover topics like arithmetic operations, fractions, time, money, measurement, and word problems. For each question there are 4 possible answer choices labelled A, B, C, or D.
Ncert solutions for class 7 maths chapter 1 integers exercise 1.4iprepkumar
The document provides solutions to 7 questions from an NCERT Class 7 math exercise on integers. Some key details:
- Question 1 involves evaluating integer expressions by dividing integers.
- Question 2 shows that a ÷ (b + c) is not always equal to (a ÷ b) + (a ÷ c) through counter examples.
- Question 3 fills in missing values to make integer division statements true.
- Question 4 gives examples of integer pairs where the first number divided by the second equals -3.
- Question 5 calculates the time when temperature reaches a given value, given an initial temperature and rate of change.
- Questions 6 and 7 involve word problems on test scoring and elevator descent time
The document provides solutions to 10 questions from an aptitude test. The solutions cover topics like roots of equations, logical reasoning, palindromes, functional dependencies in databases, probability, and functions. For each question, the full question text is included along with an explanation of the answer. Technical topics covered in some solutions include cache replacement policies, relational data models, and formal language definitions.
This document contains a sample GATE paper with questions from various subjects like mathematics, physics, chemistry and general aptitude. The questions include multiple choice, numerical answer type and explanation type questions. Some questions test concepts like differential equations, complex numbers, Laplace transforms, electric circuits etc. The document also contains information about an online portal for GATE preparation that has trained over 1 lakh students across India.
This document contains a sample paper for the GATE exam with 25 multiple choice questions covering topics such as matrices, complex numbers, differential equations, mechanics, and heat transfer. The questions test concepts like limits, determinants, vector relationships, properties of materials, kinematics of rolling objects, and definitions of terms like Poisson's ratio and Biot number. An explanation or working is provided for each question to explain the reasoning behind the correct answer.
1of 20Use the formula for the sum of the first n terms of a geom.docxhyacinthshackley2629
1of 20
Use the formula for the sum of the first n terms of a geometric sequence to solve the following.
Find the sum of the first 12 terms of the geometric sequence: 2, 6, 18, 54 . . .
A. 531,440
B. 535,450
C. 535,445
D. 431,440
2 of 20
5.0 Points
The following are defined using recursion formulas. Write the first four terms of each sequence.
a1 = 7 and an = an-1 + 5 for n ≥ 2
A. 8, 13, 21, 22
B. 7, 12, 17, 22
C. 6, 14, 18, 21
D. 4, 11, 17, 20
3 of 20
5.0 Points
How large a group is needed to give a 0.5 chance of at least two people having the same birthday?
A. 13 people
B. 23 people
C. 47 people
D. 28 people
4 of 20
5.0 Points
Write the first six terms of the following arithmetic sequence.
an = an-1 + 6, a1 = -9
A. -9, -3, 3, 9, 15, 21
B. -11, -4, 3, 9, 17, 21
C. -8, -3, 3, 9, 16, 22
D. -9, -5, 3, 11, 15, 27
5 of 20
5.0 Points
Write the first four terms of the following sequence whose general term is given.
an = 3n
A. 3, 9, 27, 81
B. 4, 10, 23, 91
C. 5, 9, 17, 31
D. 4, 10, 22, 41
6 of 20
5.0 Points
If 20 people are selected at random, find the probability that at least 2 of them have the same birthday.
A. ≈ 0.31
B. ≈ 0.42
C. ≈ 0.45
D. ≈ 0.41
7 of 20
5.0 Points
Consider the statement "2 is a factor of n2 + 3n."
If n = 1, the statement is "2 is a factor of __________."
If n = 2, the statement is "2 is a factor of __________."
If n = 3, the statement is "2 is a factor of __________."
If n = k + 1, the statement before the algebra is simplified is "2 is a factor of __________."
If n = k + 1, the statement after the algebra is simplified is "2 is a factor of __________."
A.4; 15; 28; (k + 1)2 + 3(k + 1); k2 + 5k + 8
B.4; 20; 28; (k + 1)2 + 3(k + 1); k2 + 5k + 7
C.4; 10; 18; (k + 1)2 + 3(k + 1); k2 + 5k + 4
D.4; 15; 18; (k + 1)2 + 3(k + 1); k2 + 5k + 6
8 of 20
5.0 Points
k2 + 3k + 2 = (k2 + k) + 2 ( __________ )
A. k + 5
B. k + 1
C. k + 3
D. k + 2
9 of 20
5.0 Points
The following are defined using recursion formulas. Write the first four terms of each sequence.
a1 = 3 and an = 4an-1 for n ≥ 2
A. 3, 12, 48, 192
B. 4, 11, 58, 92
C. 3, 14, 79, 123
D. 5, 14, 47, 177
10 of 20
5.0 Points
To win at LOTTO in the state of Florida, one must correctly select 6 numbers from a collection of 53 numbers (1 through 53). The order in which the selection is made does not matter. How many different selections are possible?
A. 32,957,326 selections
B. 22,957,480 selections
C. 28,957,680 selections
D. 225,857,480 selections
11 of 20
5.0 Points
Write the first six terms of the following arithmetic sequence.
an = an-1 - 0.4, a1 = 1.6
A. 1.6, 1.2, 0.8, 0.4, 0, -0.4
B. 1.6, 1.4, 0.9, 0.3, 0, -0.3
C. 1.6, 2.2, 1.8, 1.4, 0, -1.4
D. 1.3, 1.5, 0.8, 0.6, 0, -0.6
12 of 20
5.0 Points
You volunteer to help drive children at a charity event to the zoo, but you can fit only 8 of the 17 children present in your van. How many different groups of 8 children can you drive?
A. 32.
Grade 10 Math - Second Quarter Summative Testrobengie monera
This document appears to be a summative test for a 10th grade mathematics class covering topics in polynomials and geometry. It contains 45 multiple choice questions testing students' understanding of polynomial functions, properties of circles, coordinate geometry, and solving geometric problems using coordinates. The test includes questions on identifying the degree and leading term of polynomials, graphing polynomial functions, properties of secants, tangents, and circles, finding distances and areas using coordinates, and identifying geometric shapes from their vertices.
The document contains a math worksheet with multiple-choice and word problems involving integers, addition, subtraction, multiplication, division, and other basic math operations. Some example problems include representing situations with integers, comparing and ordering integers, and solving word problems about money, temperatures, distances, and rates. The document provides the steps to solve each problem.
The document provides information about GATE Solutions Electronics and Communication from 1987-2017. It contains solved questions and answers from Network Theory chapter covering topics like network analysis, DC transients, resonance, network theorems, two port networks, network functions and network graphs. The preface explains that the booklet aims to provide detailed explanations to previous year GATE questions to help students understand basic concepts and apply them to solve other questions. It is intended to be an indispensable resource for students preparing for the GATE exam.
This document contains a mock CAT exam with 26 multiple choice questions covering topics in mathematics, data interpretation, and logical reasoning. The questions are from sections including numbers, algebra, geometry, data sufficiency, and data interpretation from tables and graphs. The level of difficulty of the questions ranges from easy to moderate.
This document contains a mock CAT exam with multiple choice questions and explanations. It consists of two pages. The first page lists 60 multiple choice questions with answer options A-D. The second page provides explanations for the questions and solutions to problems. It discusses topics like probability, ratios, geometry, time/speed/distance word problems, and data interpretation from graphs.
This document contains a sample paper for the GATE exam with questions in General Aptitude and Mechanical Engineering sections. The General Aptitude section contains 10 multiple choice questions testing logical reasoning and English language skills. The Mechanical Engineering section contains 25 multiple choice questions covering topics such as materials, fluid mechanics, thermodynamics, and manufacturing processes. This sample paper is intended to help students prepare for the GATE exam by familiarizing them with the question types and difficulty level.
This document contains a sample paper for the GATE exam with questions in General Aptitude and Mechanical Engineering sections. The General Aptitude section contains 10 multiple choice questions testing logical reasoning and English language skills. The Mechanical Engineering section contains 25 multiple choice questions covering topics such as materials, fluid mechanics, thermodynamics, and manufacturing processes. This sample paper is intended to help students prepare for the GATE exam by familiarizing them with the question types and difficulty level.
For helpful CXC Maths Multiple Choice Videos please click below
These videos are very helpful
https://oke.io/dUqlSrd
https://oke.io/UWfOCCP
https://oke.io/FrCDQ
Dear Students/Parents
APEX INSTITUTE has been established with sincere and positive resolve to do something rewarding for ENGG. / PRE-MEDICAL aspirants. For this the APEX INSTITUTE has been instituted to provide a relentlessly motivating and competitive atmosphere.
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
The document provides information about an unsolved CAT paper from 2007, including 10 multiple choice questions from Section 1 and directions for answering questions involving two given statements. It also provides information about costs incurred by a company from 2002-2006 to produce a product, including a table with cost breakdowns. This information is then used to answer 4 questions about projecting costs for the company in 2007 under different production scenarios.
The document provides examples of linear equations in two variables and their solutions. It begins by giving examples of writing linear equations from word problems and expressing equations in the form ax + by + c = 0. It then gives examples of finding solutions to different linear equations by substitution. The document also demonstrates drawing graphs of various linear equations. It concludes by asking students to identify linear equations from their graphs.
Here are the steps to solve this problem:
1) Complete the table of values:
x -2 -1 0 1 2 3
y = x3 – 2x2 -8 -1 0 1 4 9
2) Sketch the graphs in the coordinate planes:
3) Describe the movements of each graph:
- The graph of y = x3 – 2x2 rises from left to right as x increases. It is always increasing.
- The graph of y = x4 rises from left to right as x increases. It is always increasing. The rate of increase gets larger as x increases.
4) Identify any noticeable points:
- The graph of y = x3
Here are the steps to solve this problem:
1) Complete the table of values:
x -2 -1 0 1 2 3
y = x3 – 2x2 -8 -1 0 1 4 9
2) Sketch the graphs in the coordinate planes:
3) Describe the movements of each graph:
- The graph of y = x3 – 2x2 rises from left to right as x increases. It is always increasing.
- The graph of y = x4 rises from left to right as x increases. It is always increasing. The rate of increase gets larger as x increases.
4) Identify any noticeable points on the graphs:
- The graph of y
K to 12 - Grade 8 Math Learners Module Quarter 2Nico Granada
Here are the completed statements based on the conclusions:
1. n(A × B) = n(B × A).
2. A × B ≠ B × A.
The key conclusions are:
1. The cardinalities of the Cartesian products A × B and B × A are equal, since n(A × B) = n(B × A).
2. However, the sets A × B and B × A are not equal, since the ordered pairs will be arranged differently, so A × B ≠ B × A.
The document introduces the topic of relations and functions and outlines 3 lessons that will be covered - rectangular coordinate system, representations of relations and functions, and linear functions and applications. It provides learning objectives for each lesson and examples of how concepts like slope, intercepts, and graphs will be explored. A pre-assessment with 20 multiple choice questions is also included to gauge students' prior knowledge.
The document introduces the key concepts that will be covered in a module on relations and functions, including the rectangular coordinate system, representations of relations and functions, and linear functions and their applications. It outlines 3 lessons that will examine how to predict the value of a quantity given the rate of change, and provides sample problems to assess students' prior knowledge on these topics before beginning the lessons.
Use PyCharm for remote debugging of WSL on a Windo cf5c162d672e4e58b4dde5d797...shadow0702a
This document serves as a comprehensive step-by-step guide on how to effectively use PyCharm for remote debugging of the Windows Subsystem for Linux (WSL) on a local Windows machine. It meticulously outlines several critical steps in the process, starting with the crucial task of enabling permissions, followed by the installation and configuration of WSL.
The guide then proceeds to explain how to set up the SSH service within the WSL environment, an integral part of the process. Alongside this, it also provides detailed instructions on how to modify the inbound rules of the Windows firewall to facilitate the process, ensuring that there are no connectivity issues that could potentially hinder the debugging process.
The document further emphasizes on the importance of checking the connection between the Windows and WSL environments, providing instructions on how to ensure that the connection is optimal and ready for remote debugging.
It also offers an in-depth guide on how to configure the WSL interpreter and files within the PyCharm environment. This is essential for ensuring that the debugging process is set up correctly and that the program can be run effectively within the WSL terminal.
Additionally, the document provides guidance on how to set up breakpoints for debugging, a fundamental aspect of the debugging process which allows the developer to stop the execution of their code at certain points and inspect their program at those stages.
Finally, the document concludes by providing a link to a reference blog. This blog offers additional information and guidance on configuring the remote Python interpreter in PyCharm, providing the reader with a well-rounded understanding of the process.
Discover the latest insights on Data Driven Maintenance with our comprehensive webinar presentation. Learn about traditional maintenance challenges, the right approach to utilizing data, and the benefits of adopting a Data Driven Maintenance strategy. Explore real-world examples, industry best practices, and innovative solutions like FMECA and the D3M model. This presentation, led by expert Jules Oudmans, is essential for asset owners looking to optimize their maintenance processes and leverage digital technologies for improved efficiency and performance. Download now to stay ahead in the evolving maintenance landscape.
Software Engineering and Project Management - Introduction, Modeling Concepts...Prakhyath Rai
Introduction, Modeling Concepts and Class Modeling: What is Object orientation? What is OO development? OO Themes; Evidence for usefulness of OO development; OO modeling history. Modeling
as Design technique: Modeling, abstraction, The Three models. Class Modeling: Object and Class Concept, Link and associations concepts, Generalization and Inheritance, A sample class model, Navigation of class models, and UML diagrams
Building the Analysis Models: Requirement Analysis, Analysis Model Approaches, Data modeling Concepts, Object Oriented Analysis, Scenario-Based Modeling, Flow-Oriented Modeling, class Based Modeling, Creating a Behavioral Model.
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024Sinan KOZAK
Sinan from the Delivery Hero mobile infrastructure engineering team shares a deep dive into performance acceleration with Gradle build cache optimizations. Sinan shares their journey into solving complex build-cache problems that affect Gradle builds. By understanding the challenges and solutions found in our journey, we aim to demonstrate the possibilities for faster builds. The case study reveals how overlapping outputs and cache misconfigurations led to significant increases in build times, especially as the project scaled up with numerous modules using Paparazzi tests. The journey from diagnosing to defeating cache issues offers invaluable lessons on maintaining cache integrity without sacrificing functionality.
Null Bangalore | Pentesters Approach to AWS IAMDivyanshu
#Abstract:
- Learn more about the real-world methods for auditing AWS IAM (Identity and Access Management) as a pentester. So let us proceed with a brief discussion of IAM as well as some typical misconfigurations and their potential exploits in order to reinforce the understanding of IAM security best practices.
- Gain actionable insights into AWS IAM policies and roles, using hands on approach.
#Prerequisites:
- Basic understanding of AWS services and architecture
- Familiarity with cloud security concepts
- Experience using the AWS Management Console or AWS CLI.
- For hands on lab create account on [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
# Scenario Covered:
- Basics of IAM in AWS
- Implementing IAM Policies with Least Privilege to Manage S3 Bucket
- Objective: Create an S3 bucket with least privilege IAM policy and validate access.
- Steps:
- Create S3 bucket.
- Attach least privilege policy to IAM user.
- Validate access.
- Exploiting IAM PassRole Misconfiguration
-Allows a user to pass a specific IAM role to an AWS service (ec2), typically used for service access delegation. Then exploit PassRole Misconfiguration granting unauthorized access to sensitive resources.
- Objective: Demonstrate how a PassRole misconfiguration can grant unauthorized access.
- Steps:
- Allow user to pass IAM role to EC2.
- Exploit misconfiguration for unauthorized access.
- Access sensitive resources.
- Exploiting IAM AssumeRole Misconfiguration with Overly Permissive Role
- An overly permissive IAM role configuration can lead to privilege escalation by creating a role with administrative privileges and allow a user to assume this role.
- Objective: Show how overly permissive IAM roles can lead to privilege escalation.
- Steps:
- Create role with administrative privileges.
- Allow user to assume the role.
- Perform administrative actions.
- Differentiation between PassRole vs AssumeRole
Try at [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
2. Solution
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APTITUDE
1. The movie was funny and I _____.
(a) could help laughing
(b) couldn’t help laughed
(c) couldn’t help laughing
(d) could helped laughed
Sol: (c)
2.
1 1 1
x : y : z = : :
2 3 4
What is the value of
x + z – y
y
?
(a) 0.75 (b) 1.25
(c) 2.25 (d) 3.25
Sol: (b)
Let x =
K
2
y =
K
3
z =
K
4
So, the value of
x + z – y
y
=
K K K
+ –
2 4 3
K
3
=
5K 3 5
× =
12 K 4
3. Both the numerator and the denominator of 3/4
are increased by a positive integer, x, and those
of 15/17 are decreased by the same integer. This
operation results in the same value for both the
fractions.
What is the value of x ?
(a) 1 (b) 2
(c) 3 (d) 4
Sol: (c)
3 + x
4 + x
=
15 – x
17 – x
Put x = 1, 2, 3, 4
Only x = 3 is satisfying.
4. A survey of 450 students about their subjects of
interest resulted in the following outcome.
• 150 students are interested in Mathematics.
• 200 students are interested in Physics.
• 175 students are interested in Chemistry.
• 50 students are interested in Mathematics and
Physics.
• 60 students are interested in Physics and
Chemistry.
• 40 students are interested in Mathematics and
Chemistry.
• 30 students are interested in Mathematics,
Physics and Chemistry.
• Remaining students are interested in
Humanities.
Based on the above information, the number of
students interested in Humanities is
(a) 10 (b) 30
(c) 40 (d) 45
Sol: (d)
n M P C = n M +n P + n C – n M P
–n P C – n C M +n M P C
n M P C =150+200+175 – 50 – 60 – 40+30
n M P C = 405
n(Humanities) = 450 – 405
= 45
5.
For the picture shown above, which one of the
following is the correct picture representing
reflection with respect to the mirror shown as the
dotted line ?
(a)
3. Solution
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(b)
(c)
(d)
Sol: (a)
6. In the last few years, several new shopping malls
were opened in the city. The total number of visitors
in the malls is impressive. However, the total
revenue generated through sales in the shops in
these malls is generally low.
Which one of the following is the CORRECT logical
inference based on the information in the above
passage ?
(a) Fewer people are visiting the malls but
spending more
(b) More people are visiting the malls but not
spending enough
(c) More people are visiting the malls and spending
more
(d) Fewer people are visiting the malls and not
spending enough
Sol: (b)
More people visiting are the walls but not spending
more. This can be conferred from the low revenue
generation through sales in the shops in these
malls.
7. In a partnership business the monthly investment
by three friends for the first six months is in the
ratio 3: 4: 5. After six months, they had to increase
their monthly investments by 10%, 15% and 20%,
respectively, of their initial monthly investment.
The new investment ratio was kept constant for
the next six months.
What is the ratio of their shares in the total profit
(in the same order) at the end of the year such
that the share is proportional to their individual
total investment over the year?
(a) 22 : 23 : 24 (b) 22 : 33 : 50
(c) 33 : 46 : 60 (d) 63 : 86 : 110
Sol: (d)
Let capitals invested by three friends for 6 months
are
C1 = 3x
C2 = 4x
C3 = 5x
P1 : P2 : P3 =
3x×6 +1.1×3x× 6
: 4x ×6 +1.15× 4x× 6
: 5x× 6 +1.2×5x×6
= 6.3 : 8.6 :11
= 63 : 86 :110
8. Consider the following equations of straight lines:
Line L1: 2x – 3y = 5
Line L2: 3x + 2y = 8
Line L3: 4x – 6y = 5
Line L4: 6x – 9y = 6
Which one among the following is the correct
statement ?
(a) L1 is parallel to L2 and L1 is perpendicular to
L3
(b) L2 is parallel to L4 and L2 is perpendicular to
L1
(c) L3 is perpendicular to L4 and L3 is parallel to
L2
(d) L4 is perpendicular to L2 and L4 is parallel to
L3
4.
5. Solution
12-02-2022 | AFTERNOON SESSION
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Sol: (d)
1
2 5 2
L y = + x – slope
3 3 3
2
3
3
L y = – x + 4 slope –
2 2
3
2x 5 2
L y = + + slope –
3 6 3
4
2x 2 2
L y = + – slope
3 3 3
L4 is perpendicular to L2.
2
–3
×
2 3
= –1
L4 is parallel to L3
Slopes are equal.
9. Given below are two statements and four
conclusions drawn based on the statements.
Statement 1: Some soaps are clean.
Statement 2: All clean objects are wet.
Conclusion I: Some clean objects are soaps.
Conclusion II: No clean object is a soap.
Conclusion III: Some wet objects are soaps.
Conclusion IV: All wet objects are soaps.
Which one of the following options can be logically
inferred?
(a) Only conclusion I is correct
(b) Either conclusion I or conclusion II is correct
(c) Either conclusion III or conclusion IV is correct
(d) Only conclusion I and conclusion III are correct
Sol: (d)
Probability (1)
Soaps
Clean
Wet
Probability (2)
Soaps clean
Wet
Conclusion (I) followed in both possibilities.
Conclusion (II) not followed.
Conclusion (III) followed in both possibilities.
Conclusion (IV) not followed.
10. An ant walks in a straight line on a plane leaving
behind a trace of its movement. The initial position
of the ant is at point P facing east.
The ant first turns 72° anticlockwise at P, and
then does the following two steps in sequence
exactly FIVE times before halting.
East
North
1. moves forward for 10 cm.
2. turns 144° clockwise.
The pattern made by the trace left behind by the
ant is
(a)
S
T R
P Q
PQ = QR = RS = ST = TP =10 cm
(b)
T S
U R
P Q
PQ = QR = RS ST = TU = UP =10cm
(c)
S
T R
P Q
SQ = QT = TR = RP = PS = 10cm
(d)
U T
W R
P
Q
S
SW = WR =RP =PT = TQ = QU=US =10 cm
Sol: (d)
Road map:
6. Solution
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72°
144°
72°
x axis
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36°
144°
y axis
S
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TECHNICAL
11. The function f(x, y) satisfies the Laplace equation
2
f x,y = 0
on a circular domain of radius r = 1 with its
center at point P with coordinates x = 0, y = 0.
The value of this function on the circular boundary
of this domain is equal to 3.
The numerical value f(0, 0) is :
(a) 0 (b) 2
(c) 3 (d) 1
Sol: (c)
12.
2 3 4
x x x
x – + – +.... dx
2 3 4
is equal to
(a)
1
+ Constant
1+ x
(b) 2
1
+ Constant
1+ x
(c)
1
– + Constant
1– x
(d) 2
1
– + Constant
1– x
Sol: (*)
I =
2 3 4
x x x
x – + – +.... dx
2 3 4
I =
2 3 4 5
x x x x
– + – +....
2 6 12 20
Option (a)
1
1+ x
=
–1 2 3
1+ x =1– x + x – x .....
So, its incorrect.
Option (b)
2
1
1+ x
= (1 + x2)–1 =
2 4 6
1– x + x – x +...
So, its incorrect.
Similarly option (c) and (d) both are incorrect.
No-correct choice given.
13. For a linear elastic and isotropic material, the
correct relationship among Young’s modulus of
elasticity (E), Poisson’s ratio (v), and shear
modulus (G) is
(a)
E
G =
2 1+ v
(b)
E
G =
2 1+ 2v
(c)
G
E =
2 1+ v
(d)
G
E =
2 1+ 2v
Sol: (a)
14. Read the following statements relating to flexure
of reinforced concrete beams:
I. In over-reinforced sections, the failure strain in
concrete reaches earlier than the yield strain
in steel.
II. In under-reinforced sections, steel reaches
yielding at a load lower than the load at which
the concrete reaches failure strain.
III. Over-reinforced beams are recommended in
practice as compared to the under-reinforced
beams.
IV. In balanced sections, the concrete reaches
failure strain earlier than the yield strain in
tensile steel.
Each of the above statements is either True or
False.
Which one of the following combinations is
correct ?
(a) I (True), II (True), III (False), IV (False)
(b) I (True), II (True), III (False), IV (True)
(c) I (False), II (False), III (True), IV (False)
(d) I (False), II (True), III (True), IV (False)
Sol: (a)
The question is based on LSM design principle
as it is describing different conditions related to
strain.
Depending on amount of reinforcement in a cross-
section, here ca be three types of sections viz.
balanced, under reinforced and over reinforced.
Balanced section is a section that is expected to
result in a balanced failure. It means at the ultimate
limit state in flexure, the concrete will attain a
limiting compressive strain of 0.0035 and steel
will attain minimum specified tensile strain of
7.
8. Solution
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s
0.87f
0.002
E
.
Under reinforced section is a section in which
steel yield before collapse.
Over reinforced section is a section in which
crushing of concrete in compression i.e. attainment
of compressive strain of 0.0035 occurs prior to
yielding of steel.
In case of over reinforced section the deflection,
crack width remain relatively low and failure occurs
without any sign of warning and hence over
reinforced flexural members are not recommended
by IS code.
Based on the above information:
Statement I is true.
Statement II is true.
Statement III is false.
Statement IV is false.
15. Match all the possible combinations between
Column X (Cement compounds) and Column Y
(Cement properties) :
Column X Column Y
(i) C3S (P) Early age strength
(ii) C2S (Q) Later age strength
(iii) C3A (R) Flash setting
(S) Highest heat of hydration
(T) Lowest heat of hydration
Which one of the following combinations is
correct?
(a) (i) - (P), (ii) - (Q) and (T), (iii) - (R) and (S)
(b) (i) - (Q) and (T), (ii) - (P) and (S), (iii) - (R)
(c) (i) - (P), (ii) - (Q) and (R), (iii) - (T)
(d) (i) - (T), (ii) - (S), (iii) - (P) and (Q)
Sol: (a)
C3S - Responsible for early age strength
C2S - Responsible for later age strength and
lowest heat of hydration
C3A - Flash setting and highest heat of hydration.
(i) - (P)
(ii) - Q & T
(iii) - R & S
16. Consider a beam PQ fixed at P, hinged at Q, and
subjected to a load F as shown in figure (not
drawn to scale). The static and kinematic degrees
of indeterminacy, respectively, are
P
F
Q
(a) 2 and 1 (b) 2 and 0
(c) 1 and 2 (d) 2 and 2
Sol: (a)
F
P
Q
MP
RP
HP
RQ
HQ
Unknown reactions = 5
Equilibrium equation = 3
Ds = 5 – 3
s
D = 2
k
D = 1 i.e., rotation ( ) at Q.
17. Read the following statements:
(P) While designing a shallow footing in sandy
soil, monsoon season is considered for critical
design in terms of bearing capacity.
(Q) For slope stability of an earthen dam, sudden
drawdown is never a critical condition.
(R) In a sandy sea beach, quicksand condition
can arise only if the critical hydraulic gradient
exceeds the existing hydraulic gradient.
(S) The active earth thrust on a rigid retaining wall
supporting homogeneous cohesionless backfill
will reduce with the lowering of water table in
the backfill.
Which one of the following combinations is
correct?
(a) (P)-True, (Q)-False, (R)-False, (S)-False
(b) (P)-False, (Q)-True, (R)-True, (S)-True
(c) (P)-True, (Q)-False, (R)-True, (S)-True
(d) (P)-False, (Q)-True, (R)-False, (S)-False
9. Solution
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18. Stresses acting on an infinitesimal soil element
are shown in the figure (with
z x ). The major
and minor principal stresses are 1 and 3 ,
respectively. Considering the compressive stresses
as positive, which one of the following expressions
correctly represents the angle between the major
principal stress plane and the horizontal plane ?
z
zx
zx
x x
zx
zx
z
(a)
–1 zx
1 x
tan
–
(b)
–1 zx
3 x
tan
–
(c)
–1 zx
1 x
tan
+
(d)
–1 zx
1 3
tan
+
Sol: (b)
z
x x
zx
z
Mohr’s circle by considering compressive stresses
positive and shear giving anticlockwise rotation
as positive.
3, 0
A
C O
B
180 – 2p
1, 0
z xz
Q ,
p
p
x xz
P , –
2p
From triangle ACP
A
C
P
P
xy
x 3
–
P
tan =
xy
x 3
–
Since, nothing is mentioned about sign convention
of shear stress, so one can adopt negative sign
for given shear stress. In that case
P
tan =
xz
3 x
–
19. Match Column X with Column Y:
Column X Column Y
(P) Horton equation (I) Design of alluvial channel
(Q) Penman method (II) Maximum flood discharge
(R) Chezy’s formula (III) Evapotranspiration
(S) Lacey’s theory (IV) Infiltration
(T) Dicken’s formula (V) Flow velocity
Which one of the following combinations is
correct ?
(a) (P)-(IV), (Q)-(III), (R)-(V), (S)-(I), (T)-(II)
(b) (P)-(III), (Q)-(IV), (R)-(V), (S)-(I), (T)-(II)
(c) (P)-(IV), (Q)-(III), (R)-(II), (S)-(I), (T)-(V)
(d) (P)-(III), (Q)-(IV), (R)-(I), (S)-(V), (T)-(II)
Sol: (a)
20. In a certain month, the reference crop
evapotranspiration at a location is 6 mm/day. If
the crop coefficient and soil coefficient are 1.2
and 0.8, respectively, the actual evapotranspiration
in mm/day is
(a) 5.76 (b) 7.20
(c) 6.80 (d) 8.00
Sol: (a)
Actual evapotranspiration (ETC)
= KS × KC × Reference evapotranspiration (ET0)
= 0.8 × 1.2 × 6
= 5.76 mm
10.
11.
12. Solution
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21. The dimension of dynamic viscosity is :
(a) –1 –1
ML T (b) –1 –2
ML T
(c) –2 –2
ML T (d) 0 –1
ML T
Sol: (a)
For Newtonian fluid,
The shear stress ( ) is experssed as
=
du
, where
dy
du
dy
= shear strain rate, and
= dynamic (or absolute)
viscosity of the fluid.
=
Force
Area
=
du Length 1
×
dy Time Length
=
2
2
Force
Length Force×Time
=
1 Length
Time
[Force] = Mass × Acceleration
=
Velocity
Mass×
Time
=
Length
Time
Mass×
Time
[Force] = –2
MLT
Dimension of dynamic viscosity =
=
–2
–1 –1
2
MLT ×T
= ML T
L
22. A process equipment emits 5 kg/h of volatile
organic compounds (VOCs). If a hood placed over
the process equipment captures 95% of the
VOCs, then the fugitive emission in kg/h is
(a) 0.25 (b) 4.75
(c) 2.50 (d) 0.48
Sol: (a)
VOC emission = 5 Kg/h
Capturing Efficiency = 95%
Fugitive emission =
5
×5 Kg h
100
= 0.25 Kg/h
23. Match the following attributes of a city with the
appropriate scale of measurements.
Attribute Scale of
measurement
(P) Average temperature (I) Interval
(°C) of a city
(Q) Name of a city (II) Ordinal
(R) Population density (III) Nominal
of a city
(S) Ranking of a city (IV) Ratio
based on ease of
business
Which one of the following combinations is
correct?
(a) (P)-(I), (Q)-(III), (R)-(IV), (S)-(II)
(b) (P)-(II), (Q)-(I), (R)-(IV), (S)-(III)
(c) (P)-(II), (Q)-(III), (R)-(IV), (S)-(I)
(d) (P)-(I), (Q)-(II), (R)-(III), (S)-(IV)
Sol: (b)
24. If the magnetic bearing of the Sun at a place at
noon is S 2° E, the magnetic declination (in
degrees) at that place is
(a) 2°E (b) 2°W
(c) 4°E (d) 4°W
Sol: (a)
MB =
S2 E = 180 – 2 = 178
TB = 180°
Declination, = TB – MB
= 180 – 178
= 2° or 2°E
25. P and Q are two square matrices of the same
order. Which of the following statement(s) is/are
correct ?
(a) If P and Q are invertible, then –1 –1 –1
PQ = Q P
(b) If P and Q are invertible, then –1 –1 –1
QP = P Q
(c) If P and Q are invertible, then –1 –1 –1
PQ = P Q
(d) If P and Q are not invertible, then
–1 –1 –1
PQ = Q P
Sol: (a, b)
If P and Q are invertible then (PQ)–1 = Q–1P–1 is
correct.
Let, PQ = C
13. Solution
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Q Q = –1 –1
Q P C
I = –1 –1
Q P C
IC–1 = –1 –1 –1
Q P CC
C–1 = –1 –1
Q P
–1
PQ = –1 –1
Q P
Hence, proved.
Similarly, we can prove if P, Q are invertible then
–1
QP = –1 –1
P Q
26. In a solid waste handling facility, the moisture
contents (MC) of food waste, paper waste, and
glass waste were found to be MCf, MCp, and
MCg, respectively. Similarly, the energy contents
(EC) of plastic waste, food waste, and glass waste
were found to be ECpp, ECf, and ECg,
respectively. Which of the following statement(s)
is/are correct ?
(a) MCf > MCp > MCg
(b) ECpp > ECf > ECg
(c) MCf < MCp < MCg
(d) ECpp < ECf < ECg
Sol: (a, b)
Typical specific weight and moisture content data
for residential, commercial, industrial, and
agricultural wastes.
Type of waste
Specifif weight, lb/yd
3
Moisture content,
% by weight
Range Typical Typical
Range
Residential (uncomopacted)
*Food wastes (mixed)
*Paper
Cardboard
Plastics
Textiles
Rubber
Leather
yard wastes
Wood
*Glass
Tin cans
Aluminum
Other metals
Dirt, ashes, etc.
Ashes
Rubbish
Residential yard wastes
Leaves (loose and dry)
Green grass (loose and moist)
Green grass (wet and compacted)
yard waste (shredded)
Yard waste (composted)
Municipal
In compactor truck
In landfill
Normally compacted
Well compacted
Commercial
Food wastes (wet)
Appliances
220-810
70-220
70-135
70-220
70-170
170-340
170-440
100-380
220-540
270-810
85-270
110-405
220-1940
540-1685
1095-1400
150-305
50-250
350-500
1000-1400
450-600
450-650
300-760
610-840
995-1250
800-1600
250-340
490
150
85
110
110
220
270
170
400
330
150
270
540
810
1255
220
100
400
1000
500
550
500
760
1010
910
305
50-80
4-10
4-8
1-4
6-15
1-4
8-12
30-80
15-40
1-4
2-4
2-4
2-4
6-12
6-12
5-20
20-40
40-80
50-90
20-70
40-60
70
6
5
2
10
2
10
60
20
2
3
2
3
8
6
15
30
60
80
50
50
20
25
25
25
70
1
15-40
15-40
15-40
50-80
0-2
Moisture content
Mcf > Mcp > Mcg
Typical values for short residue and energy conent
of residentail MSW.
Component
Inert residue percent
a
Range Typical
Energy , Btu/lb
b
Range Typical
*Food wastes
*Paper
Cardboard
Plastics
Textiles
Rubber
Leather
Yard wastes
Wood
Misc. organics
Organic
Inorganic
*Glass
Tin cans
Aluminum
Other metal
Dirt, ashes, etc.
Municipal solid wastes
2-8
4-8
3-6
6-20
2-4
8-20
8-20
2-6
0.6-2
–
96-99+
96-99+
90-99+
94-99+
60-80
5.0
6.0
5.0
10.0
2.5
10.0
10.0
4.5
1.5
–
98.0
98.0
96.0
98.0
70.0
1,500-3,000
5,000-8,000
6,000-7,500
12,000-16,000
6,500-8,000
9,000-12,000
6,500-8,500
1,000-8,000
7,500-8,500
–
50-100c
100-500
c
–
100–500
c
1,000-5,000
4,000-6,000
2,000
7,200
7,000
14,000
7,500
10,000
7,500
2,800
8,000
–
60
300
–
300
3,000
5,000
After complete combustion
As discarded basis
Energy content is from coatings, labels and attached materials.
a
b
c
Note : Btu/lb × 2.326 = kJ/kg
Energy content
ECpp > ECf > ECg
27. To design an optimum municipal solid waste
collection route, which of the following is/are NOT
desired :
(a) Collection vehicle should not travel twice down
the same street in a day.
(b) Waste collection on congested roads should
not occur during rush hours in morning or
evening.
(c) Collection should occur in the uphill direction.
(d) The last collection point on a route should be
as close as possible to the waste disposal
facility.
Sol: (c)
Some heuristic guidelines that should be taken
into consideration when laying out routes are as
follows :
1. Existing policies and regulations related to
such items as the point of collection and
frequency of collection must be identified.
2. Existing system characteristics such as crew
size and vehicle types must be coordinated.
3. Wherever possible, routes should be laid out
so that they begin and end near arterial
streets, using topographical and physical
barriers as route boundaries.
4. In hilly area, routes should start at the top of
the grade and proceed downhill as the vehicle
becomes loaded.
14. Solution
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5. Routes should be laid out so that the last
container to be collected on the route is
located nearest to the disposal site.
6. Waste generated at traffic-congested locations
should be collected as early in the day as
possible.
7. Sources at which extremely large quantities
of waste are generated should be serviced
during the first part of the days.
8. Scattered pickup points (where small quantities
of solid waste are generated) that receive the
same collection frequency should, if possible,
be serviced during one trip or on the same
day.
28. For a traffic stream, v is the space mean speed,
k is the density, q is the flow, vf is the free flow
speed, and kj is the jam density. Assume that
the speed decreases linearly with density.
Which of the following relation(s) is/are correct ?
(a)
j 2
j
f
k
q = k k – k
v
(b)
2
f
f
j
v
q = v k – k
k
(c)
2
f
f
j
v
q = v v – v
k
(d)
j 2
j
f
k
q = k v – v
v
Sol: (b)
As per Greenshield’s
V =
f
j
K
V 1–
K
and q = K × V
=
f
j
K
K 1– V
K
q =
2
f
f
j
V
V K – K
K
29. The error in measuring the radius of a 5 cm circular
rod was 0.2%. If the cross-sectional area of the
rod was calculated using this measurement, then
the resulting absolute percentage error in the
computed area is______.
(round off to two decimal places)
Sol: (0.40)
r = 5 cm
er =
0.2
×5cm = 0.01cm
100
A = 2
r
eA = r
2 r.e
Absolute perecentage error in computed area
= A
e
×100
A
=
r
2
2 r.e
×100
r
=
r
e
2× ×100
r
= 2×0.2% = 0.4%
30. The components of pure shear strain in a sheared
material are given in the matrix form :
1 1
=
1 –1
Here, Trace() = 0. Given, P= Trace (8 ) and Q
= Trace (
11 ).
The numerical value of (P+Q) is ________. (in
integer)
Sol: (32)
A – I = 0
1– 1
1 –1–
= 0
1– 1– –1 = 0
2
– 1 – 1 = 0
= 2
Eigen values of are 2 and – 2
Eigen values 8
are
8
2 and
8
–2
Eigen values of 11
are
11
2 and
11
– 2 .
P = Trace 8
( ) = Sum of Eigen values
=
8 8
2 + – 2
= 16 + 16 = 32
Q = Trace 11
( ) = Sum of Eigen values
=
11 11
2 + – 2
= 0
P + Q = 32 + 0
= 32
31. The inside diameter of a sampler tube is 50 mm.
The inside diameter of the cutting edge is kept
15.
16. Solution
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such that the Inside Clearance Ratio (ICR) is 1.0
% to minimize the friction on the sample as the
sampler tube enters into the soil.
The inside diameter (in mm) of the cutting edge
is _________________. (round off to two decimal
places)
Sol: (25)
D4
D3
D1
D2
Sampling tube
Cutting edge
D3 = 50 mm, ICR = 1%
ICR =
3 1
1
D – D
×100
D
1 =
1
1
50 – D
D
D1 = 25 mm
32. A concentrically loaded isolated square footing of
size 2 m × 2 m carries a concentrated vertical
load of 1000 kN. Considering Boussinesq’s theory
of stress distribution, the maximum depth (in m)
of the pressure bulb corresponding to 10 % of the
vertical load intensity will be ___________. (round
off to two decimal places)
Sol: (4.37)
Considering Boussinesq’s theory of stress
distribution
x =
5/2
2 2
3Q 1
2 z r
1+
z
For r = 0,
z 2
3Q
=
2 z
z = 2 2
Q 0.1
0.1q = 0.1× = Q
B 2
1
Q
40
=
2
3Q
2 z
z2 =
3× 40
z = 4.37m
2
33. In a triaxial unconsolidated undrained (UU) test
on a saturated clay sample, the cell pressure
was 100 kPa. If the deviatoric stress at failure
was 150 kPa, then the undrained shear strength
of the soil is _________ kPa. (in integer)
Sol: (75)
3 = c =100kPa
d =
1 3
– = 150Kpa
For UU test
Cuu
3 1
1 3
–
Undrained shear strength
=
1 3
uu f
–
C = =
2
=
150
2
= 75 KPa
34. A flood control structure having an expected life
of n years is designed by considering a flood of
return period T years. When T = n, and
n ,
the structure’s hydrologic risk of failure in
percentage is ____.
(round off to one decimal place)
Sol: (0.6)
Risk of failure = n
1– q
= n
1– 1 – p
=
n
1
1– 1–
T
For T =
n
Risk of failure =
1
1– = 0.632
e
35. The base length of the runway at the mean sea
level (MSL) is 1500 m. If the runway is located at
an altitude of 300 m above the MSL, the actual
length (in m) of the runway to be provided is
____________. (round off to the nearest integer)
Sol: (1604 to 1606)
17. Solution
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Correction for elevation = It should increase at a
rate of 7% per 300 m rise in elevation from MSL.
Given that,
Basic runway length at MSL = 1500 m
Elevation = 300 m
Correction =
7 300
× ×1500
100× 300
= 105 m
The actual length of runway
= 1500 + 105 = 1605 m
36. Consider the polynomial f(x) = x3 – 6x2 + 11x –
6 on the domain S given by
1 x 3 . The first
and second derivatives are
f (x) and f (x).
Consider the following statements:
I. The given polynomial is zero at the boundary
points x = 1 and x = 3.
II. There exists one local maxima of f(x) within
the domain S.
III. The second derivative
f (x) 0 throughout the
domain S.
IV. There exists one local minima of f(x) within
the domain S.
The correct option is:
(a) Only statements I, II and III are correct
(b) Only statements I, II and IV are correct
(c) Only statements I and IV are correct
(d) Only statements II and IV are correct
Sol: (b)
f x = 3 2
x – 6x +11x – 6
f(1) = 1– 6 +11– 6 = 0
f(3) = 3 2
3 – 6×3 +11×3 – 6 = 0
Statement (I) is correct.
f x = 2
3x – 12x +11
f x =
1
0 x = 2
3
Point of maxima
Point of minima
+ +
–
1 2 3
1
2 –
3
1
2+
3
–
f(x) has local maxima at
1
x = 2 –
3
Statement (II) is also true.
Now,
f x = 6x – 12
f x > 0
6x – 12 > 0
x > 2
Statement (III) is incorrect statement (IV) is also
correct.
x =
1
2 +
3
is point of minima.
37. An undamped spring-mass system with mass m
and spring stiffness k is shown in the figure. The
natural frequency and natural period of this system
are rad/s and T s, respectively. If the stiffness
of the spring is doubled and the mass is halved,
then the natural frequency and the natural period
of the modified system, respectively are
m
k
(a)
2 rad/s and T/2 s
(b) /2 rad/s and 2T s
(c)
4 rad/s and T/4 s
(d) rad/s and T s
Sol: (a)
m
k
This is question of undamped free vibration of
sinle degree of freedom system.
For the above figure it is given that natural
frequency and natural period are w rad/s and T s,
respectively.
The equation for soof system is
=
k
m
and T =
2
18. Solution
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For the modified system, stiffness of the spring is
doubled and the mass is halved.
Let us assume k' = 2k and m' =
m
,
2
here k' and
m' represent the stiffness and mass of the
modified system respectively.
=
k 2k 4k
= =
m
m m
2
=
k k
2 = 2 =
m m
and T' =
2 2 1 2 T
= = =
2 2 2
2
T =
and T are natural frequency and natural period
of the modified system.
Hence, the natural frequency of the modified
system is doubled and natural period is halved
compared to the original given system. Hence
option (a) is correct.
38. For the square steel beam cross-section shown
in the figure, the shape factor about Z–Z axis is
S and the plastic moment capacity is MP. Consider
yield stress fy = 250 MPa and a = 100 mm.
K
a a
a
a
z z
J
H
I
The values of S and MP (rounded-off to one decimal
place) are
(a) S = 2.0, MP = 58.9 kN-m
(b) S = 2.0, MP = 100.2 kN-m
(c) S = 1.5, MP = 58.9 kN-m
(d) S = 1.5, MP = 100.2 kN-m
Sol: (a)
K
a a
a
a
z z
J
H
I
Shape factor for diamond shaped section = 2
S =
y P
P
y y e
f Z
M
=
M f Z
or, MP = S.My = S.fy.Ze
K
a a
a
a
z z
J
H
I
2a
2a
2
IZZ =
4
a
12
Zzz =
4 3
3
a × 2 2a
= mm
12×a 12
So, MP =
3
–6
2× 100
2×250× ×10 kNm
12
= 58.93 kNm
39. A post-tensioned concrete member of span 15 m
and cross-section of 450 mm × 450 mm is
prestressed with three steel tendons, each of
cross-sectional area 200 mm2. The tendons are
tensioned one after another to a stress of 1500
MPa. All the tendons are straight and located at
125 mm from the bottom of the member. Assume
the prestress to be the same in all tendons and
the modular ratio to be 6. The average loss of
prestress, due to elastic deformation of concrete,
considering all three tendons is
(a) 14.16 MPa (b) 7.08 MPa
(c) 28.32 MPa (d) 42.48 MPa
Sol: (a)
This is a question of calculation of elastic
shortening loss in post tensioned member.
Given data,
length = 15 m
b × D = 450 mm × 450 mm
Numberof tendons = 3
Cross-section area of each tendon (As)
= 200 mm2.
Prestress = 1500 MPa
Modular ratio (m) = 6
19.
20. Solution
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15 m
(L-section) (Cross-section)
125 mm
e
C
L
450 mm
450
mm
From the given data, eccentricity (e) =
450
– 125 = 100 mm
2
Force in each cable (P) = 1500 × 200 × 10–3 =
300 kN
The tendons are tensioned one after another, and
hence when tendon (1) is pulled no loss in tenson
(1)
When tendon (2) is pulled, loss in tendon (1) but
no loss in tendon (2).
When tendon (3) is pulled, loss in tendon (1) and
(2), but no loss in tendon (3).
Hence, there will be 2 times losses in tendon (1),
time loss in tendon (1) and no loss in tendon (3).
While calculating elastic shortening loss, self
weight of the structure is neglected to be on the
conservative side.
Consider tensioning of tendon-1
No loss in tendon (1)
Consider tensioning of tendon-2
e
C
L
P
P
Stress in concrete at the level of prestressing
tendon
fc =
P Pe
+
A I
e =
3 3
3
300×10 300×10 ×100×100
+
450× 450 450× 450
12
= 2.36 MPa
e
P
C
L
I = moment of inertial of the section about the
centroidal axis.
As the tendons are horizontal and at the same
level fc,avg = fc.
Loss due to elastic deformation = mfc
= 6 × 2.36 = 14.16 MPa.
Considering tensioning of tendon 3
Loss due to elastic deformation in (1)
= mfc = 6 × 2.36 = 14.16 MPa
Loss due to elastic deformation in (2)
= mfc = 6 × 2.36 = 14.16 MPa
Total loss in tendon (1) = 2 × 14.16
= 28.32 MPa
Total loss in tendon (1)
= 2 × 14.16 = 28.32 MPa
In tendon (2) = 14.16 MPa
In tendon (3) = 0
Average loss of pre-stress, considering all
three tendons is
=
28.32 +14.16 + 0
3
= 14.16 MPa
40. Match the following in Column X with Column Y:
Column X
(P) In a triaxial compression test, with increase
of axial strain in loose sand under drained
shear condition, the volumetric strain
(Q) In a triaxial compression test, with increase
of axial strain in loose sand under undrained
shear condition, the excess pore water
pressure
(R) In a triaxial compression test, the pore
pressure parameter “B” for a saturated soil
(S) For shallow strip footing in pure saturated
clay, Terzaghi’s bearing capacity factor (Nq)
due to surcharge
Column Y
(I) Decreases
(II) Increases
(III) Remains same
(IV) is always 0.0
(V) is always 1.0
(VI) is always 0.5
21. Solution
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Which one of the following combinations is
correct?
(a) (P)-(I), (Q)-(II), (R)-(V), (S)-(V)
(b) (P)-(II), (Q)-(I), (R)-(IV), (S)-(V)
(c) (P)-(I), (Q)-(III), (R)-(VI), (S)-(IV)
(d) (P)-(I), (Q)-(II), (R)-(V), (S)-(VI)
Sol: (a)
(P) In a triaxial compression test, with increase
of axial strain in loose sand under drained
shear condition, the volumetric strain
decreases. As the sample is compressing.
Hence, volume is decreasing.
Volume
change
+
–
Dense sand
Loose sand
(Q) In a triaxial compression test, with increase
of axial strain in loose sand under undrained
shear condition, the excess pore water
pressure increases.
(R) In a triaxial compression test, the pore pressure
parameter “B” for a saturated soil is 1.
(S) For shallow strip footing in pure saturated
clay, Terzaghi’s bearing capacity factor (Nq)
due to surcharge is 1.
41. A soil sample is underlying a water column of
height h1, as shown in the figure. The vertical
effective stresses at points A, B and C are
A B C
, and , respectively. Let
sat and be
the saturated and submerged unit weights of the
soil sample, respectively, and w be the unit
weight of water. Which one of the following
expressions correctly represents the sum
A B C
( + ) ?
h3
h2
h1
Water level
Closed valve
A
B
C
Saturated soil
Water
(a)
2 3
(2h h ) (b)
1 2 3
(h h h )
(c)
2 3 sat w
(h h )( ) (d)
1 2 3 sat
(h h h )
Sol: (a)
h3
h2
h1
Water level
Closed valve
A
B
C
Saturated soil
Water
A = 0
B =
2
h
C =
2 3
h +h
A B c
+ + =
2 3
2h +h
42. A 100 mg of HNO3 (strong acid) is added to
water, bringing the final volume to 1.0 liter.
Consider the atomic weights of H, N, and O, as
1 g/mol, 14 g/mol, and 16 g/mol, respectively.
The final pH of this water is (Ignore the dissociation
of water.)
(a) 2.8 (b) 6.5
(c) 3.8 (d) 8.5
Sol: (a)
+ –
3 3
HNO H +NO
1 mole of HNO3 = 1 mole of H+
No. of moles of HNO3 =
100
m mol
1+14 +16×3
= 1.587 m mol
[H+] in 1 lit. of water= 1.587 m mol Lit
= –3
1.587×10 mol lit
pH =
+
–log H
=
–3
–log 1.587×10
= 2.8
43. In a city, the chemical formula of biodegradable
fraction of municipal solid waste (MSW) is
C100H250O80N. The waste has to be treated by
forced-aeration composting process for which air
requirement has to be estimated.
22. Solution
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Assume oxygen in air (by weight) = 23%, and
density of air = 1.3 kg/m3.
Atomic mass: C = 12, H = 1, O = 16, N = 14.
C and H are oxidized completely whereas N is
converted only into NH3 during oxidation.
For oxidative degradation of 1 tonne of the waste,
the required theoretical volume of air (in m3/tonne)
will be (round off to the nearest integer)
(a) 4749 (b) 8025
(c) 1418 (d) 1092
Sol: (a)
100 250 80 2 2 2 3
247
C H O N+xO 100CO + H O+NH
2
80 + x × 2 =
247
200 +
2
x = 121.75
1 mole of MSW required 121.75 mole of oxygen
No. of moles of MSW
=
1 ton
100×12 + 250 + 80×16 +14 g
No. of moles of O2 required =
1 tonne
121.75×
2744g
wt. of O2 =
121.74
×32tonne
2744
Mass of air =
3
1.42
×10 Kg
0.23
= 3
6.174×10 kg
Density of air = 1.3 Kg/m3
Volume of air =
3 3
6.174
×10 m
1.3
= 4749.2 m3 per tonne of MSW
44. A single-lane highway has a traffic density of 40
vehicles/km. If the time-mean speed and space-
mean speed are 40 kmph and 30 kmph,
respectively, the average headway (in seconds)
between the vehicles is
(a) 3.00 (b) 2.25
(c) 8.33 × 10–4 (d) 6.25 × 10–4
Sol: (a)
Given that,
Traffic density = 40 Veh/Km.
Time mean speed = 40 Kmph
Space mean speed = 30 kmph
Time Headway (sec) = ?
We know that traffic volume
= Density × Space mean speed
= 40×30 =1200 Veh hr.
and we also know that
q =
t
3600
H
where Ht - average headway (sc)
1200 =
t
3600
H
Ht =
3600
= 3 secs.
1200
45. Let y be a non-zero vector of size 2022 × 1.
Which of the following statement(s) is/are TRUE?
(a) yyT is a symmetric matrix
(b) yTy is an eigenvalue of yyT
(c) yyT has a rank of 2022
(d) yyT is invertible
Sol: (a,b)
Let vector y =
3 1
4
4
4
yT =
1 3
4 4 4
yyT =
4
4 4 4 4
4
=
3 3
16 16 16
16 16 16
16 16 16
yTy = [42 + 42 + 42]1 × 1
= [48]1 × 1
(y) =
T T T
(y ) (yy ) (y y) 1
From above information
yyT is asymmetric.
yTy is an eigen value of yyT.
yyT has rank 1
det (yyT) = 0 so, yyT is not invertible
23. Solution
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46. Which of the following statement(s) is/are correct?
(a) If a linearly elastic structure is subjected to
a set of loads, the partial derivative of the
total strain energy with respect to the
deflection at any point is equal to the load
applied at that point.
(b) If a linearly elastic structure is subjected to
a set of loads, the partial derivative of the
total strain energy with respect to the load at
any point is equal to the deflection at that
point.
(c) If a structure is acted upon by two force
system Pa and Pb , in equilibrium separately,
the external virtual work done by a system of
forces Pb during the deformations caused by
another system of forces Pa is equal to the
external virtual work done by the Pa system
during the deformation caused by the Pb
system.
(d) The shear force in a conjugate beam loaded
by the M/EI diagram of the real beam is equal
to the corresponding deflection of the real
beam.
Sol: (a,b,c)
47. Water is flowing in a horizontal, frictionless,
rectangular channel. A smooth hump is built on
the channel floor at a section and its height is
gradually increased to reach choked condition in
the channel. The depth of water at this section is
y2 and that at its upstream section is y1. The
correct statement(s) for the choked and unchoked
conditions in the channel is/are
(a) In choked condition, y1 decreases if the flow
is supercritical and increases if the flow is
subcritical.
(b) In choked condition, y2 is equal to the critical
depth if the flow is supercritical or subcritical.
(c) In unchoked condition, y1 remains unaffected
when the flow is supercritical or subcritical.
(d) In choked condition, y1 increases if the flow
is supercritical and decreases if the flow is
subcritical.
Sol: (a,b,c)
E1 y1
1
1
y2
E2
Z
Hump
Energy line
2
2
Horizontal, frictionless, rectangular channel.
Depth
y
y1
y2
yc
C
R P
Z
E2
E1
Zm
Specific
energy
R
P
Q = const.
1
2
Case-I: Subcritical flow at section 1-1:
For a subcritical flow, a decrease in specific
energy is associated with a decrease in depth
of flow and increase in velocity.
When max
0 Z Z , upstream water level
remains stationary at y1 while the depth of
flow at section 2 decreases with y2 reaching
a minimum value of yc at max
Z = Z (choking
condition).
With further increase in the value of Z (i.e.,
max
Z Z ), y2 will continue to remain at yc
and y1 will increase to
1
y , to have higher
specific energy
1
E .
Zmax
Z
y and y
1 2
Depth y1
Depth y2
yc
Case-II: Supercritical flow at section 1-1:
For a supercritical flow, a decrease in specific
energy is associated with an increase in the
depth of flow and decrease in velocity head.
24. Solution
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When max
0 Z Z depth of flow y1 at
section 1, is constant while the depth of flow
y2 at section 2 increase upto critical depth yc
at max
Z = Z (choking condition).
For max
Z Z , the depth of flow over hump
y2 = yc will remain constant and the upstream
depth y1 decreases to
1
y , to have higher
specific energy
1
E .
Z
yc
Depth y2
Depth y1
y and y
1 2
Zmax
48. The concentration s(x, t) of pollutants in a one-
dimensional reservoir at position x and time t
satisfies the diffusion equation
2
2
s(x, t) s(x, t)
D
t x
on the domain
0 x L, , where D is the diffusion
coefficient of the pollutants. The initial condition
s(x, 0) is defined by the step-function shown in
the figure.
x
L
0.4L
0
0
s0
s(x, t = 0)
The boundary conditions of the problem are given
by
s(x, t)
0
x
at the boundary points x = 0 and
x = L at all times. Consider D = 0.1 m2/s, s0 =
5 mol/m, and L = 10 m.
The steady-state concentration
L L
s s ,
2 2
at the center x =
L
2
of the reservoir (in mol/m)
is _______ (in integer)
Sol: (*)
49. A pair of six-faced dice is rolled thrice. The
probability that the sum of the outcomes in each
roll equals 4 in exactly two of the three attempts
is ______. (round off to three decimal places)
Sol: (0.02)
Event, E = {(1, 3)(3, 1)(2, 2)}
n(E) = 3
n(S) = 36
p = P(E) =
3 1
36 12
q =
1 11
P(E) 1
12 12
P(x) = 3C2(p2)(q1)
=
2
1 11
3
12 12
= 0.01909 = 0.02
50. Consider two linearly elastic rods HI and IJ, each
of length b, as shown in the figure. The rods are
co-linear, and confined between two fixed supports
at H and J. Both the rods are initially stress free.
The coefficient of linear thermal expansion is
for both the rods. The temperature of the rod IJ is
raised by T , whereas the temperature of rod HI
remains unchanged. An external horizontal force
P is now applied at node I. It is given that
6 1
10 C , T 50 C, b = 2m, AE = 106 N.
The axial rigidities of the rods HI and IJ are 2AE
and AE, respectively.
H
b
P
b
AE
J
2AE I
To make the axial force in rod HI equal to zero,
the value of the external force P (in N) is
____________ (round off to the nearest integer)
Sol: (50)
b
b
AE
J
H 2AE
P
I
For axial force to be zero in rod HI, axial
deformation will be zero i.e., force P balances
the expansion due to temperature in rod IJ.
25. Solution
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For rod IJ
P
P
net =
axial load temp
+ = 0
P
– T
AE
= 0
P =
AE T
P = 106(10–6 × 50)
P = 50N
51. The linearly elastic planar structure shown in the
figure is acted upon by two vertical concentrated
forces. The horizontal beams UV and WX are
connected with the help of the vertical linear spring
with spring constant k = 20 kN/m. The fixed
supports are provided at U and X. It is given that
flexural rigidity EI = 105 kN-m2, P = 100 kN, and
a = 5 m. Force Q is applied at the center of
beam WX such that the force in the spring VW
becomes zero.
W 2EI X
a
a
Q
k
V
U 2a
EI
P
The magnitude of force Q (in kN) is _______
(round off to the nearest integer)
Sol: (640)
U
2a
EI
P
V
W
a a
2 EI
X
Q
If force in spring is zero, there will be no
deformation in spring i.e., deflection of point V
will be equal to deflection of point W
2a
EI
P
V
2 EI
W
a a
Q
v = w
3
P 2a
3EI
=
3 2
Q a Qa
+ + a
3 2EI 2 2EI
8
P
3
=
1 1
+ Q
6 4
Q =
32
P = 640kN
5
52. A uniform rod KJ of weight w shown in the figure
rests against a frictionless vertical wall at the
point K and a rough horizontal surface at point J.
It is given that w = 10 kN, a = 4 m and b = 3 m.
b
J
K
a
The minimum coefficient of static friction that is
required at the point J to hold the rod in equilibrium
is _______ (round off to three decimal places)
Sol: (0.375)
K
J
3m
4m
FBD
4 m
J
RJ
s J
R
10 kN
Rk
K 1.5 m 1.5 m
x
y
v
F = 0
J
R = 10kN
K
M = 0
26.
27. Solution
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s J J
10×1.5 + R × 4 – R ×3 = 0
s =
10×3 – 10×1.5
= 0.375
10× 4
53. The activities of a project are given in the following
table along with their durations and dependency.
Activities Duration (days) Depends on
A 10 –
B 12 –
C 5 A
D 14 B
E 10 B, C
The total float of the activity E (in days) is
_________ (in integer).
Sol: (1)
1
2 4
3 5
A C
E
D
B
10 5
10
14
(0, 0)
(12, 12)
(10, 11) (15, 16)
(26, 26)
For activity E,
Total float (FT) =
5 4 E
L L
T – T – t
= 26 – 15 – 10 =1day
54. A group of total 16 piles are arranged in a square
grid format. The center-to-center spacing (s)
between adjacent piles is 3 m. The diameter (d)
and length of embedment of each pile are 1 m
and 20 m, respectively. The design capacity of
each pile is 1000 kN in the vertical downward
direction. The pile group efficiency g
( ) is given
by
g
(n 1) m (m 1) n
1–
90 mn
where m and n are number of rows and columns
in the plan grid of pile arrangement, and
1 d
tan .
s
The design value of the pile group capacity (in
kN) in the vertical downward direction is
__________ (round off to the nearest integer)
Sol: (11088)
S=3m
n = 4, m = 4
S = 3m, d = 1 m
L = 20 m
=
–1 –1
d 1
tan = tan =18.435
s 3
ng =
n – 1 m+ m – 1 n
1–
90 mn
=
18.435 4 –1 ×4 + 4 –1 ×4
1–
90 4×4
= 0.693
Design value of pile group capacity
= 16×1000×0.693
= 11088 KN
55. A saturated compressible clay layer of thickness
h is sandwiched between two sand layers, as
shown in the figure. Initially, the total vertical stress
and pore water pressure at point P, which is
located at the mid-depth of the clay layer, were
150 kPa and 25 kPa, respectively. Construction
of a building caused an additional total vertical
stress of 100 kPa at P. When the vertical effective
stress at P is 175 kPa, the percentage of
consolidation in the clay layer at P is __________.
(in integer)
Sand
Sand
h/2
h/2
P clay
Sol: (50)
28. Solution
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Initial effective stress at P
1
P = 150 – 25
= 125 kPa
After construction,
2
P = 125 + 100 = 225 kPa
100% consolidation will occur when effective stress
at P reaches 225 kPa.
Percentage of consolidation when vertical effective
stress is at P is 175 kPa.
=
175 –125
×100
225 – 125
=
50
×100
100
= 50
56. A hydraulic jump takes place in a 6 m wide
rectangular channel at a point where the upstream
depth is 0.5 m (just before the jump). If the
discharge in the channel is 30 m3/s and the
energy loss in the jump is 1.6 m, then the Froude
number computed at the end of the jump is
___________. (round off to two decimal places)
(Consider the acceleration due to gravity as 10
m/s2).
Sol: (0.3 to 0.4)
Given, in a rectangular channel
y
B
y1
y2
Q = 30 m3/sec,
B = 6 m
y1 = 0.5 m
EL = 1.6 m
q =
3
Q 30
= = 5 m / sec/ m
B 6
Method-I
We know that,
EL =
3
2 1
1 2
y – y
= 1.6m
4 y y
...(i)
3
2
2
y – 0.5
4× y ×0.5
= 1.6
3 2
2 2 2
y – 1.5y + 0.75y – 0.125 = 3.2y2
y2 = 2.5 m, – 0.0527 m, – 0.947 m
Hence, y2 = 2.5 m
Post jump Froude’s No. (F2) =
2
2
V
gy
=
30
6× 2.5
= 0.4
10×2.5
This problem can also be solved by using other
given data:
Method-II:
For rectangular horizontal channel,
2
2q
g
=
1 2 1 2
y + y × y y
2
2× 5
10
=
2 2
0.5 + y ×0.5× y
2
2 2
0.5y + 0.25y – 5 = 0
y2 = 2.922 m, – 3.422 m
Adopt, y2 = 2.922 m
Post jump Froude’s No. (F2) =
2
2
V
gy
=
30
2.922× 6
10× 2.922
= 0.316
Note: From Method-II result,
EL =
3 3
2 1
1 2
y – y 2.922 – 0.5
=
4y y 4×2.922×0.5
= 2.431 m
Thus, it is clearly a case of discrepancy in
given data in the question.
29. Solution
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57. A pump with an efficiency of 80% is used to draw
groundwater from a well for irrigating a flat field of
area 108 hectares. The base period and delta for
paddy crop on this field are 120 days and 144
cm, respectively. Water application efficiency in
the field is 80%. The lowest level of water in the
well is 10 m below the ground. The minimum
required horse power (h.p.) of the pump is
________. (round off to two decimal places)
(Consider 1 h.p. = 746 W; unit weight of water =
9810 N/m3)
Sol: (30.82)
Base period (B) = 120 days
Delta
= 144 cm = 1.44 m
Duty (D) =
8.64B 8.64×120
=
1.44
D = 720
hec
cumec
Q =
3
Area 108 3
= = m /s
D 720 20
Qapplied =
a
Q 3 3
= =
20×0.8 16
Power required =
mgh
t
=
w
volume
× g×h
t
=
wQgh = Qh
=
3 73575
9810× ×10 = watt
16 4
Horse power of pump =
Power required
746×efficiency
=
73575
= 30.82 hp
4×746×0.8
58. Two discrete spherical particles (P and Q) of equal
mass density are independently released in water.
Particle P and particle Q have diameters of 0.5
mm and 1.0 mm, respectively. Assume Stokes’
law is valid.
The drag force on particle Q will be________ times
the drag force on particle P. (round off to the
nearest integer)
Sol: (8 to 8)
In case of discrete particle settling and Stoke’s
law valid, at terminal velocity, since there is no
change in velocity, the net force on the body
is zero. Hence,
weight of sphere
–buoyant force
= Drag force (FD)
Vterminal
drag
force (F )
D
s
FB
Buoyant force
Weight
fluid f
= ( )
FD =
3 3
s f
D g – D g
6 6
3
D s f
F = gD –
6
For density of medium
f and mass density
of sphere
s constant,
Drag force (FD) D3
For particle P, diameter (DP) = 0.5 mm
For particle Q, diameter (DQ) = 1 mm
D Q
D P
F
F =
3 3
Q
3
P
D 1
= = 8
0.5
D
(FD)Q = 8 × (FD)P
59. At a municipal waste handling facility, 30 metric
ton mixture of food waste, yard waste, and paper
waste was available. The moisture content of this
mixture was found to be 10%. The ideal moisture
content for composting this mixture is 50%. The
amount of water to be added to this mixture to
bring its moisture content to the ideal condition is
_______metric ton. (in integer)
Sol: (12)
MSW = 30 metric ton
Moisture content of MSW = 10%
Amount of water present
30. Solution
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=
10
30× = 3metric ton
100
Ideal moisture content of MSW = 50%
Amount of water should be present for composting
=
50
30× =15 metric ton
100
Amount of water to be added to mixture to bring
its M/C to ideal condition
= 15 – 3 =12 metric ton
60. A sewage treatment plant receives sewage at a
flow rate of 5000 m3/day. The total suspended
solids (TSS) concentration in the sewage at the
inlet of primary clarifier is 200 mg/L. After the
primary treatment, the TSS concentration in
sewage is reduced by 60%. The sludge from the
primary clarifier contains 2% solids concentration.
Subsequently, the sludge is subjected to gravity
thickening process to achieve a solids
concentration of 6%. Assume that the density of
sludge, before and after thickening, is 1000 kg/
m3.
The daily volume of the thickened sludge (in m3/
day) will be_________. (round off to the nearest
integer)
Sol: (10)
PST
Q=5000 m /d
3
TSS = 200 mg/L
40% TSS
Gravity
Thickening
60% TSS
2% solids
wt. of TSS at inlet of PST
=
3 –6
L Kg
5000×10 × 200×10
d L
= 1000 Kg/d
wt. of solids in sludge from PST = 0.6×1000
= 600 Kg/d
wt. of sludge before thickening =
600
Kg d
2
100
= 30000 Kg/d
wt. of sludge after thickening =
600
Kg d
6
100
= 10000 Kg/d
Daily volume of thickened sludge
=
2
10000
m
1000
= 10 m3
61. A sample of air analyzed at 25°C and 1 atm
pressure is reported to contain 0.04 ppm of SO2.
Atomic mass of S = 32, O = 16.
The equivalent SO2 concentration (in g/m3) will
be__________. (round off to the nearest integer)
Sol: (157)
Concentration of SO2 in ppm = 0.04
Let’s equivalent concentration in
3
g m is x.
x g of SO2 present in 1 m3 of air.
.
We know, 1 mole of SO2 (at 0°C, 1 atm) has
volume of 22.4 L.
at 25° C and 1 atm, volume of SO2
=
22.4
× 273 + 25
273 + 0
= 24.45 lit
–6
x×10
96
mole of SO2 has volume
=
–6
24.45× x×10
lit.
96
in 1m3 of air
–3 3
0.255x×10 m of SO2 present in 6 3
10 m of
air.
–3
0.255x ×10 = 0.04
x =
156.86 157
0.04 ppm of SO2 =
3
2
157 g m of SO
62. A parabolic vertical crest curve connects two road
segments with grades +1.0% and –2.0%. If a 200
m stopping sight distance is needed for a driver
at a height of 1.2 m to avoid an obstacle of height
0.15 m, then the minimum curve length should be
______ m. (round off to the nearest integer)
Sol: (271 to 274)
Given that, 1
n = +1% and 2
n = –2%
n = 1 2
n – n
31. Solution
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=
1– –2 = 3%
SSD = 200 m
and h1 = 1.2 m and h2 = 0.15 m
as given N1 up gradient, and n2 - down gradient.
So curve is summit curve.
Assume L > SSD
L =
2
2
1 2
NS
2 h + h
=
2
2
3 200
×
100 2× 1.2 + 0.15
L = 272.91 m > 200 m
L = 272.91 m
63. Assuming that traffic on a highway obeys the
Greenshields model, the speed of a shockwave
between two traffic streams (P) and (Q) as shown
in the schematic is _______ kmph. (in integer)
Direction of traffic
(P)Flow = 1200
vehicles/hour
Speed = 60
kmph
(Q)Flow = 1800
vehicles/hour
Speed = 30 kmph
Sol: (14.5 to 15.5)
Speed of shock wave =
Change in flow
Change in density
Flow = Speed × Density
Density =
Flow
Speed
=
Q P
Q P
Q P
q – q
q q
–
V V
=
1800 –1200
1800 1200
–
30 60
=
600 600
= =15 Kmph
60 – 20 40
64. It is given that an aggregate mix has 260 grams
of coarse aggregates and 240 grams of fine
aggregates. The specific gravities of the coarse
and fine aggregates are 2.6 and 2.4, respectively.
The bulk specific gravity of the mix is 2.3.
The percentage air voids in the mix is
____________. (round off to the nearest integer)
Sol: (7.5 to 8.5)
Given that,
Coarse aggregate = 260 gms
Fine aggregate = 240 gms
GCA = 2.6
GFA = 2.4
Bulk specific gravity (Gm) = 2.3
Percentage air voids in the mix = ?
Gt (Theoretical specific gravity)
=
W
W
G
=
260 + 240
= 2.5
260 240
+
2.6 2.4
% air voids (VV) =
t m
t
G – G
×100
G
=
2.5 – 2.3
×100 = 8%
2.5
VV = 8%
65. The lane configuration with lane volumes in
vehicles per hour of a four-arm signalized
intersection is shown in the figure. There are only
two phases: the first phase is for the East-West
and the West-East through movements, and the
second phase is for the North-South and the
South-North through movements. There are no
turning movements. Assume that the saturation
flow is 1800 vehicles per hour per lane for each
lane and the total lost time for the first and the
second phases together is 9 seconds.
440
460
N
504
504
360
396
The optimum cycle length (in seconds), as per
the Webster’s method, is ____________. (round
off to the nearest integer)
Sol: (36 to 38)
From figure,
32. Solution
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N–S
y =
max
360 396
,
1800 1800
N–S
y =
396
= 0.22
1800
E–W
y =
max
504 + 504 440 460
, ,
2×18.0 1800 1800
= 0.28
Optimum Cycle length
=
1.5L + 5
1– y
=
N–S E–W
1.5×9 + 5
1– y + y
=
1.5×9 + 5
1– 0.22 + 0.28
C0 = 37 secs.