Key Pedagogical Features
and its angle a with the X-axis is given by
Chapter Highlights
List of important topics that are covered in chapter.
CHAPTER HIGHLIGHTS
☞ Introduction
☞ Newtonian mechanics
☞ Deformation of body
☞ Force
☞ Resolution of a force into a force and a couple
☞ Resultant of a system of coplanar forces
Distributed force system: Distributed forces (or loads) are those force that act over a length, area, or volume of a body. On the other hand, a concentrated force (point load) is a force which acts on a point.
☞ Resultant of multiple forces acting at a point
☞ Triangle law of forces
☞ Coplanar concurrent force system
☞ Coplanar non-concurrent, non-parallel force system
☞ Moment of a force
☞ Moment of a couple
☞ Equilibrium equations for different coplanar force systems
☞ Analysis of a system of forces in space
Example 1
Introduction
SOLVED EXAMPLES
In physics, the branch which deals with the study of state of rest or motion caused by the action of forces on the bodies
Engineering mechanics applies the principles and laws of mechanics to solve the problems of common engineering
The resultant of two concurrent forces ‘3P’ and ‘2P’ is R. If the first force is doubled, the resultant is also doubled. Determine the angle between the forces.
Solution
Exercises
Practice problems for students to master the concepts studied in chapter.
Newtonian mechanics or classical mechanics deals with the study of the motion of macroscopic objects under the action
It is the study of forces and conditions of equilibrium of bodies at rest subjected to the action
It is the branch of mechanics which deals with the study of motion of rigid bodies and the corelation with the forces causing and affecting their motion.
3. Kinematics: Kinematics deals with space, time relationship of a given motion of body and not at all with the forces that cause the motion.
Solved Examples
4. Kinetics: The study of the laws of motion of material bodies under the action of forces or kinetics is the study of the relationship between the forces and the resulting motion.
Some of the definitions of the idealizations used in engineering mechanics are as follows:
Solved problems are given topic-wise to learn and to apply the concepts learned in a particular section as per examination pattern.
Example 3
1. Continuum: It is defined as continuous nonspacial whole which has no empty spaces, and no part is distinct from the adjacent parts. Considering objects, in this way, ignores that the matter present in the object is made of atoms and molecules.
2. Particle: A particle is a body which has finite mass, but the dimensions can be neglected.
An electric fixture weighing 18 N hangs from a point C by two strings AC and BC as shown in the following figure. The string AC is inclined to the vertical wall at 40° and BC is inclined to the horizontal ceiling at 50°. Determine the forces in the strings.
3. System of particles: When a group of particles which are inter-related are dealt together for studying the behaviour, it is called a system of particles.
4. Rigid body: A solid body which does not undergo any deformations under the application of forces is
1. A weight of 1900 N is supported by two chains of lengths of 4 m and 3 m as shown in figure. Determine the tension in each chain.
(A) 1200 N, 1300 N (B) 1100 N, 100 N (C) 1100 N, 1200 N (D) 1520 N, 1140 N
2. Four forces of magnitudes 20 N, 40 N, 60 N and 80 N are acting respectively along the four sides of a square ABCD as shown in figure. Determine magnitude of resultant.
(A) 40 2N (B)
the graph is 3, the distance travelled by the body in 6 seconds would be (A) 40 m (B) 60 m (C) 78 m (D) 80 m
6. Match the following:
a. Two parallel forces acting on a body moving with uniform velocity 1. Collision
b. A moving particle 2. Forces in equilibrium
c. Two coplanar forces equal in magnitude but opposite in direction 3. Kinetic energy
d. Co-efficient of restitution 4. Couple
Codes: a b c d a b c d (A) 4 3 2 1 (B) 1 2 3 4 (C) 2 3 4 1 (D) None of these
7. Two forces form a couple only when (A) magnitude is same have parallel lines of action and same sense.
(B) magnitude is different, have parallel lines of action but same sense.
(C) magnitude is same have non parallel lines of action but same sense.
3. Match the following:
(D) magnitude is same and have parallel lines of action and opposite sense.
8. 45° 30° A P C
Previous Years’ Questions
1. Consider a truss PQR loaded at P with a force F as shown in the figure. The tension in the member QR is [GATE, 2008]
(A) 0.5 F (B) 0.63 F (C) 0.73 F (D) 0.87 F
2. For the truss shown in the figure, the forces F1 and F2 are 9 kN and 3 kN, respectively. The force (in kN) in the member QS is (All dimensions are in m) [GATE, 2014]
3. For the truss shown in the figure, the magnitude of the force in member PR and the support reaction at R are respectively [GATE, 2015]
and 50 kN
Previous Years’ Questions
Contains previous 10 years’ GATE Questions at the end of every chapter that help students to get an idea about the type of problems asked in GATE and prepare accordingly.
3 numbers
Hints/solutions
Exercises
1. If x = 2 and y = 3 , x + y – xy = 2 + 3 – 6
In this case, x + y – xy is irrational. If x = 2 and y = – 2 , x + y – xy = 2 + ( 2 ) – ( 2 ) (– 2 ) = 2
In this case, x + y – xy is rational.
\ We can only conclude that x + y – xy is real (Q Any real number is one which is either rational or irrational)
Hence, the correct option is (A).
2. Choice (A)
851 = 302 – 72 = (23) (37)
\ Choice (A) is not prime Choice (B)
589 = 252 – 62 = (19) (31)
\ Choice (B) is not prime. Choice (C) is divisible by 3. Choice (D) is prime. Hence, the correct option is (D).
Practice Tests
3. Twin primes are prime numbers, which differ by 2. In Choice (A), 133 is divisible by 7 and hence it is not a prime
Time-bound test provided at the end of each unit for assessment of topics leaned in the unit.
7. If the odd natural number is more than or equal to 3 its factorial’s parity would be even 1! = 1.\ 1! is the only odd number satisfying the given condition.
Hence, the correct option is (A).
8. For any perfect number, the sum of its factors is twice the number.
Hence, the correct option is (B).
9. Let x = 0. 2550 25 =
10 x = 25 (7)
100 x = 25 5 (8)
Subtracting (7) from (8)
x = 23 90
Hence, the correct option is (A).
10. Let x = 0. 321
10 x = 321 (9)
1000 x = 321 21 (10)
3.824 | Part III ■ Unit 9 ■ Water Resources Engineering Test
Water Resources Engineering
In Choice (B), the numbers are twin primes. In Choice (C), 159 is divisible by 3 and hence it is not prime.
Hence, the correct option is (D).
4. Choice (A)
Sum of the digits in the odd places= 32
Sum of the digits in the even places = 21 (Sum of the digits in the odd places) – (Sum of the digits in the even places) is divisible by 11.
\ Choice (A) is divisible by 11.
Choices (B) and (C) are not divisible by 11. Hence, the correct option is (A).
5. The number formed by the last three digits of a number must be divisible by 8 for the number to be divisible by 8. The least natural number which should be added to the number formed by the last 3 digits of the given number to make it divisible by 8 is 3.
Hence, the correct option is (A).
Hints/Solutions
This section gives complete solutions of all the unsolved questions given in the chapter. The Hints/ Solutions are included in the CD accompanying the book.
1. The region where air coming from the pole (cooler and denser) and the air of the middle cell (warmer and lighter) meet is called ________.
(A) cold front (B) warm front (C) polar front (D) occluded front
2. The intensity-duration-frequency curve from the following is (Where a < b < c) (A)
6. The product of any N consecutive natural numbers is divisible by N!, any for all values of N \ When N = 7, any such product is divisible by
7! 5040.
Subtracting (9) from (10), x = 318 53 = Hence, the correct option is (D).
The number of numbers less than N and coprime to it
Time: 60 Minutes
4. The infiltration capacity curves which are developed from infiltrometer tests or the hydrograph analysis methods are used to estimate ______ from a given storm.
(A) infiltration (B) rainfall
(C) run-off (D) All of these
5. ______ hydrograph is independent of rainfall duration. (A) Instantaneous unit hydrograph
(B) Synthetic unit hydrograph
(C) Direct run-off hydrograph (D) Unit hydrograph
6. When the seepage takes place from the stream into the ground, it is called ______ stream.
(A) perennial stream (B) influent stream (C) effluent stream (D) ephemeral stream
7. A structure with a useful life period of 100 years is designed for a 50-year flood. Then the risk in the design is given by _______.
(A) 0.68 (B) 0.71
(C) 0.87 (D) 0.99
8. The peak flow in outflow hydrographs in a channel routing occurs at ______.
(A) intersection point of inflow and outflow
CHapter
Syllabus: Civil Engineering
Section 1: Engineering Mathematics
Linear Algebra: Matrix algebra; Systems of linear equations; Eigen values and Eigen vectors.
Calculus: Functions of single variable; Limit, continuity and differentiability; Mean value theorems, local maxima and minima, Taylor and Maclaurin series; Evaluation of definite and indefinite integrals, application of definite integral to obtain area and volume; Partial derivatives; Total derivative; Gradient, Divergence and Curl, Vector identities, Directional derivatives, Line, Surface and Volume integrals, Stokes, Gauss and Green’s theorems.
Ordinary Differential Equation (ODE): First order (linear and non-linear) equations; higher order linear equations with constant coefficients; Euler-Cauchy equations; Laplace transform and its application in solving linear ODEs; initial and boundary value problems.
Partial Differential Equation (PDE): Fourier series; separation of variables; solutions of one-dimensional diffusion equation; first and second order one-dimensional wave equation and two-dimensional Laplace equation.
Probability and Statistics: Definitions of probability and sampling theorems; Conditional probability; Discrete Random variables: Poisson and Binomial distributions; Continuous random variables: normal and exponential distributions; Descriptive statistics - Mean, median, mode and standard deviation; Hypothesis testing.
Numerical Methods: Accuracy and precision; error analysis. Numerical solutions of linear and non-linear algebraic equations; Least square approximation, Newton’s and Lagrange polynomials, numerical differentiation, Integration by trapezoidal and Simpson’s rule, single and multi-step methods for first order differential equations.
Section 2: Structural Engineering
Engineering Mechanics: System of forces, free-body diagrams, equilibrium equations; Internal forces in structures; Friction and its applications; Kinematics of point mass and rigid body; Centre of mass; Euler’s equations of motion; Impulse-momentum; Energy methods; Principles of virtual work.
Solid Mechanics: Bending moment and shear force in statically determinate beams; Simple stress and strain relationships; Theories of failures; Simple bending theory, flexural and shear stresses, shear centre; Uniform torsion, buckling of column, combined and direct bending stresses.
Structural Analysis: Statically determinate and indeterminate structures by force/energy methods; Method of superposition; Analysis of trusses, arches, beams, cables and frames; Displacement methods: Slope deflection and moment distribution methods; Influence lines; Stiffness and flexibility methods of structural analysis.
Construction Materials and Management: Construction Materials: Structural steel-composition, material properties and behaviour; Concrete—constituents, mix design, short-term and long-term properties; Bricks and mortar; Timber; Bitumen. Construction Management: Types of construction projects; Tendering and construction contracts; Rate analysis and standard specifications; Cost estimation; Project planning and network analysis—PERT and CPM.
Concrete Structures: Working stress, Limit state and Ultimate load design concepts; Design of beams, slabs, columns; Bond and development length; Prestressed concrete; Analysis of beam sections at transfer and service loads.
Steel Structures: Working stress and Limit state design concepts; Design of tension and compression members, beams and beam-columns, column bases; Connections—simple and eccentric, beam-column connections, plate girders and trusses; Plastic analysis of beams and frames.
Section 3: Geotechnical Engineering
Soil Mechanics: Origin of soils, soil structure and fabric; Three-phase system and phase relationships, index properties; Unified and Indian standard soil classification system; Permeability—one dimensional flow, Darcy’s law; Seepage through soils—two-dimensional flow, flow nets, uplift pressure, piping; Principle of effective stress, capillarity, seepage force and quicksand condition; Compaction in laboratory and field conditions; One-dimensional consolidation, time rate of consolidation; Mohr’s circle, stress paths, effective and total shear strength parameters, characteristics of clays and sand.
Foundation Engineering: Sub-surface investigations—scope, drilling bore holes, sampling, plate load test, standard penetration and cone penetration tests; Earth pressure theories—Rankine and Coulomb; Stability of slopes—finite and infinite slopes, method of slices and Bishop’s method; Stress distribution in soils—Boussinesq’s and Westergaard’s theories, pressure bulbs; Shallow foundations—Terzaghi’s and Meyerhoff’s bearing capacity theories, effect of water table; Combined footing and raft foundation; Contact pressure; Settlement analysis in sands and clays; Deep foundations—types of piles, dynamic and static formulae, load capacity of piles in sands and clays, pile load test, negative skin friction.
Section 4: Water Resources Engineering
Fluid Mechanics: Properties of fluids, fluid statics; Continuity, momentum, energy and corresponding equations; Potential flow, applications of momentum and energy equations; Laminar and turbulent flow; Flow in pipes, pipe networks; Concept of boundary layer and its growth.
Hydraulics: Forces on immersed bodies; Flow measurement in channels and pipes; Dimensional analysis and hydraulic similitude; Kinematics of flow, velocity triangles; Basics of hydraulic machines, specific speed of pumps and turbines; Channel Hydraulics—Energy-depth relationships, specific energy, critical flow, slope profile, hydraulic jump, uniform flow and gradually varied flow.
Hydrology: Hydrologic cycle, precipitation, evaporation, evapo-transpiration, watershed, infiltration, unit hydrographs, hydrograph analysis, flood estimation and routing, reservoir capacity, reservoir and channel routing, surface run-off models, ground water hydrology - steady state well hydraulics and aquifers; Application of Darcy’s law.
Irrigation: Duty, delta, estimation of evapo-transpiration; Crop water requirements; Design of lined and unlined canals, head works, gravity dams and spillways; Design of weirs on permeable foundation; Types of irrigation systems, irrigation methods; Water logging and drainage; Canal regulatory works, cross-drainage structures, outlets and escapes.
Section 5: Environmental Engineering
Water and Waste Water: Quality standards, basic unit processes and operations for water treatment. Drinking water standards, water requirements, basic unit operations and unit processes for surface water treatment, distribution of water. Sewage and sewerage treatment, quantity and characteristics of wastewater. Primary, secondary and tertiary treatment of wastewater, effluent discharge standards. Domestic wastewater treatment, quantity of characteristics of domestic wastewater, primary and secondary treatment. Unit operations and unit processes of domestic wastewater, sludge disposal.
Air Pollution: Types of pollutants, their sources and impacts, air pollution meteorology, air pollution control, air quality standards and limits.
Municipal Solid Wastes: Characteristics, generation, collection and transportation of solid wastes, engineered systems for solid waste management (reuse/recycle, energy recovery, treatment and disposal).
Noise Pollution: Impacts of noise, permissible limits of noise pollution, measurement of noise and control of noise pollution.
Section 6: Transportation Engineering
Transportation Infrastructure: Highway alignment and engineering surveys; Geometric design of highways—crosssectional elements, sight distances, horizontal and vertical alignments; Geometric design of railway track; Airport runway length, taxiway and exit taxiway design.
Highway Pavements: Highway materials—desirable properties and quality control tests; Design of bituminous paving mixes; Design factors for flexible and rigid pavements; Design of flexible pavement using IRC: 37—2012; Design of rigid pavements using IRC: 58—2011; Distresses in concrete pavements.
Traffic Engineering: Traffic studies on flow, speed, travel time—delay and O-D study, PCU, peak hour factor, parking study, accident study and analysis, statistical analysis of traffic data; Microscopic and macroscopic parameters of traffic flow, fundamental relationships; Control devices, signal design by Webster’s method; Types of intersections and channelization; Highway capacity and level of service of rural highways and urban roads.
Section 7: Geomatics Engineering
Principles of surveying; Errors and their adjustment; Maps scale, coordinate system; Distance and angle measurement - Levelling and trigonometric levelling; Traversing and triangulation survey; Total station; Horizontal and vertical curves. Photogrammetry scale, flying height; Remote sensing basics, platform and sensors, visual image interpretation; Basics of Geographical information system (GIS) and Geographical Positioning system (GPS).
Important Tips for Gate
The followings are some important tips that would be helpful for students to prepare for GATE examination:
1. Go through the pattern (using previous year GATE paper) and syllabus of the exam and start preparing accordingly.
2. Preparation time for GATE depends on many factors, such as, individual’s aptitude, attitude, fundamentals, and concentration level. Generally, rigorous preparation for 4 to 6 months is considered good but it may vary from student to student.
3. Make a list of books that cover complete syllabus, solved previous years questions, and mock tests for practice based on latest GATE pattern. Purchase these books and start your preparation.
4. Make a list of topics that needs to be studied and make priority list for studying every topic based upon the marks for which that particular topic is asked in GATE examination. Make a timetable for study of topics and follow the timetable strictly.
5. While preparing any topics, highlight important points that can be revised during the last minute preparation.
6. Solve questions (numerical) based on latest exam pattern as much as possible, keeping weightage of that topic in mind. Whatever topics you decide to study, make sure that you know everything about it.
7. Go through previous year papers (say last ten years) to check your knowledge and note the distribution of different topics.
8. Finish your detailed study of topics one and a half month before your exam, and during the last month, revise all the topics once again and clear leftover doubts.
Number of Questions: 65
CE: Civil Engineering
Set - I
Total Marks: 100.0
WronganswerforMCQwillresultinnegativemarks,( -1/3)for1-markquestionsand( -2/3)for2-markquestions.
Question Number 1 Correct: 1; Wrong: -0.33
The matrix P is the inverse of a matrix Q. If I denotes the identity matrix, which one of the following options is correct?
(A) PQ = I but QP ≠ I
(B) QP = I but PQ ≠ I
(C) PQ = I and QP = I
(D) PQ - QP=I
Solution: Given that the matrix P is the inverse of a matrix Q, i.e., Q-1 = P
We know that, QQ-1 = Q-1Q = I
⇒ QP = PQ = I
Hence, the correct answer is option (C).
Question Number 2 Correct: 1; Wrong: -0.33
The number of parameters in the univariate exponential and Gaussian distributions, respectively, are
(A) 2 and 2 (B) 1 and 2
(C) 2 and 1 (D) 1 and 1
Solution: In exponential distribution, the probability density function is fxex () = λ λ
Here, l is the parameter i.e., only one parameter.
In Gaussian distribution (normal distribution), the probability density function is
fxex xex () ; =−∞≤ ≤∞
Now, gxfxdxedxex x () () == ∫ ∫
∴= ∫ gxedxex x () (1)
Let etedxd xxt =⇒ −=
⇒= edxd xt
∴ Eq. (1) becomes, gxedt e e t t ex () ()=−
Hence, the correct answer is option (B).
Question Number 4 Correct: 1; Wrong: -0.33
An elastic bar of length L. uniform cross-sectional area A, coefficient of thermal expansion a, and Young’s modulus E is fixed at the two ends. The temperature of the bar is increased by T, resulting in an axial stress σ. Keeping all other parameters unchanged, if the length of the bar is doubled, the axial stress would be
(A) σ (B) 2σ (C) 0.5σ (D) 0.25 aσ
Solution:
fxNe
() (, ) ==
Here, µ and σ are the parameters i.e., there are two parameters. Hence, the correct answer is option (B).
Question Number 3 Correct: 1; Wrong: -0.33
Let x be a continuous variable defined over the interval (-∞, ∞), and f(x) = exex The integral gxfxdx () () = ∫ is equal to (A) e ex (B) eex (C) eex (D) ex
Solution: Given the probability density function ƒ(x) of a continuous random variable x is
(a) Single length R1
Reaction ‘R’ induces stress ‘σ’. From thermal elongation, δα
δ le = Constrained
, . R A
(b) Doubled length
Thermal elongation
δα
The redundant reaction, P2 = Reaction (loud) From elongation (axial) bar,
Hence, the correct answer is option (A).
Question Number 5 Correct: 1; Wrong: -0.33
A simply supported beam is subjected to a uniformly distributed load. Which one of the following statements is true?
(A) Maximum or minimum shear force occurs where the curvature is zero.
(B) Maximum or minimum bending moment occurs where the shear force is zero.
(C) Maximum or minimum bending moment occurs where the curvature is zero.
(D) Maximum bending moment and maximum shear force occur at the same section.
Solution: A simply supported team is subjected to uniformly distributed load; then, the maximum (or) minimum bending moment occurs where the shear force is zero. Hence, the correct answer is option (b).
Question Number 6 Correct: 1; Wrong: -0.33
According to IS:456-2000, which one of the following statements about the depth of neutral axis xu ,bal for a balanced reinforced concrete section is correct?
(A) xu ,bal depends on the grade of concrete only.
(B) xu ,bal depends on the grade of steel only.
(C) xu ,bal depends on both the grade of concrete and grade of steel.
(D) xu ,bal does not depend on the grade of concrete and grade of steel.
Solution: According to IS:456-2000
x fy ud , . bal = + × 700 1100 087
∴ x u,bal depends upon grade of steel only. Hence, the correct answer is option (b).
Question Number 7 Correct: 1; Wrong: -0.33
The figure shows a two-hinged parabolic arch of span L subjected to a uniformly distributed load of intensity q per unit length. q per unit length
The maximum bending moment in the arch is equal to (A) qL2 8 (B) qL2 12 (C) zero (D) qL2
Solution: If a two-hinged (or) three-hinged parabolic arch is subjected to UDL throughout is length, then the bending moment is zero everywhere. Hence, the correct answer is option (D).
Question Number 8 Correct: 1; Wrong: -0.33
List I lists the type of gain or loss of strength in soils. List II lists the property of process responsible for the loss or gain of strength in soils.
List I
List II
P. Regain of strength with time 1. Boiling
Q. Loss of strength due to cyclic loading 2. Liquefaction
R. Loss of strength due to upward seepage 3. Thixotropy
S. Loss of strength due to remoulding 4. Sensitivity
The correct match between List I and List II is:
Codes:
(A) P–4, Q–1, R–2, S–3 (B) P–3, Q–1, R–2, S–4 (C) P–3, Q–2, R–1, S–4 (D) P–4, Q–2, R–1, S–3
Solution: Regain of strength with time—Thixotropy
Loss of strength due to cyclic loading—Liquefaction
Loss of strength due to upward seepage—Boiling
Loss of strength due to remoulding—Sensitivity
∴ P–3, Q–2, R–1, S–4.
Hence, the correct answer is option (C).
Question Number 9 Correct: 1; Wrong: -0.33
A soil sample is subjected to a hydrostatic pressure, σ. The Mohr circle for any point in the soil sample would be
(A) a circle of radius σ and center at the origin
(B) a circle of radius σ and center at a distance σ from the origin
(C) a point at a distance σ from the origin
(D) a circle of diameter σ and center at the origin.
Solution: Under hydrostatic pressure for a point (object) in fluid, the pressure would be same in all directions by Pascal’s law
Question Number 10 Correct: 1; Wrong: –0.33
A strip footing is resting on the ground surface of a pure clay be having an undrained cohesion c u . The ultimate bearing capacity of the footing is equal to
(A) 2pc u (B) pc u (C) (p + 1)c u (D) (p + 2)c u
Solution: Footing is at surface. Hence, Df = 0
qv = CN c + gDfNq + 0.5 gBN r
⇒ for clay
N r = 0, N q = 1
∴ qu = CN c
As per Terzaghi, N c = 5.7, and as per Meyerhoff and Prondtl, N c = 5.14
qu = (A + 2)C u = 5.14C u Hence, the correct answer is option (D).
Question Number 11 Correct: 1; Wrong: –0.33
Mohr’s circle for a state of stress is as follows:
A uniformly distributed line load of 500 kN/m is acting on the ground surface. Based on Boussinesq’s theory, the ratio of vertical stress at a depth 2 m to that at 4 m, right below the line of loading, is
(A) 0.25 (B) 0.5 (C) 2.0 (D) 4.0
Solution: Uniformly distributed load = 500 kN/m Ratio of vertical stress at depth 2 m to that at 4 m below the line of loading is, in general,
Circle is identified by following:
(i) Centre (ii) Radius
Centre of Mohr’s circle,
Circle
, 0)
Radius of Mohr’s circle,
If it is just below the line of action of load (x = 0) Then,
Point at a distance σ from origin. Hence, the correct answer is option (C).
Hence, the correct answer is option (C).
Question Number 12 Correct: 1; Wrong: –0.33
For a steady incompressible laminar flow between two infinite parallel stationary plates, the shear stress variation is
(A) linear with zero value at the plates
(B) linear with zero value at the centre
(C) quadratic with zero value at the plates
(D) quadratic with zero value at the centre
Solution: The stress variation between parallel stationary plates is as follows
Hence, the correct answer is option (B).
Question Number 13 Correct: 1; Wrong: -0.33
The reaction rate involving reactants A and B is given by kAB [] [] αβ Which one of the following statements is valid for the reaction to be a first-order reaction?
(A) a = 0 and b = 0 (B) a = 1 and b = 0
(C) a = 1 and b = 1 (D) a = 1 and b = 2
Solution: Reaction rate = r = K(A)a(B)b, where order of the reaction = a + b
For first-order reaction, a + b = 1
Among the options given, this is possible only when a = 1, b = 0.
Hence, the correct answer is option (B).
Question Number 14 Correct: 1; Wrong: –0.33
The wastewater from a city, containing a high concentration of biodegradable organics, is being steadily discharged into a flowing river at a location S. If the rate of aeration of the river water is lower than the rate of degradation of organics, then the dissolved oxygen of the river water
(A) is lowest at the location S
(B) is lowest at a point upstream of the location S
(C) remains constant all along the length of the river.
(D) is lowest at a point downstream of the location S
Solution: As the given rate of aeration is less than the rate of degradation, which decreases with time (or) distance, minimum DO is observed downstream of point of disposal ‘S’, where both rate of reaction and degradation become equal.
Question Number 15 Correct: 1; Wrong: –0.33
Which one of the following is NOT present in the acid rain?
(A) HNO3 (B) H2SO4
(C) H2CO3 (D) CH3COOH
Solution: CH3COOH is a base which does not harm environment. HNO3, H2SO4, and H2CO3 are formed from NO x, SOx, and CO2. Hence, they form acid rain.
Hence, the correct answer is option (D).
Question Number 16 Correct: 1 Wrong : –0.33
A super-elevation e is provided on a circular horizontal curve such that a vehicle can be stopped on the curve without sliding. Assuming a design speed v and maximum coefficient of side friction fmax, which one of the following criteria should be satisfied?
(A) ef ≤ max
(B) e > fmax
(C) No limit on e can be set
(D) e f f = 1 2 () max max
Solution: For sliding to occur, e > f and v gRf 2 >
For overturning to occur, e < f and e v gR < 2
Hence, the correct answer is option (A).
Question Number 17 Correct: 1; Wrong: -0.33
A runway is being constructed in a new airport as per the International Civil Aviation Organization (ICAO) recommendations. The elevation and the airport reference temperature of this airport are 535 m above the mean sea level and 22.65°C, respectively. Consider the effective gradient of runway as 1%. The length of runway required for a designaircaft under the standard conditions is 2000 m. Within the framework of applying sequential corrections as per the ICAO recommendations, the length of runway corrected for the temperature is:
(A) 2223 m (B) 2250 m
(C) 2500 m (D) 2750 m
Solution: Correction for elevation = 7% increase per 300 m
So, correction = 7 100 535 300 2000 249 66 ×× = .m
Corrected length = 2000 + 249.6 = 2249.6 m
Correction for temperature = 15 - 0.0005 × 535 = 11.5225°C
Rise of temp = 22.65°C - 11.523°C = 11.127°C
Correction =× = 2249 6 100 11 127 250 320 . .. m
Hence, the correct answer is option (D).
Correct length = 2249.66 + 250.32 = 2499.9 m ≈ 2500 m. Hence, the correct answer is option (C).
Question Number 18
Correct: 1; Wrong: –0.33
The accuracy of an Electronic Distance Measuring Instrument (EDMI) is specified as ±(a mm + b ppm). Which one of the following statement is correct?
(A) Both a and b remain constant, irrespective of the distance being measured.
(B) a remains constant and b varies in proportion to the distance being measured.
(C) a varies in proportion to the distance being measured and b remains constant.
(D) Both a and b vary in proportion to the distance being measured.
Solution: Accuracy of EDMI is generally stated in terms of constant instruments error and a measuring error proportional to the distance being measured: ±(a mm + b ppm). The first part indicates a constant instrument error that is independent of length of line measured, while the second component is a distance-related error. Hence, the correct answer is option (B).
Question Number 19 Correct: 1; Wrong: -0.33
The number of spectral bands in the Enhanced Thematic Mapper sensor on the remote sensing satellite Landsat-7 is (A) 64 (B) 10 (C) 8 (D) 15
Solution: Total number of spectoral bands in an enhanced thematic mapper sensor is 8.
Hence, the correct answer is option (C).
Question Number 20
Correct: 1; Wrong: 0
Consider the following partial differential equation:
For this equation to be classified as parabolic, the value of B2 must be ______.
Solution: Given partial differential equation is
A second-order partial differential equation
From Eq. (1),
Hence, the correct answer is
Question Number 21 Correct: 1; Wrong: 0
Solution:
Hence, the correct answer is -1.
Question Number 22 Correct: 1; Wrong: 0
A 3 m thick clay layer is subjected to an initial uniform pore pressure of 145 kPa as shown in the figure.
For the given ground conditions, the time (in days, rounded to the nearest integer) required for 90% consolidation would be _______.
Solution: As per Terzaghi’s 1D consolidation theory
Hence, the correct answer is 1771 days.
Question Number 23 Correct: 1; Wrong: 0
A triangular pipe network is shown in the figure.
The head loss in each pipe is given by hf = rQ1.8, with the variables expressed in a consistent set of units. The value of
r for the pipe AB is 1 and for the pipe BC is 2. If the discharge supplied at the point A (i.e., 100) is equally divided between the pipes AB and AC, the value of r (up to two decimal places) for the pipe AC should be ______.
Solution: Given hf = r ⋅ Q1.8
Under condition of an equal discharge distribution in pipes AB and AC, the discharges in AB and AC will be 50 and 50.
For a closed loop ABCA
Σhf = 0
⇒ 1(50)1.8 - 2(20)1.8 - r(50)1.8 = 0
⇒ 1143.26 - 439.42 = r(50)1.8
⇒ r = 0.62
Hence, the correct answer is 0.62.
Question Number 24
Correct: 1; Wrong: 0
The ordinates of a 2-hours unit hydrograph for a catchment are given as
The ordinate (in m3/s) of a 4-hours unit hydrograph for this catchment at the time of 3 hours would be ________.
Solution: Time (Hour) Ordinate of 2-hours Unit Hydrograph Ordinate of 2-hours Unit Hydrograph Lag by 2 Hours Ordinate of 4-hours DRH Ordinate of 4-hours Unit
Ordinate of 4-hours UH at 3 hours duration = 15 m3/s.
Hence, the correct answer is 15 m3/s.
Question Number 25
Correct: 1; Wrong: 0
Vehicles arriving at an intersection from one of the approach roads follow the poisson distribution. The mean rate of arrival is 900 vehicles per hour. If a gap is greater than 8 seconds is _______.
Solution: The mean rate of arrival = l = 900 veh/h 900 60 60 025 × = veh/s veh/s ∵ λ
Probability that the gap is greater than 8 seconds
= P(t ≥ 8) = e -8l
= e -8 × 0.25
= e -2
= 0.1353
Hence, the correct answer is 0.1353.
Question Number 26 Correct: 2; Wrong: -0.66
For the function fxabxx () ,, =+ ≤≤ 0 1 to be a valid probability density functions, which one of the following statement is correct?
(a) a = 1, b = 4 (B) a = 0.5, b = 1
(c) a = 0, b =1 (d) a = 1, b = -1
Solution: Given fxabxx () ; =+ ≤≤ 01
= Ordinate of DR H 2cm
For ƒ(x) to be a probability density function, fxdx () =
1
⇒+ = a b 2 1 (1)
Among the pairs of values of a and b given in the options, only the values a = 0.5 and b = 1 of option (B) are satisfying Eq. (1).
∴ a = 0.5, b = 1.
Hence, the correct answer is option (B).
Question Number 27 Correct: 2; Wrong: -0.66
The solution of the equation dQdtQ+= 1 with Q = 0 at t = 0 is
(A) Qtet () =− 1 (B) Qtet () =+ 1 (C) Qte t () =+ 1 (D) Qtet () =− 1
Solution: Given the differential equation is dQdtQ+= 1 (1) where Qt==00 at From Eq. (1),
Integrating on both the sides, 1
∴The solution of given differential equation is Q = 1 - e -t . Hence, the correct answer is option (D).
Question Number 28 Correct: 2; Wrong: -0.66
Consider the matrix 51 42
Which one of the following statement is TRUE for the eigenvalues and eigenvectors of this matrix?
(A) Eigenvalue 3 has multiplicity of 2, and only one independent eigenvector exists.
(B) Eigenvalue 3 has a multiplicity of 2, and two independent eigenvectors exist.
(C) Eigenvalue 3 has a multiplicity of 2, and no independent eigenvector exists.
(D) Eigenvalues are 3 and -3. And two independent eigenvectors exist.
Solution: Let A =
51 41
The characteristic equation of A is,
∴l = 3 is eigenvalue of A of multiplicity 2.
Rank of (A - 3I) = 1.
∴ If l is an eigenvalue of a matrix A of order n and r is the rank of A - lI, then the number of linearly independent eigenvectors of A corresponding to the eigen value l is x - r.
Here, n = 2 and r = 1
∴ n - r = 2 - 1 = 1.
∴ The number of linearly independent eigen vectors of A is 1.
Hence, the correct answer is option (A).
Question Number 29 Correct: 2; Wrong: -0.66 A planar truss tower structure is shown in the figure.
Consider the following statements about the external and internal determinacies of the truss.
(P) Externally Determinate
(Q) External Static Indeterminacy = 1
(R) External Static Indeterminacy = 2
(S) Internal Determinate
(T) Internal Static Indeterminacy = 1
(U) Internal Static Indeterminacy = 2
Which one of the following options is correct?
(A) P-False; Q-True; R-False; S-False; T-False; UTrue
(B) P-False; Q-True; R-False; S-False; T-True; UFalse
(C) P-False; Q-False; R-True; S-False; T-False; UTrue
(D) P-True; Q-True; R-False; S-True; T-False; UTrue
Solution: D se = r - s = 4 - 3 = 1°
Dsi = m - (2j - 3) = 15 - (2 × 8 - 3) = 2°
Hence, the correct answer is option (A).
Question Number 30 Correct: 2; Wrong: –0.66
List I contains three broad classes of irrigation supply canal outlets. List II presents hydraulic performance attributes.
List I List II
P. Non-modular outlet 1. Outlet discharge depends on the water levels in both the supply canal as well as the receiving water course
Q. Semi-modular outlet 2. Outlets discharge is fixed and is independent of the water levels in both the supply canal as well as the receiving water course
R. Modular outlet 3. Outlet discharge depends only on the water level in the supply canal
The correct match of the items is List I with the items in List II is
Codes:
(A) P–1; Q–2; R–3 (B) P–3; Q–1; R–2 (C) P–2; Q–3; R–1 (D) P–1; Q–3; R–2
Solution:
P—Nonmodular outlet
Q—Semimodular outlet
Outlet discharge depends on water levels in both the supply canal as well as receiving water course
Outlet discharge is fixed and is independent of water levels in both the supply canal as well as the receiving water course
R—Modular outlet Outlet discharge depends only on the water level in supply canal
Hence, the correct answer is option (D).
Question Number 31 Correct: 2; Wrong: -0.66
A 1 m wide rectangular channel has bed slope of 0.0016 and the Manning’s roughness coefficient is 0.04. Uniform flow takes place in the channel at a flow depth of 0.5 m. At a particular section, gradually varied flow (GVF) is observed and the flow depth is measured as 0.6 m. The GVF profile at that section is classified as (A) S1 (B) S2 (C) M1 (D) M2
Solution:
For rectangular channel:
Question Number 32 Correct: 2; Wrong: -0.66
The following observations are made while testing aggregate for its suitability in pavement construction:
(i) Mass of oven-dry aggregate in air = 1000 gm
(ii) Mass of saturated surface-dry aggregate in air = 1025 gm
(iii) Mass of saturated surface-dry aggregate under water = 625 gm
Based on the above observations, the correct statement is
(A) bulk specific gravity of aggregate = 2.5 and Water absorption = 2.5%
(B) bulk specific gravity of aggregate = 2.5 and Water absorption = 2.4%
(C) apparent specific gravity of aggregate = 2.5 and Water absorption = 2.5%
(D) apparent specific gravity of aggregate = 2.5 and water absorption= 2.4%
Solution: Mass of saturated surface dry aggregate = w a + w w w w = 1025 - 1000 = 25 g
Mass of saturated surface dry aggregate under water = 625 g
Since yn > yc. (Thus, it is a mild slope) Hence, the correct answer is option (C).
Hence, the correct answer is option (A).
Question Number 33 Correct: 2; Wrong: -0.66
The queue length (in number of vehciles) versus time (in seconds) plot for an approach to a signalized intersection with the cycle length of 96 seconds is shown in the figure (not drawn to scale).
Queue length
At time t = 0, the light has just turned red. The effective green time is 36 seconds, during which vehicles discharge at the saturation flow rate, s (in veh/hour). Vehicle arrive at a uniform rate, v (in veh/hour), throughout the cycle. Which one of the following statements is TRUE?
(A) v = 600 veh/hour, and for this cycle, the average stopped delay per vehicle = 30 seconds.
(B) s = 1800 veh/hour, and for this cycle, the average stopped delay per vehicle = 28.125 seconds.
(C) v = 600 veh/hour, and for this cycle, the average stopped delay per vehicle = 45 seconds.
(D) s = 1200 veh/hour, and for this cycle, the average stopped delay per vehicle = 28.125 seconds.
Solution:
Average stopped delay = + = 060 2 30 seconds .
Hence, the correct answer is option (A).
Question Number 34 Correct: 2; Wrong: -0.66
The radius of a horizontal circular curve on a highway is 120 m. The design speed is 60 km/h, and the design coefficient of lateral friction between the tyre and the road surface is 0.15. The estimated value of superelevation required (if full lateral friction is assumed to develop), and the value of coefficient of friction needed (if no superelevation is provided) will, respectively, be (A) 1 11 6 010 . and (B) 1 10 5 037 . and (C) 1 11 6 024 . and (D) 1 12 9 024 . and
Solution: R = 120 m Vdesign = 60 km/h f = 0.15
. . (when full fraction is considered)
⇒ f = 0.24 (when no super elevation is considered). Hence, the correct answer is option (C).
Question Number 35 Correct: 2; Wrong: -0.66
The observed bearing of a traverse are given below:
46o15′
108o15′ RQ 286o15′
201o30′ SR 20o30′
The station(s) most likely to be affected by the local attraction is/are.
(A) Only R (B) Only S
(C) R and S (D) P and Q
Solution: Line Bearing PQ 46°15′ QR 108°15′ RS 201°15′ ST 321°15′ QP 226°15′ RQ 286°15′ SR 20°30′ TS 141°45′
Fore bearing - Back bearing = 180°
PQ, QR, and ST have this difference of 180°
∴Points P, Q, S, and T are free from local attraction. Hence, only R is affected by local attraction. Hence, the correct answer is option (A).
Question Number 36 Correct: 2; Wrong: -0.66
The laboratory tests on a soil sample yields the following results: natural moisture content = 18%, liquid limit = 60%, plastic limit = 25%, percentage of clay sized fraction = 25%.
The liquidity index and activity (as per the expression proposed by Skempton) of the soil, respectively, are
(A) -0.2 and 1.4 (B) 0.2 and 1.4 (C) -1.2 and 0.714 (D) 1.2 and 0.714
Solution:
Hence, the correct answer is option (A).
Question Number 37 Correct: 2; Wrong: 0
Consider the equation du dt t =+31 2 with u = 0 at t = 0.
This is numerically solved by using the forward Euler method with a step size, Dt = 2. The absolute error in the solution at the end of the first time step is ______.
Solution: Given differential equation is du dt t =+31 2 (1)
where u = 0 at t = 0.
Solution by Euler’s method: du dt
By Euler’s method,
∴= = ut22 at (2)
Solution by analytical method:
⇒= ++ uttc 3 (3)
Givenat
⇒= ++
⇒= 00 0 0 3 c c (FromEq. (3))
∴From Eq. (3), utt =+ 3
Now, u at t = 2 is
ut at = =+ = 2 3 22 10
∴= u ()210 (4)
From Eqs. (2) and (4),
∴Error = 10 - 2 = 8. Hence, the correct answer is 8.
Question Number 38 Correct: 2; Wrong: 0
A pre-tensioned rectangular concrete beam 150 mm wide and 300 mm depth is prestressed with three straight tendons, each having a cross-sectional area of 50 mm2, to an initial stress of 1200 N/mm2. The tendons are located at 100 mm from the soffit of the beam. If the modular ratio is 6, the loss of prestressing force (in kN, up to one decimal place) due to the elastic deformation of concrete only is _______.
Solution: Prestressing force, P = 3 × 50 × 1200 = 180000 N
Stress in concrete at the location of steel:
Loss of stress = m × fc = 6 × 5.33 = 32 N/mm2
Loss of prestressing force:
30 50 32 1000 48 ×× = .kN.
Hence, the correct answer is 4.8 kN.
Question Number 39
Correct: 2; Wrong: 0
Consider the stepped bar made with a linear elastic material and subjected to an axial load of 1 kN, as shown in the figure.
Segments 1 and 2 have cross-sectional area of 100 mm2 and 60 mm2, Young’s modulus of 2 × 105 MPa and 3 × 105 MPa, and length of 400 mm and 900 mm, respectively. The strain energy (in N-mm, up to one decimal place) in the bar due to the axial load is _____.
A1 = 100 mm2
E1 = 2 × 105 MPa
L1 = 400 mm
A2 = 60 mm2
E2 = 3 × 105 MPa
L2 = 900 mm
Strain energy, U PL EA
U N-mm
The value of support reaction (in kN) at B should be equal to _________.
Solution: Two spans AB and BC
Fixed end moment (FEM):
L1 = 400 mm A1 = 100 mm2 L2 = 900 mm A2 = 60 mm2
E1 = 2 × 105 MPa P = 1 kN
E2 = 3 × 105 MPa
Hence, the correct answer is 26.5 N-mm.
Question Number 40 Correct: 2; Wrong: 0
The value of M in the beam ABC shown in the figure is such that the joint B does not rotate.
Span BC, MBC = 0
MBC = 0
MI = 0
[ No load on beam]
Stiffness at joint B.
By moment distribution method:
From given data,
θB = 0
End moments, MKBABAB =⋅ θ
⇒ MBA = 0
Moreover, MBA = 0.33M + 13.2 = 0
M = 13 2 033 . .
M = -40
Support reaction at B: Individual span analysis M C M MBA MBC = + Span AB, M MBA RBA RAB MAB
Moment equilibrium at AB: About A, ΣMA = 0
⇒
Moment equilibrium at BC: About C, ΣMC = 0
⇒+ ×=
⇒− −× −+ = =−
⇒= MR R R R BCBC BC BC BC 60 13 50 33 40 60 06 0 (. .( ))
Reaction, RB = RBA + RBC
RB = 50.
Hence, the correct answer is 50.
Question Number 41
For a moving concentrated load of 50 kN on the beam, the magnitude of the maximum bending moment (in kN-m) obtained at the support C will be equal to ________.
Solution: A B C D
AB = BC = 4 m, CD = 10 m
Introduce a moment hinge at C with disturbing boundary conditions: A B C ymax D
Correct: 2; Wrong: 0
Consider the beam ABCD shown in the figure. A internal hinge B C D
AB = BC = 4 m CD = 10 m
This is all for the unit moving load.
Total moving load is 50 kN.
M max = 50 × 4
M max = 200 kN-m.
Hence, the correct answer is 200 kN-m.
Question Number 42
Correct: 2; Wrong: 0
Consider two axially loaded columns, namely, 1 and 2, made of a linear elastic material with Young’s modulus 2 × 105 MPa, square cross-section with side 10 mm, and length 1 m. For column 1, one end is fixed and the other end is free. For column 2, one end is fixed and the other end is pinned. Based on the Euler’s theory, the ratio (up to one decimal place) of the buckling load of column 2 to the buckling load of column 1 is ______.
Solution: E = 2 × 105 MPa
L = 1 m = 1000 mm
b = 10 mm, d = 10 mm
I = × = 10 10 12 833 3 3 .mm 4
Two columns
Effective length L e
Column (1) Le 1 = 2L = 200 mm
Column (2) L L e 2 2 1000 2 ==
Le 2 = 707 1 mm.
Euler’s buckling load, Pcr EI
Ratio of buckling loads,
Hence, the correct answer is 8.
Question Number 43
Correct: 2; Wrong: 0
A column is subjected to a load through a bracket as shown in the figure.
Hence, the correct answer is 5.5 m/s.
Question Number 45 Correct: 2; Wrong: 0
The activity details of a project are given below:
The resultant force (in kN, up to one decimal place) in the bolt 1 is _____.
The estimated minimum time (in days) for the completion of the project will be __________.
Solution:
Hence, the correct answer is 6
Question Number 44
Correct: 2; Wrong: 0
A particle of mass 2 kg is travelling at a velocity of 1.5 m/s.
A force f(t) = 3t2 (in N) is applied to it in the direction of motion for a duration of 2 seconds, where t denotes time in seconds. The velocity (in m/s, up to one decimal place) of the particle immediately after the removal of the force is ______.
On the basis of the data given, the activity on arrow diagram can be drawn as below
along path
= 49 days
Time along path
= 46 days
= 51 days
Minimum time for completion of project = 51 days Hence, the correct answer is 51 days.
