## Concrete Technology

 Question 1
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)
 A 2 B 4 C 8 D 16
GATE CE 2022 SET-2   RCC Structures
Question 1 Explanation:
Given that,
Coarse aggregate = 260 gms
Fine aggregate = 240 gms
$G_{CA}=2.6$
$G_{FA}=2.4$
Bulk specific gravity $G_m=2.3$
Percentage air voids in the mix = ?
$G_t$ (Theoretical specific gravity)

\begin{aligned} &=\frac{\Sigma W}{\Sigma \frac{W}{G}}\\ &=\frac{260+240}{\frac{260}{2.6}+\frac{240}{2.4}}\\ &=2.5 \end{aligned}
\begin{aligned} % \text{ air voids} (V_V)&=\frac{G_t-G_m}{G_t} \times 100\\ &=\frac{2.5-2.3}{2.3}\times 100\\ V_V&=8% \end{aligned}
 Question 2
Match all the possible combinations between Column X (Cement compounds) and Column Y (Cement properties):
$\begin{array}{|c|l|}\hline \text{Column X}&\text{Column Y} \\ \hline (i) C_3S & \text{(P) Early age strength} \\ \hline (ii) C_2S & \text{(Q) Later age strength}\\ \hline (iii) C_3A& \text{(R) Flash setting}\\ \hline & \text{(S) Highest heat of hydration}\\ \hline & \text{(T) Lowest heat of hydration}\\ \hline \end{array}$
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)
GATE CE 2022 SET-2   RCC Structures
Question 2 Explanation:
$C_3S-$ Responsible for early age strength
$C_2S -$ Responsible for later age strength and lowest heat of hydration
$C_3A-$ Flash setting and highest heat of hydration
 Question 3
Which of the following equations is correct for the Pozzolanic reaction?
 A $Ca(OH)_2$ + Reactive Superplasticiser + $H_2O \rightarrow C-S-H$ B $Ca(OH)_2$ + Reactive Silicon dioxide + $H_2O \rightarrow C-S-H$ C $Ca(OH)_2$ + Reactive Sulphates + $H_2O \rightarrow C-S-H$ D $Ca(OH)_2$ + Reactive Sulphur + $H_2O \rightarrow C-S-H$
GATE CE 2022 SET-1   RCC Structures
Question 3 Explanation:
Pozzolanic materials have no cementing properties itself but have the property of combining with lime to produce stable compound.
Pozzolana is considered as siliceous and aluminous materials and when added in cement it have $SiO_2$ and $Al_2O_3$ form.
So, pozzolanic reaction :
$H_2O$ + Reactive slilica-di-oxide + $H_2O \rightarrow C-S-H$ gel or tobermonite gel
There are 3 questions to complete.

## Friction

 Question 1
A horizontal force of P kN is applied to a homogeneous body of weight 25 kN, as shown in the figure. The coefficient of friction between the body and the floor is 0.3. Which of the following statement(s) is/are correct?

 A The motion of the body will occur by overturning. B Sliding of the body never occurs. C No motion occurs for $P \leq 6 kN$. D The motion of the body will occur by sliding only.
GATE CE 2022 SET-1   Solid Mechanics
Question 1 Explanation:

Minimum force for sliding
$(P_{min})_{sliding}=(f_s)_{max}$ ...(i)
Applying equilibrium equation in vertical direction
Normal reaction = Weight
N=mg=25 kN ...(ii)
Using equation (i) and (ii)
$(P_{min})_{sliding}=\mu N =0.3 \times 25=7.5 kN$
Minimum force for overturning

At the verge of overturning
$(P_{min})_{oberturning} \times 2=W \times 2$
$(P_{min})_{oberturning}=\frac{25 \times 0.5} {2}=6.25 kN$
Here, $(P_{min})_{oberturning} \lt (P_{min})_{sliding}$
First overtuning will take place.
Sliding will not take place.
There is 1 question to complete.

## GATE Civil Engineering 2022 SET-2

 Question 1
The function $f(x, y)$ satisfies the Laplace equation
$\triangledown ^2f(x,y)=0$
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 of $f(0, 0)$ is:
 A 0 B 2 C 3 D 1
Engineering Mathematics   Partial Differential Equation
Question 1 Explanation:
According to given condition given function f(x,y) is nothing but constant function i.e. f(x,y)=3 because this is the only function whose value is 3 at any point on the boundary of unit circle and it is also satisfying Laplace equation, so
f(0,0)=3
 Question 2
$\int \left ( x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+... \right )dx$ is equil to
 A $\frac{1}{1+x}+constant$ B $\frac{1}{1+x^2}+constant$ C $-\frac{1}{1-x}+constant$ D $-\frac{1}{1-x^2}+constant$
Engineering Mathematics   Calculus
Question 2 Explanation:
MTA- Marks to All
$I=\int \left ( x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+...\infty \right )dx$
$I=\frac{x^2}{2}-\frac{x^3}{6}+\frac{x^4}{12}-\frac{x^5}{20}+...$
Option (A)
$\frac{1}{1+x}=(1+x)^{-1}=1-x+x^2-x^3...\infty$
So, its incorrect.
Option (B)
$\frac{1}{1+x^2}=(1+x^2)^{-1}=1-x^2+x^4-x^6...\infty$
So, its incorrect.
Similarly option (C) and (D) both are incorrect.
No-correct choice given.
 Question 3
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 $G=\frac{E}{2(1+v)}$ B $G=\frac{E}{(1+2v)}$ C $E=\frac{G}{2(1+v)}$ D $E=\frac{G}{(1+2v)}$
Solid Mechanics   Properties of Metals, Stress and Strain
Question 3 Explanation:
$E=2G(1+\mu )$
$G=$ Shear modulas
$\mu =$Poission's ratio
$E=$ Young's modulus
 Question 4
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)
RCC Structures   Footing, Columns, Beams and Slabs
Question 4 Explanation:
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 $0.002+\frac{0.87f_y}{E_s}$
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.
 Question 5
Match all the possible combinations between Column X (Cement compounds) and Column Y (Cement properties):
$\begin{array}{|c|l|}\hline \text{Column X}&\text{Column Y} \\ \hline (i) C_3S & \text{(P) Early age strength} \\ \hline (ii) C_2S & \text{(Q) Later age strength}\\ \hline (iii) C_3A& \text{(R) Flash setting}\\ \hline & \text{(S) Highest heat of hydration}\\ \hline & \text{(T) Lowest heat of hydration}\\ \hline \end{array}$
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)
RCC Structures   Concreate Technology
Question 5 Explanation:
$C_3S-$ Responsible for early age strength
$C_2S -$ Responsible for later age strength and lowest heat of hydration
$C_3A-$ Flash setting and highest heat of hydration
 Question 6
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

 A 2 and 1 B 2 and 0 C 1 and 2 D 2 and 2
Structural Analysis   Determinacy and Indeterminacy
Question 6 Explanation:

Static indeterminacy, SI=r-3=(3+2)-3=2
Kinematic indeterminacy=0+1=1
 Question 7
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
Geotechnical Engineering   Shallow Foundation and Bearing Capacity
Question 7 Explanation:
In monsoon season sand will be fully saturated hence this will be critical condition in designing of shallow foundation.
In case of sudden drawdown flow direction reverses hence for slope stability, it will be critical condition.
In sandy sea beach, quicksand condition can arise only if existing hydraulic gradient exceeds the critical hydraulic gradient.
 Question 8
Stresses acting on an infinitesimal soil element are shown in the figure (with $\sigma _z \gt \sigma _x$). The major and minor principal stresses are $\sigma _1$ and $\sigma _3$, 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?

 A $\tan ^{-1}\left ( \frac{\tau _{zx}}{\sigma _1-\sigma _x} \right )$ B $\tan ^{-1}\left ( \frac{\tau _{zx}}{\sigma _3-\sigma _x} \right )$ C $\tan ^{-1}\left ( \frac{\tau _{zx}}{\sigma _1+\sigma _x} \right )$ D $\tan ^{-1}\left ( \frac{\tau _{zx}}{\sigma _1+\sigma _3} \right )$
Solid Mechanics   Principal Stress and Principal Strain
Question 8 Explanation:

\begin{aligned} \Sigma F_x &=0 \\ \sigma _x(BC)-\tau _Z \times (AB)\sigma _1 \sin \theta &= 0\\ \sigma _x\left ( \frac{AC \sin \theta }{\cos \theta } \right )+\tau _{zx}\left ( \frac{AC \cos \alpha }{\cos \theta } \right ) &=\sigma _1 \frac{AC \sin \theta }{\cos \theta }\\ \sigma _x \tan \theta +\tau _{zx} &=\sigma _1 \tan \theta \\ \tan \theta(\sigma _1-\sigma _2) &= \tau _{zx}\\ \tan \theta &= \left ( \frac{\tau _{zx}}{\sigma _1-\sigma _x} \right ) \end{aligned}
 Question 9
Match Column X with Column Y:
$\begin{array}{|l|l|}\hline \text{Column X}&\text{Column Y} \\ \hline \text{(P) Horton equation} & \text{((I) Design of alluvial channel} \\ \hline \text{(Q) Penman method} & \text{(II) Maximum flood discharge}\\ \hline \text{(R) Chezys formula}& \text{(III) Evapotranspiration}\\ \hline \text{(S) Lacey's theory}& \text{(IV) Infiltration}\\ \hline \text{(T) Dicken's formula}& \text{(V) Flow velocity}\\ \hline \end{array}$
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)
Fluid Mechanics and Hydraulics   Fluid Dynamics and Flow Measurements
 Question 10
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 7.2 C 6.8 D 8
Engineering Hydrology   Evaporation, Transpiration and Stream Flow Measurement
Question 10 Explanation:
Actual evapotranspiration $(ET_C)$
$=K_S \times K_C \times$ Reference evapotranspiration $(ET_0)$
$=0.8 \times 1.2 \times 6=5.76mm$
There are 10 questions to complete.

## GATE Civil Engineering 2022 SET-1

 Question 1
Consider the following expression:
$z=\sin(y+it)+\cos(y-it)$
where $z, y,$ and $t$ are variables, and $i=\sqrt{-1}$ is a complex number. The partial differential equation derived from the above expression is
 A $\frac{\partial^2 z}{\partial t^2}+\frac{\partial^2 z}{\partial y^2}=0$ B $\frac{\partial^2 z}{\partial t^2}-\frac{\partial^2 z}{\partial y^2}=0$ C $\frac{\partial z}{\partial t}-i\frac{\partial z}{\partial y}=0$ D $\frac{\partial z}{\partial t}+i\frac{\partial z}{\partial y}=0$
Engineering Mathematics   Partial Differential Equation
Question 1 Explanation:
\begin{aligned} z&=\sin(y+it)+ \cos (y-it)\\ \frac{\partial z}{\partial y}&=\cos (y+it)-\sin (y-it)\\ \frac{\partial ^2 z}{\partial ^2 y^2}&=-\sin(y+it)- \cos (y-it)\\ \frac{\partial ^2 z}{\partial ^2 y^2}&=-z \;\;...(i)\\ \frac{\partial z}{\partial t}&=i \cos (y+it)+i\sin (y-it)\\ \frac{\partial ^2 z}{\partial ^2 t^2}&=+\sin(y+it)+ \cos (y-it)\\ \frac{\partial ^2 z}{\partial ^2 t^2}&=z\;\;...(ii)\\ &\text{Adding (i) and (ii)}\\ &\frac{\partial ^2 z}{\partial ^2 y^2}+\frac{\partial ^2 z}{\partial ^2 t^2}=0 \end{aligned}
 Question 2
For the equation
$\frac{d^3y}{dx^3}+x\left ( \frac{dy}{dx} \right )^{\frac{3}{2}}+x^2y=0$
the correct description is
 A an ordinary differential equation of order 3 and degree 2. B an ordinary differential equation of order 3 and degree 3. C an ordinary differential equation of order 2 and degree 3. D an ordinary differential equation of order 3 and degree 3/2.
Engineering Mathematics   Ordinary Differential Equation
Question 2 Explanation:
$\frac{d^3y}{dx^3}+x\left ( \frac{dy}{dx} \right )^{3/2}+x^2y=0$
Power of $\left ( \frac{dy}{dx} \right )$ is fractional, make it integer.
$\frac{d^3y}{dx^3}+x^2y=-x\left ( \frac{dy}{dx} \right )^{3/2}$
$\left (\frac{d^3y}{dx^3}+x^2y \right )^2=x^2\left ( \frac{dy}{dx} \right )^{3}$
Now order = 3 and degree = 2
 Question 3
The hoop stress at a point on the surface of a thin cylindrical pressure vessel is computed to be 30.0 MPa. The value of maximum shear stress at this point is
 A 7.5 MPa B 15.0 MPa C 30.0 MPa D 22.5 MPa
Solid Mechanics   Bending and Shear Stresses
Question 3 Explanation:
Given,
Hoop stress $(\sigma _h)=\frac{pd}{2t}=30MPa$
Maximum shear stress in plane $(\tau _{max})_{\text{in plane}}=\frac{\frac{pd}{2t}-\frac{pd}{4t}}{2}=7.5MPa$
 Question 4
In the context of elastic theory of reinforced concrete, the modular ratio is defined as the ratio of
 A Young's modulus of elasticity of reinforcement material to Young?s modulus of elasticity of concrete. B Youngs modulus of elasticity of concrete to Young?s modulus of elasticity of reinforcement material. C shear modulus of reinforcement material to the shear modulus of concrete. D Young's modulus of elasticity of reinforcement material to the shear modulus of concrete.
RCC Structures   Working Stress and Limit State Method
Question 4 Explanation:
This is a question of working stress method i.e. elastic theory.
Modular ratio
$=\frac{E_s}{E_c}=\frac{\text{Young's modulus of steel}}{\text{Young's modulus of concrete}}$
 Question 5
Which of the following equations is correct for the Pozzolanic reaction?
 A $Ca(OH)_2$ + Reactive Superplasticiser + $H_2O \rightarrow C-S-H$ B $Ca(OH)_2$ + Reactive Silicon dioxide + $H_2O \rightarrow C-S-H$ C $Ca(OH)_2$ + Reactive Sulphates + $H_2O \rightarrow C-S-H$ D $Ca(OH)_2$ + Reactive Sulphur + $H_2O \rightarrow C-S-H$
RCC Structures   Concrete Technology
Question 5 Explanation:
Pozzolanic materials have no cementing properties itself but have the property of combining with lime to produce stable compound.
Pozzolana is considered as siliceous and aluminous materials and when added in cement it have $SiO_2$ and $Al_2O_3$ form.
So, pozzolanic reaction :
$H_2O$ + Reactive slilica-di-oxide + $H_2O \rightarrow C-S-H$ gel or tobermonite gel
 Question 6
Consider the cross-section of a beam made up of thin uniform elements having thickness $t(t \lt \lt a)$ shown in the figure. The $(x,y)$ coordinates of the points along the center-line of the cross-section are given in the figure.

The coordinates of the shear center of this cross-section are:
 A x = 0, y = 3a B x = 2a, y = 2a C x = -a, y = 2a D x = -2a, y = a
Solid Mechanics   Theory of Columns and Shear Centre
Question 6 Explanation:
Shear centre of section consisting of two intersecting narrow rectangles always lies at the intersection of centrelines of two rectangles.

Coordinate of shear centre (0, 3a).
 Question 7
Four different soils are classified as CH, ML, SP, and SW, as per the Unified Soil Classification System. Which one of the following options correctly represents their arrangement in the decreasing order of hydraulic conductivity?
 A SW, SP, ML, CH B SW, SP, ML, CH C SP, SW, CH, ML D ML, SP, CH, SW
Geotechnical Engineering   Classification of Soils and Clay Minerals
Question 7 Explanation:
Hydraulic conductivity Order.
Gravel $\gt$ Sand $\gt$ silt $\gt$ lay
 Question 8
Let $\sigma _v'$ and $\sigma _h'$ denote the effective vertical stress and effective horizontal stress, respectively. Which one of the following conditions must be satisfied for a soil element to reach the failure state under Rankine?s passive earth pressure condition?
 A $\sigma ' _v \lt\sigma ' _h$ B $\sigma ' _v \gt\sigma ' _h$ C $\sigma ' _v = \sigma ' _h$ D $\sigma ' _v + \sigma ' _h =0$
Geotechnical Engineering   Retaining Wall-Earth Pressure Theories
Question 8 Explanation:
We know, $\sigma _h'=K\sigma _v'$
For passive earth pressure,
\begin{aligned} k&=K_P \gt 1\\ \Rightarrow \frac{\sigma _h'}{\sigma _v'}&=K_P \gt 1\\ \Rightarrow \sigma _h' \gt \sigma _v' \end{aligned}
 Question 9
With respect to fluid flow, match the following in Column X with Column Y:
$\begin{array}{|l|l|}\hline \text{Column X}& \text{Column Y}\\ \hline \text{(P) Viscosity} & \text{(I) Mach number}\\ \hline \text{(Q) Gravity}&\text{(II) Reynolds number}\\ \hline \text{(R) Compressibility}&\text{(III) Euler number}\\ \hline \text{(S) Pressure} &\text{(IV) Froude number}\\ \hline \end{array}$
Which one of the following combinations is correct?
 A (P) - (II), (Q) - (IV), (R) - (I), (S) - (III) B (P) - (III), (Q) - (IV), (R) - (I), (S) - (II) C (P) - (IV), (Q) - (II), (R) - (I), (S) - (III) D (P) - (II), (Q) - (IV), (R) - (III), (S) - (I)
Fluid Mechanics and Hydraulics   Flow Through Pipes
Question 9 Explanation:
Reynold's number ($R_e$) is defined when apart from inertial force, viscous forces are dominant.
$R_e=\frac{\text{Inertial force}}{\text{Viscous force}}$
Froude?s number ($F_e$): It is used when in addition to inertial force, gravity forces are important.
$F_e=\frac{\text{Inertial force}}{\text{Gravity force}}$
Euler number ($E_u$): It is used when apart from inertial force, only pressure forces are dominant.
$E_u=\frac{\text{Inertial force}}{\text{Pressure force}}$
Mach number ($M$): It is used when in addition to inertial force, compressibility forces are dominant
$M=\frac{\text{Inertial force}}{\text{Elastic force}}$
 Question 10
Let $\psi$ represent soil suction head and $K$ represent hydraulic conductivity of the soil. If the soil moisture content $\theta$ increases, which one of the following statements is TRUE?
 A $\psi$ decreases and K increases. B $\psi$ increases and K decreases. C Both $\psi$ and K decrease. D Both $\psi$ and K increase.
Geotechnical Engineering   Properties of Soils
Question 10 Explanation:
$h_c\propto \frac{1}{R}$
$K\propto S$
Water content $\uparrow \rightarrow R\uparrow \rightarrow h_c \downarrow \rightarrow \psi \downarrow$
Water content $\uparrow \rightarrow S\uparrow \rightarrow K \uparrow$
There are 10 questions to complete.

## General Aptitude

 Question 1
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 $72^{\circ}$ anticlockwise at P, and then does the following two steps in sequence exactly FIVE times before halting.
1. moves forward for 10 cm.
2. turns $144^{\circ}$ clockwise.

The pattern made by the trace left behind by the ant is

 A A B B C C D D
GATE CE 2022 SET-2      Numerical Ability
Question 1 Explanation:
MTA - Marks to All
 Question 2
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
GATE CE 2022 SET-2      Verbal Ability
 Question 3
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
GATE CE 2022 SET-2      Numerical Ability
 Question 4
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
GATE CE 2022 SET-2      Numerical Ability
 Question 5
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
GATE CE 2022 SET-2      Verbal Ability
 Question 6

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 A B B C C D D
GATE CE 2022 SET-2      Verbal Ability
 Question 7
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
GATE CE 2022 SET-2      Verbal Ability
 Question 8
Both the numerator and the denominator of $frac{3}{4}$ are increased by a positive integer, x, and those of $frac{15}{17}$ 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
GATE CE 2022 SET-2      Numerical Ability
 Question 9
$x:y:z=\frac{1}{2}:\frac{1}{3}:\frac{1}{4}$
What is the value of $frac{x+z-y}{y}$
 A 0.75 B 1.25 C 2.25 D 3.25
GATE CE 2022 SET-2      Numerical Ability
 Question 10
The movie was funny and I _________.
 A could help laughing B couldn't help laughed C couldn't help laughing D could helped laughed
GATE CE 2022 SET-2      Verbal Ability

There are 10 questions to complete.

## Verbal Ability

 Question 1
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
GATE CE 2022 SET-2   General Aptitude
 Question 2
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
GATE CE 2022 SET-2   General Aptitude
 Question 3

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 A B B C C D D
GATE CE 2022 SET-2   General Aptitude
 Question 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
GATE CE 2022 SET-2   General Aptitude
 Question 5
The movie was funny and I _________.
 A could help laughing B couldn't help laughed C couldn't help laughing D could helped laughed
GATE CE 2022 SET-2   General Aptitude
 Question 6

Consider a cube made by folding a single sheet of paper of appropriate shape. The interior faces of the cube are all blank. However, the exterior faces that are not visible in the above view may not be blank.
Which one of the following represents a possible unfolding of the cube?

 A A B B C C D D
GATE CE 2022 SET-1   General Aptitude
Question 6 Explanation:
MTA - Marks to All
 Question 7
Healthy eating is a critical component of healthy aging. When should one start eating healthy? It turns out that it is never too early. For example, babies who start eating healthy in the first year are more likely to have better overall health as they get older.
Which one of the following is the CORRECT logical inference based on the information in the above passage?
 A Healthy eating is important for those with good health conditions, but not for others B Eating healthy can be started at any age, earlier the better C Eating healthy and better overall health are more correlated at a young age, but not older age D Healthy eating is more important for adults than kids
GATE CE 2022 SET-1   General Aptitude
 Question 8
In the following diagram, the point R is the center of the circle. The lines PQ and ZV are tangential to the circle. The relation among the areas of the squares, PXWR, RUVZ and SPQT is

 A Area of SPQT = Area of RUVZ = Area of PXWR B Area of SPQT = Area of PXWR - Area of RUVZ C Area of PXWR = Area of SPQT - Area of RUVZ D Area of PXWR = Area of RUVZ - Area of SPQT
GATE CE 2022 SET-1   General Aptitude
Question 8 Explanation:
From $\Delta$PQR by Pythagoras theorem,
$PR^2=PQ^2+QR^2$
Also, QR = RZ,
$\Rightarrow$ $PR^2=PQ^2+RZ^2$
$\Rightarrow$ Area of PXWR = Area of SPQT + Area of RUVZ
$\Rightarrow$ Area of SPQT = Area of PXWR - Area of RUVZ
 Question 9
Given the statements:

P is the sister of Q.
Q is the husband of R.
R is the mother of S.
T is the husband of P.

Based on the above information, T is ______ of S.
 A the grandfather B an uncle C the father D a brother
GATE CE 2022 SET-1   General Aptitude
Question 9 Explanation:

 Question 10
If
p:q = 1:2
q:r = 4:3
r:s = 4:5
and u is 50% more than s, what is the ratio p?u?
 A 2 : 15 B 16 : 15 C 1 : 5 D 16 : 45
GATE CE 2022 SET-1   General Aptitude
Question 10 Explanation:
Given,
$\frac{P}{Q}=\frac{1}{2}, \frac{Q}{R}=\frac{4}{3},\frac{R}{S}=\frac{4}{5}$
$\frac{P}{Q}=\frac{1 \times 8}{2 \times 8}, \frac{Q}{R}=\frac{4\times 4}{3\times 4},\frac{R}{S}=\frac{4 \times 3}{5 \times 3}$
\begin{aligned} P&:&Q&:&R&:&S\\ 8&:&16&:&12&:&15 \end{aligned}
$U=15 \times 1.5=22.5$
$\frac{P}{U}=\frac{ 8}{22.5}= \frac{16}{45}$
There are 10 questions to complete.

## Numerical Ability

 Question 1
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 $72^{\circ}$ anticlockwise at P, and then does the following two steps in sequence exactly FIVE times before halting.
1. moves forward for 10 cm.
2. turns $144^{\circ}$ clockwise.

The pattern made by the trace left behind by the ant is

 A A B B C C D D
GATE CE 2022 SET-2   General Aptitude
Question 1 Explanation:
MTA - Marks to All
 Question 2
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
GATE CE 2022 SET-2   General Aptitude
 Question 3
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
GATE CE 2022 SET-2   General Aptitude
 Question 4
Both the numerator and the denominator of $frac{3}{4}$ are increased by a positive integer, x, and those of $frac{15}{17}$ 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
GATE CE 2022 SET-2   General Aptitude
 Question 5
$x:y:z=\frac{1}{2}:\frac{1}{3}:\frac{1}{4}$
What is the value of $frac{x+z-y}{y}$
 A 0.75 B 1.25 C 2.25 D 3.25
GATE CE 2022 SET-2   General Aptitude
 Question 6
In the square grid shown on the left, a person standing at P2 position is required to move to P5 position.
The only movement allowed for a step involves, "two moves along one direction followed by one move in a perpendicular direction". The permissible directions for movement are shown as dotted arrows in the right.
For example, a person at a given position Y can move only to the positions marked X on the right.
Without occupying any of the shaded squares at the end of each step, the minimum number of steps required to go from P2 to P5 is

 A 4 B 5 C 6 D 7
GATE CE 2022 SET-1   General Aptitude
Question 6 Explanation:
Minimum number of steps required are:
P2 -> Q4 -> S3 -> T5 -> R4 -> P5
 Question 7

The above frequency chart shows the frequency distribution of marks obtained by a set of students in an exam.
From the data presented above, which one of the following is CORRECT?
 A mean $\gt$ mode $\gt$ median B mode $\gt$ median $\gt$ mean C mode $\gt$ mean $\gt$ median D median $\gt$ mode $\gt$ mean
GATE CE 2022 SET-1   General Aptitude
Question 7 Explanation:
$Mean=\frac{3 \times 3+9\times 4+11\times 5+7\times 6+14\times 7+2\times 8+4\times 9}{3+9+11+7+14+2+4}=5.84$
\begin{aligned} Median&=\frac{\left ( \frac{n}{2} \right )^{th}+\left ( \frac{n}{2} +1\right )^{th}}{2}\\ &=\frac{\left ( \frac{50}{2} \right )^{th}+\left ( \frac{50}{2} +1\right )^{th}}{2}\\ &=\frac{25^{th}+26^{th}}{2}\\ &=\frac{6+6}{2}=6 \end{aligned}
Mode = frequently occured observation (data with high frequency)=7
 Question 8
P invested Rs. 5000 per month for 6 months of a year and Q invested Rs. $x$ per month for 8 months of the year in a partnership business. The profit is shared in proportion to the total investment made in that year.
If at the end of that investment year, Q receives $\frac{4}{9}$ of the total profit, what is the value of $x$ (in Rs)?
 A 2500 B 3000 C 4687 D 8437
GATE CE 2022 SET-1   General Aptitude
Question 8 Explanation:
Let total profit $=K$
So, profit by$Q=\frac{4}{9}K$
profit by $P=K-\frac{4}{9}K=\frac{5}{9}K$
\begin{aligned} \frac{\text{Profit by P}}{\text{Profit by Q}}&=\frac{5}{4}\\ \frac{5}{4}&=\frac{C_PT_P}{C_QT_Q}\\ &=\frac{5000 \times 6 }{x \times 8}\\ x&=\frac{5000 \times 6 \times 4}{8 \times 5}\\ x&=3000 \end{aligned}
 Question 9
Two straight lines pass through the origin $(x_0,y_0)=(0,0)$. One of them passes through the point $(x_1,y_1)=(1,3)$ and the other passes through the point $(x_2,y_2)=(1,2)$.
What is the area enclosed between the straight lines in the interval $[0,1]$ on the x-axis?
 A 0.5 B 1 C 1.5 D 2
GATE CE 2022 SET-1   General Aptitude
Question 9 Explanation:
Equation of first straight line passing through (0, 0) and (1,3)
\begin{aligned} y-y_1&=\left ( \frac{y_2-y_1}{x_2-x_1} \right )(x-x_1) \\ y-0&=\frac{3-0}{1-0}(x-0) \\ y&=3x \end{aligned}
Equation of second stragith line passing through (0,0) and (1,2)
\begin{aligned} y-y_1&=\left ( \frac{y_2-y_1}{x_2-x_1} \right )(x-x_1) \\ y-0&=\frac{2-0}{1-0}(x-0) \\ y&=2x \end{aligned}

$Area=\int_{0}^{1}(3x-2x)dx=\left ( \frac{3x^2}{2}-x^2 \right )_0^1=\frac{1}{2}=0.5$$Area=\int_{0}^{1}(3x-2x)dx=\left ( \frac{3x^2}{2}-x^2 \right )_0^1=\frac{1}{2}=0.5$
 Question 10
In an equilateral triangle PQR, side PQ is divided into four equal parts, side QR is divided into six equal parts and side PR is divided into eight equals parts. The length of each subdivided part in cm is an integer. The minimum area of the triangle PQR possible, in $cm^2$, is
 A 18 B 24 C $48 \sqrt{3}$ D $144 \sqrt{3}$
GATE CE 2021 SET-2   General Aptitude
Question 10 Explanation:

For $\left(\frac{a}{4}, \frac{a}{6}, \frac{a}{8}\right)$ to be integer, a must be LCM of 4, 6 and 8. So a = 24
$\text { Area }=\frac{\sqrt{3}}{4} a^{2}=\frac{\sqrt{3}}{4} \times 24^{2}=144 \sqrt{3}$

There are 10 questions to complete.

## GATE Civil Engineering 2021 SET-1

 Question 1
The rank of matrix $\left[\begin{array}{llll} 1 & 2 & 2 & 3 \\ 3 & 4 & 2 & 5 \\ 5 & 6 & 2 & 7 \\ 7 & 8 & 2 & 9 \end{array}\right]$ is
 A 1 B 2 C 3 D 4
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Using $R_{2} \rightarrow R_{2} \rightarrow 3 R_{1}, R_{3} \rightarrow R_{3}-5 R_{1}, R_{4} \rightarrow R_{4}-7 R_{1}$
$A=\left[\begin{array}{cccc} 1 & 2 & 2 & 3 \\ 0 & -2 & -4 & -4 \\ 0 & -4 & -8 & -8 \\ 0 & -6 & -12 & -12 \end{array}\right]$
Using $R_{3} \rightarrow R_{3}-2 R_{2}, R_{4} \rightarrow R_{4}-3 R_{2}$
$A=\left[\begin{array}{cccc} 1 & 2 & 2 & 3 \\ 0 & -2 & -4 & -4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$
So, $\rho(A)=$ No. of non-zero rows = 2.
 Question 2
If $P=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right]$ and $Q=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]$ then $Q^{T} P^{T}$ is
 A $\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right]$ B $\left[\begin{array}{ll} 1 & 3 \\ 2 & 4 \end{array}\right]$ C $\left[\begin{array}{ll} 2 & 1 \\ 4 & 3 \end{array}\right]$ D $\left[\begin{array}{ll} 2 & 4 \\ 1 & 3 \end{array}\right]$
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
$\begin{array}{l} \quad P Q=\left[\begin{array}{ll} 1 & 3 \\ 2 & 4 \end{array}\right]\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]=\left[\begin{array}{ll} 2 & 4 \\ 1 & 3 \end{array}\right] \\ (P Q)^{\top}=\left[\begin{array}{ll} 2 & 4 \\ 1 & 3 \end{array}\right] \end{array}$
Now using Reversal law
$Q^{\top} P^{\top}=(P Q) T=\left[\begin{array}{ll} 2 & 4 \\ 1 & 3 \end{array}\right]$
 Question 3
The shape of the cumulative distribution function of Gaussian distribution is
 A Horizontal line B Straight line at 45 degree angle C Bell-shaped D S-shaped
Engineering Mathematics   Probability and Statistics
Question 3 Explanation:

$PDF:f(x)=\frac{1}{\sigma \sqrt{2 \pi}}e^{-(x-\mu )^2/(2\sigma ^2)}$
$CDF:F(x)=\frac{1}{2}\left [ 1+eff\left ( \frac{x-\mu }{\sigma \sqrt{2}} \right ) \right ]$
 Question 4
A propped cantilever beam EF is subjected to a unit moving load as shown in the figure (not to scale). The sign convention for positive shear force at the left and right sides of any section is also shown.

The CORRECT qualitative nature of the influence line diagram for shear force at G is
 A B C D
Structural Analysis   Influence Line Diagram and Rolling Loads
Question 4 Explanation:

As per Muller Breslau principle ILD for stress function (shear $-V_{G}$) will be a combination of curves ($3^{\circ}$ curves).
 Question 5
Gypsum is typically added in cement to
 A prevent quick setting B enhance hardening C increase workability D decrease heat of hydration
Construction Materials and Management
Question 5 Explanation:
The Gypsum is added to cement at the end of grinding clinker it is added to prevent quick setting.
 Question 6
The direct and indirect costs estimated by a contractor for bidding a project is Rs.160000 and Rs.20000 respectively. If the mark up applied is 10% of the bid price, the quoted price (in Rs.) of the contractor is
 A 200000 B 198000 C 196000 D 182000
Construction Materials and Management
Question 6 Explanation:
Direct Costs = 160000
Indirect costs = 20000
Mark up applied is 10% of the Bid price
Bid price = Direct cost + Indirect cost + Markup
Bid price = Direct cost + Indirect cost + $\frac{10}{100} \times$ Bid price
$\text{Bid price}\left (1-\frac{10}{100} \right )=DC+IC$
$\text{Bid price}=\frac{160000+20000}{0.9}=2,00,000$
 Question 7
In an Oedometer apparatus, a specimen of fully saturated clay has been consolidated under a vertical pressure of $50 \mathrm{kN} / \mathrm{m}^{2}$ and is presently at equilibrium. The effective stress and pore water pressure immediately on increasing the vertical stress to $150 \mathrm{kN} / \mathrm{m}^{2}$, respectively are
 A $150 \mathrm{kN} / \mathrm{m}^{2}$ and 0 B $100 \mathrm{kN} / \mathrm{m}^{2}$ and $50 \mathrm{kN} / \mathrm{m}^{2}$ C $50 \mathrm{kN} / \mathrm{m}^{2}$ and $100 \mathrm{kN} / \mathrm{m}^{2}$ D 0 and $150 \mathrm{kN} / \mathrm{m}^{2}$
Geotechnical Engineering   Effective Stress and Permeability
Question 7 Explanation:
Stress is increased suddenly, hence entire change will be taken by water $\Delta \bar{\sigma}=\Delta U=100 \mathrm{kPa}$.
There will be no change in effective stress
$\therefore \qquad \qquad\bar{\sigma}=50 \mathrm{kPa}$
 Question 8
A partially-saturated soil sample has natural moisture content of 25% and bulk unit weight of $18.5 \mathrm{kN} / \mathrm{m}^{3}$. The specific gravity of soil solids is 2.65 and unit weight of water is $9.81 \mathrm{kN} / \mathrm{m}^{3}$. The unit weight of the soil sample on full saturation is
 A $21.12 \mathrm{kN} / \mathrm{m}^{3}$ B $19.03 \mathrm{kN} / \mathrm{m}^{3}$ C $20.12 \mathrm{kN} / \mathrm{m}^{3}$ D $18.50 \mathrm{kN} / \mathrm{m}^{3}$
Geotechnical Engineering   Properties of Soils
Question 8 Explanation:
\begin{aligned} \mathrm{w} &=0.25, \gamma_{\mathrm{t}}=18.5 \mathrm{kN} / \mathrm{m}^{3} \\ \mathrm{G}_{\mathrm{s}} &=2.65, \gamma_{\mathrm{w}}=9.81 \\ \gamma_{\mathrm{t}} &=\frac{G_{\mathrm{S}} \gamma_{W}(1+w)}{1+e} \\ \Rightarrow \qquad \qquad \qquad \qquad \mathrm{e} &=\frac{2.65 \times 9.81 \times 1.25}{18.5}-1\\ \Rightarrow \qquad \qquad \qquad \qquad e&=0.756\\ \text{At full saturation}, \quad \mathrm{S}&=1\\ \Rightarrow \qquad \qquad \qquad \quad \gamma_{\mathrm{sat}}&=\frac{\left(G_{\mathrm{S}}+e\right) \gamma_{\mathrm{W}}}{1+e}\\ \gamma_{\text {sat }} &=\frac{(2.65+0.756) \times 9.81}{1.756} \\ &=19.03 \mathrm{kN} / \mathrm{m}^{3} \end{aligned}
 Question 9
If water is flowing at the same depth in most hydraulically efficient triangular and rectangular channel sections then the ratio of hydraulic radius of triangular section to that of rectangular section is
 A $\frac{1}{\sqrt{2}}$ B $\sqrt{2}$ C 1 D 2
Fluid Mechanics and Hydraulics   Open Channel Flow
Question 9 Explanation:
Efficient channel section

\begin{aligned} A & =y^{2} & & A=2 y^{2} \\ P & =2 \sqrt{2} y & P & =4 y \\ R_{I} & =\frac{y}{2 \sqrt{2}} & R_{I I}&=\frac{y}{2} \\ \therefore \qquad \qquad \qquad \frac{R_{I}}{R_{I I}} & =\frac{1}{\sqrt{2}} & \end{aligned}
 Question 10
Kinematic viscosity' is dimensionally represented as
 A $\frac{M}{LT}$ B $\frac{M}{L^{2} T}$ C $\frac{T^{2}}{L}$ D $\frac{L^{2}}{T}$
Fluid Mechanics and Hydraulics   Dimensional Analysis
Question 10 Explanation:
Kinematic viscosity
$v=\frac{\mu }{\rho }=\frac{kg/m\cdot s}{kg/m^3}=m^2/s$
$[v]=\frac{m^2}{s }=\frac{L^2}{T}$
There are 10 questions to complete.

## GATE Civil Engineering 2021 SET-2

 Question 1
The value of $\lim _{x \rightarrow \infty} \frac{x \ln (x)}{1+x^{2}}$ is
 A 0 B 1 C 0.5 D $\infty$
Engineering Mathematics   Calculus
Question 1 Explanation:
\begin{aligned} &\lim _{x \rightarrow \infty}\left(\frac{x \ln x}{x^{2}+1}\right) \qquad \qquad \qquad \qquad \qquad \left(\frac{\infty}{\infty} \text { form }\right)\\ &=\lim _{x \rightarrow \infty}\left(\frac{x\left(\frac{1}{x}\right)+\ln x}{2 x}\right) \qquad \qquad \qquad \left(\frac{\infty}{\infty} \text { form }\right)\\ \lim _{x \rightarrow \infty}\left(\frac{0+\frac{1}{x}}{2}\right)&=\lim _{x \rightarrow \infty}\left(\frac{1}{2 x}\right)=\frac{1}{2 \times \infty}=0 \end{aligned}
 Question 2
The rank of the matrix $\left[\begin{array}{cccc} 5 & 0 & -5 & 0 \\ 0 & 2 & 0 & 1 \\ -5 & 0 & 5 & 0 \\ 0 & 1 & 0 & 2 \end{array}\right]$ is
 A 1 B 2 C 3 D 4
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
\begin{aligned} \left[\begin{array}{cccc} 5 & 0 & 1 & 0 \\ 0 & 2 & 0 & 1 \\ -5 & 0 & -1 & 0 \\ 0 & 1 & 0 & 2 \end{array}\right] & \stackrel{R_{1} \longleftrightarrow R_{1}+R_{3}}{\longrightarrow}\left[\begin{array}{llll} 5 & 0 & 1 & 0 \\ 0 & 2 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 2 \end{array}\right] \\ & \stackrel{R_{4} \longleftrightarrow R_{4}-\frac{1}{2} R_{2}}{\longrightarrow}\left[\begin{array}{llll} 5 & 0 & 1 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{3}{2} \end{array}\right]\\ &R_{3} \longleftrightarrow R_{4}\left[\begin{array}{llll}5 & 0 & 1 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & \frac{3}{2} \\ 0 & 0 & 0 & 0\end{array}\right] \end{aligned}
Rank(A) = 3
 Question 3
The unit normal vector to the surface $X^{2}+Y^{2}+Z^{2}-48=0$ at the point (4,4,4) is
 A $\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}$ B $\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$ C $\frac{2}{\sqrt{2}}, \frac{2}{\sqrt{2}}, \frac{2}{\sqrt{2}}$ D $\frac{1}{\sqrt{5}}, \frac{1}{\sqrt{5}}, \frac{1}{\sqrt{5}}$
Engineering Mathematics   Calculus
Question 3 Explanation:
\begin{aligned} \phi &=x^{2}+y^{2}+z^{2}-48, P(4,4,4) \\ \operatorname{grad} \phi &=\vec{\nabla} \phi=\hat{i} \frac{\partial \phi}{\partial x}+\hat{j} \frac{\partial \phi}{\partial y}+\hat{k} \frac{\partial \phi}{\partial z} \\ &=(2 x) \hat{i}+(2 y) \hat{j}+(2 z) \hat{k} \\ \vec{n} &=(\operatorname{grad} \phi)_{P}=8 \hat{i}+8 \hat{j}+8 \hat{k} \\ \hat{n} &=\frac{\vec{n}}{|\vec{n}|}=\frac{8 \hat{i}+8 \hat{j}+8 \hat{k}}{\sqrt{64+64+64}}=\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}} \\ & \simeq\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}},\right) \end{aligned}
 Question 4
If A is a square matrix then orthogonality property mandates
 A $A A^{T}=I$ B $A A^{T}=0$ C $A A^{T}=A^{-1}$ D $A A^{T}=A^{2}$
Engineering Mathematics   Linear Algebra
Question 4 Explanation:
$\text { If, } \qquad \qquad A A^{\top}=I \quad \text { or } A^{-1}=A^{T}$
The matrix is orthogonal.
 Question 5
In general, the CORRECT sequence of surveying operations is
 A Field observations$\rightarrow$ Reconnaissance$\rightarrow$ Data analysis$\rightarrow$ Map making B Data analysis$\rightarrow$ Reconnaissance$\rightarrow$ Field observations $\rightarrow$ Map making C Reconnaissance$\rightarrow$ Field observations $\rightarrow$ Data analysis $\rightarrow$ Map making D Reconnaissance$\rightarrow$ Data analysis $\rightarrow$ Field observations $\rightarrow$ Map making
Geometics Engineering   Fundamental Concepts of Surveying
Question 5 Explanation:
Reconnaissance$\rightarrow$Field observations$\rightarrow$Data analysis$\rightarrow$Map making
 Question 6
Strain hardening of structural steel means
 A experiencing higher stress than yield stress with increased deformation B strengthening steel member externally for reducing strain experienced C strain occurring before plastic flow of steel material D decrease in the stress experienced with increasing strain
Solid Mechanics   Properties of Metals, Stress and Strain
Question 6 Explanation:
Strain hardening is experiencing higher stress than yield stress with increased deformation
In the figure AB = Strain hardening zone
OA = Linear elastic zone
Stress corresponding to point 'A' is yield stress.

 Question 7
A single story building model is shown in the figure. The rigid bar of mass 'm' is supported by three massless elastic columns whose ends are fixed against rotation. For each of the columns, the applied lateral force (P) and corresponding moment (M) are also shown in the figure. The lateral deflection $(\delta)$ of the bar is given by $\delta=\frac{P L^{3}}{12 E I}$, where L is the effective length of the column, E is the Young's modulus of elasticity and I is the area moment of inertia of the column cross-section with respect to its neutral axis.

For the lateral deflection profile of the columns as shown in the figure, the natural frequency of the system for horizontal oscillation is
 A $6 \sqrt{\frac{E I}{m L^{3}}} \mathrm{rad} / \mathrm{s}$ B $\frac{1}{L} \sqrt{\frac{2 E I}{m}} \mathrm{rad} / \mathrm{s}$ C $6 \sqrt{\frac{6 E I}{m L^{3}}} \mathrm{rad} / \mathrm{s}$ D $\frac{2}{L} \sqrt{\frac{E I}{m}} \mathrm{rad} / \mathrm{s}$
Solid Mechanics   Deflection of Beams
Question 7 Explanation:

As the deflection will be same in all the 3 columns, so it represents a parallel connection.

\begin{aligned} k_{e q} &=3 k=\frac{36 E I}{L^{3}} \\ \text { Natural frequency }(\omega) &=\sqrt{\frac{k}{m}} \\ &=\sqrt{\frac{36 E I}{m L^{3}}}=6 \sqrt{\frac{E I}{m L^{3}}} \mathrm{rad} / \mathrm{s} \end{aligned}
 Question 8
Seasoning of timber for use in construction is done essentially to
 A increase strength and durability B smoothen timber surfaces C remove knots from timber logs D cut timber in right season and geometry
Construction Materials and Management
Question 8 Explanation:
Option 1 Increase strength and durability.
The process of drying of timber is known as seasoning.
Natural tree has more the 50% weight of water of its dry weight.
If we directly use this timber the because of irregular drying internal stresses will develop between fibres of timber and it will develop lots of defects (warps, shakes etc).
 Question 9
In case of bids in Two-Envelop System, the correct option is
 A Technical bid is opened first B Financial bid is opened first C Both (Technical and Financial) bids are opened simultaneously D Either of the two (Technical and Financial) bids can be opened first
Construction Materials and Management
Question 9 Explanation:
Option 1 technical bid is opened first

Opening of Tender
First technical bid is opened and after ensuring that all the technical aspects of a contractor are in order than only financial bid is opened
1. Envelope 1 ( Technical bid )

1. Cover letter
2. Registration Details
3. Pre-qualification documents
4. Earnest money deposit
5. Assumptions & Deviations in making of tender
6. Drawings

2. Envelope 2 (Financial Bid)

1. Forms of tender
 Question 10
The most appropriate triaxial test to assess the long-term stability of an excavated clay slope is
 A consolidated drained test B unconsolidated undrained test C consolidated undrained test D unconfined compression test
Geotechnical Engineering   Shear Strength of Soil
Question 10 Explanation:
To assess the long term stability of clayey soil, the results of consolidated drained (CD) test are used.
There are 10 questions to complete.