## GATE Civil Engineering 2023 SET-2

 Question 1
Let $\phi$ be a scalar field, and $\mathbf{u}$ be a vector field. Which of the following identities is true for $div(\phi \mathbf{u})$ ?
 A $\operatorname{div}(\phi \mathbf{u})=\phi \operatorname{div}(\mathbf{u})+\mathbf{u} \cdot \operatorname{grad}(\phi)$ B $\operatorname{div}(\phi \mathbf{u})=\phi \operatorname{div}(\mathbf{u})+\mathbf{u} \times \operatorname{grad}(\phi)$ C $\operatorname{div}(\phi \mathbf{u})=\phi \operatorname{grad}(\mathbf{u})+\mathbf{u} \cdot \operatorname{grad}(\phi)$ D $\operatorname{div}(\phi \mathbf{u})=\phi \operatorname{grad}(\mathbf{u})+\mathbf{u} \times \operatorname{grad}(\phi)$
Engineering Mathematics   Calculus
Question 1 Explanation:
$div(\phi \mathrm{u})=\phi div(\mu)+ugrad(\phi)$
 Question 2
Which of the following probability distribution functions (PDFs) has the mean greater than the median? A Function 1 B Function 2 C Function 3 D Function 4
Engineering Mathematics   Probability and Statistics
Question 2 Explanation: Option (B) is correct.

 Question 3
A remote village has exactly 1000 vehicles with sequential registration numbers starting from 1000 . Out of the total vehicles, $30 \%$ are without pollution clearance certificate. Further, even- and oddnumbered vehicles are operated on even- and oddnumbered dates, respectively.
If 100 vehicles are chosen at random on an evennumbered date, the number of vehicles expected without pollution clearance certificate is.
 A 15 B 30 C 50 D 70
Engineering Mathematics   Probability and Statistics
Question 3 Explanation:
Since $30 \%$ of the total vehicles are without pollution clearance certificate.
Out of the 100 chosen vehicle, $30 \%$ i.e. $100 \times 0.3=30$ vehicle are expected to be without pollution clearance certificate.
 Question 4
A circular solid shaft of span $L=5 \mathrm{~m}$ is fixed at one end and free at the other end. $A$ torque $T=$ $100 \mathrm{kN} . \mathrm{m}$ is applied at the free end. The shear modulus and polar moment of inertia of the section are denoted as $\mathrm{G}$ and $\mathrm{J}$, respectively. The torsional rigidity $\mathrm{GJ}$ is $50,000 \mathrm{kN} . \mathrm{m}^ 2 / \mathrm{rad}$. The following are reported for this shaft:

Statement i) The rotation at the free end is 0.01 $\mathrm{rad}$
Statement ii) The torsional strain energy is 1.0 kN.m

With reference to the above statements, which of the following is true?
 A Both the statements are correct B Statement i) is correct, but Statement ii) is wrong C Statement i) is wrong, but Statement ii) is correct D Both the statements are wrong
Solid Mechanics   Torsion of Shafts and Pressure Vessels
Question 4 Explanation: $\phi_{\mathrm{BA}}=\frac{\text { T.L. }}{\mathrm{GJ}}=\frac{(100)^{*} 5}{50000}=0.01 \mathrm{rad}$
$\Rightarrow$ Torsional strain energy $(U)$
\begin{aligned} & U=\frac{T^{2} L}{2 G J}=\frac{1}{2} \times T * \phi_{B A} \\ & U=\frac{1}{2} * 100 * 0.01=0.5 \mathrm{kN}-\mathrm{m} \end{aligned}
Hence, statement (i) correct and statement (ii) is incorrect.
 Question 5
M20 concrete as per IS 456: 2000 refers to concrete with a design mix having
 A an average cube strength of $20 \mathrm{MPa}$ B an average cylinder strength of $20 \mathrm{MPa}$ C a 5-percentile cube strength of $20 \mathrm{MPa}$ D a 5-percentile cylinder strength of $20 \mathrm{MPa}$
RCC Structures   Shear, Torsion, Bond, Anchorage and Development Length
Question 5 Explanation:
In M20, M refers to mix and 20 to characteristic cube strength. As per clause no. 6.1.1, IS456: 2000 characteristic strength is defined as the strength below which not more than 5 percent of the test results are expected to fall.
Hence, correct option is (C).

There are 5 questions to complete.

## GATE Civil Engineering 2023 SET-1

 Question 1
For the integral

$\mathrm{I}=\int_{-1}^{1} \frac{1}{\mathrm{x}^{2}} \mathrm{dx}$

which of the following statements is TRUE?
 A $\quad \mathrm{I}=0$ B $\quad \mathrm{I}=2$ C $\quad \mathrm{I}=-2$ D The integral does not converge
Engineering Mathematics   Calculus
Question 1 Explanation:
\begin{aligned} I & =\int_{-1}^{1} \frac{1}{x^{2}} d x \\ & =2 \int_{0}^{1} \frac{1}{x^{2}} d x \quad(\because \quad f(-x)=f(x)) \\ & =2 \lim _{t \rightarrow 0^{+}} \int_{t}^{1} \frac{d x}{x^{2}} \\ & =2 \lim _{t \rightarrow 0^{+}}\left(\frac{-1}{x}\right)_{t}^{1} \\ & =-2 \lim _{t \rightarrow 0^{+}}\left(1-\frac{1}{t}\right) \\ & =2 \lim _{t \rightarrow 0^{+}}\left(\frac{1}{t}-1\right) \\ & =2 \lim _{h \rightarrow 0^{+}}\left(\frac{1}{0+h}-1\right)\\ & =2(\infty-1) \\ & =\infty \;\;\; (Divergent) \end{aligned}
 Question 2
A hanger is made of two bars of different sizes. Each bar has a square cross-section. The hanger is loaded by three-point loads in the mid vertical plane as shown in the figure. Ignore the self-weight of the hanger. What is the maximum tensile stress in $\mathrm{N} / \mathrm{mm}^{2}$ anywhere in the hanger without considering stress concentration effects? A 15 B 25 C 35 D 45
Solid Mechanics   Principal Stress and Principal Strain
Question 2 Explanation: $\sigma_{A B}=\frac{P_{A B}}{A_{A B}}=\frac{250 \times 10^{3}}{100 \times 100}=25 \mathrm{~N} / \mathrm{mm}^{2}$
$\sigma_{B C}=\frac{P_{B C}}{A_{B C}}=\frac{50 \times 10^3}{50 \times 50}=20 \mathrm{N} / \mathrm{mm}^{2}$
$\sigma_{\max }=\sigma_{\mathrm{AB}}=25 \mathrm{~N} / \mathrm{mm}^{2}$

 Question 3
Creep of concrete under compression is defined as the
 A increase in the magnitude of strain under constant stress B increase in the magnitude of stress under constant strain C decrease in the magnitude of strain under constant stress D decrease in the magnitude of stress under constant strain
RCC Structures   Working Stress and Limit State Method
Question 3 Explanation:
Under sustained compressive loading, deformation in concrete increases with time even through the applied stress level is not changed. The time dependent component of strain is called creep.
 Question 4
A singly reinforced concrete beam of balanced section is made of M20 grade concrete and $\mathrm{Fe} 415$ grade steel bars. The magnitudes of the maximum compressive strain in concrete and the tensile strain in the bars at ultimate state under flexure, as per IS 456: 2000 are _______ respectively. (round off to four decimal places)
 A 0.0035 and 0.0038 B 0.0020 and 0.0018 C 0.0035 and 0.0041 D 0.0020 and 0.0031
RCC Structures   Prestressed Concrete Beams
Question 4 Explanation:
Given data,
Balanced section, singly reinforced beam.
As per Clause No. 38.1, IS $456: 2000$,
Maximum strain in concrete at the outermost compression fibre $=0.0035$
and strain in the tension reinforcement for balanced section at ultimate state under flexure
\begin{aligned} & =0.002+\frac{f_{y}}{1.15 E_{s}} \\ & =0.002+\frac{415}{1.15 \times 2 \times 10^{5}}=0.0038 \end{aligned}
 Question 5
In cement concrete mix design, with the increase in water-cement ratio, which one of the following statements is TRUE?
 A Compressive strength decreases but workability increase B Compressive strength increases but workability decreases C Both compressive strength and workability decrease D Both compressive strength and workability increase
RCC Structures   Concrete Technology
Question 5 Explanation:
As the water-cement ratio increases, the porosity in the hardened concrete increases and hence the strength decreases.
Also, as water-cement ratio increases, ducts higher water availability, the workability increases.

There are 5 questions to complete.

## Concrete Technology

 Question 1
In cement concrete mix design, with the increase in water-cement ratio, which one of the following statements is TRUE?
 A Compressive strength decreases but workability increase B Compressive strength increases but workability decreases C Both compressive strength and workability decrease D Both compressive strength and workability increase
GATE CE 2023 SET-1   RCC Structures
Question 1 Explanation:
As the water-cement ratio increases, the porosity in the hardened concrete increases and hence the strength decreases.
Also, as water-cement ratio increases, ducts higher water availability, the workability increases.
 Question 2
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 2 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 3
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 3 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 4
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 4 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 4 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

There are 5 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

There are 5 questions to complete.

## General Aptitude

 Question 1
Consider a spherical globe rotating about an axis passing through its poles. There are three points $P$, $\mathrm{Q}$, and $\mathrm{R}$ situated respectively on the equator, the north pole, and midway between the equator and the north pole in the northern hemisphere. Let $P$, $\mathrm{Q}$, and $\mathrm{R}$ move with speeds $v_{P}, v_{Q}$, and $v_{R}$, respectively.
Which one of the following options is CORRECT?
 A $v_{P} \lt v_{R} \lt v_{Q}$ B $v_{P} \lt v_{Q} \lt v_{R}$ C $v_{P}\gt v_{R} \gt v_{Q}$ D $v_{P}=v_{R} \neq v_{Q}$
GATE CE 2023 SET-2      Numerical Ability
Question 1 Explanation: Velocity, $V=\omega r$.
Here, $\omega=$ constant.
Hence, more is the distance from the axis of rotation more will be the velocity.
$\therefore \quad \mathrm{V}_{\mathrm{P}} \gt \mathrm{V}_{\mathrm{R}} \gt \mathrm{V}_{\mathrm{Q}}$
 Question 2
If $x$ satisfies the equation $4^{8^{x}}=256$, then $x$ is equal to
 A $\frac{1}{2}$ B $\log _{16} 8$ C $\frac{2}{3}$ D $\log _{4} 8$
GATE CE 2023 SET-2      Numerical Ability
Question 2 Explanation:
$4^{8^{x}}=256$
$\rightarrow$ Taking log to the base 4 . on both side.
$8^{x}=\log _{4} 256=4$
Taking lot to the base 8 on both sides, we get
\begin{aligned} x & =\log _{8} 4 \\ & =\log _{2^{3}} 2^{2} \\ x & =\frac{2}{3} \end{aligned}

 Question 3
Based only on the following passage, which one of the options can be inferred with certainty?
When the congregation sang together, Apenyo would also join, though her little screams were not quite audible because of the group singing. But whenever there was a special number, trouble would begin; Apenyo would try singing along, much to the embarrassment of her mother. After two or three such mortifying Sunday evenings, the mother stopped going to church altogether until Apenyo became older and learnt to behave.
At home too, Apenyo never kept quiet; she hummed or made up silly songs to sing by herself, which annoyed her mother at times but most often made her become pensive. She was by now convinced that her daughter had inherited her love of singing from her father who had died unexpectedly away from home.

[Excerpt from These Hills Called Home by Temsula Ao]
 A The mother was embarrassed about her daughter's singing at home. B The mother's feelings about her daughter's singing at home were only of annoyance C The mother was not sure if Apenyo had inherited her love of singing from her father. D When Apenyo hummed at home, her mother tended to become thoughtful.
GATE CE 2023 SET-2      Verbal Ability
 Question 4
Three husband-wife pairs are to be seated at a circular table that has six identical chairs. Seating arrangements are defined only by the relative position of the people. How many seating arrangements are possible such that every husband sits next to his wife?
 A 16 B 4 C 120 D 720
GATE CE 2023 SET-2      Numerical Ability
Question 4 Explanation:
Let us form the pairs of Husband-wife. Now these pairs can be arranged around circular table in
\begin{aligned} & =(3-1) ! \text { ways } \\ & =2 \text { ways } \end{aligned}
Their possible internal arrangement $\mathrm{s}$
\begin{aligned} & =2 ! \times 2 ! \times 2 ! \\ & =8 \end{aligned}
Hence, total seating arrangement.
$=2 \times 8=16$
 Question 5
Elvesland is a country that has peculiar beliefs and practices. They express almost all their emotions by gifting flowers. For instance, if anyone gifts a white flower to someone, then it is always taken to be a declaration of one's love for that person. In a similar manner, the gifting of a yellow flower to someone often means that one is angry with that person.

Based only on the information provided above, which one of the following sets of statement(s) can be logically inferred with certainty?

(i) In Elvesland, one always declares one's love by gifting a white flower.
(ii) In Elvesland, all emotions are declared by gifting flowers.
(iii) In Elvesland, sometimes one expresses one's anger by gifting a flower that is not yellow.
(iv) In Elvesland, sometimes one expresses one's love by gifting a white flower.
 A only (ii) B (i), (ii) and (iii) C (i), (iii) and (iv) D only (iv)
GATE CE 2023 SET-2      Verbal Ability

There are 5 questions to complete.

## Verbal Ability

 Question 1
Based only on the following passage, which one of the options can be inferred with certainty?
When the congregation sang together, Apenyo would also join, though her little screams were not quite audible because of the group singing. But whenever there was a special number, trouble would begin; Apenyo would try singing along, much to the embarrassment of her mother. After two or three such mortifying Sunday evenings, the mother stopped going to church altogether until Apenyo became older and learnt to behave.
At home too, Apenyo never kept quiet; she hummed or made up silly songs to sing by herself, which annoyed her mother at times but most often made her become pensive. She was by now convinced that her daughter had inherited her love of singing from her father who had died unexpectedly away from home.

[Excerpt from These Hills Called Home by Temsula Ao]
 A The mother was embarrassed about her daughter's singing at home. B The mother's feelings about her daughter's singing at home were only of annoyance C The mother was not sure if Apenyo had inherited her love of singing from her father. D When Apenyo hummed at home, her mother tended to become thoughtful.
GATE CE 2023 SET-2   General Aptitude
 Question 2
Elvesland is a country that has peculiar beliefs and practices. They express almost all their emotions by gifting flowers. For instance, if anyone gifts a white flower to someone, then it is always taken to be a declaration of one's love for that person. In a similar manner, the gifting of a yellow flower to someone often means that one is angry with that person.

Based only on the information provided above, which one of the following sets of statement(s) can be logically inferred with certainty?

(i) In Elvesland, one always declares one's love by gifting a white flower.
(ii) In Elvesland, all emotions are declared by gifting flowers.
(iii) In Elvesland, sometimes one expresses one's anger by gifting a flower that is not yellow.
(iv) In Elvesland, sometimes one expresses one's love by gifting a white flower.
 A only (ii) B (i), (ii) and (iii) C (i), (iii) and (iv) D only (iv)
GATE CE 2023 SET-2   General Aptitude

 Question 3
Kind : _______ : : Often : Seldom
 A Cruel B Variety C Type D Kindred
GATE CE 2023 SET-2   General Aptitude
Question 3 Explanation:
Often and seldom are antonyms. Hence, correct option will be the antonym of kind i.e. Cruel.
 Question 4
The line ran ______ the page, right through the centre, and divided the page into two.
 A across B of C between D about
GATE CE 2023 SET-2   General Aptitude
 Question 5
The James Webb telescope, recently launched in space, is given humankind unprecedented access to the depths of time by imaging very old stars formed almost 13 billion years ago. Astrophysicists and cosmologists believe that this odyssey in space may even shed light on the existence of dark matter. Dark matter is supposed to interact only via the gravitational interaction and not through the electromagnetic-, the weak - or the stronginteraction. This may justify the epithet "dark" in dark matter.
Based on the above paragraph, which one of the following statements is FALSE?
 A No other telescope has captured images of starts older than those captured by the James Webb telescope. B People other than astrophysicists and cosmologists may also believe in the existence of dark matter. C The James Webb telescope could be of use in the research on dark matter. D If dark matter was known to interact via the strong-interaction, then the epithet "dark" would be justified.
GATE CE 2023 SET-1   General Aptitude

There are 5 questions to complete.

## Numerical Ability

 Question 1
Consider a spherical globe rotating about an axis passing through its poles. There are three points $P$, $\mathrm{Q}$, and $\mathrm{R}$ situated respectively on the equator, the north pole, and midway between the equator and the north pole in the northern hemisphere. Let $P$, $\mathrm{Q}$, and $\mathrm{R}$ move with speeds $v_{P}, v_{Q}$, and $v_{R}$, respectively.
Which one of the following options is CORRECT?
 A $v_{P} \lt v_{R} \lt v_{Q}$ B $v_{P} \lt v_{Q} \lt v_{R}$ C $v_{P}\gt v_{R} \gt v_{Q}$ D $v_{P}=v_{R} \neq v_{Q}$
GATE CE 2023 SET-2   General Aptitude
Question 1 Explanation: Velocity, $V=\omega r$.
Here, $\omega=$ constant.
Hence, more is the distance from the axis of rotation more will be the velocity.
$\therefore \quad \mathrm{V}_{\mathrm{P}} \gt \mathrm{V}_{\mathrm{R}} \gt \mathrm{V}_{\mathrm{Q}}$
 Question 2
If $x$ satisfies the equation $4^{8^{x}}=256$, then $x$ is equal to
 A $\frac{1}{2}$ B $\log _{16} 8$ C $\frac{2}{3}$ D $\log _{4} 8$
GATE CE 2023 SET-2   General Aptitude
Question 2 Explanation:
$4^{8^{x}}=256$
$\rightarrow$ Taking log to the base 4 . on both side.
$8^{x}=\log _{4} 256=4$
Taking lot to the base 8 on both sides, we get
\begin{aligned} x & =\log _{8} 4 \\ & =\log _{2^{3}} 2^{2} \\ x & =\frac{2}{3} \end{aligned}

 Question 3
Three husband-wife pairs are to be seated at a circular table that has six identical chairs. Seating arrangements are defined only by the relative position of the people. How many seating arrangements are possible such that every husband sits next to his wife?
 A 16 B 4 C 120 D 720
GATE CE 2023 SET-2   General Aptitude
Question 3 Explanation:
Let us form the pairs of Husband-wife. Now these pairs can be arranged around circular table in
\begin{aligned} & =(3-1) ! \text { ways } \\ & =2 \text { ways } \end{aligned}
Their possible internal arrangement $\mathrm{s}$
\begin{aligned} & =2 ! \times 2 ! \times 2 ! \\ & =8 \end{aligned}
Hence, total seating arrangement.
$=2 \times 8=16$
 Question 4
Consider a circle with its centre at the origin (O), as shown. Two operations are allowed on the circle.

Operation 1: Scale independently along the x and y axes.
Operation 2: Rotation in any direction about the origin.

Which figure among the options can be achieved through a combination of these two operations on the given circle? A A B B C C D D
GATE CE 2023 SET-2   General Aptitude
 Question 5
There are 4 red, 5 green, and 6 blue balls inside a box. If ?? number of balls are picked simultaneously, what is the smallest value of ?? that guarantees there will be at least two balls of the same colour?
One cannot see the colour of the balls until they are picked.
 A 4 B 15 C 5 D 2
GATE CE 2023 SET-2   General Aptitude

There are 5 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.

There are 5 questions to complete.