## General Aptitude

 Question 1
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      Numerical Ability
Question 1 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
$\text { Area }=\frac{\sqrt{3}}{4} a^{2}=\frac{\sqrt{3}}{4} \times 24^{2}=144 \sqrt{3}$
 Question 2

In the figure shown above, PQRS is a square. The shaded portion is formed by the intersection of sectors of circles with radius equal to the side of the square and centers at S and Q.
The probability that any point picked randomly within the square falls in the shaded area is __________
 A $4-\frac{\pi}{2}$ B $\frac{1}{2}$ C $\frac{\pi}{2}-1$ D $\frac{\pi}{4}$
GATE CE 2021 SET-2      Numerical Ability
Question 2 Explanation:
\begin{aligned} \text { Probability } &=\frac{f A}{T A} \\ f A &=\left(\frac{\pi r^{2}}{4}-\frac{r^{2}}{2}\right) \times 2 \\ \frac{f A}{T A} &=\frac{\left(\frac{\pi r^{2}}{4}-\frac{r^{2}}{2}\right) \times 2}{r^{2}}=\left(\frac{\pi}{2}-1\right) \end{aligned}
 Question 3
1.Some football players play cricket.
2.All cricket players play hockey.
Among the options given below, the statement that logically follows from the two statements 1 and 2 above, is :
 A No football player plays hockey B Some football players play hockey C All football players play hockey D All hockey players play football
GATE CE 2021 SET-2      Verbal Ability
Question 3 Explanation:

 Question 4
The author said, "Musicians rehearse before their concerts. Actors rehearse their roles before the opening of a new play. On the other hand, I find it strange that many public speakers think they can just walk onto the stage and start speaking. In my opinion, it is no less important for public speaker to rehearse their talks."
Based on the above passage., which one of the following is TRUE?
 A The author is of the opinion that rehearsing is important for musicians, actors and public speakers B The author is of the opinion that rehearsing is less important for public speakers than for musicians and actors C The author is of the opinion that rehearsing is more important only for musicians than public speakers D The author is of the opinion that rehearsal is more important for actors than musicians
GATE CE 2021 SET-2      Verbal Ability
Question 4 Explanation:
The last sentence of the passage decides the answer with the key words "No Less Important".
 Question 5
On a planar field, you travelled 3 units East from a point O. Next you travelled 4 units South to arrive at point P. Then you travelled from P in the North-East direction such that you arrive at a point that is 6 units East of point O. Next, you travelled in the North-West direction, so that you arrive at point Q that is 8 units North of point P.
The distance of point Q to point O, in the same units, should be _____________
 A 3 B 4 C 5 D 6
GATE CE 2021 SET-2      Numerical Ability
Question 5 Explanation:

$O Q=\sqrt{3^{2}+4^{2}}=5$
 Question 6
Four persons P, Q, R and S are to be seated in a row. R should not be seated at the second position from the left end of the row. The number of distinct seating arrangements possible is:
 A 6 B 9 C 18 D 24
GATE CE 2021 SET-2      Numerical Ability
Question 6 Explanation:
Number of arrangements $=3 \times 3 !=18$
 Question 7
$\oplus$ and $\odot$ are two operators on numbers p and q such that $p \odot q=p-q$, and $p \oplus q=p \times q$
Then, $(9 \odot(6 \oplus 7)) \odot(7 \oplus(6 \odot 5))=$
 A 40 B -26 C -33 D -40
GATE CE 2021 SET-2      Numerical Ability
Question 7 Explanation:
\begin{aligned} [9-(6 \times 7)]-[7 \times 1] &=-33-7 \\ &=-40 \end{aligned}
 Question 8
Two identical cube shaped dice each with faces numbered 1 to 6 are rolled simultaneously. The probability that an even number is rolled out on each dice is:
 A $\frac{1}{36}$ B $\frac{1}{12}$ C $\frac{1}{8}$ D $\frac{1}{4}$
GATE CE 2021 SET-2      Numerical Ability
Question 8 Explanation:
Probability of getting even number on a dice$=\frac{3}{6}=\frac{1}{2}$
$\therefore$Two dice are rolled simultaneously,
Hence required probability $=\frac{1}{2} \times \frac{1}{2}=\frac{1}{4}$
 Question 9

The mirror image of the above text about X-axis is

 A A B B C C D D
GATE CE 2021 SET-2      Numerical Ability
 Question 10
i. Arun and Aparna are here.
ii. Arun and Aparna is here.
iii. Arun's families is here.
iv. Arun's family is here.

Which of the above sentences are grammatically CORRECT?
 A (i) and (ii) B (i) and (iv) C (ii) and (iv) D (iii) and (iv)
GATE CE 2021 SET-2      Verbal Ability
Question 10 Explanation:
Two subject joined with 'and' become plural and hence plural verb is there in first statement, in fourth sentence the subject is family which is singular and takes singular verb.

There are 10 questions to complete.

## Verbal Ability

 Question 1
1.Some football players play cricket.
2.All cricket players play hockey.
Among the options given below, the statement that logically follows from the two statements 1 and 2 above, is :
 A No football player plays hockey B Some football players play hockey C All football players play hockey D All hockey players play football
GATE CE 2021 SET-2   General Aptitude
Question 1 Explanation:

 Question 2
The author said, "Musicians rehearse before their concerts. Actors rehearse their roles before the opening of a new play. On the other hand, I find it strange that many public speakers think they can just walk onto the stage and start speaking. In my opinion, it is no less important for public speaker to rehearse their talks."
Based on the above passage., which one of the following is TRUE?
 A The author is of the opinion that rehearsing is important for musicians, actors and public speakers B The author is of the opinion that rehearsing is less important for public speakers than for musicians and actors C The author is of the opinion that rehearsing is more important only for musicians than public speakers D The author is of the opinion that rehearsal is more important for actors than musicians
GATE CE 2021 SET-2   General Aptitude
Question 2 Explanation:
The last sentence of the passage decides the answer with the key words "No Less Important".
 Question 3
i. Arun and Aparna are here.
ii. Arun and Aparna is here.
iii. Arun's families is here.
iv. Arun's family is here.

Which of the above sentences are grammatically CORRECT?
 A (i) and (ii) B (i) and (iv) C (ii) and (iv) D (iii) and (iv)
GATE CE 2021 SET-2   General Aptitude
Question 3 Explanation:
Two subject joined with 'and' become plural and hence plural verb is there in first statement, in fourth sentence the subject is family which is singular and takes singular verb.
 Question 4
Humans have the ability to construct worlds entirely in their minds, which don't exist in the physical world. So far as we know, no other species possesses this ability. This skill is so important that we have different words to refer to its different flavors, such as imagination, invention and innovation.
Based on the above passage, which one of the following is TRUE?
 A No species possess the ability to construct worlds in their minds B The terms imagination, invention and innovation refer to unrelated skills C We do not know of any species other than humans who possess the ability to construct mental worlds D Imagination, invention and innovation are unrelated to the ability to construct mental worlds
GATE CE 2021 SET-1   General Aptitude
Question 4 Explanation:
Option (b) and (d) are weekend by the word 'UNRELATED SKILLS'. Option (c) is weekend by the expression, no species posses the ability.
Hence answer is option (a) which reflects the information given in the passage.
 Question 5
Statement: Either P marries Q or X marries Y
Among the options below, the logical NEGATION of the above statement is :
 A P does not marry Q and X marries Y B Neither P marries Q nor X marries Y C X does not marry Y and P marries Q D P marries Q and X marries Y
GATE CE 2021 SET-1   General Aptitude
Question 5 Explanation:
The statement says only one of these two action will happen, it's NEGATION should be a confirmed action, hence option (c) is the answer.
 Question 6
Getting to the top is _____ than staying on top.
 A more easy B much easy C easiest D easier
GATE CE 2021 SET-1   General Aptitude
Question 6 Explanation:
When the comparison is between two things we use the second degree of the adjective.The degree form of easy are: (easy - easier - easiest)
 Question 7
Nominal interest rate is defined as the amount paid by the borrower to the lender for using the borrowed amount for a specific period of time. Real interest rate calculated on the basis of actual value (inflation-adjusted), is approximately equal to the difference between nominal rate and expected rate of inflation in the economy.
Which of the following assertions is best supported by the above information?
 A Under high inflation, real interest rate is low and borrowers get benefited B Under low inflation, real interest rate is high and borrowers get benefited C Under high inflation, real interest rate is low and lenders get benefited D Under low inflation, real interest rate is low and borrowers get benefited
GATE CE 2020 SET-2   General Aptitude
 Question 8
After the inauguration of the new building, the Head of the Department (HoD) collated faculty preferences for office space. P wanted a room adjacent to the lab. Q wanted to be close to the lift. R wanted a view of the playground and S wanted a corner office.
Assuming that everyone was satisfied, which among the following shows a possible allocation?

 A A B B C C D D
GATE CE 2020 SET-2   General Aptitude
 Question 9
Select the word that fits the analogy:
Partial : Impartial :: Popular: _______
 A Impopular B Dispopular C Mispopular D Unpopular
GATE CE 2020 SET-2   General Aptitude
 Question 10
Select the most appropriate word that can replace the underlined word without changing the meaning of the sentence:
Now-a-days, most children have a tendency to belittle the legitimate concerns of their parents.
 A disparage B applaud C reduce D begrudge
GATE CE 2020 SET-2   General Aptitude
There are 10 questions to complete.

## Numerical Ability

 Question 1
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 1 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
$\text { Area }=\frac{\sqrt{3}}{4} a^{2}=\frac{\sqrt{3}}{4} \times 24^{2}=144 \sqrt{3}$
 Question 2

In the figure shown above, PQRS is a square. The shaded portion is formed by the intersection of sectors of circles with radius equal to the side of the square and centers at S and Q.
The probability that any point picked randomly within the square falls in the shaded area is __________
 A $4-\frac{\pi}{2}$ B $\frac{1}{2}$ C $\frac{\pi}{2}-1$ D $\frac{\pi}{4}$
GATE CE 2021 SET-2   General Aptitude
Question 2 Explanation:
\begin{aligned} \text { Probability } &=\frac{f A}{T A} \\ f A &=\left(\frac{\pi r^{2}}{4}-\frac{r^{2}}{2}\right) \times 2 \\ \frac{f A}{T A} &=\frac{\left(\frac{\pi r^{2}}{4}-\frac{r^{2}}{2}\right) \times 2}{r^{2}}=\left(\frac{\pi}{2}-1\right) \end{aligned}
 Question 3
On a planar field, you travelled 3 units East from a point O. Next you travelled 4 units South to arrive at point P. Then you travelled from P in the North-East direction such that you arrive at a point that is 6 units East of point O. Next, you travelled in the North-West direction, so that you arrive at point Q that is 8 units North of point P.
The distance of point Q to point O, in the same units, should be _____________
 A 3 B 4 C 5 D 6
GATE CE 2021 SET-2   General Aptitude
Question 3 Explanation:

$O Q=\sqrt{3^{2}+4^{2}}=5$
 Question 4
Four persons P, Q, R and S are to be seated in a row. R should not be seated at the second position from the left end of the row. The number of distinct seating arrangements possible is:
 A 6 B 9 C 18 D 24
GATE CE 2021 SET-2   General Aptitude
Question 4 Explanation:
Number of arrangements $=3 \times 3 !=18$
 Question 5
$\oplus$ and $\odot$ are two operators on numbers p and q such that $p \odot q=p-q$, and $p \oplus q=p \times q$
Then, $(9 \odot(6 \oplus 7)) \odot(7 \oplus(6 \odot 5))=$
 A 40 B -26 C -33 D -40
GATE CE 2021 SET-2   General Aptitude
Question 5 Explanation:
\begin{aligned} [9-(6 \times 7)]-[7 \times 1] &=-33-7 \\ &=-40 \end{aligned}
 Question 6
Two identical cube shaped dice each with faces numbered 1 to 6 are rolled simultaneously. The probability that an even number is rolled out on each dice is:
 A $\frac{1}{36}$ B $\frac{1}{12}$ C $\frac{1}{8}$ D $\frac{1}{4}$
GATE CE 2021 SET-2   General Aptitude
Question 6 Explanation:
Probability of getting even number on a dice$=\frac{3}{6}=\frac{1}{2}$
$\therefore$Two dice are rolled simultaneously,
Hence required probability $=\frac{1}{2} \times \frac{1}{2}=\frac{1}{4}$
 Question 7

The mirror image of the above text about X-axis is

 A A B B C C D D
GATE CE 2021 SET-2   General Aptitude
 Question 8
A function, $\lambda$, is defined by
$\lambda(p, q)=\left\{\begin{array}{cl} (p-q)^{2}, & \text { if } p \geq q \\ p+q, & \text { if } p \lt q \end{array}\right.$
The value of the expression $\frac{\lambda(-(-3+2),(-2+3))}{(-(-2+1))}$ is:
 A -1 B 0 C $\frac{16}{3}$ D 16
GATE CE 2021 SET-1   General Aptitude
Question 8 Explanation:
$\frac{\lambda(-(-3+2),(-2+3))}{(-(2+1))}=\lambda \frac{(1,1)}{1}=\lambda(1,1)$
So, 1st definition will be applicable as p = q.
$\text { Hence, } \qquad \lambda(1,1)=(1-1)^{2}=0$
 Question 9

Five line segments of equal lengths, PR, PS, QS, QT and RT are used to form a star as shown in the figure above.
The value of $\theta$, in degrees, is ________
 A 36 B 45 C 72 D 108
GATE CE 2021 SET-1   General Aptitude
Question 9 Explanation:

Sum of angle formed at the pentagon = $540^{\circ}$
Each angle of $=\frac{540}{5}=108^{\circ}$
$\angle x=180-108=72^{\circ}$
Sum of angle of triangle $=180^{\circ}$
\begin{aligned} 72^{\circ}+72^{\circ}+\theta &=180^{\circ} \\ \theta &=36^{\circ} \end{aligned}
 Question 10
Consider two rectangular sheets, Sheet M and Sheet N of dimensions 6cm x 4cm each.
Folding operation 1: The sheet is folded into half by joining the short edges of the current shape.
Folding operation 2: The sheet is folded into half by joining the long edges of the current shape.
Folding operation 1 is carried out on Sheet M three times.
Folding operation 2 is carried out on Sheet N three times.
The ratio of perimeters of the final folded shape of Sheet N to the final folded shape of Sheet M is ____.
 A 0.546528 B 0.126389 C 0.295139 D 0.217361
GATE CE 2021 SET-1   General Aptitude
Question 10 Explanation:

$(\text { Perimeter })_{M}=2(2+1.5)=7$

$\text { (Perimeter })_{N}=2(0.5+6)=13$
Required ratio $=\frac{13}{7}$

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.

## GATE 2022 Civil Engineering Syllabus

Revised syllabus of GATE 2022 Civil Engineering by IIT.

Practice GATE Civil Engineering previous year questions

Download the GATE 2022 Civil Engineering Syllabus pdf from the official site of IIT Bombay. Analyze the GATE 2022 revised syllabus for Civil Engineering.

## GATE CE 2017 SET-1

 Question 1
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 $PQ = I \text{ but }QP \neq I$ B $QP = I \text{ but }PQ \neq I$ C $PQ = I \text{ and } QP = I$ D $PQ - QP = I$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Given that P is inverse of Q.
\begin{aligned} P&=Q^{-1} & P&=Q^{-1} \\ PQ&=Q^{-1}Q & QP&=QQ^{-1} \\ PQ&=I & QP&=I \\ \therefore PQ&=QP=I \end{aligned}
 Question 2
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
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
In exponential,
$f\left ( x \right )=\lambda e^{-\lambda x}$; $x=0$
The parameter is $\lambda$
In Gaussian, $f(x)=\frac{1}{\sigma \sqrt{2\pi }}e^{-\frac{1}{2}\left ( \frac{x-\mu }{\sigma } \right )^{2}}; \;\; -\infty \lt x\lt \infty$
The parameters are $\mu$ and $\sigma$.
 Question 3
Let x be a continuous variable defined over the interval ($-\infty ,\infty$), and $f(x)=e^{-x-e^{-x}}$. The integral $g(x)=\int f(x)dx$ is equal to
 A $e^{e^{-x}}$ B $e^{-e^{-x}}$ C $e^{-e^{x}}$ D $e^{-x}$
Engineering Mathematics   Calculus
Question 3 Explanation:
\begin{aligned} f\left ( x \right )&=e^{-x-e^{-x}}= e^{-x}.e^{-e^{-x}} \\ y\left ( x \right )&=\int f\left ( x \right )dx=\int e^{-x}.e^{-e^{-x}}dx\\ \text{Let } e^{-x}&=t \\ -e^{-x}dx&=dt \\ \int f\left ( x \right )dx&=\int e^{-t}.\left ( -dt \right ) \\ &=\frac{e^{-t}}{-1}.\left ( -d \right ) \\ &=e^{-t} \\ &=e^{-\left ( e^{-x} \right )} \\ &=e^{-e^{-x}} \end{aligned}
 Question 4
An elastic bar of length L, uniform cross sectional area A, coefficient of thermal expansion $\alpha$, 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 $\sigma$. Keeping all other parameters unchanged, if the length of the bar is doubled, the axial stress would be
 A $\sigma$ B 2$\sigma$ C 0.5$\sigma$ D 0.25$\alpha \sigma$
Solid Mechanics   Properties of Metals, Stress and Strain
Question 4 Explanation:

$\sigma=\alpha T E$
$\therefore \;$ Length have no effect on thermal stress.
$\therefore \;$ Axial stress is only $\sigma$.
 Question 5
A simply supported beam is subjected to 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.
Solid Mechanics   Shear Force and Bending Moment
 Question 6
According to IS 456-2000, which one of the following statements about the depth of neutral axis $\chi _{u,bal}$ for a balanced reinforced concrete section is correct?
 A $\chi _{u,bal}$ depends on the grade of concrete only. B $\chi _{u,bal}$ depends on the grade of steel only. C $\chi _{u,bal}$ depends on both the grade of concrete and grade of steel. D $\chi _{u,bal}$ bal does not depend on the grade of concrete and grade of steel.
RCC Structures   Working Stress and Limit State Method
Question 6 Explanation:
$x_{u, \text { bal }}=\frac{700}{1100+0.87 f_{y}} \times d$
So it depends upon grade of steel only.
 Question 7
The figure shows a two-hinged parabolic arch of span L subjected to a uniformly distributed load of intensity q per unit length

The maximum bending moment in the arch is equal to
 A $\frac{qL^{2}}{8}$ B $\frac{qL^{2}}{12}$ C Zero D $\frac{qL^{2}}{10}$
Structural Analysis   Arches
Question 7 Explanation:
If a two hinged or three hinged parabolic arch is subjected to UDL throughout its length, bending moment is zero everywhere.
 Question 8
Group I lists the type of gain or loss of strength in soils. Group II lists the property or process responsible for the loss or gain of strength in soils.

The correct match between Group I and Group II is
 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
Geotechnical Engineering   Properties of Soils
Question 8 Explanation:
Loss in strength of soil due to remoulding at same water content is termed as sensitivity.
Over a period of time soil regain a part of its lost strength is termed as thixotropy. When seepage takes place in upward direction, seepage pressure acts in upward direction and effective stress is reduced consequently shear strength is reduced.
In liquefaction, due to dynamic/cyclic loading in loose saturated sand, effective stress decreases and decrease in shear strength is recorded.
 Question 9
A soil sample is subjected to a hydrostatic pressure, $\sigma$. The Mohr circle for any point in the soil sample would be
 A a circle of radius $\sigma$ and center at the origin B a circle of radius $\sigma$ and center at a distance $\sigma$ from the origin C a point at a distance $\sigma$ from the origin D a circle of diameter $\sigma$ and center at the origin
Geotechnical Engineering   Shear Strength of Soil
Question 9 Explanation:
Hydrostatic pressure acts equally in all directions.

 Question 10
A strip footing is resting on the ground surface of a pure clay bed having an undrainedcohesion $C_{u}$. The ultimate bearing capacity of the footing is equal to
 A $2\pi C_{u}$ B $\pi C_{u}$ C $(\pi+1) C_{u}$ D $(\pi+2) C_{u}$
Geotechnical Engineering   Shallow Foundation and Bearing Capacity
Question 10 Explanation:
Footing is at surface Hence,
\begin{aligned} \text{Hence},\quad D_{t}&=0 \\ q_{u}&=C N_{c}+\gamma D_{f} N_{q}+0.5 \mathrm{B} \gamma N_{\gamma} \end{aligned}
$\Rightarrow \quad$For clay
\begin{aligned} N_{Y} &=0, N_{q}=1 \\ \therefore \quad q_{U}&=C N_{c} \end{aligned}
As per Terzaghi
$N_{c}=5.7$
and as per Meyerhoff and Prandtl.
\begin{aligned} N_{c} &=5.14 \\ \therefore \quad q_{u} &=(\pi+2) C_{u} \end{aligned}
There are 10 questions to complete.

## GATE CE 2017 SET-2

 Question 1
Consider the following simultaneous equations (with $c_{1} \; and \; c_2$ being constants):
$3x_{1}+2x_{2}=c_{1}$
$4x_{1}+x_{2}=c_{2}$
The characteristic equation for these simultaneous equations is
 A $\lambda ^{2}-4\lambda -5=0$ B $\lambda ^{2}-4\lambda+5=0$ C $\lambda ^{2}+4\lambda-5=0$ D $\lambda ^{2}+4\lambda+5=0$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
\begin{aligned} \left [ A \right ]&=\begin{bmatrix} 3 & 2\\ 4 & 1 \end{bmatrix} \\ \left [ A-\lambda I \right ]&=\begin{bmatrix} 3-\lambda & 2\\ 4 & 1-\lambda \end{bmatrix} \\ \left | A-\lambda I \right |&=0 \\ \left ( 3-\lambda \right )\left ( 1-\lambda \right )-8&=0 \\ 3-4\lambda +\lambda ^{2}-8&=0 \\ \lambda ^{2}-4\lambda -5&=0 \end{aligned}
 Question 2
Let w=f(x,y), where x and y are functions of t. Then, according to the chain rule, $\frac{dw}{dt}$ is equal to
 A $\frac{dw}{dx}\frac{dx}{dt}+\frac{dw}{dy}\frac{dt}{dt}$ B $\frac{\partial w}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial w}{\partial y}\frac{\partial y}{\partial t}$ C $\frac{\partial w}{\partial x}\frac{dx}{dt}+\frac{\partial w}{\partial y}\frac{dy}{dt}$ D $\frac{dw}{dx}\frac{\partial x}{\partial t}+\frac{dw}{dy}\frac{\partial y}{\partial t}$
Engineering Mathematics   Calculus
Question 2 Explanation:
$w=f\left ( x, y \right )$
By Chain rule,
$\frac{dw}{dt}=\frac{\partial w}{\partial x}\times \frac{dx}{dt}+\frac{\partial w}{\partial y}\times \frac{dy}{dt}$
 Question 3
Given that the scope of the construction work is well-defined with all its drawings, specifications, quantities and estimates, which one of the following types of contract would be most preferred?
 A EPC contract B Percentage rate contract C Item rate contract D Lump sum contract
Construction Materials and Management
Question 3 Explanation:
Scope of construction work is well-defined with all its drawings, specification quantities and estimates, then lump sum contract is used.
 Question 4
Let G be the specific gravity of soil solids, w the water content in the soil sample,$\gamma _{w}$ the unit weight of water, and $\gamma _{d}$ the dry unit weight of the soil. The equation for the zero air voids line in a compaction test plot is
 A $\gamma _{d}=\frac{G\gamma _{w}}{1+Gw}$ B $\gamma _{d}=\frac{G\gamma _{w}}{Gw}$ C $\gamma _{d}=\frac{Gw}{1+\gamma _{w}}$ D $\gamma _{d}=\frac{Gw}{1-\gamma _{w}}$
Geotechnical Engineering   Properties of Soils
Question 4 Explanation:
Percentage air void line is relation between dry unit weight and water content at constant air void. Hence equation of zero air void line is
\begin{aligned} \gamma_{\mathrm{d}}&=\left(1-n_{\mathrm{a}}\right) \frac{G \gamma_{\mathrm{w}}}{1+w G} &\left(n_{\mathrm{a}}=0\right) \\ \gamma_{\mathrm{d}}&=\frac{G \gamma_{\mathrm{w}}}{1+w G} \end{aligned}
 Question 5
Consider the following statements related to the pore pressure parameters, A and B:
P. A always lies between 0 and 1.0
Q. A can be less than 0 or greater than 1.0
R. B always lies between 0 and 1.0
S. B can be less than 0 or greater than 1.0

For these statements, which one of the following options is correct?
 A P and R B P and S C Q and R D Q and S
Geotechnical Engineering   Consolidation of Soils
Question 5 Explanation:
Pore pressure parameter B lies in between 0 to 1 and pore pressure parameter A may be as low as -0.5 and may be as high as 3.
 Question 6
Consider a rigid retaining wall with partially submerged cohesionless backfill with a surcharge. Which one of the following diagrams closely represents the Rankine's active earth pressure distribution against this wall?
 A A B B C C D D
Geotechnical Engineering   Retaining Wall-Earth Pressure Theories
Question 6 Explanation:

 Question 7
If a centrifugal pump has an impeller speed of N (in rpm), discharge Q (in $m^{3}/s$ ) and the total head H (in m), the expression for the specific speed $N_{s}$ of the pump is given by
 A $N_{s}=\frac{NQ^{0.5}}{H^{0.5}}$ B $N_{s}=\frac{NQ^{0.5}}{H}$ C $N_{s}=\frac{NQ^{0.5}}{H^{0.75}}$ D $N_{s}=\frac{NQ}{H^{0.75}}$
Fluid Mechanics and Hydraulics   Hydraulic Pumps
Question 7 Explanation:
Specific speed of pump.
$N_{\mathrm{s}}=\frac{N \sqrt{Q}}{H^{3 / 4}}=\frac{N Q^{0.5}}{H^{0.75} }$
 Question 8
As per Noise Pollution (Regulation and Control) Rules 2000 of India, the day time noise limit for a residential zone, expressed in dB(A) $L_{eq}$, is
 A 55 B 65 C 75 D 85
Environmental Engineering   Air and Noise Pollution
 Question 9
Following observations have been made for the elevation and temperature to ascertain the stability of the atmosphere:

The atmosphere is classified as
 A Stable B Unstable C Neutral D Inverse
Environmental Engineering   Air and Noise Pollution
Question 9 Explanation:
$ELR_{1-2}=\frac{15.5-15}{\left ( 60-10 \right )\times 10^{-3}}=10^{\circ}C/km$
$ELR_{2-3}=\frac{15-14.3}{\left ( 130-60 \right )\times 10^{-3}}=10^{\circ}C/km$
Since $ELR\gt ALR(9.8^{\circ}C/km)$
Atmosphere is unstable.
 Question 10
The most important type of species involved in the degradation of organic matter in the case of activated sludge process is
 A Autotrophs B Heterotrophs C Prototrophs D Photo-autotrophs
Environmental Engineering   Treatment of Waste Water
Question 10 Explanation:
Activated sludge process is designed primarily for satisfaction of carbonaceous BOD which is done by heterotrophs.
There are 10 questions to complete.

## GATE CE 2018 SET-1

 Question 1
Which one of the following matrices is singular?
 A $\begin{bmatrix} 2 &5 \\ 1 & 3 \end{bmatrix}$ B $\begin{bmatrix} 3 &2 \\ 2 & 3 \end{bmatrix}$ C $\begin{bmatrix} 2 &4 \\ 3 &6 \end{bmatrix}$ D $\begin{bmatrix} 4 &3 \\ 6 &2 \end{bmatrix}$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Option (A): $\left | A \right |=6-5=1$
Option (B): $\left | A \right |=9-4=5$
Option (C): $\left | A \right |=12-12=0$
Option (D): $\left | A \right |=8-18=-10$
Hence matrix (C) is singular.
 Question 2
For the given orthogonal matrix Q,
$Q=\begin{bmatrix} 3/7 &2/7 &6/7\\ -6/7 &3/7 &2/7\\ 2/7 &6/7 &-3/7 \end{bmatrix}$
The inverse is
 A $\begin{bmatrix} 3/7 &2/7 &6/7\\ -6/7 &3/7 &2/7\\ 2/7 &6/7 &-3/7 \end{bmatrix}$ B $\begin{bmatrix} -3/7 &-2/7 &-6/7\\ 6/7 &-3/7 &-2/7\\ -2/7 &-6/7 &3/7 \end{bmatrix}$ C $\begin{bmatrix} 3/7 &-6/7 &2/7\\ 2/7 &3/7 &6/7\\ 6/7 &2/7 &-3/7 \end{bmatrix}$ D $\begin{bmatrix} -3/7 &6/7 &-2/7\\ -6/7 &-3/7 &-6/7\\ -2/7 &-2/7 &3/7 \end{bmatrix}$
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
\begin{aligned} \left | Q \right |&=\frac{3}{7}\left ( -\frac{9}{49}-\frac{12}{49} \right )-\frac{2}{7}\left ( \frac{18}{49}-\frac{4}{49} \right )+\frac{6}{7}\left ( \frac{-36}{49}-\frac{6}{49} \right ) \\ &=-1 \\ Adj. \; Q&=\begin{bmatrix} -\frac{21}{49} & \frac{42}{49} &-\frac{14}{49} \\ -\frac{14}{49}& -\frac{21}{49} & -\frac{42}{49}\\ -\frac{42}{49} & -\frac{14}{49} & \frac{21}{42} \end{bmatrix} \\ \therefore\;\; Q^{-1}&=\frac{Adj\: Q}{\left | Q \right |}=\begin{bmatrix} \frac{3}{7} & -\frac{6}{7} & \frac{2}{7}\\ \frac{2}{7} & \frac{3}{7} & \frac{6}{7}\\ \frac{6}{7} & \frac{2}{7} & -\frac{3}{7} \end{bmatrix} \end{aligned}
Or $\because$ Q is orthogonal
$\therefore \;\; Q^{-1}=Q^{T}$
 Question 3
At the point x= 0, the function $f(x)=x^{3}$ has
 A local maximum B local minimum C both local maximum and minimum D neither local maximum nor local minimum
Engineering Mathematics   Calculus
Question 3 Explanation:
$f\left ( x \right )=x^{3}$ at $x=0$

At $x=0$, the function $y=x^{3}$ has neither minima nor maxima.
 Question 4
A column of height h with a rectangular cross-section of size ax2a has a buckling load of P. If the cross-section is changed to 0.5a x 3a and its height changed to 1.5h, the buckling load of the redesigned column will be
 A P/12 B P/4 C P/2 D 3P/4
RCC Structures   Footing, Columns, Beams and Slabs
Question 4 Explanation:
\begin{aligned} \text { For column, } & P=\frac{\pi^{2} E I_{\min }}{L^{2}} \\ &=\frac{\pi^{2} E\left(\frac{2 a \times a^{3}}{12}\right)}{h^{2}}=\frac{\pi^{2} E a^{4}}{6 h^{2}}\\ \text{For new column, }P&=\frac{\pi^{2} E\left[\frac{3 a \times(0.5 a)^{3}}{12}\right]}{(1.5 h)^{2}} \\ &=\frac{1}{12} \times \frac{\pi^{2} E a^{4}}{6 h^{2}}=\frac{P}{12} \end{aligned}
 Question 5
A steel column of ISHB 350 @72.4 kg/m is subjected to a factored axial compressive load of 2000 kN. The load is transferred to a concrete pedestal of grade M20 through a square base plate. Consider bearing strength of concrete as 0.45$f_{ck}$, where $f_{ck}$ is the characteristic strength of concrete. Using limit state method and neglecting the self weight of base plate and steel column, the length of a side of the base plate to be provided is
 A 39 cm B 42 cm C 45 cm D 48 cm
Design of Steel Structures   Beams
Question 5 Explanation:
\begin{aligned} &\text{Area required for base plate}\\ &=\frac{\text{Factored load}}{\text{Bearing capacity of concrete}}\\ &=\frac{2000\times 10^{3}}{0.45\times 20}=222222.222mm^{2} \end{aligned}
So, side of base plate $=\sqrt{\text{Area}}=471.4 mm= 47.14 cm$
Since, provided area must be more than required
So, answer should be 48 cm.
 Question 6
The Le Chatelier apparatus is used to determine
 A compressive strength of cement B fineness of cement C setting time of cement D soundness of cement
Construction Materials and Management
Question 6 Explanation:
Le Chatelier Apparatus is used to detemine the soundness of cement as per IS code4031 (part 3); this cement testing procedure is called Le Chatelier test for determining the unsoundness properties of cement due to presence of "free lime".
 Question 7
 A creep B hydration C segregation D shrinkage
RCC Structures   Working Stress and Limit State Method
Question 7 Explanation:
 Question 8
A solid circular beam with radius of 0.25 m and length of 2 m is subjected to a twisting moment of 20 kNm about the z-axis at the free end, which is the only load acting as shown in the figure. The shear stress component $\tau_{xy}$ at Point 'M' in the cross-section of the beam at a distance of 1 m from the fixed end is
 A 0.0 MPa B 0.51 MPa C 0.815 MPa D 2.0 MPa
Solid Mechanics   Torsion of Shafts and Pressure Vessels
Question 8 Explanation:

The only non-zero stresses are $\tau_{\theta z}=\tau_{z \theta}=\tau$
if $\theta$ is $90^{\circ}$ then $\theta=y$
\begin{aligned} \text{Hence}\quad\tau_{z y} &=\tau_{y z}=\tau_{\max } \\ &=16 T / \pi \mathrm{d}^{3}=0.815 \mathrm{MPa} \end{aligned}
But in rest of the planes shear stresses are zero, hence, $\tau_{x y}=\tau_{y x}=0$
 Question 9
Two rectangular under-reinforced concrete beam sections X and Y are similar in all aspects except that the longitudinal compression reinforcement in section Y is 10% more. Which one of the following is the correct statement?
 A Section X has less flexural strength and is less ductile than section Y B Section X has less flexural strength but is more ductile than section Y C Sections X and Y have equal flexural strength but different ductility D Sections X and Y have equal flexural strength and ductility
RCC Structures   Footing, Columns, Beams and Slabs
Question 9 Explanation:

Due to presence of more compression steel in section Y, NA of section of Y is above than as of X. It means Y is more under-reinforced than X so ductility of Y is more.
Since compression steel of Y is more so flexure resistance of X is less than as of Y.
 Question 10
The percent reduction in the bearing capacity of a strip footing resting on sand under flooding condition (water level at the base of the footing) when compared to the situation where the water level is at a depth much greater than the width of footing, is approximately
 A 0 B 25 C 50 D 100
Geotechnical Engineering   Shallow Foundation and Bearing Capacity
Question 10 Explanation:
For strip footing on sand (c=0)
$q_{u}=\gamma D_{f} N_{q}+0.5 \mathrm{B} \gamma N_{\gamma}$
In flooding condition water level rises to base of footing hence IIIrd term unit weight of soil will change and IInd term unit weight will be unaffected.
$\begin{array}{ll} \therefore & q_{u}=\gamma D_{f} N_{q}+0.5 B \gamma N_{\gamma} \\ \because & \gamma \simeq \frac{1}{2} \gamma_{s a t} \end{array}$
Hence third term reduced and second term will be same thereby percentage reduction will not be 50%.
According to option approach answer should be 25%
Note: If water table rises to ground level then both $\gamma$ will reduce to $\gamma$. Hence, percentage reduction would be approximately 50%.
There are 10 questions to complete.