## 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 2015 SET-2

 Question 1
While minimizing the function $f(x)$, necessary and sufficient conditions for a point, $x_{0}$ to be a minima are:
 A ${f}'(x_{0}) \gt 0\: and\: \: {f}''(x_{0})=0$ B ${f}'(x_{0}) \lt 0\: and\: \: {f}''(x_{0})=0$ C ${f}'(x_{0})=0\: and\: \: {f}''(x_{0}) \lt 0$ D ${f}'(x_{0})=0\: and\: \: {f}''(x_{0}) \gt 0$
Engineering Mathematics   Calculus
Question 1 Explanation:
$f(x)$ has a local minimum at $x=x_{0}$
if ${f}'\left ( x_{0} \right )=0$
and ${f}''\left ( x_{0} \right ) \gt 0$
 Question 2
In Newton-Raphson iterative method, the initial guess value ( $x_{ini}$) is considered as zero while finding the roots of the euation:$f(x)=-2+6x-4x^{2}+0.5x^{3}$. The correction, $\Delta x$, to be added to $x_{ini}$ in the first iteration is____________.
 A 0.5 B 0.33 C 0.8 D 0.2
Engineering Mathematics   Numerical Methods
Question 2 Explanation:
\begin{aligned} f\left ( x \right )&=-2+6x-4x^{2}+0.5x^{3} \\ {f}'\left ( x \right )&=6-8x+1.5x^{2} \\ x_{ini}&=0\\ \text{By Newton }&\text{Raphson Method,}\\ x_{1}&=x_{ini}-\frac{f\left ( x_{ini} \right )}{{f}'\left ( x_{ini} \right )} \\ &= 0-\frac{-2}{6} \\ \Rightarrow \;\;x_{1}&=\frac{1}{3} \\ \therefore \;\; \Delta x&=x_{1}-x_{ini}=\frac{1}{3} \end{aligned}
 Question 3
Given , $i=\sqrt{-1}$, the value of the definite integral, $I=\int_{0}^{\pi/2}\frac{\cos x+i\sin x}{\cos x-i\sin x}dx$ is:
 A 1 B -1 C i D -i
Engineering Mathematics   Calculus
Question 3 Explanation:
\begin{aligned} I&=\int_{0}^{\pi /2}\frac{\cos x+i\sin x}{\cos x-i\sin x}dx \\ &=\int_{0}^{\pi /2}\frac{e^{ix}}{e^{-ix}}dx=e^{2ix}dx \\ &=\frac{1}{2i}\left [ e^{2ix} \right ]_{0}^{\pi /2} \\ &=\frac{1}{2i}\left [ e^{i\pi }-1 \right ] \\ &=\frac{1}{2i}\left ( -1-1 \right ) \\ &=-\frac{1}{i}=i \end{aligned}
 Question 4
$\lim_{x\rightarrow \infty }\left ( 1+\frac{1}{x} \right )^{2x}$ is equal to
 A $e^{-2}$ B e C 1 D $e^{2}$
Engineering Mathematics   Calculus
Question 4 Explanation:
$y=\: \lim_{x\rightarrow \infty }\left ( 1+\frac{1}{x} \right )^{2x}$
$\log y = \lim_{x\rightarrow \infty }2x\log \left ( 1+\frac{1}{x} \right )$
Which is in the form of $\infty \times 0$.
To convert this into $\frac{0}{0}$ form, we rewrite as,
$\Rightarrow \log y= \lim_{x\rightarrow \infty }\frac{2\log \left ( 1+\frac{1}{x} \right )}{1/x}$
Now is in $\frac{0}{0}$ form.
Using L'Hospital's Rule,
\begin{aligned} \log y&=\lim_{x\rightarrow \infty }\frac{\frac{2\times -\frac{1}{x^{2}}}{1+\frac{1}{x}}}{-\frac{1}{x^{2}}} \\ \log y&=\lim_{x\rightarrow \infty }\frac{2}{1+\frac{1}{x}}=2 \\ \therefore \;\; y&= e^{2}\end{aligned}
 Question 5
Let $\mathbf{A}=\left [ a_{ij} \right ],\; \; \; 1\leq i, \; j\leq n$ with $n\geq 3\; and\; a_{ij}=i.j$. The rank of A is:
 A 0 B 1 C n-1 D n
Engineering Mathematics   Linear Algebra
Question 5 Explanation:
Rank of A=1
Because each row will be scalar multiple of first row.So we will get only one-zero row in row Echeleaon form of A.

Alternative:
Rank of A=1
Because all the minors of order greater than 1 will be zero.
 Question 6
A horizontal beam ABC is loaded as shown in the figure below. The distance of the point of contraflexure from end A (in m) is _________.
 A 1.25 B 0.25 C 0.5 D 0.75
Solid Mechanics   Shear Force and Bending Moment
Question 6 Explanation:
Reaction at B

\begin{aligned} \Delta_{\mathrm{B}}&=0 \text { (Compatibility condition) }\\ \frac{10 \times(0.75)^{3}}{3 E I}&+\frac{2.5 \times 0.75^{2}}{2 E I}-\frac{R_{B} \times 0.75^{3}}{3 E I}=0 \\ \frac{9 R_{B}}{64 E I}&=\frac{135}{64 E I}\\ \therefore \quad \mathrm{R}_{\mathrm{B}}&=15 \mathrm{kN} \end{aligned}
BM at a distance x from free end
Reaction at B

\begin{aligned} & & \mathrm{BM}_{x} &=10 \times x-15 \times(x-0.25)=0 \\ \Rightarrow & & 10 x &=15 x-3.75 \\ \Rightarrow & & 5 x &=3.75 \\ \therefore & & x &=0.75 \mathrm{m} \end{aligned}
$\therefore$ From end A, distance is 0.25m
 Question 7
For the plane stress situation shown in the figure, the maximum shear stress and the plane on which it acts are:
 A -50 MPa, on a plane 45$^{\circ}$ clockwise w.r.t. x-axis B -50 MPa, on a plane 45$^{\circ}$ anti-clockwise w.r.t. x-axis C 50 MPa, at all orientations D Zero, at all orientations
Solid Mechanics   Principal Stress and Principal Strain
Question 7 Explanation:
Under hydrostatic loading condition, stresses at a point in all directions are equal and hence no shear stress.
Alternatively,
$\tau=\frac{\sigma_{1}-\sigma_{2}}{2}=\frac{50-50}{2}=0$
Thus, Mohr's circle reduces to a point.
Hence shear stress at all orientations is zero.
 Question 8
A guided support as shown in the figure below is represented by three springs (horizontal, vertical and rotational) with stiffness $k_{x},k_{y}\: and \: k_{\theta }$ respectively. The limiting values of $k_{x},k_{y}\: and \: k_{\theta }$ are:
 A $\infty ,0,\infty$ B $\infty ,\infty ,\infty$ C $0 ,\infty ,\infty$ D $\infty ,\infty ,0$
Structural Analysis   Determinacy and Indeterminacy
 Question 9
A column of size 450 mm x 600 mm has unsupported length of 3.0 m and is braced against side sway in both directions. According to IS 456:2000, the minimum eccentricities (in mm) with respect to major and minor principle axes are:
 A 20.0 and 20.0 B 26.0 and 21.0 C 26.0 and 20.0 D 21.0 and 15.0
RCC Structures   Footing, Columns, Beams and Slabs
Question 9 Explanation:

$e_{\min }=\operatorname{maximum}\left\{\begin{array}{l} \frac{L}{500}+\frac{D}{30} \\ 20 \mathrm{mm} \end{array}\right.$
x-x will be major axis and y-y will be minor axis.
\begin{aligned} \therefore \quad e_{\min y y}&=\frac{3000}{500}+\frac{450}{30}=21 \mathrm{mm} \\ \text{and }\quad e_{\min xx}&=\frac{3000}{500}+\frac{600}{30}=26 \mathrm{mm} \end{aligned}
 Question 10
Prying forces are:
 A shearing forces on the bolts because of the joints B tensile forces due to the flexibility of connected parts C bending forces on the bolts because of the joints D forces due the friction between connected parts
Design of Steel Structures   Structural Fasteners
There are 10 questions to complete.

## GATE CE 2016 SET-1

 Question 1
Newton-Raphson method is to be used to find root of equation $3x-e^{x}+\sin x=0$. If the initial trial value for the root is taken as 0.333, the next approximation for the root would be _________
 A 0.33 B 0.54 C 0.36 D 0.76
Engineering Mathematics   Numerical Methods
Question 1 Explanation:
According to Newton-Raphson Method:
\begin{aligned} x_{N+1}&=X_{N}-\frac{f\left ( X_{N} \right )}{{F}'\left ( X_{N} \right )} \\ f\left ( x \right )&=3x-e^{x}+\sin x \\ {f}'\left ( x \right )&=3-e^{x}+\cos x \\ \Rightarrow \;\; X_{1}&=X_{0}-\frac{f\left ( 0.333 \right )}{{f}'\left ( 0.333 \right )} \\ &=0.333-\frac{3\times 0.333-e^{0.333}+\sin 0.333}{3-e^{0.333}+\cos 0.333} \\ \therefore \;\; X_{1}&=0.36 \end{aligned}
 Question 2
The type of partial differential equation $\frac{\partial^2 P}{\partial x^2} + \frac{\partial^2 P}{\partial y^2} +3\frac{\partial^2 P}{\partial x \partial y } +2\frac{\partial P}{\partial x}-\frac{\partial P}{\partial y}=0$ is
 A elliptic B parabolic C hyperbolic D none of these
Engineering Mathematics   Ordinary Differential Equation
Question 2 Explanation:
Comparing the given equation with the general form of second order partial differential equation, we have A=1, B=3, C=1 $\Rightarrow\; B^{2}-4AC=5\gt0$
$\therefore$ PDE is Hyperbola.
 Question 3
If the entries in each column of a square matrix M add up to 1, then an eigen value of M is
 A 4 B 3 C 2 D 1
Engineering Mathematics   Linear Algebra
Question 3 Explanation:
Consider the '$2\times 2$' square matrix $M=\begin{bmatrix} a & b\\ c & d \end{bmatrix}$
$\Rightarrow \;\; \lambda ^{2}-\left ( a+d \right )\lambda +\left ( ad-bc \right )=0 \;\;...(i)$
Putting $\lambda =1$, we get
0=0 which is true.
$\therefore \;\;\lambda =1$ satisfied the eq.(i) but $\lambda =2,3,4$ does not satisfy the eq.(i). For all possible values of a, d.
 Question 4
Type II error in hypothesis testing is
 A acceptance of the null hypothesis when it is false and should be rejected B rejection of the null hypothesis when it is true and should be accepted C rejection of the null hypothesis when it is false and should be rejected D acceptance of the null hypothesis when it is true and should be accepted
Engineering Mathematics   Probability and Statistics
 Question 5
The solution of the partial differential equation $\frac{\partial u}{\partial t}=\alpha \frac{\partial^2 u}{\partial x^2}$ is of the form
 A $C\cos \left ( kt \right )\left \lfloor C_{1}e^{(\sqrt{k/\alpha })x} + C_{2}e^{-(\sqrt{k/\alpha })x} \right \rfloor$ B $Ce^{kt}\left \lfloor C_{1}e^{(\sqrt{k/\alpha })x} + C_{2}e^{-(\sqrt{k/\alpha })x} \right \rfloor$ C $Ce^{kt}\left \lfloor C_{1}\cos (\sqrt{k/\alpha } )x+ C_{2}\sin {(-\sqrt{k/\alpha })x} \right \rfloor$ D $C\sin (kt)\left \lfloor C_{1}\cos (\sqrt{k/\alpha } )x+ C_{2}\sin {(-\sqrt{k/\alpha })x} \right \rfloor$
Engineering Mathematics   Ordinary Differential Equation
Question 5 Explanation:
The PDE $\frac{\partial u}{\partial t}=\alpha \frac{\partial^2 u}{\partial x^2} \;\;...(i)$
Solution of (i) is,
$u\left ( x,t \right )=\left ( A\cos px+B\sin px \right )Ce^{-p^{2}\alpha t}$
Put $-p^{2}\alpha =k$
$\Rightarrow \;\; p=\sqrt{-\frac{k}{\alpha }}=\sqrt{\frac{k}{\alpha }}i$
Putting value of p in eq.(i),
\begin{aligned}u\left ( x,t \right )&=\left ( A\cos \sqrt{\frac{k}{\alpha }}x+b\sin h\sqrt{\frac{k}{\alpha }}x \right )Ce^{kt} \\ &=Ce^{kt}\left [ A\left \{ \frac{e^{\sqrt{\frac{k}{\alpha }}x}+e^{-\sqrt{\frac{k}{\alpha }}x}}{2} \right \}+B\left \{ \frac{e^{\sqrt{\frac{k}{\alpha }}x}-e^{-\sqrt{\frac{k}{\alpha }}x}}{2} \right \} \right ] \\ &=Ce^{kt}\left [ e^{\sqrt{\frac{k}{\alpha }}x}\left \{ \frac{A+B}{2} \right \}+e^{-\sqrt{\frac{k}{\alpha }}x}\left \{ \frac{A-B}{2} \right \} \right ] \\ &=Ce^{kt}\left [ c_{1}e^{\sqrt{\frac{k}{\alpha }}x}+c_{2}e^{-\sqrt{\frac{k}{\alpha }}x} \right ] \end{aligned}
 Question 6
Consider the plane truss with load P as shown in the figure. Let the horizontal and vertical reactions at the joint B be $H_{B}$ and $V_{B}$, respectively and $V_{C}$ be the vertical reaction at the joint C.

Which one of the following sets gives the correct values of $V_{B}, H_{B}$ and $V_{C}$ ?
 A $V_{B}=0;H_{B}=0;V_{C}=P$ B $V_{B}=P/2;H_{B}=0;V_{C}=P/2$ C $V_{B}=P/2;H_{B}=P\left (\sin 60^{^{\circ} }\ \right);V_{C}=P/2$ D $V_{B}=P;H_{B}=P\left (\cos 60^{^{\circ} }\ \right);V_{C}=0$
Structural Analysis   Trusses
Question 6 Explanation:

\begin{aligned} V_{B}+V_{C} &=P &\ldots(i)\\ \Sigma M_{B} &=0\\ V_{C} \times 2 L-P \times 2 L&=0 \\ \Rightarrow \quad V_{C}&=P \\ \text{From(i), }\quad V_{B}&=0 \\ \Sigma F_{x}&=0 \\ \Rightarrow \quad H_{B}&=0 \end{aligned}
 Question 7
In shear design of an RC beam, other than the allowable shear strength of concrete $(\tau _{c})$ there is also an additional check suggested in IS 456-2000 with respect to the maximum permissible shear stress $(\tau _{cmax})$. The check for $(\tau _{cmax})$ is required to take care of
 A additional shear resistance from reinforcing steel B additional shear stress that comes from accidental loading C possibility of failure of concrete by diagonal tension D possibility of crushing of concrete by diagonal compression
RCC Structures   Shear, Torsion, Bond, Anchorage and Development Length
 Question 8
The semi-compact section of a laterally unsupported steel beam has an elastic section modulus, plastic section modulus and design bending compressive stress of $500 \; cm^3, 650 \; cm^3$ and 200MPa, respectively. The design flexural capacity (expressed in kNm) of the section is ____.
 A 1000 B 100 C 120 D 1200
Design of Steel Structures   Beams
Question 8 Explanation:
Design flexural capacity,
$M_{d}= \beta _{b}Z_{p}\frac{f_{y}}{\gamma _{m0}}$
$\beta _{b}= 1.0$ (for plastic and compact sections)
$\; \; \; \; = \frac{Z_{e}}{Z_{p}}$ (for semi-compact sections)
$Z_{p}=\,$ Plastic section modulus
$Z_{e}=\,$ elastic section modulus
$f_{y}=\,$ yeild stress of steel provided
$\gamma_{m0}=\,$ material factor of safety of steel against yeilding
As, design bending stress is directly given,
\begin{aligned} M_{d}&= \beta _{b}Z_{p}f_{d} \\ &= \frac{Z_{e}}{Z_{p}}\times Z_{p}\times f_{d} \\ &= Z_{e}\times f_{d} \\ &= 500\times 10^{3}\times 200\times 10^{-6} \\ &= 100 kN-m \end{aligned}
 Question 9
Bull's trench kiln is used in the manufacturing of
 A Lime B cement C bricks D none of these
Construction Materials and Management
Question 9 Explanation:
Bull Trench kiln is a continuous kiln generally oval in plan. It is 50 to 100 m long and 1.5-2.5 m deep below ground level. It is devided into 8-12 sections which is used for manufacturing of bricks.
 Question 10
The compound which is largely responsible for initial setting and early strength gain of Ordinary Portland Cement is
 A $C_{3}A$ B $C_{3}S$ C $C_{2}S$ D $C_{4}AF$
Construction Materials and Management
Question 10 Explanation:
Tricalcium Silicate $\left ( C_{3}S \right )$ hardens rapidly and is largely responsible for initial set and early strength. In general, the early strength of potland cement concrete is higher with incresed percentage of $\left ( C_{3}S \right )$
There are 10 questions to complete.

## GATE CE 2016 SET-2

 Question 1
The spot speeds (expressed in km/hr) observed at a road section are 66, 62, 45, 79, 32, 51, 56, 60, 53, and 49. The median speed (expressed in km/hr) is ________.
(Note: answer with one decimal accuracy)
 A 54.5 B 51.5 C 53.5 D 56
Engineering Mathematics   Probability and Statistics
Question 1 Explanation:
Median speed is the speed at the middle value in series of spot speeds that are arranged in ascending order. 50% of speed values will be greater than the median 50% will be less than the median.
Ascending order order of spot speed studies are 32, 39, 45, 51, 53, 56, 60, 62, 66, 79
Median speed=$\frac{53+56}{2}$=54.5 km/hr
 Question 2
The optimum value of the function $f(x)=x^{2}-4x+2$ is
 A 2 (maximum) B 2 (minimum) C $-2$ (maximum) D $-2$ (minimum)
Engineering Mathematics   Calculus
Question 2 Explanation:
\begin{aligned} {f}'&=0 \\ \Rightarrow \;\; 2x-4&=0 \\ \Rightarrow \;\; x&=2 \text{ (stationary point)}\\ {f}''\left ( x \right )&=2 \gt 0 \\ \Rightarrow\;\; f(x)& \text{ is minimum at } x=2\end{aligned}
i.e., $\left ( 2 \right )^{2}-4\left ( 2 \right )+2=-2$
$\therefore$ The optimum value of $f(x)$ is $-2$ (minimum).
 Question 3
The Fourier series of the function,
$\begin{matrix} f(x) & =0 & -\pi \lt x \leq 0 \\ f(x) &=\pi-x & 0 \lt x \lt \pi \end{matrix}$

in the interval $[-\pi ,\pi ]$ is

$f(x)=\frac{\pi }{4}+\frac{2}{\pi }\left [ \frac{\cos x}{1^{2}}+\frac{\cos 3x}{3^{2}}+\cdots\: \cdots \ \cdots \right ]$ $+ \left [ \frac{\sin x}{1}+\frac{\sin 2x}{2}+\frac{\sin 3x}{3}+\cdots \: \cdots\: \cdot \right ]$

The convergence of the above Fourier series at x = 0 gives
 A $\sum_{n=1}^{\infty }\frac{1}{n^{2}}=\frac{\pi ^{2}}{6}$ B $\sum_{n=1}^{\infty }\frac{(-1)^{n+1}}{n^{2}}=\frac{\pi ^{2}}{12}$ C $\sum_{n=1}^{\infty }\frac{1}{(2n-1)^{2}}=\frac{\pi ^{2}}{8}$ D $\sum_{n=1}^{\infty }\frac{(-1)^{n+1}}{(2n-1)}=\frac{\pi}{4}$
Engineering Mathematics   Partial Differential Equation
Question 3 Explanation:
The function is $f(x)=0$
$-p\lt x\leq 0$
$=p-x,\, 0 \lt x \lt \pi$
And Fourier series is,
$f\left ( x \right )=\frac{\pi }{4}+\frac{2}{\pi }\left [ \frac{\cos x}{1^{2}}+\frac{\cos 3x}{3^{2}}+\frac{\cos 5x}{5^{2}}+... \right ]+\left [ \frac{\sin x}{1}+\frac{\sin 2x}{2}+\frac{\sin 3x}{3}+... \right ] ...\left ( i \right )$
At x=0, (a point of discontinuity), the fourier series converges to $\frac{1}{2}\left [ f\left ( 0^{-1} \right )+f\left ( 0^{+} \right ) \right ]$
where $f\left ( 0^{-} \right )=\lim_{x\rightarrow 0}\left ( \pi -x \right )=\pi$
Hence, eq. (i), we get,
$\frac{\pi }{2}=\frac{\pi }{4}+\frac{2}{\pi }\left [ \frac{1}{1^{2}}+\frac{1}{3^{2}}+... \right ]$
$\Rightarrow \;\; \frac{1}{1}+\frac{1}{3^{2}}+\frac{1}{5^{2}}+...\frac{\pi ^{2}}{8}$
 Question 4
X and Y are two random independent events. It is known that $P(X)=0.40$ and $P(X\cup Y^{C})=0.7$. Which one of the following is the value of $P(X\cup Y)$ ?
 A 0.7 B 0.5 C 0.4 D 0.3
Engineering Mathematics   Probability and Statistics
Question 4 Explanation:
$\; \; \; \; P\left ( X\: \cup \: Y^{c} \right )=0.7$
$\Rightarrow \; \; P\left ( X \right )+P\left ( Y^{c} \right )-P\left ( X \right )P\left ( Y^{c} \right )=0.7$
(Since X, Y are independent events)
$\Rightarrow \; \; P\left ( X \right )+1-P\left ( Y \right )-P\left ( X \right )\left \{ 1-P\left ( Y \right ) \right \}=0$
$\Rightarrow \; \; P\left ( X \right )-P\left ( X\: \cap \: Y \right )=0.3\; \; \; \; \; \; ...\left ( i \right )$
$\; \; \; \; P\left ( X\: \cup \: Y \right )=P\left ( X \right )+P\left ( Y \right )-P\left ( X\: \cap \: Y \right )$
$\; \; \; \; =0.4+0.3=0.7$
 Question 5
What is the value of $\lim_{\begin{matrix} x\rightarrow 0\\ y\rightarrow 0 \end{matrix}} \frac{xy}{x^{2}+y^{2}}$ ?
 A 1 B -1 C 0 D Limit does not exist
Engineering Mathematics   Calculus
Question 5 Explanation:
(i) $\lim_{x\rightarrow \infty }\frac{xy}{x^{2}+y^{2}}\lim_{y\rightarrow \infty }\left ( \frac{0}{0^{2}+y^{2}} \right )=0$
(i.e., put $x=0$ and then $y=0$)
(ii) $\lim_{x\rightarrow 0 y\rightarrow 0}\frac{xy}{x^{2}+y^{2}}\lim_{x\rightarrow 0}\left ( \frac{0}{x^{2}+0} \right )=0$
( i.e., put $y=0$ and then $x=0$)
(iii)$\lim_{x\rightarrow 0 y\rightarrow 0}\frac{xy}{x^{2}+y^{2}}\lim_{x\rightarrow 0}\frac{x\left ( mx \right )}{x^{2}+m^{2}x^{2}}$
(i.e., put $y=mx$)
$\lim_{x\rightarrow \infty }\left ( \frac{m}{1+m^{2}} \right )=\frac{m}{1+m^{2}}$
which depends on m.
 Question 6
The kinematic indeterminacy of the plane truss shown in the figure is
 A 11 B 8 C 3 D 0
Structural Analysis   Determinacy and Indeterminacy
Question 6 Explanation:
Kinematic indeterminacy,
\begin{aligned} D_{k}&=2 j-r_{e} \\ &=2 \times 7-3=11 \end{aligned}
 Question 7
As per IS 456-2000 for the design of reinforced concrete beam, the maximum allowable shear stress $\tau _{cmax}$ depends on the
RCC Structures   Shear, Torsion, Bond, Anchorage and Development Length
 Question 8
An assembly made of a rigid arm A-B-C hinged at end A and supported by an elastic rope C-D at end C is shown in the figure. The members may be assumed to be weightless and the lengths of the respective members are as shown in the figure.

Under the action of a concentrated load P at C as shown, the magnitude of tension developed in the rope is
 A $\frac{3P}{\sqrt{2}}$ B $\frac{P}{\sqrt{2}}$ C $\frac{3P}{8}$ D $\sqrt{2}P$
Engineering Mechanics
Question 8 Explanation:

\begin{aligned} \sum M_{A}&=0\\ \Rightarrow \; R_{D}\times 2 L-P\times L&=0\\ \Rightarrow R_{D}&=\frac{P}{2} \end{aligned}
At joint D:

\begin{aligned} \sum F_{y}&=0\\ \Rightarrow \; T \cos 45^{\circ}&=\frac{P}{2}\\ \therefore \; T&=\frac{P}{\sqrt{2}} \end{aligned}
 Question 9
As per Indian standards for bricks, minimum acceptable compressive strength of any class of burnt clay bricks in dry state is
 A 10.0MPa B 7.5MPa C 5.0MPa D 3.5MPa
Construction Materials and Management
Question 9 Explanation:
As per IS 1077 : 1992 clause 4.1, minimum strength of burnt clay bricks is 3.5 Mpa
 Question 10
A construction Project consists of twelve activities.The estimated duration (in days) required to complete each of the activities along with the corresponding network diagram is shown below.

Total floats (in days) for the activities 5-7 and 11-12 for the project are, respectively,
 A 25 and 1 B 1 and 1 C 0 and 0 D 81 and 0
Construction Materials and Management
Question 10 Explanation:
Total float can be determined once the activity times i.e. EST, EFT, LST and LFT are known.
total float,\begin{aligned} F_{T}&=LST-EST\\ &=LFT-EFT \end{aligned}

For activity 5-7,
\begin{aligned} EST&=38\\ EFT&=63\\ LFT&=63\\ LST&=38\\ F_{T}&=0 \end{aligned}
for activity 11-12,
\begin{aligned} EST&=80\\ EFT&=81\\ LFT&=81 \\ LST&=80 \\ F_T&=0 \end{aligned}
$\textbf{Note}$:It can be seen directly that since the slack of all events are zero, there is not margin left for the occurence of events and therefore.
Maximum available line=Time required for completion of activity
$\therefore \; F_{t}$ for all activities is zero.
There are 10 questions to complete.