## Verbal Ability

 Question 1
The world is going through the worst pandemic in the past hundred years. The air travel industry is facing a crisis, as the resulting quarantine requirement for travelers led to weak demand.
In relation to the first sentence above, what does the second sentence do?
 A Restates an idea from the first sentence. B Second sentence entirely contradicts the first sentence. C The two statements are unrelated. D States an effect of the first sentence.
GATE ME 2021 SET-2   General Aptitude
Question 1 Explanation:
First option is wrong because second sentence does not contradict the first sentence. Third option is wrong because two sentences are related. Fourth option is wrong because the second sentence does not repeat the first one.
Hence second option is correct which shows the result of the cause.
 Question 2
Given below are two statements 1 and 2, and two conclusions I and II.

Statement 1: All entrepreneurs are wealthy.
Statement 2: All wealthy are risk seekers.

Conclusion I: All risk seekers are wealthy.
Conclusion II: Only some entrepreneurs are risk seekers.

Based on the above statements and conclusions, which one of the following options is CORRECT?
 A Only conclusion I is correct B Only conclusion II is correct C Neither conclusion I nor II is correct D Both conclusions I and II are correct
GATE ME 2021 SET-2   General Aptitude
Question 2 Explanation:
Possible cases are:

Conclusion-I is incorrect becaue some risk seeker are wealthy.
Conclusion-II is also incorrect because all the entrepreneurs are risk seeker as well as wealthy
 Question 3
Consider the following sentences:

(i) The number of candidates who appear for the GATE examination is staggering.
(ii) A number of candidates from my class are appearing for the GATE examination.
(iii) The number of candidates who appear for the GATE examination are staggering.
(iv) A number of candidates from my class is appearing for the GATE examination.

Which of the above sentences are grammatically CORRECT?
 A (i) and (ii) B (i) and (iii) C (ii) and (iii) D (ii) and (iv)
GATE ME 2021 SET-2   General Aptitude
Question 3 Explanation:
"The number of" is singular and it takes singular verb. "A number of " is plural and it takes plural verb.
 Question 4
Oxpeckers and rhinos manifest a symbiotic relationship in the wild. The oxpeckers warn the rhinos about approaching poachers, thus possibly saving the lives of the rhinos. Oxpeckers also feed on the parasitic ticks found on rhinos.
In the symbiotic relationship described above, the primary benefits for oxpeckers and rhinos respectively are,
 A Oxpeckers get a food source, rhinos have no benefit. B Oxpeckers save their habitat from poachers while the rhinos have no benefit. C Oxpeckers get a food source, rhinos may be saved from the poachers. D Oxpeckers save the lives of poachers, rhinos save their own lives.
GATE ME 2021 SET-1   General Aptitude
Question 4 Explanation:
Option (A) and (B) are weekend by expression 'rhinos have no benefit'. Oxpeckers do not save life of poachers, so option (D) is incorrect.
Hence, option (C) is correct.
 Question 5
"The increased consumption of leafy vegetables in the recent months is a clear indication that the people in the state have begun to lead a healthy lifestyle"

Which of the following can be logically inferred from the information presented in the above statement?
 A The people in the state did not consume leafy vegetables earlier. B Consumption of leafy vegetables may not be the only indicator of healthy lifestyle. C Leading a healthy lifestyle is related to a diet with leafy vegetables. D The people in the state have increased awareness of health hazards causing by consumption of junk foods.
GATE ME 2021 SET-1   General Aptitude
Question 5 Explanation:
The last sentence of the passage is reflecting in option (c) only.
 Question 6
Ms. X came out of a building through its front door to find her shadow due to the morning sun falling to her right side with the building to her back. From this, it can be inferred that building is facing _____
 A North B East C West D South
GATE ME 2021 SET-1   General Aptitude
Question 6 Explanation:
Morning sun is falling from east then the shadow will fall to the west. So west should be on the right side of Ms. X. So Ms. X came out towards south. Hence, her building is facing south.
 Question 7
Consider the following sentences:

(i) After his surgery, Raja hardly could walk.
(ii) After his surgery, Raja could barely walk.
(iii) After his surgery, Raja barely could walk.
(iv) After his surgery, Raja could hardly walk.

Which of the above sentences are grammatically CORRECT
 A (i) and (ii) B (i) and (iii) C (iii) and (iv) D (ii) and (iv)
GATE ME 2021 SET-1   General Aptitude
Question 7 Explanation:
Hardly/Scarcely/Barely have same sense that is negative and they are used after the verb (could barely and could hardly).
 Question 8
Climate change and resilience deal with two aspects - reduction of sources of non-renewable energy resources and reducing vulnerability of climate change aspects. The terms 'mitigation' and 'adaptation' are used to refer to these aspects, respectively.
Which of the following assertions is best supported by the above information?
 A Mitigation deals with consequences of climate change B Adaptation deals with causes of climate change C Mitigation deals with actions taken to reduce the use of fossil fuels. D Adaptation deals with actions taken to combat green-house gas emissions
GATE ME 2020 SET-2   General Aptitude
Question 8 Explanation:
 Question 9
Select the word that fits the analogy:
White: Whitening : : Light : _____
 A Lightning B Lightening C CLighting D Enlightening
GATE ME 2020 SET-2   General Aptitude
Question 9 Explanation:
 Question 10
The recent measures to improve the output would ______ the level of production to our satisfaction.
 A increase B decrease C Cspeed D equalise
GATE ME 2020 SET-2   General Aptitude
Question 10 Explanation:

There are 10 questions to complete.

## Numerical Ability

 Question 1
Consider a square sheet of side 1 unit. The sheet is first folded along the main diagonal. This is followed by a fold along its line of symmetry. The resulting folded shape is again folded along its line of symmetry. The area of each face of the final folded shape, in square units, equal to _____
 A $\frac{1}{4}$ B $\frac{1}{8}$ C $\frac{1}{16}$ D $\frac{1}{32}$
GATE ME 2021 SET-2   General Aptitude
Question 1 Explanation:

Area$=\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}=\frac{1}{8}$
 Question 2

The ratio of the area of the inscribed circle to the area of the circumscribed circle of an equilateral triangle is
 A $\frac{1}{8}$ B $\frac{1}{6}$ C $\frac{1}{4}$ D $\frac{1}{2}$
GATE ME 2021 SET-2   General Aptitude
Question 2 Explanation:

\begin{aligned} \sin 30^{\circ} &=\frac{r}{R} \\ \text { Area ratio } &=\frac{\pi r^{2}}{\pi R^{2}}=\sin ^{2} 30=\frac{1}{4} \end{aligned}
 Question 3
A box contains 15 blue balls and 45 black balls. If 2 balls are selected randomly, without replacement, the probability of an outcome in which the first selected is a blue ball and the second selected is a black ball, is ____
 A $\frac{3}{16}$ B $\frac{45}{236}$ C $\frac{1}{4}$ D $\frac{3}{4}$
GATE ME 2021 SET-2   General Aptitude
Question 3 Explanation:
The probability of first ball is blue and second ball is black is given as,
$P=\frac{15}{60} \times \frac{45}{59}=\frac{45}{236}$
 Question 4
The front door of Mr. X's house faces East. Mr. X leaves the house, walking 50 m straight from the back door that is situated directly opposite to the front door. He then turns to his right, walks for another 50 m and stops. The direction of the point Mr. X is now located at with respect to the starting point is ____
 A South-East B North-East C West D North-West
GATE ME 2021 SET-2   General Aptitude
Question 4 Explanation:

 Question 5
If $\oplus \div \odot =2;\;\oplus \div \triangle =3;\;\odot +\triangle =5;\;\triangle \times \otimes =10$,
Then, the value of $( \otimes - \oplus)^2$ is
 A 0 B 1 C 4 D 16
GATE ME 2021 SET-2   General Aptitude
Question 5 Explanation:
\begin{aligned} \frac{\oplus}{\odot}&=2, \frac{\oplus}{\Delta}=3 \\ \therefore \qquad\frac{\Delta}{\odot}&=\frac{2}{3} &\ldots(1)\\ \odot+\Delta&=5 &\ldots(2)\\ \text{From (1) and (2)}\\ \Delta&=2, \odot=3\\ \text{and}\\ \oplus &=6,2 \times \otimes=10 \\ \otimes &=5 \\ \Rightarrow \quad(\otimes-\oplus)^{2}&=(5-6)^{2} =1 \end{aligned}
 Question 6
A digital watch X beeps every 30 seconds while watch Y beeps every 32 seconds. They beeped together at 10 AM. The immediate next time that they will beep together is ____
 A 10.08 AM B 10.42 AM C 11.00 AM D 10.00 PM
GATE ME 2021 SET-2   General Aptitude
Question 6 Explanation:
LCM of (30 and 32) is 480
480 seconds = 8 minutes
Hence, time will be 10.08 pm
 Question 7
Five persons P, Q, R, S and T are to be seated in a row, all facing the same direction, but not necessarily in the same order. P and T cannot be seated at either end of the row. P should not be seated adjacent to S. R is to be seated at the second position from the left end of the row. The number of distinct seating arrangements possible is:
 A 2 B 3 C 4 D 5
GATE ME 2021 SET-2   General Aptitude
Question 7 Explanation:
The possible distinct arrangement are
S R P T A, A R P T S, S R T P A
Hence, number of distinct sitting arrangement. = 3
 Question 8
Five persons P, Q, R, S and T are sitting in a row not necessarily in the same order. Q and R are separated by one person, and S should not be seated adjacent to Q.
The number of distinct seating arrangements possible is:
 A 4 B 8 C 10 D 16
GATE ME 2021 SET-1   General Aptitude
Question 8 Explanation:
The possible seating arrangements are
QPRST, QTRSP
QPRTS, QTRPS
PQTRS, TQPRS
SPQTR, STQPR
RPQTS, RTQPS
SRPQT, SRTQP
SPRTQ, STRPQ
PSRTQ, TSRPQ
Hence, total seating arrangements are 16
 Question 9

The distribution of employees at the rank of executives, across different companies C1, C2,... ,C6 is presented in the chart given above. The ratio of executives with a management degree to those without a management degree in each of these companies is provided in the table above. The total number of executives across all companies is 10,000.

The total number of management degree holders among the executives in companies C2 and C5 together is .
 A 225 B 600 C 1900 D 2500
GATE ME 2021 SET-1   General Aptitude
Question 9 Explanation:
Number of employee in $C2$ company $=\frac{5}{100} \times 10000=500$
Number of management degree holder employee in $\mathrm{C} 2=\frac{1}{5} \times 500=100$
Number of employee in $\mathrm{C} 5$ company $=\frac{20}{100} \times 10000=2000$
Number of management degree holder employee in $\mathrm{C} 5=\frac{9}{10} \times 2000=1800$
Total management degree holder employee $=100+1800=1900$
 Question 10
The number of hens, ducks and goats in farm P are 65, 91 and 169, respectively. The total number of hens, ducks and goats in a nearby farm Q is 416. The ratio of hens:ducks:goats in farm Q is 5:14:13. All the hens, ducks and goats are sent from farm Q to farm P.
The new ratio of hens:ducks:goats in farm P is
 A 5:07:13 B 5:14:13 C 10:21:26 D 21:10:26
GATE ME 2021 SET-1   General Aptitude
Question 10 Explanation:
In farm P,
Hens =65, Ducks =91, Goats =169
In farm Q,
Hens : Ducks : Goats
5: 14: 13
\begin{aligned} \text { Hens } &=\frac{5}{32} \times 416=65 \\ \text { Ducks }&=\frac{14}{32} \times 416=182\\ \text { Goats }&=\frac{13}{32} \times 416=169 \end{aligned}
$\because$From farm d, hens, ducks and goats are sent to farm P.
\begin{aligned} \therefore \text{ Total hens }&=65+65=130\\ \text { Total ducks } &=91+182=273 \\ \text { Total goats } &=169+169=338 \\ \text { New ratio } &=130: 273: 338 \\ &=10: 21: 26 \end{aligned}

There are 10 questions to complete.

## General Aptitude

 Question 1
The world is going through the worst pandemic in the past hundred years. The air travel industry is facing a crisis, as the resulting quarantine requirement for travelers led to weak demand.
In relation to the first sentence above, what does the second sentence do?
 A Restates an idea from the first sentence. B Second sentence entirely contradicts the first sentence. C The two statements are unrelated. D States an effect of the first sentence.
GATE ME 2021 SET-2      Verbal Ability
Question 1 Explanation:
First option is wrong because second sentence does not contradict the first sentence. Third option is wrong because two sentences are related. Fourth option is wrong because the second sentence does not repeat the first one.
Hence second option is correct which shows the result of the cause.
 Question 2
Consider a square sheet of side 1 unit. The sheet is first folded along the main diagonal. This is followed by a fold along its line of symmetry. The resulting folded shape is again folded along its line of symmetry. The area of each face of the final folded shape, in square units, equal to _____
 A $\frac{1}{4}$ B $\frac{1}{8}$ C $\frac{1}{16}$ D $\frac{1}{32}$
GATE ME 2021 SET-2      Numerical Ability
Question 2 Explanation:

Area$=\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}=\frac{1}{8}$
 Question 3

The ratio of the area of the inscribed circle to the area of the circumscribed circle of an equilateral triangle is
 A $\frac{1}{8}$ B $\frac{1}{6}$ C $\frac{1}{4}$ D $\frac{1}{2}$
GATE ME 2021 SET-2      Numerical Ability
Question 3 Explanation:

\begin{aligned} \sin 30^{\circ} &=\frac{r}{R} \\ \text { Area ratio } &=\frac{\pi r^{2}}{\pi R^{2}}=\sin ^{2} 30=\frac{1}{4} \end{aligned}
 Question 4
A box contains 15 blue balls and 45 black balls. If 2 balls are selected randomly, without replacement, the probability of an outcome in which the first selected is a blue ball and the second selected is a black ball, is ____
 A $\frac{3}{16}$ B $\frac{45}{236}$ C $\frac{1}{4}$ D $\frac{3}{4}$
GATE ME 2021 SET-2      Numerical Ability
Question 4 Explanation:
The probability of first ball is blue and second ball is black is given as,
$P=\frac{15}{60} \times \frac{45}{59}=\frac{45}{236}$
 Question 5
Given below are two statements 1 and 2, and two conclusions I and II.

Statement 1: All entrepreneurs are wealthy.
Statement 2: All wealthy are risk seekers.

Conclusion I: All risk seekers are wealthy.
Conclusion II: Only some entrepreneurs are risk seekers.

Based on the above statements and conclusions, which one of the following options is CORRECT?
 A Only conclusion I is correct B Only conclusion II is correct C Neither conclusion I nor II is correct D Both conclusions I and II are correct
GATE ME 2021 SET-2      Verbal Ability
Question 5 Explanation:
Possible cases are:

Conclusion-I is incorrect becaue some risk seeker are wealthy.
Conclusion-II is also incorrect because all the entrepreneurs are risk seeker as well as wealthy
 Question 6
The front door of Mr. X's house faces East. Mr. X leaves the house, walking 50 m straight from the back door that is situated directly opposite to the front door. He then turns to his right, walks for another 50 m and stops. The direction of the point Mr. X is now located at with respect to the starting point is ____
 A South-East B North-East C West D North-West
GATE ME 2021 SET-2      Numerical Ability
Question 6 Explanation:

 Question 7
If $\oplus \div \odot =2;\;\oplus \div \triangle =3;\;\odot +\triangle =5;\;\triangle \times \otimes =10$,
Then, the value of $( \otimes - \oplus)^2$ is
 A 0 B 1 C 4 D 16
GATE ME 2021 SET-2      Numerical Ability
Question 7 Explanation:
\begin{aligned} \frac{\oplus}{\odot}&=2, \frac{\oplus}{\Delta}=3 \\ \therefore \qquad\frac{\Delta}{\odot}&=\frac{2}{3} &\ldots(1)\\ \odot+\Delta&=5 &\ldots(2)\\ \text{From (1) and (2)}\\ \Delta&=2, \odot=3\\ \text{and}\\ \oplus &=6,2 \times \otimes=10 \\ \otimes &=5 \\ \Rightarrow \quad(\otimes-\oplus)^{2}&=(5-6)^{2} =1 \end{aligned}
 Question 8
A digital watch X beeps every 30 seconds while watch Y beeps every 32 seconds. They beeped together at 10 AM. The immediate next time that they will beep together is ____
 A 10.08 AM B 10.42 AM C 11.00 AM D 10.00 PM
GATE ME 2021 SET-2      Numerical Ability
Question 8 Explanation:
LCM of (30 and 32) is 480
480 seconds = 8 minutes
Hence, time will be 10.08 pm
 Question 9
Consider the following sentences:

(i) The number of candidates who appear for the GATE examination is staggering.
(ii) A number of candidates from my class are appearing for the GATE examination.
(iii) The number of candidates who appear for the GATE examination are staggering.
(iv) A number of candidates from my class is appearing for the GATE examination.

Which of the above sentences are grammatically CORRECT?
 A (i) and (ii) B (i) and (iii) C (ii) and (iii) D (ii) and (iv)
GATE ME 2021 SET-2      Verbal Ability
Question 9 Explanation:
"The number of" is singular and it takes singular verb. "A number of " is plural and it takes plural verb.
 Question 10
Five persons P, Q, R, S and T are to be seated in a row, all facing the same direction, but not necessarily in the same order. P and T cannot be seated at either end of the row. P should not be seated adjacent to S. R is to be seated at the second position from the left end of the row. The number of distinct seating arrangements possible is:
 A 2 B 3 C 4 D 5
GATE ME 2021 SET-2      Numerical Ability
Question 10 Explanation:
The possible distinct arrangement are
S R P T A, A R P T S, S R T P A
Hence, number of distinct sitting arrangement. = 3

There are 10 questions to complete.

## GATE Mechanical Engineering 2021 SET-2

 Question 1
Consider an n x n matrix $A$ and a non-zero n x 1 vector $p$. Their product $Ap=\alpha ^2p$, where $\alpha \in \mathbb{R}$ and $\alpha \notin \{-1,0,1\}$. Based on the given information, the eigen value of $A^2$ is:
 A $\alpha$ B $\alpha ^2$ C $\sqrt{\alpha }$ D $\alpha ^4$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Given, $A P=\alpha^{2} P$
By comparison with $A X=\lambda X \Rightarrow$
$\Rightarrow \quad \lambda=\alpha^{2}$
Hence, eigen value of A is $\alpha^{2}$, so eigen value of $A^{2}$ is $\alpha^{4}$.
 Question 2
If the Laplace transform of a function $f(t)$ is given by $\frac{s+3}{(s+1)(s+2)}$ , then $f(0)$ is
 A 0 B $\frac{1}{2}$ C 1 D $\frac{3}{2}$
Engineering Mathematics   Differential Equations
Question 2 Explanation:
By using partial fraction concept.
\begin{aligned} f(t) &=L^{-1}\left[\frac{s+3}{(s+1)(s+2)}\right] \\ &=L^{-1}\left[\frac{2}{s+1}-\frac{1}{s+2}\right] \\ \Rightarrow \qquad f(t) &=2 e^{-t}-e^{-2 t} \\ \text { So, } \qquad f(c)&=2 e^{0}-e^{0}=2-1=1 \end{aligned}
 Question 3
The mean and variance, respectively, of a binomial distribution for $n$ independent trials with the probability of success as $p$, are
 A $\sqrt{np},np(1-2p)$ B $\sqrt{np}, \sqrt{np(1-p)}$ C $np,np$ D $np,np(1-p)$
Engineering Mathematics   Probability and Statistics
Question 3 Explanation:
Mean= np
Variance = npq = np(1 - p)
 Question 4
The Cast Iron which possesses all the carbon in the combined form as cementite is known as
 A Grey Cast Iron B Spheroidal Cast Iron C Malleable Cast Iron D White Cast Iron
Manufacturing Engineering   Engineering Materials
Question 4 Explanation:
On the basis of nature of carbon present in cast iron, it may be divided into white cast iron and gray cast iron.
In the gray cast iron, carbon is present in free form as graphite. Under very slow rate of cooling during solidification, carbon atoms get sufficient time to separate out in pure form as graphite. In addition, certain elements promote decomposition of cementite. Silicon and nickel are two commonly used graphitizing elements.
In white cast iron, carbon is present in the form of combined form as cementite. In normal conditions, carbon has a tendency to combine with iron to form cementite.
 Question 5
The size distribution of the powder particles used in Powder Metallurgy process can be determined by
 A Laser scattering B Laser reflection C Laser absorption D Laser penetration
Manufacturing Engineering   Forming Process
Question 5 Explanation:
Particle Size, Shape, and Distribution:
Particle size is generally controlled by screening, that is, by passing the metal powder through screens (sieves) of various mesh sizes. Several other methods also are available for particle-size analysis:
1. Sedimentation, which involves measuring the rate at which particles settle in a fluid.
2. Microscopic analysis, which may include the use of transmission and scanning- electron microscopy.
3. Light scattering from a laser that illuminates a sample, consisting of particles suspended in a liquid medium; the particles cause the light to be scattered, and a detector then digitizes the signals and computes the particle-size distribution.
4. Optical methods, such as particles blocking a beam of light, in which the particle is sensed by a photocell.
5. Suspending particles in a liquid and detecting particle size and distribution by electrical sensors.
 Question 6
In a CNC machine tool, the function of an interpolator is to generate
 A signal for the lubrication pump during machining B error signal for tool radius compensation during machining C NC code from the part drawing during post processing D reference signal prescribing the shape of the part to be machined
Manufacturing Engineering   Computer Integrated Manufacturing
Question 6 Explanation:
In contouring systems the machining path is usually constructed from a combination of linear and circular segments. It is only necessary to specify the coordinates of the initial and final points of each segment, and the feed rate. The operation of producing the required shape based on this information is termed interpolation and the corresponding unit is the "interpolator". The interpolator coordinates the motion along the machine axes, which are separately driven, by providing reference positions instant by instant for the position-and velocity control loops, to generate the required machining path. Typical interpolators are capable of generating linear and circular paths.
 Question 7
The machining process that involves ablation is
 A Abrasive Jet Machining B Chemical Machining C Electrochemical Machining D Laser Beam Machining
Manufacturing Engineering   Machining and Machine Tool Operation
Question 7 Explanation:
Laser beam machining (LBM) is a nonconventional machining process, which broadly refers to the process of material removal, accomplished through the interactions between the laser and target materials. The processes can include laser drilling, cutting, grooving, writing, scribing, ablation, welding, cladding, milling, and so on. LBM is a thermal process, and unlike conventional mechanical processes, LBM removes material without mechanical engagement. In general, the workpiece is heated to melting or boiling point and removed by melt ejection, vaporization, or ablation.
 Question 8
A PERT network has 9 activities on its critical path. The standard deviation of each activity on the critical path is 3. The standard deviation of the critical path is
 A 3 B 9 C 27 D 81
Industrial Engineering   PERT and CPM
Question 8 Explanation:
In CPM,
$\begin{array}{l} \sigma=\sqrt{\text { sum of variance along critical path }} \\ \sigma=\sqrt{\sigma^{2}+\sigma^{2}+\ldots .+\sigma^{2}} \\ \sigma=\sqrt{9 \sigma^{2}}=\sqrt{9 \times 9}=9 \end{array}$
 Question 9
The allowance provided in between a hole and a shaft is calculated from the difference between
 A lower limit of the shaft and the upper limit of the hole B upper limit of the shaft and the upper limit of the hole C upper limit of the shaft and the lower limit of the hole D lower limit of the shaft and the lower limit of the hole
Manufacturing Engineering   Metrology and Inspection
Question 9 Explanation:
It is minimum clearance or maximum interference. It is the intentional difference between the basic dimensions of the mating parts. The allowance may be positive or negative.

 Question 10
In forced convective heat transfer, Stanton number (St), Nusselt number (Nu), Reynolds number (Re) and Prandtl number (Pr) are related as
 A $\text{St}=\frac{\text{Nu}}{\text{Re Pr}}$ B $\text{St}=\frac{\text{Nu Pr}}{\text{Re}}$ C $\text{St}=\text{Nu Pr Re}$ D $\text{St}=\frac{\text{Nu Re}}{\text{Pr}}$
Heat Transfer   Free and Forced Convection
Question 10 Explanation:
$S t=\frac{N u}{R e \times P r}$
There are 10 questions to complete.

## GATE Mechanical Engineering 2021 SET-1

 Question 1
If $y(x)$ satisfies the differential equation

$(\sin x)\frac{dy}{dx}+y \cos x =1$

subject to the condition $y(\pi /2)=\pi /2$, then $y(\pi /6)$ is
 A 0 B $\frac{\pi}{6}$ C $\frac{\pi}{3}$ D $\frac{\pi}{2}$
Engineering Mathematics   Differential Equations
Question 1 Explanation:
\begin{aligned} \frac{d y}{d x}+y \cot x&=\text{cosec} x\\ 1.F. \qquad&=e^{\int \cot x d x}=e^{\log \sin x}=\sin x\\ \Rightarrow \quad y(\sin x)&=\int \text{cosec} x \sin x d x+c\\ \Rightarrow \qquad y \sin x&=x+c\\ \Rightarrow \qquad \frac{\pi}{2} \sin \frac{\pi}{2} & =\frac{\pi}{2}+c \\ \Rightarrow \qquad \frac{\pi}{2} & =\frac{\pi}{2}+c \quad \Rightarrow c=0 \\ \Rightarrow \qquad y \sin x & =x\\ \Rightarrow \qquad y \sin \frac{\pi}{6}&=\frac{\pi}{6}\\ \Rightarrow \qquad y\left(\frac{1}{2}\right) &=\frac{\pi}{6} \\ \Rightarrow y &=\frac{\pi}{3} \end{aligned}
 Question 2
The value of $\lim_{x \to 0}\left ( \frac{1- \cos x}{x^2} \right )$ is
 A $\frac{1}{4}$ B $\frac{1}{3}$ C $\frac{1}{2}$ D 1
Engineering Mathematics   Calculus
Question 2 Explanation:
\begin{aligned} \lim _{x \rightarrow 0}\left(\frac{1-\cos x}{x^{2}}\right)&=? \;\;\;\;\;\;\left(\frac{0}{0} \text { form }\right) \\ \text { Applying } L \cdot H \text { rule } & =\lim _{x \rightarrow 0} \frac{\sin x}{2 x}\left(\frac{0}{0}\right)=\lim _{x \rightarrow 0} \frac{\cos x}{2}=\frac{1}{2} \end{aligned}
 Question 3
The Dirac-delta function $(\delta (t-t_0)) \text{ for }t,t_0 \in \mathbb{R},$ has the following property

$\int_{a}^{b}\varphi (t)\delta (t-t_0)dt=\left\{\begin{matrix} \varphi (t_0) & a \lt t_0 \lt b\\ 0 &\text{otherwise} \end{matrix}\right.$

The Laplace transform of the Dirac-delta function $\delta (t-a)$ for $a \gt 0$;
$\mathcal{L} (\delta (t-a))=F(s)$ is
 A 0 B $\infty$ C $e^{sa}$ D $e^{-sa}$
Engineering Mathematics   Differential Equations
Question 3 Explanation:
\begin{aligned} \because \qquad \int_{0}^{-} f(t) \delta(t-a) d t&=f(a) \\ \therefore \qquad L\{\delta(t-a)\}&=\int_{0}^{-} e^{-s t} \delta(t-a) d t=e^{-a s} \end{aligned}
 Question 4
The ordinary differential equation $\frac{dy}{dt}=-\pi y$ subject to an initial condition $y(0)=1$ is solved numerically using the following scheme:

$\frac{y(t_{n+1})-y(t_n)}{h}=-\pi y(t_n)$

where $h$ is the time step, $t_n=nh,$ and $n=0,1,2,...$. This numerical scheme is stable for all values of $h$ in the interval.
 A $0 \lt h \lt \frac{2}{\pi}$ B $0 \lt h \lt 1$ C $0 \lt h \lt \frac{\pi}{2}$ D for all $h \gt 0$
Engineering Mathematics   Differential Equations
Question 4 Explanation:
\begin{aligned} \frac{y\left(t_{n+1}\right)-y\left(t_{n}\right)}{h} &=-\pi y\left(t_{n}\right) \\ y_{n+1} &=-\pi / y_{n}+y_{n}=(-\pi h+1) y_{n} \end{aligned}
It is recursion relation between $y_{n+1}$ and $y_{n}$
So solution will be stable if
\begin{aligned} |-\pi h+1| & \lt 1 \\ -1 \lt -\pi h+1 & \lt 1 \\ -2 \lt -\pi h & \lt 0 \\ 0 & \lt \pi h \lt 2 \\ 0 & \lt h \lt \frac{2}{\pi} \end{aligned}
Therefore option (A) is correct.
 Question 5
Consider a binomial random variable $X$. If $X_1,X_2,..., X_n$ are independent and identically distributed samples from the distribution of $X$ with sum $Y=\sum_{i=1}^{n}X_i$, then the distribution of $Y$ as $n\rightarrow \infty$ can be approximated as
 A Exponential B Bernoulli C Binomial D Normal
Engineering Mathematics   Probability and Statistics
 Question 6
The loading and unloading response of a metal is shown in the figure. The elastic and plastic strains corresponding to 200 MPa stress, respectively, are

 A 0.01 and 0.01 B 0.02 and 0.01 C 0.01 and 0.02 D 0.02 and 0.02
Strength of Materials   Stress and Strain
Question 6 Explanation:
Elastic strain : Which can be recovered $= 0.03 - 0.01 = 0.02$
Plastic strain : Permanent strain $= 0.01$
 Question 7
In a machining operation, if a cutting tool traces the workpiece such that the directrix is perpendicular to the plane of the generatrix as shown in figure, the surface generated is

 A plane B cylindrical C spherical D a surface of revolution
Manufacturing Engineering   Machining and Machine Tool Operation
Question 7 Explanation:

 Question 8
The correct sequence of machining operations to be performed to finish a large diameter through hole is
 A drilling, boring, reaming B boring, drilling, reaming C drilling, reaming, boring D boring, reaming, drilling
Manufacturing Engineering   Machining and Machine Tool Operation
Question 8 Explanation:
Drilling: to produce a hole, which then may be followed by boring it to improve its dimensional accuracy and surface finish.

Boring: to enlarge a hole or cylindrical cavity made by a previous process or to produce circular internal grooves.

Reaming: is an operation used to (a) make an existing hole dimensionally more accurate than can br achived by drilling alone and (b) improve its surface finish. The most accurate holes in workpieces generally are produced by the following sequence of operation.

Centering -> Drilling -> Boring -> Reaming.
 Question 9
In modern CNC machine tools, the backlash has been eliminated by
 A preloaded ballscrews B rack and pinion C ratchet and pinion D slider crank mechanism
Manufacturing Engineering   Computer Integrated Manufacturing
Question 9 Explanation:

 Question 10
Consider the surface roughness profile as shown in the figure.

The center line average roughness ($R_a \text{ in }\mu m$) of the measured length (L) is
 A 0 B 1 C 2 D 4
Manufacturing Engineering   Metrology and Inspection
Question 10 Explanation:
$R_{G}=\frac{\sum_{i=1}^{n} y}{n}=\frac{4}{4}=1$
There are 10 questions to complete.

## GATE 2022 Mechanical Engineering Syllabus

Revised syllabus of GATE 2022 Mechanical Engineering by IIT.

Practice GATE Mechanical Engineering previous year questions

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

## GATE ME 2014 SET-4

 Question 1
Which one of the following equations is a correct identity for arbitrary 3x3 real matrices P, Q and R?
 A $P(Q+R)=PQ+RP$ B $(P-Q)^{2}=P^{2}-2PQ+Q^{2}$ C $det(P+Q)=det \; P+det \; Q$ D $(P+Q)^{2}=P^{2}+PQ+QP+Q^{2}$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
\begin{aligned} (P+Q)^{2} &=P^{2}+P Q+Q P+Q^{2} \\ &=P \cdot P+P \cdot Q+Q \cdot P+Q \cdot Q \\ &=P^{2}+P Q+Q P+Q^{2} \end{aligned}
 Question 2
The value of the integral $\int_{0}^{2}\frac{(x-1)^{2}\sin(x-1)}{(x-1)^{2}+\cos(x-1)}dx$ is
 A 3 B 0 C -1 D -2
Engineering Mathematics   Calculus
Question 2 Explanation:
\begin{aligned} I &= \int_{0}^{2}\frac{(x-1)^2 \sin (x-1)}{(x-1)^2 + \cos (x-1)}dx\\ \text{Taking}\; x-1=z&\Rightarrow dx=dz \\ \text{for}\; x=0,&z\rightarrow -1\; \text{and}\; x=2, z \rightarrow 1 \\ \therefore \; I &=\int_{-1}^{1} \frac{z^2 \sin z}{z^2+\cos z}dz\\ \text{let}\;\; f(z)&=\frac{z^2 \sin z}{z^2+ \cos z} dz\\ f(-z) &= \frac{z^2 \sin z}{z^2+ \cos z}\\ f(z)&= -f(z) \; \text{function is ODD}\\ \therefore \;\; I&=0 \end{aligned}
 Question 3
The solution of the initial value problem $\frac{\mathrm{d} y}{\mathrm{d} x}= -2xy;$ y(0)=2 is
 A $1+e^{-x^{2}}$ B $2e^{-x^{2}}$ C $1+e^{x^{2}}$ D $2e^{x^{2}}$
Engineering Mathematics   Differential Equations
Question 3 Explanation:
$\frac{d y}{d x}=2 x y=0 \qquad ...(1)$
$I F.=e^{\int 2 x d x}=e^{x^{2}}$
Multiplying I.F. to both side of equation (1)
$e^{x^{2}}\left[\frac{d y}{d x}+2 x y\right]=0$
$\Rightarrow \frac{d}{d x}\left(e^{x^{2}} y\right)=0$
$e^{x^{2}} y=c$
from the given boundary condition, C=2
$\therefore e^{x^{2}} y=2$
$y=2 e^{-x^{2}}$
 Question 4
A nationalized bank has found that the daily balance available in its savings accounts follows a normal distribution with a mean of Rs. 500 and a standard deviation of Rs. 50. The percentage of savings account holders, who maintain an average daily balance more than Rs. 500 is _______
 A 12% B 65% C 98% D 50%
Engineering Mathematics   Probability and Statistics
Question 4 Explanation:
49 to 51
$Z=\frac{x-\mu}{\sigma}$
$\text { Given, } x=500, \mu=500, \sigma=50$
$\therefore \quad z=\frac{500-500}{50}=0$
$\therefore P(0)=50 \%$
 Question 5
Laplace transform of $cos(\omega t)$ is $\frac{s}{s^{2}+\omega ^{2}}$. The Laplace transform of $e^{-2t}cos(4t)$ is
 A $\frac{s-2}{(s-2)^{2}+16}$ B $\frac{s+2}{(s-2)^{2}+16}$ C $\frac{s-2}{(s+2)^{2}+16}$ D $\frac{s+2}{(s+2)^{2}+16}$
Engineering Mathematics   Differential Equations
Question 5 Explanation:
Given $L \cos (\omega t)=\frac{s}{s^{2}+\omega^{2}}$
$\Rightarrow L\left(e^{-2 t} \cos 4 t\right)=?$
By the given formula
$\Rightarrow \quad L(\cos 4 t)=\frac{s}{s^{2}+16}$
$\Rightarrow L\left(e^{2 t} \cos 4 t\right)=\frac{s+2}{(s+2)^{2}+16}$
( $\because$ By using first shifting property of Laplace)
Hence option should be (D).
 Question 6
In a statically determinate plane truss, the number of joints (j) and the number of members (m) are related by
 A $j=2m-3$ B $m=2j+1$ C $m=2j-3$ D $m=2j-1$
Theory of Machine   Planar Mechanisms
Question 6 Explanation:
A simple truss is formed by enlarging the basic truss element which contains three members and three joints, by adding two additional members for each additional joint, so the total number of members m in a simple truss is given by
$m=3+2(j-3)=2 j-3$
where m= number of members
j= total number of joints (including those attached to the supports)
 Question 7
If the Poisson's ratio of an elastic material is 0.4, the ratio of modulus of rigidity to Young's modulus is _______
 A 0.357 B 0.123 C 0.2658 D 1.354
Strength of Materials   Stress-strain Relationship and Elastic Constants
Question 7 Explanation:
$(0.35 \text { to } 0.36)$
$E=2 G(1+v)$
$\frac{G}{E}=\frac{1}{2(1+v)}=\frac{1}{2 \times 1.4}=\frac{1}{2.8}=0.357$
 Question 8
Which one of the following is used to convert a rotational motion into a translational motion?
 A Bevel gears B Double helical gears C Worm gears D Rack and pinion gears
Machine Design   Gears
Question 8 Explanation:
Bevel gears: Rotational motion transfer between axes at right angle.
Worm gears: For large reduction ratio in a single stage.
Double helical gears: Rotational motion transfer between parallel axes.
Rack and Pinion gears: Rotational to linear motion conversion.
 Question 9
The number of independent elastic constants required to define the stress-strain relationship for an isotropic elastic solid is _______
 A 0.5 B 1 C 1.5 D 2
Strength of Materials   Stress-strain Relationship and Elastic Constants
Question 9 Explanation:
(1.9 to 2.1)
Either E or G, 2 independent constant
Either G or K, 2 independent constant
Either E or K, 2 independent constant
 Question 10
A point mass is executing simple harmonic motion with an amplitude of 10 mm and frequency of 4 Hz. The maximum acceleration (m/$s^2$) of the mass is _______
 A 6.78 B 9.54 C 6.31 D 3.98
Theory of Machine   Vibration
Question 10 Explanation:
\begin{aligned} f_{n} &=\frac{\omega_{n}}{2 \times \pi} \\ \Rightarrow \omega_{n} &=2 \pi f=2 \times 3.14 \times 4=25.12 \mathrm{rad} / \mathrm{s} \\ a_{\max } &=\omega_{n}^{2} x=(25.12)^{2} \times 10 \times 10^{-3} \\ &=6.31 \mathrm{m} / \mathrm{s}^{2} \end{aligned}
There are 10 questions to complete.

## GATE ME 2015 SET-1

 Question 1
If any two columns of a determinant P= $\begin{vmatrix} 4 & 7 & 8\\ 3 & 1 & 5\\ 9& 6 & 2 \end{vmatrix}$ are interchanged, which one of the following statements regarding the value of the determinant is CORRECT?
 A Absolute value remains unchanged but sign will change. B Both absolute value and sign will change. C Absolute value will change but sign will not change D Both absolute value and sign will remain unchanged
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Property of determinant: If any two row or column are interchanged, then mangnitude of determinant remains same but sign changes.
 Question 2
Among the four normal distributions with probability density functions as shown below, which one has the lowest variance?
 A I B II C III D IV
Engineering Mathematics   Probability and Statistics
Question 2 Explanation:
We know that probability density function of normal distribution is given by
$F(x)=\frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{1}{2}\left ( \frac{x-\mu }{\sigma } \right )^2}$
For having lowest variance $(\sigma ^2)$, standard deviation $(\sigma )$ will be lowest. From the density function $f(x)$, we can say that as σ decreases $F(x)$ will increase, so curve having highest peak has lowest standard deviation and variance.
 Question 3
Simpson's $1/3$ rule is used to integrate the function $\frac{3}{5}x^{2}+\frac{9}{5}$ between x = 0 and x = 1 using the least number of equal sub -intervals. The value of the integral is _____________
 A 1 B 2 C 3 D 4
Engineering Mathematics   Numerical Methods
Question 3 Explanation:
$f(x)=\frac{3}{5} x^{2}+\frac{9}{5}$
$\begin{array}{|c|c|c|c|}\hline x & 0 & 0.5 & 1 \\\hline f(x) & 1.8 & 1.95 & 2.4 \\\hline\end{array}\\ \Rightarrow\int_{0}^{1} f(x)=\frac{h}{3}\left[y_{0}+4 y_{1}+y_{2}\right]$
$=\frac{0.5}{3}[1.8+4(1.95)+2.4]=2$
 Question 4
The value of $\lim_{x\rightarrow 0}\frac{1-cos(x^{2})}{2x^{4}}$ is
 A 0 B $1/2$ C $1/4$ D undefined
Engineering Mathematics   Calculus
Question 4 Explanation:
$\lim _{x \rightarrow 0} \frac{1-\cos \left(x^{2}\right)}{2 x^{4}} \quad$ putting the x $\rightarrow 0$
we get $\frac{0}{0}$ form
Applying L' Hospital rule
$\Rightarrow \lim _{x \rightarrow 0} \frac{2 x \sin \left(x^{2}\right)}{8 x^{3}} \quad$
$\Rightarrow \lim _{x \rightarrow 0} \frac{\sin \left(x^{2}\right)}{4 x^{2}}$
$\Rightarrow \frac{1}{4} \lim _{x \rightarrow 0} \frac{\sin \left(x^{2}\right)}{x^{2}}$
$\Rightarrow \frac{1}{4} \lim _{x^{2} \rightarrow 0} \frac{\sin \left(x^{2}\right)}{x^{2}}=\frac{1}{4} \times 1=\frac{1}{4}$
 Question 5
Given two complex numbers $z_{1}=5+(5\sqrt{3})i$ and $z_{2}=2/\sqrt{3}+2i$, the argument of $z_{1}/z_{2}$ in degrees is
 A 0 B 30 C 60 D 90
Engineering Mathematics   Complex Variables
Question 5 Explanation:
\begin{aligned} z_{1} &=5+(5 \sqrt{3}) i \\ z_{2} &=\frac{2}{\sqrt{3}}+2 i \\ \arg \left(z_{1}\right) &=\theta_{1}=\tan ^{-1}\left(\frac{5 \sqrt{3}}{5}\right) \\ \theta_{1} &=60^{\circ} \\ \arg \left(z_{2}\right) &=\theta_{2}=\tan ^{-1}\left(\frac{2}{2 \sqrt{3}}\right) \\ \theta_{2} &=60^{\circ} \\ \arg \left(\frac{z_{1}}{z_{2}}\right) &=\arg \left(z_{1}\right)-\arg \left(z_{2}\right) \\ &=60-60=0^{\circ} \end{aligned}
 Question 6
Consider fully developed flow in a circular pipe with negligible entrance length effects. Assuming the mass flow rate, density and friction factor to be constant, if the length of the pipe is doubled and the diameter is halved, the head loss due to friction will increase by a factor of
 A 4 B 16 C 32 D 64
Fluid Mechanics   Flow Through Pipes
Question 6 Explanation:
\begin{aligned} h_{f}&=\frac{f L V^{2}}{2 g d}\\ &\text{Mass flow rate,}\\ m&=\rho A V=\rho \times \frac{\pi}{4} d^{2} V\\ or\quad V&=\frac{4 m}{\rho \pi d^{2}} \\ \therefore \quad h_{f}&=\frac{f L}{2 g d} \times \frac{16 m^{2}}{\rho^{2} \pi^{2} d^{4}}\\ &=\frac{8 f m^{2}}{\rho^{2} \pi^{2}} \times \frac{L}{d^{5}} \\ &=C \times \frac{L}{d^{5}} \text { where } C=\frac{8 f m^{2}}{\rho^{2} \pi^{2}} \\ h_{f_{1}}&=C \times \frac{L}{d^{5}}\\ and\quad h_{2} &=C \times \frac{2 L}{(d / 2)^{5}}=\frac{C \times 2 L}{\frac{d^{5}}{2^{5}}} \\ &=\frac{C \times 2 \times 2^{5} L}{d^{5}}=\frac{C \times 64 L}{d^{5}} \\ h_{1 / 2} &=64 h_{11} \end{aligned}
The head loss due to friction will increase by a factor of 64
 Question 7
The Blasius equation related to boundary layer theory is a
 A third-order linear partial differential equation B third-order nonlinear partial differential equation C second-order nonlinear ordinary differential equation D third-order nonlinear ordinary differential equation
Heat Transfer   Heat Transfer in Flow Over Plates and Pipes
Question 7 Explanation:
$2 \frac{d^{3} f}{d \eta^{3}}+f \frac{d^{2} f}{d \eta^{2}}=0$
third-order non-linear differential equation.
 Question 8
For flow of viscous fluid over a flat plate, if the fluid temperature is the same as the plate temperature, the thermal boundary layer is
 A thinner than the velocity boundary layer B thicker than the velocity boundary layer C of the same thickness as the velocity boundary layer D not formed at all
Heat Transfer   Heat Transfer in Flow Over Plates and Pipes
Question 8 Explanation:
For the flow of viscous fluid over a flat plate if the fluid temperature is the same as the plate temperature the thermal boundary layer is not formed at all because boundary layer formation took place if there is some difference in fluid property of no slip layer and remaining fluid.
As here given that fluid is viscous and flowing over a flat plate. Here, definitely a kinematic boundary layer will be formed but as there is no temperature difference so no formation of thermal boundary layer.
 Question 9
For an ideal gas with constant values of specific heats, for calculation of the specific enthalpy,
 A it is sufficient to know only the temperature B both temperature and pressure are required to be known C both temperature and volume are required to be known D both temperature and mass are required to be known
Thermodynamics   First Law, Heat, Work and Energy
Question 9 Explanation:
For constant valves of specific heats

dh = F(T) only

So for calculation of specific enthalpy it is sufficient to know only the temperature.
 Question 10
A Carnot engine (CE-1) works between two temperature reservoirs A and B, where $T_{A}$ = 900 K and $T_{B}$ = 500 K. A second Carnot engine (CE-2) works between temperature reservoirs B and C, where $T_{C}$ = 300 K. In each cycle of CE-1 and CE-2, all the heat rejected by CE-1 to reservoir B is used by CE-2. For one cycle of operation, if the net Q absorbed by CE-1 from reservoir A is 150 MJ, the net heat rejected to reservoir C by CE-2 (in MJ) is ______________
 A 20 B 30 C 50 D 60
Thermodynamics   Second Law, Carnot Cycle and Entropy
Question 10 Explanation:

For Carnot engine (CE-2)
\begin{aligned} \eta_{2}&=1-\frac{T_{C}}{T_{B}}\\ \text{also}\qquad \eta_{2} &=1-\frac{Q_{C}}{Q_{B}} \\ \therefore \quad 1-\frac{T_{C}}{T_{B}} &=1-\frac{Q_{C}}{Q_{B}}\\ \text{or}\qquad\frac{T_{C}}{T_{B}} &=\frac{Q_{C}}{Q_{B}} \\ \frac{300}{500} &=\frac{Q_{C}}{\frac{5 \times 150}{9}}\\ \text{or}\qquad Q_{c}&=50 \mathrm{MJ} \end{aligned}
For Carnot engine (CE-2)
\begin{aligned} \eta_{2}&=1-\frac{T_{C}}{T_{B}}\\ \text{also}\qquad \eta_{2} &=1-\frac{Q_{C}}{Q_{B}} \\ \therefore \quad 1-\frac{T_{C}}{T_{B}} &=1-\frac{Q_{C}}{Q_{B}}\\ \text{or}\qquad\frac{T_{C}}{T_{B}} &=\frac{Q_{C}}{Q_{B}} \\ \frac{300}{500} &=\frac{Q_{C}}{\frac{5 \times 150}{9}}\\ \text{or}\qquad Q_{c}&=50 \mathrm{MJ} \end{aligned}
There are 10 questions to complete.

## GATE ME 2015 SET-2

 Question 1
At least one eigenvalue of a singular matrix is
 A positive B zero C negative D imaginary
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
For singular matrix
$|A|=0$
According to properties of eigen value
Product of eigen values $=|A|=0$
$\Rightarrow$ Atleast one of the eigen value is zero.
 Question 2
At x = 0, the function $f(x)=\left | x \right |$ has
 A a minimum B a maximum C a point of inflexion D neither a maximum nor minimum
Engineering Mathematics   Calculus
Question 2 Explanation:
The graph of |x| is

from the graph we can say that
|x| has minimum at x=0
 Question 3
Curl of vector $V(x,y,z)= 2x^{2}i+3z^{2}j+y^{3}k$ at $x=y=z=1$ is
 A $-3i$ B $3i$ C $3i-4j$ D $3i-6k$
Engineering Mathematics   Calculus
Question 3 Explanation:
Curl of vector$=\left|\begin{array}{ccc} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ 2 x^{2} & 3 z^{2} & y^{3} \end{array}\right|$
$=i\left[\frac{\partial}{\partial x}\left(y^{3}\right) \frac{\partial}{\partial z}\left(3 z^{2}\right)\right]+j\left[\frac{\partial}{\partial x}\left(y^{3}\right) \frac{\partial}{\partial z}\left(2 x^{2}\right)\right]$
$+k\left[\frac{\partial}{\partial x}\left(3z^{2}\right) \frac{\partial}{\partial z}\left(2 x^{2}\right)\right]$
$=i\left[3 y^{2}-6 z\right]-[10]+k[0+0]$
$\text { At } x=1, y=1 \text { and } z=1$
$\text { Curl }=i\left(3 \times 1^{2}-6 \times 1\right)=-3 i$
 Question 4
The Laplace transform of $e^{i5t}$ where $i=\sqrt{-1}$ is
 A $(s-5i)/(s^{2}-25)$ B $(s+5i)/(s^{2}+25)$ C $(s+5i)/(s^{2}-25)$ D $(s-5i)/(s^{2}+25)$
Engineering Mathematics   Differential Equations
Question 4 Explanation:
\begin{aligned} e^{j 5 t} &=\cos 5 t+i \sin 5 t \\ L\left\{e^{(i 5 f)}\right\}&=\frac{s}{s^{2}+25}+\frac{5 i}{s^{2}+25} \\ &=\frac{s+5 i}{s^{2}+25} 2 e^{-x^{2}} \end{aligned}
 Question 5
Three vendors were asked to supply a very high precision component. The respective probabilities of their meeting the strict design specifications are 0.8, 0.7 and 0.5. Each vendor supplies one component. The probability that out of total three components supplied by the vendors, at least one will meet the design specification is ___________
 A 0.12 B 0.97 C 0.65 D 1
Engineering Mathematics   Probability and Statistics
Question 5 Explanation:
Probability of atleast one meet the specification
\begin{aligned} &=1-(\bar{A} \cap \bar{B} \cap \bar{C}) \\ &=1-(0.2 \times 0.3 \times 0.5) \\ &=0.97 \end{aligned}
 Question 6
A small ball of mass 1 kg moving with a velocity of 12 m/s undergoes a direct central impact with a stationary ball of mass 2 kg. The impact is perfectly elastic. The speed (in m/s) of 2 kg mass ball after the impact will be ____________
 A 8m/s B 9m/s C 9m/s D 4m/s
Engineering Mechanics   Impulse and Momentum, Energy Formulations
Question 6 Explanation:

1. Conserving linear momentum
\begin{aligned} 1 \times 12 &=1 \times V_{1}+2 \times V_{2} \\ 12 &=V_{1}+2 V_{2}\qquad \ldots(i) \end{aligned}
2. Velocity of approach = Velocity of seperation
$12-0=v_{2}-v_{1}$
$V_{2}-V_{1}=12 \qquad \ldots(ii)$
From (i) and (ii), we get
$v_{2}=8 \mathrm{m} / \mathrm{s}$
 Question 7
A rod is subjected to a uni-axial load within linear elastic limit. When the change in the stress is 200 MPa, the change in the strain is 0.001. If the Poisson's ratio of the rod is 0.3, the modulus of rigidity (in GPa) is ________________
 A 76.9230GPa B 12.35698GPa C 19.2365GPa D 98.1458GPa
Strength of Materials   Stress-strain Relationship and Elastic Constants
Question 7 Explanation:

With in linear elastic limit
$\sigma=E \in$
$E \rightarrow \text{ slope of } \sigma \text{ vs }\in \text{ curve}$
\begin{aligned} E &=\frac{d \sigma}{d \epsilon}=\frac{200}{0.001}=200 \mathrm{GPa} \\ E &=2 G[1+\mu] \\ G &=\frac{E}{2(1+\mu)}=\frac{200}{2(1+0.3)}=\frac{100}{1.3} \\ &=76.9230 \mathrm{GPa} \end{aligned}
 Question 8
A gas is stored in a cylindrical tank of inner radius 7 m and wall thickness 50 mm. The gage pressure of the gas is 2 MPa. The maximum shear stress (in MPa) in the wall is
 A 35 B 70 C 140 D 280
Strength of Materials   Thin Cylinder
Question 8 Explanation:
Maximum shear stress in the wall
$=\frac{\sigma_{1}}{2}=\frac{p d}{4 t}=\frac{2 \times 14 \times 1000}{4 \times 50}=140 \mathrm{MPa}$
 Question 9
The number of degrees of freedom of the planetary gear train shown in the figure is
 A 0 B 1 C 2 D 3
Theory of Machine   Gear and Gear Train
Question 9 Explanation:
Degree of freedom: $F=3(l-1)-2 j-h$
$=3(4-1)-2 \times 3-1=2$
 Question 10
In a spring-mass system, the mass is m and the spring constant is k. The critical damping coefficient of the system is 0.1 kg/s. In another spring-mass system, the mass is 2m and the spring constant is 8k. The critical damping coefficient (in kg/s) of this system is ____________
 A 0.4kg/s B 1kg/s C 0.9kg/s D 2kg/s
Theory of Machine   Vibration
Question 10 Explanation:
\begin{aligned} \xi &=\frac{C}{2 \sqrt{m k}} \\ C &=2 \xi \sqrt{m k} \\ \frac{C_{2}}{C_{1}} &=\frac{2 \xi_{2} \sqrt{m_{2} k_{2}}}{2 \xi_{1} \sqrt{m_{1} k_{1}}} \\ \frac{C_{2}}{0.1} &=\frac{2 \times 1 \sqrt{2 m \times 8 k}}{2 \times 1 \sqrt{m \times k}} \\ C_{2} &=0.4 \mathrm{kg} / \mathrm{s} \end{aligned}
There are 10 questions to complete.