## 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 Mechanical Engineering 2019 SET-1

 Question 1
Consider the matrix
$P=\begin{bmatrix} 1 & 1 &0 \\ 0&1 &1 \\ 0& 0 & 1 \end{bmatrix}$
The number of distinct eigenvalues of P is
 A 0 B 1 C 2 D 3
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
$\text { Given: } A=\left[\begin{array}{lll} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right]$
It is an upper triangular matrix. It's diagonal elements are eigen values.
The eigen values of the matrix are 1, 1, 1.
$\therefore$Number of distinct eigen values = 1
Hence, option (B) is correct.
 Question 2
A parabola $x=y^2 \; with \; 0\leq x\leq 1$ is shown in the figure. The volume of the solid of rotation obtained by rotating the shaded area by 360$^{\circ}$ around the x-axis is
 A $\frac{\pi}{4}$ B $\frac{\pi}{2}$ C $\pi$ D $2 \pi$
Engineering Mathematics   Calculus
Question 2 Explanation:
$\text { Given: } y^{2}=x, 0 \leq x \leq 1$
The value of solid obtained by rotating the area bounded by the curve
\begin{aligned} \mathrm{y}^{2}&=\mathrm{x}, 0 \leq \mathrm{x} \leq 1 \text{ about x-axis is}\\ V &=\int_{a}^{b} \pi y^{2} d x \\ V &=\int_{0}^{1} \pi x d x \\ &=\left(\frac{\pi \mathrm{x}^{2}}{2}\right)_{0}^{1} \\ &=\frac{\pi}{2} \end{aligned}
 Question 3
For the equation $\frac{dy}{dx} + 7x^2 y=0$, if y(0)=3/7, then the value of y(1)is
 A $\frac{7}{3}e^{-7/3}$ B $\frac{7}{3}e^{-3/7}$ C $\frac{3}{7}e^{-7/3}$ D $\frac{3}{7}e^{-3/7}$
Engineering Mathematics   Differential Equations
Question 3 Explanation:
Given $\frac{d y}{d x}+7 x^{2} y=0 \ldots(1)$
With $y(0)=\frac{3}{7} \ldots(2)$
Now,(1) is written as
$\Rightarrow \int \frac{1}{y} d y+\int 7 x^{2} d x=C$
$\Rightarrow \log y+\frac{7 x^{3}}{3}=C$
$\Rightarrow y=e^{\frac{7 x^{3}}{3}+c} \; \; ...(3)$
Using (2),(3) becomes $\frac{3}{7}=\mathrm{e}^{0} \cdot \mathrm{e}^{\mathrm{C}}(\mathrm{or}) \mathrm{e}^{\mathrm{C}}=\frac{3}{7} \; \; ...(4)$
$\therefore$ The solution of (1) with (3) &(4) is given by
$y=y(x)=e^{\frac{-7 x^{3}}{3}+c}=e^{\frac{-7 x^{3}}{3}} \cdot e^{c}=\frac{3}{7} \cdot e^{\frac{-7 x^{3}}{3}}$
Hence, $y(1)=y=\frac{3}{7} \cdot e^{-\frac{7}{3}}$
 Question 4
The lengths of a large stock of titanium rods follow a normal distribution with a mean ($\mu$) of 440 mm and a standard deviation ($\sigma$) of 1mm. What is the percentage of rods whose lengths lie between 438 mm and 441 mm?
 A 81.85% B 68.40% C 99.75% D 86.64%
Engineering Mathematics   Probability and Statistics
Question 4 Explanation:

$\begin{array}{l} \mathrm{Z}=\frac{x-\mu}{\sigma} \\ \mathrm{Z}(\mathrm{x}=438)=\frac{438-440}{1}=-2 \\ \mathrm{P}(\mathrm{Z}=-2)=2.28 \% \\ \mathrm{Z}(\mathrm{x}=441)=\frac{441-440}{1}=1 \\ \mathrm{P}(\mathrm{Z}=1)=84.13 \% \\ \end{array}$
The percentage of rods whose lengths lie between 438 mm and 441 mm =
$\begin{array}{c} =\mathrm{P}(\mathrm{Z}=1)-\mathrm{P}(\mathrm{Z}=-2) \\ =84.13 \%-2.28 \%=81.85 \% \end{array}$
 Question 5
A flat-faced follower is driven using a circular eccentric cam rotating at a constant angular velocity $\omega$. At time t=0, the vertical position of the followeris y(0)=0, and the system is in the configuration shown below.

The vertical position of the follower face, y(t) is given by
 A $e \sin \omega t$ B $e(1+ \cos 2\omega t)$ C $e(1- \cos \omega t)$ D $e \sin 2 \omega t$
Theory of Machine   Cams
Question 5 Explanation:

\begin{aligned} \mathrm{AA}_{1}=\mathrm{y} &=\mathrm{e}(1-\cos \theta) \\ &=\mathrm{e}(1-\cos \omega \mathrm{t}) \end{aligned}
 Question 6
The natural frequencies corresponding to the spring-mass systems I and II are $\omega _I \; and \; \omega _{II}$, respectively. The ratio $\frac{\omega _I}{\omega _{II}}$ is
 A $\frac{1}{4}$ B $\frac{1}{2}$ C 2 D 4
Theory of Machine   Vibration
Question 6 Explanation:
System: I
$k_{e q}=\frac{k \cdot k}{k+k}=\frac{k}{2}$
$\omega_{\mathrm{n}_{1}}=\sqrt{\frac{\mathrm{k}}{2 \mathrm{m}}}$
System - II
$\mathrm{k}_{\mathrm{eq}}=2 \mathrm{k} \quad \omega_{\mathrm{n}_{2}}=\sqrt{\frac{2 \mathrm{k}}{\mathrm{m}}}$
$\frac{\omega_{\mathrm{n}_{1}}}{\omega_{\mathrm{n}_{2}}}=\frac{\sqrt{\frac{\mathrm{k}}{2 \mathrm{m}}}}{\sqrt{\frac{2 \mathrm{k}}{\mathrm{m}}}}=\sqrt{\frac{\mathrm{k}}{2 \mathrm{m}} \times \frac{\mathrm{m}}{2 \mathrm{k}}}=\frac{1}{2}$
 Question 7
A spur gear with 20$^{\circ}$ full depth teeth is transmitting 20 kW at 200 rad/s. The pitch circle diameter of the gear is 100 mm. The magnitude of the force applied on the gear in the radial direction is
 A 0.36 kN B 0.73 kN C 1.39 kN D 2.78kN
Theory of Machine   Gear and Gear Train
Question 7 Explanation:
$\begin{array}{l} \phi=20^{\circ}, P=20 k W, \omega=200 \mathrm{rad} / \mathrm{s}, d=100 \mathrm{mm}=0.1 \mathrm{m} \\ \text { Torque }=\text { Power } / \omega \\ \mathrm{T}=\frac{20000}{200}=100 \mathrm{Nm} \\ \text { Now, } \mathrm{T}=\mathrm{F}_{\mathrm{T}} \times \frac{\mathrm{d}}{2} \\ \Rightarrow 100=\mathrm{F}_{\mathrm{T}} \times \frac{0.1}{2} \\ \Rightarrow \mathrm{F}_{\mathrm{T}}=2000 \mathrm{N}\\ \frac{F_{R}}{F_{T}}=\tan \phi \\ \Rightarrow \mathrm{F}_{\mathrm{R}}=2000 \times \tan 20^{\circ} \\ \Rightarrow \mathrm{F}_{\mathrm{R}}=727.94 \mathrm{N}=0.73 \mathrm{kN} \end{array}$
 Question 8
During a non-flow thermodynamic process(1-2) executed by a perfect gas, the heat interaction is equal to the work interaction ($Q_{1-2}=W_{1-2}$) when the process is
 A Isentropic B Polytropic C Isothermal D Adiabatic
Thermodynamics   Thermodynamic System and Processes
Question 8 Explanation:
Given, $\mathrm{Q}_{1-2}=\mathrm{W}_{1-2}$
$\therefore \Delta \mathrm{U}_{1-2}=0$
$\Rightarrow \mathrm{c}_{\mathrm{v}}\left[\mathrm{T}_{2}-\mathrm{T}_{1}\right]=0$
$\Rightarrow \mathrm{T}_{1}=\mathrm{T}_{2}$
So, the process is isothermal.
 Question 9
For a hydrodynamically and thermally fully developed laminar flow through a circular pipe of constant cross-section, the Nusselt number at constant wall heat flux ($Nu_q$) and that at constant wall temperature ($Nu_T$) are related as
 A $Nu_q \gt Nu_T$ B $Nu_q \lt Nu_T$ C $Nu_q = Nu_T$ D $Nu_q = ( Nu_T)^2$
Heat Transfer   Heat Transfer in Flow Over Plates and Pipes
Question 9 Explanation:
For laminar flow through circular tube:
$\mathrm{Nu}_{\mathrm{q}}=4.36$ (For constant heat flux)
$\mathrm{Nu}_{\mathrm{T}}=3.66$ (For constant wall temperature)
$\mathrm{Nu}_{\mathrm{q}} \gt \mathrm{Nu}_{\mathrm{T}}$
 Question 10
As per common design practice, the three types of hydraulic turbines, in descending order of flow rate, are
 A Kaplan, Francis, Pelton B Pelton, Francis, Kaplan C Francis, Kaplan, Pelton D Pelton, Kaplan, Francis
Fluid Mechanics   Turbines and Pumps
Question 10 Explanation:
Kaplan turbine has highest flow area hence it can handle highest discharge. On the other hand, Pelton turbine has lowest flow area hence it works on low discharge.
$\therefore \mathrm{Q}_{\mathrm{Kaplan}} \gt \mathrm{Q}_{\mathrm{Francis}}>\mathrm{Q}_{\mathrm{Pclton}}$
There are 10 questions to complete.

## GATE Mechanical Engineering 2019 SET-2

 Question 1
In matrix equation [A]{X}={R},
$[A]=\begin{bmatrix} 4 & 8 & 4\\ 8& 16 & -4\\ 4& -4 & 15 \end{bmatrix}$, $\{X\}=\begin{Bmatrix} 2\\ 1\\ 4 \end{Bmatrix}$ and $\{R\}=\begin{Bmatrix} 32\\ 16\\ 64 \end{Bmatrix}$.
One of the eigenvalues of matrix [A] is
 A 4 B 8 C 15 D 16
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Given that AX=R
$\begin{array}{l} \Rightarrow\left[\begin{array}{ccc} 4 & 8 & 4 \\ 8 & 16 & -4 \\ 4 & -4 & 15 \end{array}\right]\left[\begin{array}{l} 2 \\ 1 \\ 4 \end{array}\right]=\left[\begin{array}{l} 32 \\ 16 \\ 64 \end{array}\right] \\ \Rightarrow\left[\begin{array}{ccc} 4 & 8 & 4 \\ 8 & 16 & -4 \\ 4 & -4 & 15 \end{array}\right]\left[\begin{array}{l} 2 \\ 1 \\ 4 \end{array}\right]=16\left[\begin{array}{c} 32 \\ 4 \end{array}\right](\because A X=\lambda X) \end{array}$
$\therefore$ One of eigen value of the given matrix A is given by $\lambda=16$
 Question 2
The directional derivative of the function $f(x,y)=x^2+y^2$ along aline directed from (0,0) to (1,1), evaluated at the point x=1, y=1 is
 A $\sqrt{2}$ B 2 C $2 \sqrt{2}$ D $4 \sqrt{2}$
Engineering Mathematics   Calculus
Question 2 Explanation:
Given function is $f(x, y)=x^{2}+y^{2}$
The direction vector a is given by
$\overline{\mathrm{a}}=(1,1)-(0,0)=\overline{\mathrm{i}}+\overline{\mathrm{j}}$
Let the given point be $P=(x, y)=(1,1)$
Now, the directional directive of f(x, y) in the direction of vector a at point is given by
$\mathrm{D} . \mathrm{D}=(\nabla \mathrm{f}) \cdot \cdot \frac{\overline{\mathrm{a}}}{|\mathrm{a}|}$
$\Rightarrow D . D=(2 x \bar{i}+2 x \bar{j}) \cdot \frac{(\bar{i}+\bar{j})}{\sqrt{1+1}}$
$\therefore D . D=\frac{2+2}{\sqrt{2}}=2 \sqrt{2}$
 Question 3
The differential equation $\frac{dy}{dx}+4y=5$ is valid in the domain $0\leq x\leq 1$ with y(0)=2.25. The solution of the differential equation is
 A $y=e^{-4x}+5$ B $y=e^{-4x}+1.25$ C $y=e^{4x}+5$ D $y=e^{4x}+1.25$
Engineering Mathematics   Differential Equations
Question 3 Explanation:
$\begin{array}{l} \text { Given } \frac{\mathrm{dy}}{\mathrm{dx}}+4 \mathrm{y}=5,0 \leq \mathrm{x} \leq 1 \; \; \ldots(1) \\ \because \frac{d y}{d x}+P(x, y)=Q(x) \\ \text { With } \mathrm{y}(0)=2.25\;\;\; \ldots(2) \\ \text { Here, I.F }=\mathrm{e}^{\int 4 \mathrm{dx}}=\mathrm{e}^{4 \mathrm{x}} \end{array}$
The general solution of (1) is given by
$y \cdot e^{4 x}=\int(5)\left(e^{4 x}\right) d x+c$
$\Rightarrow \mathrm{y} \cdot \mathrm{e}^{4 \mathrm{x}}=\frac{5}{4} \mathrm{e}^{4 \mathrm{x}}+\mathrm{c}\cdots(3)$
Using (2) and (3)
$(2.25)(1)=\left(\frac{5}{4}\right)(1)+c$
$c=1 \cdots(4)$
The solution of (1) from (3) and (4) is
$y \cdot e^{4 x}=\frac{5}{4} e^{4 x}+1$
or $y=\frac{5}{4}+e^{4 x}=e^{-4 x}+1.25$
 Question 4
An analytic function f(z) of complex variable z=x+iy may be written as $f(z)=u(x,y)+iv(x,y)$. Then u(x,y) and v(x,y) must satisfy
 A $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=\frac{\partial v}{\partial x}$ B $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$ C $\frac{\partial u}{\partial x}=-\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=\frac{\partial v}{\partial x}$ D $\frac{\partial u}{\partial x}=-\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$
Engineering Mathematics   Complex Variables
Question 4 Explanation:
Given that the complex function f(z)=u(x,y)+ i v(x,y) is an analytic function.
$\Rightarrow$ the Cauchy-Riemann equation will satisfy for u(x,y) & v(x,y)
$\therefore \frac{\partial \mathrm{u}}{\partial \mathrm{x}}=\frac{\partial \mathrm{v}}{\partial \mathrm{y}} \text{ and } \frac{\partial \mathrm{u}}{\partial \mathrm{x}}=-\frac{\partial \mathrm{v}}{\partial \mathrm{y}}$
 Question 5
A rigid triangular body, PQR, with sides of equal length of 1 unit moves on a flat plane. At the instant shown, edge QR is parallel to the x-axis, and the body moves such that velocities of points P and R are $V_P \; and \; V_R$, in the x and y directions, respectively. The magnitude of the angular velocity of the body is
 A $2V_R$ B $2V_P$ C $V_R/\sqrt{3}$ D $V_P/\sqrt{3}$
Theory of Machine   Planar Mechanisms
Question 5 Explanation:
$\begin{array}{l} \Rightarrow \mathrm{V}_{\mathrm{R}}=(\mathrm{IR}) \omega \\ \Rightarrow \omega=\frac{\mathrm{V}_{\mathrm{R}}}{(\mathrm{IR})} \\ \Rightarrow \omega \times \frac{\mathrm{V}_{\mathrm{R}}}{\frac{1}{2}} \\ \Rightarrow \omega=2 \mathrm{V}_{\mathrm{R}} \end{array}$

 Question 6
Consider a linear elastic rectangular thin sheet of metal, subjected to uniform uniaxial tensile stress of 100 MPa along the length direction. Assume plane stress conditions in the plane normal to the thickness. The Young's modulus E=200 MPa and Poisson's ratio v=0.3 are given. The principal strains in the plane of the sheet are
 A (0.35, -0.15) B (0.5, 0.0) C (0.5, -0.15) D (0.5, -0.5)
Strength of Materials   Stress and Strain
Question 6 Explanation:
\begin{aligned} \sigma_{\mathrm{x}}&=100 \mathrm{MPa} \\ v&=\mu=0.3 \\ \sigma_{\mathrm{y}}&=0, \sigma_{\mathrm{z}}=0, \mathrm{E}=200 \mathrm{MPa} \\ &\text { Principal strain in x-direction } \\ &=\epsilon_{1}=\epsilon_{\mathrm{x}}=\frac{\sigma_{\mathrm{x}}}{\mathrm{E}}-\mu \frac{\sigma_{\mathrm{y}}}{\mathrm{E}} \\ &=\frac{100}{200}-0=0.5\\ &\text { Principal strain in y-direction } \\ &=\epsilon_{2}=\epsilon_{\mathrm{y}}=\frac{\sigma_{\mathrm{y}}}{\mathrm{E}}-\mu \frac{\sigma_{\mathrm{x}}}{\mathrm{E}}\\ &=0-(0.3)\left(\frac{100}{200}\right)=-0.15\\ &\therefore\left(\epsilon_{\mathrm{x}}, \epsilon_{\mathrm{y}}\right)=(0.5-0.15) \end{aligned}
 Question 7
A spur gear has pitch circle diameter D and number of teeth T. The circular pitch of the gear is
 A $\frac{\pi D}{T}$ B $\frac{T}{D}$ C $\frac{D}{T}$ D $\frac{2 \pi D}{T}$
Theory of Machine   Gear and Gear Train
Question 7 Explanation:
Circular pitch : It is the distance between two similar points on adjacent teeth measured along pitch
circle circumference circular pitch
$\left(P_{c}\right)=\frac{\text { Pitch circlecircum }}{\text { Number of teeth }}=\frac{\pi D}{T}$
 Question 8
Endurance limit of a beam subjected to pure bending decreases with
 A decrease in the surface roughness and decrease in the size of the beam B increase in the surface roughness and decrease in the size of the beam C increase in the surface roughness and increase in the size of the beam D decrease in the surface roughness and increase in the size of the beam
Question 8 Explanation:
Endurance limit decreases with increase in surface roughness and with increase in size of the beam.
 Question 9
A two-dimensional incompressible frictionless flow field is given by $\vec{u}=x\hat{i}-y\hat{j}$. If $\rho$ is the density of the fluid, the expression for pressure gradient vector at any point in the flow field is given as
 A $\rho (x\hat{i}+y\hat{j})$ B $-\rho (x\hat{i}+y\hat{j})$ C $\rho (x\hat{i}-y\hat{j})$ D $-\rho (x^2 \hat{i} + y^2 \hat{j})$
Fluid Mechanics   Viscous, Turbulent Flow and Boundary Layer Theory
Question 9 Explanation:
Given, 2-D incompressible frictionless fluid flow.
$\overrightarrow{\mathrm{u}}=x \hat{\mathrm{i}}-y \hat{\mathrm{j}}$
Thus, velocity components in x and y directions are :
$\mathrm{u}=\mathrm{x} \text { and } \mathrm{v}=-\mathrm{y}$
Navier-Stokes equation for incompressible, frictionless fluid flow reduces to
$\rho \frac{\mathrm{DV}}{\mathrm{Dt}}=-\nabla \overrightarrow{\mathrm{P}}+\overrightarrow{\rho \mathrm{g}}$
There are no components of body force in x and y direction. Hence,
$\rho \frac{\mathrm{D} \overrightarrow{\mathrm{V}}}{\mathrm{Dt}}=-\nabla \overrightarrow{\mathrm{P}}$
where, $\nabla \overrightarrow{\mathrm{P}}$ is the pressure gradient vector
Hence,
\begin{aligned} \bigtriangledown \vec{P}&=\rho \left [ \left ( u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y} \right )\hat{i}+\left ( u\frac{\partial v}{\partial x} +v\frac{\partial v}{\partial y}\right ) \hat{j}\right ]\\ &=-\rho \left [ \{ x(1)+(-y)(0)\}\hat{i}+\{ x(0)+(-y)(-1)\} \hat{j}\right ]\\ &=-\rho (x\hat{i}+y\hat{j}) \end{aligned}
 Question 10
Sphere-1 with a diameter of 0.1 m is completely enclosed by another sphere-2 of diameter 0.4 m. The view factor $F_{12}$ is
 A 0.0625 B 0.25 C 0.5 D 1
Question 10 Explanation:

Given data:
$\mathrm{d}_{1}=0.1 \mathrm{m}$
$\mathrm{d}_{2}=0.4 \mathrm{m}$
$\mathrm{F}_{1-1}=0$ (from the geometry)
$\mathrm{F}_{1-1}+\mathrm{F}_{1-2}=1$
$\mathrm{F}_{1-2}=1$
There are 10 questions to complete.

## GATE Mechanical Engineering 2020 SET-1

 Question 1
Multiplication of real valued square matrices of same dimension is
 A associative B commutative C always positive definite D not always possible to compute
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Matrix multiplication is associative.
 Question 2
The value of $\lim_{x \to 1} \left ( \frac{1-e^{-c(1-x)}}{1-xe^{-c(1-x)}} \right )$ is
 A c B c+1 C $\frac{c}{c+1}$ D $\frac{c+1}{c}$
Engineering Mathematics   Calculus
Question 2 Explanation:
Applying L Hospital rule
$\lim_{x \to 1}\left ( \frac{1-e^{-c(1-x)}}{1-xe^{-c(1-x)}} \right )=\lim_{x \to 1}\left ( \frac{1-e^{-c+cx}}{-x(ce^{-c+x})-(e^{-c+cx})} \right ) =\frac{-c}{-c-1}=\frac{c}{c+1}$
 Question 3
The Laplace transform of a function f(t) is $L(f)=\frac{1}{s^2+\omega ^2}$. Then f(t) is
 A $f(t)=\frac{1}{\omega ^2}(1-\cos \omega t)$ B $f(t)=\frac{1}{\omega} \cos \omega t$ C $f(t)=\frac{1}{\omega} \sin \omega t$ D $f(t)=\frac{1}{\omega ^2}(1-\sin \omega t)$
Engineering Mathematics   Differential Equations
Question 3 Explanation:
$L(t)=\frac{1}{s^{2}+\omega ^{2}}$
$f(t)=L^{-1}\left \{ \frac{1}{s^{2}+\omega ^{2}} \right \}=\frac{1}{\omega }\sin \omega t$
 Question 4
Which of the following function f(z), of the complex variable z, is NOT analytic at all the points of the complex plane?
 A $f(z)=z^2$ B $f(z)=e^z$ C $f(z)=\sin z$ D $f(z)=\log z$
Engineering Mathematics   Complex Variables
Question 4 Explanation:
logz is not analytic at all points.
 Question 5
The members carrying zero force (i.e. zero-force members) in the truss shown in the figure, for any load $P \gt 0$ with no appreciable deformation of the truss (i.e. with no appreciable change in angles between the members), are
 A BF and DH only B BF, DH and GC only C BF, DH, GC, CD and DE only D BF, DH, GC, FG and GH only
Engineering Mechanics   FBD, Equilirbium, Plane Trusses and Virtual work
Question 5 Explanation:
If at any joint three forces are acting out of which two of them are collinear then force in third member must be zero.
For member ED look at joint E.

Similarity look for other members.
 Question 6
A single-degree-of-freedom oscillator is subjected to harmonic excitation $F(t) = F_0 \cos (\omega t)$ as shown in the figure.

The non-zero value of $\omega$, for which the amplitude of the force transmitted to the ground will be $F_0$, is
 A $\sqrt{\frac{k}{2m}}$ B $\sqrt{\frac{k}{m}}$ C $\sqrt{\frac{2k}{m}}$ D $2 \sqrt{\frac{k}{m}}$
Theory of Machine   Vibration
Question 6 Explanation:
Given,
$F_{1}=F_{0}$
then transmissibility
$\epsilon =\frac{F_{T}}{F_{0}}=\frac{F_{0}}{F_{0}}=1$
\begin{aligned} \frac{\sqrt{1+\left ( \frac{2\xi \omega }{\omega _n} \right )^2}}{\left ( 1-\left ( \frac{\omega }{\omega _n} \right )^2 \right )^2+\left ( \frac{2\xi \omega }{\omega _n} \right )^2}=1 \\ 1+ \left ( \frac{2\xi \omega }{\omega _n} \right )^2=\left ( 1-\left ( \frac{\omega }{\omega _n} \right )^2 \right )^2+ \left ( \frac{2\xi \omega }{\omega _n} \right )^2\\ 1-\left ( \frac{\omega }{\omega _n} \right )^2 =\pm 1 \\ \text{ Taking (-ve) sign,} \\ 1-\left ( \frac{\omega }{\omega _n} \right )^2 =-1 \\ \frac{\omega }{\omega _n} =\sqrt{2}\\ \omega=\sqrt{2}\omega_n=\sqrt{\frac{2k}{m}} \end{aligned}
 Question 7
The stress state at a point in a material under plane stress condition is equi-biaxial tension with a magnitude of 10 MPa. If one unit on the $\sigma -\tau$ plane is 1 MPa, the Mohr's circle representation of the state-of-stress is given by
 A a circle with a radius equal to principal stress and its center at the origin of the $\sigma -\tau$ plane B a point on the $\sigma$ axis at a distance of 10 units from the origin C a circle with a radius of 10 units on the $\sigma -\tau$ plane D a point on the $\tau$ axis at a distance of 10 units from the origin
Strength of Materials   Mohr's Circle
Question 7 Explanation:

The given state of stress is represented by a point on $\sigma -\tau$ graph which is located on $\sigma$-axis at a distance of 10 units from origin.
 Question 8
A four bar mechanism is shown below

For the mechanism to be a crank-rocker mechanism, the length of the link PQ can be
 A 80 mm B 200 mm C 300 mm D 350 mm
Theory of Machine   Displacement, Velocity and Acceleration
Question 8 Explanation:

For Crank-Rocker mechanism, shortest link must be crank and adjacent to fixed as well as Grashoff's law must be satisfied.
If $l = 80 mm$ then shortest will be = 80 mm
as well as (80 + 600) $\lt$(400 + 300)
680 $\lt$ 700
Therefore law is satisfied.
$\Rightarrow$ l = 80 mm
 Question 9
A helical gear with $20^{\circ}$ pressure angle and $30^{\circ}$ helix angle mounted at the mid-span of a shaft that is supported between two bearings at the ends. The nature of the stresses induced in the shaft is
 A normal stress due to bending only B normal stress due to bending in one plane and axial loading; shear stress due to torsion C normal stress due to bending in two planes and axial loading; shear stress due to torsion D normal stress due to bending in two planes; shear stress due to torsion
Machine Design   Gears
 Question 10
The crystal structure of $\gamma$ iron (austenite phase) is
 A BCC B FCC C HCP D BCT
Manufacturing Engineering   Engineering Materials
Question 10 Explanation:
Austenite has a cubic-close packed crystal structure, also referred to as a face-centred cubic structure with an atom at each corner and in the centre of each face of the unit cell.
There are 10 questions to complete.