## Shortest Path Algorithm Mock Test – 2

 Question 1
Suppose you have a directed acyclic graph with $n$ vertices and $O(n)$ edges, all having nonnegative weights. Running time (in form of $n$) of an efficient algorithm for finding the shortest path to each vertex from a single source is
 A $O(n)$ B $O(n \log n)$ C $O(n^2)$ D $O(n^2 \log n)$
Algorithm
Question 1 Explanation:
The given conditions allow us to use either the DAG shortest paths algorithm or Dijkstra's. DAG shortest paths runs in $\Theta (V+E)=\Theta (n)$, and Dijkstra's runs in $O((V + E) \log V )$, or $O(V \log V + E)$ if using a Fibonacci heap. In either case, Dijkstra's runs in $O(n \log n)$. DAG shortest paths is faster in this case
https://www.cs.bgu.ac.il/~ds182/wiki.files/14-shortest_paths.pdf

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 Question 2
Which of the following Statemets is/are TRUE?

S1: Consider two positively weighted graphs $G = (V; E; w)$ and $G_0 = (V; E; w_0)$ with the same vertices $V$ and edges $E$ such that, for any edge $e$ in $E$, we have $w_0(e) = w(e)^2$. For any two vertices $u, v$ in $V$ , any shortest path between $u$ and $v$ in $G_0$ is also a shortest path in $G$.

S2: If the priority queue in Dijkstra's algorithm is implemented using a sorted linked list (where the list is kept sorted after each priority queue operation), then Dijkstra's algorithm would run in $O(ElgV +V lgV)$ time.
 A S1 is True and S2 is False B S1 is False and S2 is True C S1 and S2 are False D S1 and S2 are True
Algorithm
Question 2 Explanation:
S1 is FALSE
Assume we have two paths in $G$ as (1) $path: u-x-v : 4$ with $w(u,x)=2,w(x,v)=2$ (2)path: u-v:3 with $w(u,v)=3$. The first one is shorter in $G_0$ while the second one is shorter in $G$.

S2 is FLASE
The list can take $\Theta (V)$ time to insert a node, and there are $(V)$ nodes to insert, for a running time of $\Omega (V^2)$. In addition, the $\Theta (E)$ calls to decrease-key can each take $\Theta (V)$ time for a total running time of $\Theta ((V + E)V )$.

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 Question 3
Which of the following Statemets is/are TRUE?

S1: Given a graph G = (V, E) with positive edge weights, the Bellman-Ford algorithm and Dijkstra's algorithm can produce different shortest-path trees despite always producing the same shortest-path weights.

S2: Dijkstra's algorithm may not terminate if the graph contains negative-weight
 A S1 is True and S2 is False B S1 is False and S2 is True C S1 and S2 are False D S1 and S2 are True
Algorithm
Question 3 Explanation:
S1 is TRUE
Both algorithms are guaranteed to produce the same shortestpath weight, but if there are multiple shortest paths, Dijkstra's will choose the shortest path according to the greedy strategy, and Bellman-Ford will choose the shortest path depending on the order of relaxations, and the two shortest path trees may be different.

S2 is FLASE
It always terminates after |E| relaxations and |V|+|E| priority queue operations, but may produce incorrect results.

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There are 3 questions to complete.

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Shortest Path Algorithm Mock Test – 1

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## Graph Traversal (BFS and DFS)

 Question 1
An articulation point in a connected graph is a vertex such that removing the vertex and its incident edges disconnects the graph into two or more connected components.
Let T be a DFS tree obtained by doing DFS in a connected undirected graph G.
Which of the following options is/are correct?
 A Root of T can never be an articulation point in G. B Root of T is an articulation point in G if and only if it has 2 or more children. C A leaf of T can be an articulation point in G. D If u is an articulation point in G such that x is an ancestor of u in T and y is a descendent of u in T, then all paths from x to y in G must pass through u.
GATE CSE 2021 SET-1   Algorithm
Question 1 Explanation:
 Question 2
G is an undirected graph with vertex set {v1, v2, v3, v4, v5, v6, v7} and edge set {v1v2, v1v3, v1v4 ,v2v4, v2v5, v3v4, v4v5, v4v6, v5v6, v6v7 }. A breadth first search of the graph is performed with v1 as the root node. Which of the following is a tree edge?
 A v2v4 B v1v4 C v4v5 D v3v4
ISRO CSE 2020   Algorithm
Question 2 Explanation:
 Question 3
Which of the following is application of Breath First Search on the graph?
 A Finding diameter of the graph B Finding bipartite graph C Both (A) and (B) D None of the above
ISRO CSE 2018   Algorithm
Question 3 Explanation:
 Question 4
Let G be a simple undirected graph. Let TD be a depth first search tree of G. Let TB be a breadth first search tree of G. Consider the following statements.

(I) No edge of G is a cross edge with respect to TD. (A cross edge in G is between two nodes neither of which is an ancestor of the other in TD.)
(II) For every edge (u,v) of G, if u is at depth i and v is at depth j in TB, then |i-j|=1.

Which of the statements above must necessarily be true?
 A I only B II only C Both I and II D Neither I nor II
GATE CSE 2018   Algorithm
Question 4 Explanation:
 Question 5
The Breadth First Search (BFS) algorithm has been implemented using the queue data structure. Which one of the following is a possible order of visiting the nodes in the graph below?
 A MNOPQR B NQMPOR C QMNROP D POQNMR
GATE CSE 2017 SET-2   Algorithm
Question 5 Explanation:
 Question 6
Breadth First Search(BFS) is started on a binary tree beginning from the root vertex. There is a vertex t at a distance four from the root. If t is the n-th vertex in this BFS traversal, then the maximum possible value of n is______ .
 A 16 B 15 C 31 D 32
GATE CSE 2016 SET-2   Algorithm
Question 6 Explanation:
 Question 7
Consider the following directed graph:

The number of different topological orderings of the vertices of the graph is
 A 4 B 5 C 6 D 7
GATE CSE 2016 SET-1   Algorithm
Question 7 Explanation:
 Question 8
Let G = (V, E) be a simple undirected graph, and s be a particular vertex in it called the source. For $x \in V$, let d(x) denote the shortest distance in G from s to x. A breadth first search (BFS) is performed starting at s. Let T be the resultant BFS tree. If (u,v) is an edge of G that is not in T, then which one of the following CANNOT be the value of d(u)-d(v)?
 A -1 B 0 C 1 D 2
GATE CSE 2015 SET-1   Algorithm
Question 8 Explanation:
 Question 9
Suppose depth first search is executed on the graph below starting at some unknown vertex. Assume that a recursive call to visit a vertex is made only after first checking that the vertex has not been visited earlier. Then the maximum possible recursion depth (including the initial call) is _________.
 A 16 B 19 C 17 D 20
GATE CSE 2014 SET-3   Algorithm
Question 9 Explanation:
 Question 10
Let G be a graph with n vertices and m edges. What is the tightest upper bound on the running time of Depth First Search on G, when G is represented as an adjacency matrix?
 A $\Theta (n)$ B $\Theta (n+m)$ C $\Theta (n^{2})$ D $\Theta (m^{2})$
GATE CSE 2014 SET-1   Algorithm
Question 10 Explanation:
There are 10 questions to complete.

## General Aptitude

 Question 1
In an equilateral triangle PQR, side PQ is divided into four equal parts, side QR is divided into six equal parts and side PR is divided into eight equals parts. The length of each subdivided part in cm is an integer. The minimum area of the triangle PQR possible, in $cm^2$, is
 A 18 B 24 C $48 \sqrt{3}$ D $144 \sqrt{3}$
GATE CE 2021 SET-2      Numerical Ability
Question 1 Explanation:

For $\left(\frac{a}{4}, \frac{a}{6}, \frac{a}{8}\right)$ to be integer, a must be LCM of 4, 6 and 8. So a = 24
$\text { Area }=\frac{\sqrt{3}}{4} a^{2}=\frac{\sqrt{3}}{4} \times 24^{2}=144 \sqrt{3}$
 Question 2

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

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

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

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

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

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

There are 10 questions to complete.

## Verbal Ability

 Question 1
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.

## Verbal Ability

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

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

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

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

## Numerical Ability

 Question 1
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.

## Numerical Ability

 Question 1
In an equilateral triangle PQR, side PQ is divided into four equal parts, side QR is divided into six equal parts and side PR is divided into eight equals parts. The length of each subdivided part in cm is an integer. The minimum area of the triangle PQR possible, in $cm^2$, is
 A 18 B 24 C $48 \sqrt{3}$ D $144 \sqrt{3}$
GATE CE 2021 SET-2   General Aptitude
Question 1 Explanation:

For $\left(\frac{a}{4}, \frac{a}{6}, \frac{a}{8}\right)$ to be integer, a must be LCM of 4, 6 and 8. So a = 24
$\text { Area }=\frac{\sqrt{3}}{4} a^{2}=\frac{\sqrt{3}}{4} \times 24^{2}=144 \sqrt{3}$
 Question 2

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

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

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

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

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

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

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

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

There are 10 questions to complete.

## 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.