Aptitude Mechanical previous year questions with answer
Ques 21 Gate 2022 Set-1
Humans are naturally compassionate and honest. In a study using strategically placed wallets that appear “lost”, it was found that wallets with money are more likely to be returned than wallets without money. Similarly, wallets that had a key and money are more likely to be returned than wallets with the same amount of money alone. This suggests that the primary reason for this behavior is compassion and empathy.
Which one of the following is the CORRECT logical inference based on the information in the above passage?
Eliminating wrong options:
Option A - "People do not care about money" is incorrect. The passage says wallets with money are more likely to be returned, which means people do care about the owner's loss, not that they ignore money.
Option C - "Wallets used in experiments are more likely to be returned" is not mentioned anywhere in the passage. This is outside the scope.
Option D - "Money is always more important than keys" is incorrect. The passage actually shows the opposite — a key adds more reason to return the wallet.
Option B - The passage states the primary reason is compassion and empathy. A key suggests someone is locked out and suffering. People relate to that suffering and are therefore more likely to return it. This directly aligns with the passage's conclusion.
The correct logical inference is "Wallets with a key are more likely to be returned because people relate to suffering of others." (Option B)
Ques 22 Gate 2022 Set-1
A rhombus is formed by joining the midpoints of the sides of a unit square.What is the diameter of the largest circle that can be inscribed within the rhombus?
Given:
• A unit square has sides of length 1.
• Let the vertices of the unit square be (0,0), (1,0), (1,1), and (0,1).
• The midpoints of its sides are:
• (0 + 1)/2, (0 + 0)/2 → (1/2, 0)
• (1 + 1)/2, (0 + 1)/2 → (1, 1/2)
• (1 + 0)/2, (1 + 1)/2 → (1/2, 1)
• (0 + 0)/2, (1 + 0)/2 → (0, 1/2)
Determine the Rhombus Properties:
• The rhombus is formed by connecting these four midpoints. Let's denote the vertices as A=(1/2, 0), B=(1, 1/2), C=(1/2, 1), D=(0, 1/2).
• Side length (s): We can calculate the distance between any two adjacent midpoints, for example, A and B.
s = √[(1 - 1/2)2 + (1/2 - 0)2]
s = √[(1/2)2 + (1/2)2]
s = √[1/4 + 1/4] = √[2/4] = √[1/2]
s = 1/√2
• Diagonal lengths:
• d1 (distance between A and C) = √[(1/2 - 1/2)2 + (1 - 0)2] = √[02 + 12] = 1
• d2 (distance between B and D) = √[(1 - 0)2 + (1/2 - 1/2)2] = √[12 + 02] = 1
Calculate the Area of the Rhombus:
• The area of a rhombus can be calculated using its diagonals:
Area = (d1 × d2) / 2
Area = (1 × 1) / 2 = 1/2
Determine the Diameter of the Inscribed Circle:
• The diameter of the largest circle that can be inscribed within a rhombus is equal to the altitude (or height) of the rhombus.
• The area of a rhombus can also be calculated as: Area = side × altitude (h).
• Therefore, h = Area / side.
• Substitute the calculated values:
h = (1/2) / (1/√2)
h = (1/2) × √2
h = √2 / 2
h = 1/√2
• The diameter of the inscribed circle is equal to h.
• Diameter = 1/√2.
Ques 23 Gate 2022 Set-1
An equilateral triangle, a square and a circle have equal areas.What is the ratio of the perimeters of the equilateral triangle to square to circle?
Formulas for Area and Perimeter:
• Equilateral Triangle with side length s:
Area (At) = (s2√3) / 4
Perimeter (Pt) = 3s
• Square with side length a:
Area (As) = a2
Perimeter (Ps) = 4a
• Circle with radius r:
Area (Ac) = πr2
Perimeter (Pc) = 2πr
Express Side/Radius in terms of Area (A):
Since At = As = Ac = A:
• For the equilateral triangle:
A = (s2√3) / 4
s2 = 4A / √3
s = √(4A / √3) = 2√A / (31/4)
• For the square:
A = a2
a = √A
• For the circle:
A = πr2
r2 = A / π
r = √(A / π) = √A / √π
Calculate Perimeters in terms of Area (A):
• For the equilateral triangle (Pt):
Pt = 3s = 3 × (2√A / 31/4) = 6√A / 31/4
To simplify 6 / 31/4:
6 / 31/4 = (2 × 3) / 31/4 = 2 × 31 - 1/4 = 2 × 33/4
Note that 33/4 = (33)1/4 = 271/4 = √√27.
Also, √(3√3) = (3 × 31/2)1/2 = (33/2)1/2 = 33/4.
So, Pt = 2√A × √(3√3)
• For the square (Ps):
Ps = 4a = 4√A
• For the circle (Pc):
Pc = 2πr = 2π × (√A / √π) = 2√A√π
Determine the Ratio of Perimeters:
Pt : Ps : Pc = (2√A × √(3√3)) : (4√A) : (2√A√π)
Divide all terms by 2√A (since A is a positive area, √A ≠ 0):
√(3√3) : 2 : √π
Ques 24 Gate 2022 Set-1
Given below are three conclusions drawn based on the following three
statements
Statement 1: All teachers are professors.
Statement 2: No professor is a male.
Statement 3: Some males are engineers.
Conclusion I: No engineer is a professor.
Conclusion II: Some engineers are professors.
Conclusion III: No male is a teacher.
Which one of the following options can be logically inferred?
Represent the Statements:
Let's use sets to represent the categories mentioned in the statements:
• T = Set of Teachers
• P = Set of Professors
• M = Set of Males
• E = Set of Engineers
Now, translate the given statements:
• Statement 1: All teachers are professors.
This implies that the set of Teachers is a subset of the set of Professors (T ⊆ P). If someone is a teacher, they must also be a professor.
• Statement 2: No professor is a male.
This means the set of Professors and the set of Males are disjoint (P ∩ M = ∅). There is no common element between these two sets. If someone is a professor, they cannot be male, and if someone is male, they cannot be a professor.
• Statement 3: Some males are engineers.
This indicates that there is an overlap between the set of Males and the set of Engineers (M ∩ E ≠ ∅). At least one individual exists who is both male and an engineer.
Evaluate Each Conclusion:
• Conclusion I: No engineer is a professor.
From Statement 2, we know that P ∩ M = ∅ (Professors and Males are separate).
From Statement 3, we know that M ∩ E ≠ ∅ (Some Males are Engineers).
Consider an individual who is both male and an engineer (as guaranteed by S3). Because they are male, they cannot be a professor (due to S2). So, any engineer who is male cannot be a professor.
However, the statements do not provide any information about engineers who are *not* male. It is possible for some non-male engineers to exist, and it is also possible for these non-male engineers to be professors (as they wouldn't violate S2).
Since we cannot definitively conclude that *no* engineer is a professor (there might be non-male engineers who are professors), Conclusion I is not logically inferred as correct.
• Conclusion II: Some engineers are professors.
This conclusion is the direct opposite of Conclusion I. As established above, while it's *possible* for some non-male engineers to be professors, the given statements do not provide any information that *guarantees* the existence of such individuals.
For example, we could have a scenario where all engineers are male (which would make it impossible for any engineer to be a professor, given S2), or a scenario where all non-male engineers are not professors. Neither contradicts the given statements.
Since the statements do not guarantee this overlap, Conclusion II is not logically inferred as correct.
• Conclusion III: No male is a teacher.
From Statement 1: All teachers are professors (T ⊆ P).
From Statement 2: No professor is a male (P ∩ M = ∅).
If all teachers are professors, and no professor is male, then it logically follows that no teacher can be male. If an individual is a teacher, they must be a professor (from S1). If they are a professor, they cannot be male (from S2). Therefore, if they are a teacher, they cannot be male.
This establishes that the set of Teachers and the set of Males are disjoint (T ∩ M = ∅).
Thus, Conclusion III is logically inferred as correct.
Based on the evaluation, only Conclusion III is correct.
The final answer is A) Only conclusion III is correct
Ques 25 Gate 2022 Set-1
In a 12-hour clock that runs correctly, how many times do the second, minute, and hour hands of the clock coincide, in a 12-hour duration from 3 PM in a day to 3 AM the next day?
Relative Speeds of Hour and Minute Hands:
• The minute hand completes a full circle (360 degrees) in 60 minutes. Its speed is 6 degrees per minute.
• The hour hand completes a full circle (360 degrees) in 12 hours (720 minutes). Its speed is 0.5 degrees per minute.
Equation for Coincidence:
Let t be the number of minutes past a given hour 'H'. We can set the angular positions of the hands equal to find when they coincide.
• Position of minute hand: θm = 6t (degrees from 12 o'clock, clockwise)
• Position of hour hand: θh = 30H + 0.5t (degrees from 12 o'clock, clockwise, where H is the hour number 0-11)
For the hands to coincide, their positions must be equal (modulo 360 degrees):
6t ≡ 30H + 0.5t (mod 360)
Rearranging the equation:
5.5t ≡ 30H (mod 360)
(11/2)t = 30H + 360k, for some integer k
11t = 60H + 720k
t = (60H + 720k) / 11
Coincidences in a 12-hour Cycle:
The hour and minute hands coincide 11 times in any 12-hour period. These 11 distinct coincidences occur approximately at:
1. 12:00:00 (exact)
2. ~1:05:27 (1 hour, 5 minutes, 27 seconds)
3. ~2:10:54
4. ~3:16:21
5. ~4:21:49
6. ~5:27:16
7. ~6:32:43
8. ~7:38:10
9. ~8:43:38
10. ~9:49:05
11. ~10:54:32
(The next coincidence would be exactly 12 hours after the first 12:00:00 mark).
Counting within the specified duration [3 PM to 3 AM]:
Let's list the coincidence times, converting them to PM/AM for the specified interval:
• 12:00:00 PM (noon) - Not in the interval [3 PM, 3 AM]
• ~1:05:27 PM - Not in the interval [3 PM, 3 AM]
• ~2:10:54 PM - Not in the interval [3 PM, 3 AM]
• ~3:16:21 PM - In the interval (1st occurrence)
• ~4:21:49 PM - In the interval (2nd occurrence)
• ~5:27:16 PM - In the interval (3rd occurrence)
• ~6:32:43 PM - In the interval (4th occurrence)
• ~7:38:10 PM - In the interval (5th occurrence)
• ~8:43:38 PM - In the interval (6th occurrence)
• ~9:49:05 PM - In the interval (7th occurrence)
• ~10:54:32 PM - In the interval (8th occurrence)
• 12:00:00 AM (midnight) - In the interval (9th occurrence)
• ~1:05:27 AM - In the interval (10th occurrence)
• ~2:10:54 AM - In the interval (11th occurrence)
• ~3:16:21 AM - Not in the interval (after 3 AM)
Thus, the hour and minute hands coincide 11 times within the 12-hour duration from 3 PM to 3 AM.
Ques 26 Gate 2021 Set-2
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:
Ques 27 Gate 2021 Set-2
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?
Ques 28 Gate 2021 Set-2
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
Ques 29 Gate 2021 Set-2
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
Ques 30 Gate 2021 Set-2
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