CS and IT Gate 2024 Set-2 Questions with Answer

To find the correct propositional logic representation, let us analyze the conditional structure of the statement:
• Identify the Propositions:
    • p: Fail grade can be given.
    • ¬p: Fail grade cannot be given (the negation of p).
    • q: Student scores more than 50% marks.

• Analyze the Phrase Structure:
The statement is written as: "Fail grade cannot be given when student scores more than 50% marks."
In logic, the word "when" introduces the sufficient condition (the premise or antecedent). The sentence can be rephrased into standard "If... then..." form without changing its meaning:
    "If a student scores more than 50% marks, then a fail grade cannot be given."

• Map to Logical Operators:
    • Antecedent (If part): q
    • Consequent (Then part): ¬p
    • Conditional operator: →

Combining these components gives the logical implication expression: q → ¬p.

Thus, the correct representation is option (a).

The sequence walk → jog → sprint shows an increasing level of intensity, where each action is stronger or faster than the previous one.
In the same way, we need to find a word that shows increasing emotional intensity in the sequence:
bothered → _______ → daunted

Here is what each word means:

• Bothered – slightly upset or disturbed.
• Fazed – noticeably disturbed, shaken, or upset; stronger than "bothered."
• Daunted – intimidated or overwhelmed; stronger than "fazed."

So the correct increasing order of emotional intensity is:
bothered → fazed → daunted
Therefore, the appropriate word to fill the blank is (c) fazed.

Explanation:
To find the number of trials each wizard makes, we need to calculate the total number of unique ways to arrange the four elements.
1. Identify the Permutation Problem:
• There are 4 distinct elements available: {water, air, fire, earth}.
• The problem states that they mix all four elements in all possible orders.
• Since the order of mixing matters and all items are used without repetition, this is a standard permutation problem of $n$ distinct objects.

2. Calculate the Total Combinations ($n!$):
The number of unique ways to arrange $n$ distinct objects is given by $n$ factorial ($n!$):
• For 4 elements: 4! = 4 × 3 × 2 × 1 = 24.

3. Evaluate per Wizard:
• The question asks for the number of trials each wizard makes before coming to the conclusion.
• Since they work independently and both try all possible combinations, each wizard completes the full set of 24 unique arrangements.

4. Conclusion:
Each wizard makes exactly 24 trials, making option (a) the correct choice.

Explanation:
Let us solve this problem using a Venn diagram approach.
1. Identify the Total Population Breakdown:
• Total number of students = 10,000
• Students who like neither branch = 1,500
• Therefore, the total number of students who like at least one branch (the union of both sets) is:
    10,000 - 1,500 = 8,500
2. Define the Variables:
• Let C be the total number of students who like their core branches.
• Let O be the total number of students who like other branches.
• The problem states that the number of students who like both branches is:
    C ∩ O = 500
3. Set up the Algebraic Equations:
• According to the problem, the number of students who like core branches is 1/4th of those who like other branches:
    C = O / 4 → O = 4C
• Using the standard set union formula:
    Total Union = C + O - (C ∩ O)
    8,500 = C + 4C - 500
4. Solve for C:
    8,500 + 500 = 5C
    9,000 = 5C
    C = 9,000 / 5 = 1,800

Explanation:
Let us simplify the given logarithmic equation step-by-step using standard logarithm rules.
1. Apply Logarithmic Properties:
The given equation is:
    ln((x + y) / 2) = 1/2 [ln(x) + ln(y)]

Using the log addition rule, ln(x) + ln(y) = ln(xy):
    ln((x + y) / 2) = 1/2 ln(xy)

Using the power rule, b · ln(a) = ln(ab):
    ln((x + y) / 2) = ln((xy)^1/2)

2. Remove the Natural Logarithm from Both Sides:
    (x + y) / 2 = √(xy)
    x + y = 2√(xy)

3. Square Both Sides to Eliminate the Radical:
    (x + y)² = (2√(xy))²
    x² + 2xy + y² = 4xy

Bring 4xy to the left side:
    x² - 2xy + y² = 0
    (x - y)² = 0
    x - y = 0 → x = y

4. Evaluate the Target Expression:
Since x = y, we substitute y for x (or vice versa) in the expression x/y + y/x:
    x/y + y/x = y/y + y/y = 1 + 1 = 2

5. Conclusion:
The value of the expression is exactly 2, making option (c) the correct choice.

Explanation:
Let us look at the differences between consecutive terms in the sequence to identify the underlying pattern:
• 1st term to 2nd term: 9 - 6 = 3
• 2nd term to 3rd term: 14 - 9 = 5

The difference increases by 2 each time, indicating a sequence of consecutive odd numbers (+3, +5, +7, +9, +11). Let's test this pattern to find the value of x:

• 3rd term to 4th term (x): 14 + 7 = 21

To find the correct sequence, let us analyze the logical connections and structural transitions between the sentences:
• Sentence S introduces the primary historical action—a breakaway chunk of the Antarctic Plate colliding with the Eurasian Plate. This serves as the natural starting point for the passage.
• Sentence P opens with "This fortuitous geological event," which directly references the tectonic collision described in sentence S. It explains the immediate physical result of that collision (rocks rising to a height of 4 km). Therefore, P must follow S.
• Sentence R talks about "the cooling of this massive edifice." This phrase directly connects back to the hot, rising mountains described in sentence P. It details the long-term environmental consequences of the cooling process. Therefore, R must follow P.
• Sentence Q begins with the concluding transition word "Thus" and summarizes the modern scientific perspective derived from the preceding sequence of events (viewing the Himalayas as an active event rather than a static feature). This serves as the perfect concluding thought for the passage.

Combining these connections gives the logical flow: S → P → R → Q. This perfectly matches option (d).

Let the Selling Price (SP) of each item = ₹100
For Item 1 (10% Profit):
SP = CP × (100 + Profit%) ÷ 100
100 = CP × 110 ÷ 100
CP = 100 × 100 ÷ 110
CP₁ = ₹90.91
For Item 2 (10% Loss):
SP = CP × (100 − Loss%) ÷ 100
100 = CP × 90 ÷ 100
CP = 100 × 100 ÷ 90
CP₂ = ₹111.11
Total SP = 100 + 100 = ₹200
Total CP = 90.91 + 111.11 = ₹202.02
Since Total CP > Total SP, there is a Loss
Net Loss = 202.02 − 200 = ₹2.02
Loss% = (Net Loss ÷ Total CP) × 100
Loss% = (2.02 ÷ 202.02) × 100
Loss% = 1%
Shortcut Formula:
Net Loss% = (Common%)² ÷ 100
Net Loss% = (10)² ÷ 100
Net Loss% = 100 ÷ 100
Net Loss% = 1%
The person made a net loss of 1%

From the pie charts:
Year 2007:
Hydro = 30%, Solar = 5%, Wind = 5%
Total Renewable in 2007 = 30 + 5 + 5 = 40%
Year 2023:
Hydro = 35%, Solar = 20%, Wind = 10%
Total Renewable in 2023 = 35 + 20 + 10 = 65%
Increase in renewable share = 65 − 40 = 25%
Percentage increase = (25 ÷ 40) × 100 = 62.5%
The percentage increase in the share of renewable sources = 62.5% (Option D)

To find the minimum number of cuts required to divide a cube into 8 pieces of equal size and shape, let us analyze the dimensions of the cube:
• A cube is a three-dimensional object with three distinct axes: length, width, and height.
• To ensure that all resulting pieces have the exact same size and shape, our cuts must be symmetrical along these axes.
• If we make 1 straight cut passing completely through the middle along the length, the cube is divided into 2 equal parts.
• Making a 2nd straight cut through the middle along the width divides those parts further, resulting in 4 equal pieces.
• Making a 3rd straight cut through the middle along the height cuts all existing pieces in half, resulting in exactly 8 equal pieces (each piece being a smaller, identical cube).

Since each cut must go completely from one side to the other, making 1 cut along each of the 3 coordinate planes gives us exactly 2 × 2 × 2 = 8 identical pieces. Thus, the minimum number of cuts required is 3, matching option (a).

Explanation:
Let us analyze the possible execution sequences of threads T₁ and T₂ based on which thread acquires the shared semaphore s1 (initialized to 1) first.
Case 1: T₁ executes first 1. T₁ executes wait(s1) successfully (s1 becomes 0). 2. T₁ increments x to 1 and prints 1. 3. T₁ executes wait(s2). Since s2 = 0, T₁ blocks here. 4. If a context switch occurs, T₂ tries to execute wait(s1). Since s1 = 0, T₂ also blocks.
Both threads are permanently blocked waiting for each other, resulting in a deadlock where only 1 was printed.
→ This validates Option (c).
Case 2: T₂ executes first 1. T₂ executes wait(s1) successfully (s1 becomes 0). 2. T₂ increments x to 1 and prints 1. 3. T₂ signals s2 (s2 = 1) and signals s1 (s1 = 1), completing its execution. 4. T₁ is now free to run. It passes wait(s1), increments x to 2, and prints 2. 5. T₁ passes wait(s2) without blocking because s2 was set to 1 by T₂.
Both threads finish successfully, printing 1 followed by 2.
→ This validates Option (b).
Conclusion: The possible outcomes are b and c.