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For I: The function f(t) = t + 1/t has its minimum at t = 1. Since (1/3, 2) includes 1, the function decreases then increases, so it is not always increasing.

For II: Let g(t) = √(1 + t²) / t = √(1/t² + 1). As t increases, 1/t² strictly decreases.

Since x < y, it follows that 1/x² > 1/y² ⇒ √(1/x² + 1) > √(1/y² + 1).

Thus, g(x) > g(y) is always true for the given range.

∴ The correct answer is II only.

Alternate Method

Given: 1/3 < x < y < 2

For Statement I: x + 1/x < y + 1/y

⇒ Let x = 0.5 and y = 1.0 (both in range 1/3 < x < y < 2)

⇒ x + 1/x = 0.5 + 2 = 2.5

⇒ y + 1/y = 1 + 1 = 2

⇒ 2.5 is not less than 2, so Statement I is False.

For Statement II: √(1 + y²)/y < √(1 + x²)/x

⇒ Let g(t) = √(1 + t²)/t = √((1 + t²)/t²) = √(1/t² + 1)

⇒ Since x < y ⇒ x² < y² ⇒ 1/x² > 1/y²

⇒ 1/x² + 1 > 1/y² + 1

⇒ √(1/x² + 1) > √(1/y² + 1)

⇒ Therefore, Statement II is True.

∴ The correct answer is II only.

Additional Information

Monotonic Functions

A function is monotonic if its derivative does not change sign. f(t) = t + 1/t has f'(t) = 1 − 1/t², which is negative for t < 1 and positive for t > 1.

Inequality Property

For positive values, if a > b, then 1/a < 1/b and √a > √b. These properties are essential for comparing fractional expressions.

AM-GM Inequality

For any positive number x, x + 1/x ≥ 2. The minimum value 2 occurs exactly at x = 1.

Given measuring cylinders:

1 litre, 6 litres, 25 litres and 100 litres

We need to measure 298 litres.

To minimize the number of measurements, use the largest cylinder as much as possible.

298 = 100 + 100 + 100 − 1 − 1

So, measure:

100 litre cylinder → 3 times

1 litre cylinder → 2 times

Total measurements = 3 + 2 = 5

Hence, the minimum number of measurements required is 5.

Hence, the correct answer is "Option 2".

Shortcut Trick

Total weight required = 20 kg. Available weights: 1 kg, 2 kg, 5 kg, 10 kg.

Count of 1 kg weights must be between 8 and 11:

→ Case 1: 8 × 1 kg | Remaining 12 kg → {10,2}, {5,5,2}, {2×6} (3 ways)

→ Case 2: 9 × 1 kg | Remaining 11 kg → {5,2,2,2} (1 way)

→ Case 3: 10 × 1 kg | Remaining 10 kg → {10}, {5,5}, {2×5} (3 ways)

→ Case 4: 11 × 1 kg | Remaining 9 kg → {5,2,2} (1 way)

Total ways = 3 + 1 + 3 + 1 = 8.

∴ The correct answer is 8 ways.

Alternate Method

Given:

Weights available = {1, 2, 5, 10} kg

Target weight = 20 kg

Condition = 8 ≤ (1 kg weights) ≤ 11

Calculations:

Let n₁, n₂, n₅, n₁₀ be the number of weights of 1 kg, 2 kg, 5 kg, and 10 kg respectively.

⇒ Total Weight: 1 × n₁ + 2 × n₂ + 5 × n₅ + 10 × n₁₀ = 20

⇒ Where 8 ≤ n₁ ≤ 11

⇒ If n₁ = 8: 2n₂ + 5n₅ + 10n₁₀ = 12. Solutions: (n₁₀=1, n₅=0, n₂=1); (n₁₀=0, n₅=2, n₂=1); (n₁₀=0, n₅=0, n₂=6). (3 ways)

⇒ If n₁ = 9: 2n₂ + 5n₅ + 10n₁₀ = 11. Solutions: (n₁₀=0, n₅=1, n₂=3). (1 way)

⇒ If n₁ = 10: 2n₂ + 5n₅ + 10n₁₀ = 10. Solutions: (n₁₀=1, n₅=0, n₂=0); (n₁₀=0, n₅=2, n₂=0); (n₁₀=0, n₅=0, n₂=5). (3 ways)

⇒ If n₁ = 11: 2n₂ + 5n₅ + 10n₁₀ = 9. Solutions: (n₁₀=0, n₅=1, n₂=2). (1 way)

⇒ Total ways = 3 + 1 + 3 + 1 = 8 ways.

∴ The correct answer is 8 ways.

Additional Information

Partition of a Number

The number of ways to write a positive integer n as a sum of positive integers from a given set is a classic combinatorial problem often solved using generating functions or systematic counting.

Coin Change Logic

When denominations (weights) are given, the number of ways to make a total sum is restricted by the parity (even/odd) of the target. For example, if all remaining weights are even, the remainder must be even to have any solutions.

Constraint Handling

In problems with constraints (like 8 to 11 weights), always isolate the variable with the constraint first and then calculate independent cases for each possible value of that variable.

The logic for determining the number of pieces produced by cutting a solid cube with n cuts, where pieces remain unmoved, is based on the distribution of cuts along the three axes (x, y, and z).

The total number of pieces x_n is calculated using the formula:

x_n = (c_x + 1) × (c_y + 1) × (c_z + 1)

where c_x + c_y + c_z = n (total cuts).

Condition for the third cut:

After 3 cuts (n = 3), the pieces are identical. This can happen in three possible configurations:

Analysis for the fourth cut (n = 4):

We add one additional cut to the configurations identified above to find the possible values for x₄:

1. From configuration (3, 0, 0):

• Add cut to same axis: (4, 0, 0) → x₄ = (4+1) × 1 × 1 = 5
• Add cut to new axis: (3, 1, 0) → x₄ = (3+1) × (1+1) × 1 = 8

2. From configuration (2, 1, 0):

• Add cut to first axis: (3, 1, 0) → x₄ = (3+1) × (1+1) × 1 = 8
• Add cut to second axis: (2, 2, 0) → x₄ = (2+1) × (2+1) × 1 = 9
• Add cut to third axis: (2, 1, 1) → x₄ = (2+1) × (1+1) × (1+1) = 12

3. From configuration (1, 1, 1):

• Add cut to any axis: (2, 1, 1) → x₄ = (2+1) × (1+1) × (1+1) = 12

The possible values for x₄ are 5, 8, 9, and 12. Any value higher than the maximum achievable with 4 cuts (which is 12 for orthogonal cuts) is impossible.

Therefore, 16 is not a possible value for x₄.

Hence, the correct answer is "Option 1".

Shortcut Trick

Net increase in total marks for the first correction = 100 − 0 = 100.

Since the average increase is 4, Total students (n) = 100 ÷ 4 = 25.

Second correction: 81 is replaced by 56, so total marks decrease by 81 − 56 = 25.

Net change in average for second correction = −25 ÷ 25 = −1.

Overall change in average (y − x) = 4 − 1 = 3.

∴ The correct answer is 3.

Alternate Method

Given: Initial average = x, Corrected average = y. Correction 1: 0 → 100. Correction 2: 81 → 56.

Formula Used: Average = Sum of observations ÷ Number of observations (n)

⇒ Let Total Students = n and Initial Sum = S

⇒ Initial Average x = S ÷ n

⇒ After 1^st correction: (S + 100) ÷ n = x + 4

⇒ S/n + 100/n = x + 4 ⇒ x + 100/n = x + 4 ⇒ 100/n = 4 ⇒ n = 25

⇒ Final Correction 2: New Sum = S + 100 + (56 − 81) = S + 100 − 25 = S + 75

⇒ Final Average y = (S + 75) ÷ 25 = S/25 + 75/25

⇒ y = x + 3

⇒ y − x = 3

∴ The correct answer is 3.

Additional Information

Net Change in Average

Net Change in Average = (Net Change in Sum) ÷ (Total Number of Items). If the total sum increases, the average increases, and vice versa.

Correction of Errors

When data is misread, the Correct Sum = (Wrong Sum − Wrong Value + Correct Value). This adjustment helps find the true average without recalculating every entry.

Average Property

If every observation in a set is increased or decreased by a constant value 'k', the average also increases or decreases by that same value 'k'.

Logic followed here is:

The validity of the conclusions is determined by analyzing the specific differences mentioned between HMPV (Human Metapneumovirus) and SARS-CoV-2 in terms of their severity, mortality rates, and long-term health effects.

Analysis of Statement 1:

The passage focuses on the physiological and clinical differences between the two viruses, such as the severity of illness and the presence of long-term symptoms. While it mentions SARS-CoV-2 and HMPV have different epidemiology, it does not explicitly state that the primary reason SARS-CoV-2 became a pandemic was specifically due to its novelty or lack of prior exposure. Without direct evidence in the text linking pandemic status solely to novelty, this conclusion cannot be definitively drawn from the given information. Therefore, Statement 1 is invalid.

Analysis of Statement 2:

The passage provides a clear comparison of the impacts of the two viruses:

Because HMPV causes milder illness and lacks post-viral symptoms, its impact on the individual and the healthcare system is fundamentally different from SARS-CoV-2. This disparity implies that public health responses, treatment protocols, and management strategies should be tailored to the specific threat level of each virus. Thus, they should not be dealt with in a similar manner. Therefore, Statement 2 is valid.

The pattern observed is that only the second statement logically follows from the differences in clinical outcomes described in the text.

Therefore, the correct conclusion is 2 only.

The correct answer is None of the above

Key Points

• The passage highlights the distinction between HMPV (Human Metapneumovirus) and SARS-CoV-2, emphasizing that HMPV causes milder illness, has a longer history of documentation, and lacks the long-term post-viral symptoms associated with COVID-19.
• Statement 1 is incorrect because the text focuses on describing the characteristics and historical context of the viruses rather than proposing or evolving new methodologies or frameworks for scientific analysis.
• Statement 2 is incorrect because the primary intent of the author is to contrast the severity and social impact of the two viruses to show why they should not be treated in the same manner. Merely 'establishing similarities and differences' is too broad and does not capture the specific intent of highlighting the fundamental differences in their impact on human populations.
• Statement 3 is incorrect as the passage discusses epidemiology, pathology, and symptomatology, but it does not provide information regarding remedies, medical treatments, or cures for HMPV or SARS-CoV-2.
• Since none of the provided statements accurately capture the author's intent of differentiating the severity and historical context of the two viruses, the correct choice is None of the above.

Additional Information

• Human Metapneumovirus (HMPV) was first identified in 2001 and belongs to the Pneumoviridae family (formerly Paramyxoviridae). It is a common cause of upper and lower respiratory tract infections.
• Unlike the SARS-CoV-2 pandemic, which was driven by a novel virus to which the global population had no pre-existing immunity, HMPV has been circulating for decades, leading to a level of community exposure that prevents widespread pandemic-level mortality in most healthy individuals.
• The distinction in long-term outcomes is a major differentiator; while SARS-CoV-2 can lead to Long COVID (post-acute sequelae), HMPV is generally associated with acute recovery without chronic post-viral syndromes.

Based on the clinical characteristics and the nature of the viruses described in the passage, let us analyze the statements to draw logical inferences:

Analysis of Statement 1:

The passage and context suggest that SARS-CoV-2 was a novel virus when it emerged, meaning it was previously unknown to the human immune system and medical researchers. In contrast, Human Metapneumovirus (HMPV), while less discussed in popular media, has been identified and studied for a significantly longer period in clinical history. Thus, it has a longer documentation of studies. Therefore, Statement 1 is a valid inference.

Analysis of Statement 2:

The text explicitly states that HMPV causes milder illness and that deaths are very rare. This is a direct contrast to the impact of SARS-CoV-2, which resulted in a global pandemic with high rates of severity and mortality. Because their epidemiological impacts and clinical outcomes are fundamentally different, they result in different outcomes for those affected. Therefore, Statement 2 is a valid inference.

Analysis of Statement 3:

The passage highlights that HMPV is characterized by no long-term post-viral symptoms. This is one of the most significant differentiators from SARS-CoV-2, which is widely known for causing long-term impacts (often termed as Long COVID). The presence of chronic post-viral issues in one and their absence in the other makes this a rational differentiator. Therefore, Statement 3 is a valid inference.

Analysis of Statement 4:

The provided text does not mention or provide data regarding seasonal specificity for both viruses. While many respiratory viruses are seasonal, SARS-CoV-2 has demonstrated the ability to cause waves of infection across various seasons and climates, making it difficult to infer seasonal specificity as a common factor based strictly on the logical flow of this comparison. Therefore, Statement 4 is not a supported inference.

Conclusion:

Statements 1, 2, and 3 reflect the logical and rational conclusions that can be drawn regarding the differences in novelty, clinical outcomes, and long-term impacts of the two viruses.

Therefore, the correct answer is 1, 2 and 3.

Given expression:

(x × y) + z

x, y and z take distinct values from 2, 3, 4, 5.

To find the largest value (M):

Choose the largest values for multiplication:

x = 5, y = 4, z = 3

M = (5 × 4) + 3

M = 20 + 3 = 23

To find the smallest value (N):

Choose the smallest distinct values:

x = 2, y = 3, z = 4

N = (2 × 3) + 4

N = 6 + 4 = 10

Therefore,

M − N = 23 − 10 = 13

Hence, the correct answer is "Option 3".

Shortcut Trick

Combine proportions: Since y ∝ x² and x ∝ 1/z, then y ∝ 1/z².

Using the ratio: y₁ ÷ y₂ = (z₂ ÷ z₁)²

50 ÷ 98 = (z₂ ÷ (7/25))² ⇒ 25/49 = (25z₂ ÷ 7)²

Taking square root on both sides: 5/7 = 25z₂ ÷ 7 ⇒ 25z₂ = 5 ⇒ z₂ = 1/5.

∴ The correct answer is 1/5.

Alternate Method

Given: y ∝ x², x ∝ 1/z, z = 7/25, x = 5, y = 50. Find z when y = 98.

Formula Used: y = k₁x² and x = k₂/z

Step 1: Find constants from initial values.

⇒ y = k₁x² ⇒ 50 = k₁ × 5² ⇒ 50 = 25k₁ ⇒ k₁ = 2

⇒ x = k₂/z ⇒ 5 = k₂ / (7/25) ⇒ 5 = k₂ × (25/7) ⇒ k₂ = 35/25 = 7/5

Step 2: Find new x when y = 98.

⇒ y = 2x² ⇒ 98 = 2x² ⇒ x² = 49 ⇒ x = 7 (as x > 0)

Step 3: Find new z using x = 7 and k₂ = 7/5.

⇒ x = (7/5) / z ⇒ 7 = 7 / (5z) ⇒ 35z = 7

⇒ z = 7/35 = 1/5

∴ The correct answer is 1/5.

Additional Information

Direct Variation

When two variables change in the same ratio, y ∝ x or y = kx. Here, y/x remains constant.

Inverse Variation

When one variable increases and the other decreases proportionally, x ∝ 1/z or xz = k. Their product is constant.

Joint Variation

Describes a situation where a variable depends on two or more other variables, e.g., y ∝ xz² ⇒ y = kxz².

Chain Rule of Variation

If A ∝ B and B ∝ C, then A ∝ C. If A ∝ B and B ∝ 1/C, then A ∝ 1/C.