2026 Prelims Paper 2 UPSC Question Paper

For \( \frac{1}{3} \) < x < y < 2, which of the following statements is/are always correct?
I. x + \( \frac{1}{x} \) < y + \( \frac{1}{y} \)
II. \( \dfrac{\sqrt{1+y^2}}{y} \) < \( \dfrac{\sqrt{1+x^2}}{x} \)
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What is the minimum number of times one needs to measure to get 298 litres of water from a tank, if the measuring cylinders have capacities 1 litre, 6 litres, 25 litres and 100 litres?

There are four types of weights, namely 1 kg, 2 kg, 5 kg and 10 kg. What is the maximum number of different ways one can measure 20 kg, if at least eight but not more than eleven weights of 1 kg are to be used while measuring?

A cut on a solid object divides the object into two parts where the new surfaces thus produced are plane. On the other hand, a single cut can be used to cut more than one object at a time. In an experiment, the total number of pieces produced by applying n cuts is denoted by x_n. This experiment is performed on a solid cube, where pieces remain unmoved after each cut. In this experiment, if after the third cut, the pieces are identical, then which of the following is not a possible value for x_4?

The class average x in a test increases by 4 when the score of a student is rectified, whose corrected score is 100 instead of 0. Later, the score of another student was found to have been recorded as 81 in place of 56. If there are no other corrections and the final corrected average is y, then y - x is

Directions for the next 3 (three) items:
Read the following passage and answer the items that follow. Your answers should be based solely on the passage.
Passage:
Passage: When SARS-CoV-2 was first detected in 2019, it was a truly novel virus for the world. At that time, no one in the world had been exposed to SARS-CoV-2 or had specific immunity against it. In contrast, people across the world have been exposed to HMPV for decades and the virus has been studied extensively. HMPV and SARS-CoV-2 belong to two very different virus families with fundamentally different characteristics and epidemiology, and unlike SARS-CoV-2, HMPV shows strong seasonality. Both viruses, especially over the long term, produce different severities, and the affected segments of the population do not fully overlap. Generally, HMPV causes mild disease, with deaths being very rare and the post-viral symptoms of the disease not remaining long term.

Which of the following conclusions is/are valid?
1. Though SARS-CoV-2 and HMPV are similar viruses with somewhat different epidemiology, the former (SARS-CoV-2) became a pandemic because it was novel and people in the past had not been exposed to it.
2. The basic impact of these two viruses on human population is different, and hence they should not be dealt with in a similar manner.
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In the above passage, the author's intent is reflected in which of the following?
1. To develop methodologies for an objective analysis of these two viruses.
2. To establish the epidemiological similarities and differences between these two viruses.
3. To present a better understanding of the treatments for HMPV when analysed in conjunction with SARS-CoV-2.
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Which of the following statements reflect the logical and rational inferences that can be drawn from this passage?
1. HMPV has, historically, had longer documentation of studies when compared with SARS-CoV-2.
2. The two viruses are different from each other and this results in markedly different outcomes amongst those affected.
3. Long-term impacts of the two viruses are dissimilar and this is an important differentiator between them.
4. Seasonal specificity is a common factor between these two viruses.
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Three variables x, y and z take values 2, 3, 4 or 5 such that their values are always distinct. If M and N denote the largest possible value and the smallest possible value, respectively, for the expression {(x x y) + z}, then M - N is

Suppose x, y and z are variables taking positive real numbers as their possible values. It is given that y is directly proportional to x^2 and x is inversely proportional to z. For z = \( \frac{7}{25} \), the values of x and y are 5 and 50, respectively. If y = 98, what is z equal to?