ICAR AIEEA PG Agri-Business Management 09 July 2023 (Shift 1) Paper

Relative speed calculation:

• Since the goods train passes a single point observer (the man) in 9 seconds, the distance covered equals the length of the goods train ($270\text{ m}$).
• Relative speed in m/s:
  $\text{Relative Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{270\text{ m}}{9\text{ s}} = 30\text{ m/s}$.
• Convert relative speed to km/h:
  $\text{Relative Speed} = 30 \times \frac{18}{5} = 108\text{ km/h}$.
• Because the two trains move in opposite directions, relative speed is the sum of their speeds:
  $\text{Speed}_{\text{goods}} + \text{Speed}_{\text{man's train}} = 108\text{ km/h}$.
  $\text{Speed}_{\text{goods}} + 40 = 108 \implies \text{Speed}_{\text{goods}} = 108 - 40 = 68\text{ km/h}$.

Volume displacement computation:

• Total volume of water displaced by 40 men:
  $\text{Total Volume} = 40 \times 4\text{ m}^3 = 160\text{ m}^3$.
• Base area of the rectangular tank:
  $\text{Base Area} = \text{Length} \times \text{Breadth} = 30\text{ m} \times 20\text{ m} = 600\text{ m}^2$.
• Rise in water level ($h$):
  $h = \frac{\text{Total Volume}}{\text{Base Area}} = \frac{160}{600} = \frac{4}{15}\text{ m}$.
• Convert to centimeters:
  $h = \frac{4}{15} \times 100\text{ cm} = \frac{400}{15} \approx 26.67\text{ cm} \approx 27\text{ cm}$.

Mathematical verification of number system statements:

• Statement (A) is true: The set of non-negative integers $\{0, 1, 2, 3, \dots\}$ defines the whole numbers.
• Statement (B) is false: Integers include negative whole numbers ($\{\dots, -3, -2, -1\}$), zero, and positive numbers.
• Statement (C) is false: 91 is a composite number ($7 \times 13 = 91$).
• Statement (D) is true: Any integer $n$ can be expressed as a ratio of integers $\frac{n}{1}$ (where denominator $\ne 0$), making all natural numbers, whole numbers, and integers rational numbers.