We are given the sum of the present ages of two cousins and the ratio of their ages ten years ago. We can set up a system of two linear equations based on this information and solve for their present ages.
Let the present ages of the two cousins be \(C_1\) and \(C_2\).
From the first statement, "The sum of the ages of two cousins is 35", we can write the equation:
\[ C_1 + C_2 = 35 \quad \text{(Equation 1)} \]Ten years ago, their ages would have been \(C_1 - 10\) and \(C_2 - 10\).
From the second statement, "Ten years ago, the ratio of their ages was 2 : 1", we can write:
\[ \frac{C_1 - 10}{C_2 - 10} = \frac{2}{1} \]Cross-multiply to simplify this equation:
\[ 1 \times (C_1 - 10) = 2 \times (C_2 - 10) \]\[ C_1 - 10 = 2C_2 - 20 \]Rearrange the terms to get \(C_1\) in terms of \(C_2\):
\[ C_1 = 2C_2 - 20 + 10 \]\[ C_1 = 2C_2 - 10 \quad \text{(Equation 2)} \]Now, substitute Equation 2 into Equation 1:
\[ (2C_2 - 10) + C_2 = 35 \]\[ 3C_2 - 10 = 35 \]Solve for \(C_2\):
\[ 3C_2 = 35 + 10 \]\[ 3C_2 = 45 \]\[ C_2 = \frac{45}{3} \]\[ C_2 = 15 \]Substitute the value of \(C_2\) back into Equation 2 to find \(C_1\):
\[ C_1 = 2(15) - 10 \]\[ C_1 = 30 - 10 \]\[ C_1 = 20 \]So, the present ages of the two cousins are 20 years and 15 years.
Let's verify these ages with the given conditions:
B) 20, 15 — This option matches our calculated present ages.
This question tests the understanding of Newton's Law of Universal Gravitation. We need to analyze how the gravitational force changes when the mass of one of the interacting objects is altered.
B) The force of gravitation is doubled. As derived from Newton's Law of Universal Gravitation, if one mass is doubled while the other mass and the distance remain constant, the gravitational force also doubles.
To determine the town farthest west, we need to establish the relative positions of all the towns based on the given directional clues. We can visualize this by placing the towns on an imaginary east-west line.
"The town of Paranda is located on Green lake." This gives us a reference point. Let's place Paranda (P) on our line.
"The town of Akram is to the west of Paranda." This means Akram (A) is to the left of Paranda (P).
Current order: A --- P
"Tokhada is to the east of Akram but to the west of Paranda." This places Tokhada (T) between Akram and Paranda.
Current order: A --- T --- P
"Kokran is east of Bopri but west of Tokhada and Akram." This is the crucial step. Kokran (K) is west of both Tokhada (T) and Akram (A). Since Akram is already to the west of Tokhada, Kokran must be west of Akram. Also, Bopri (B) is to the west of Kokran.
This implies the order: B --- K --- A --- T --- P
By arranging all the towns in a single line from west to east, we get:
Bopri (B) <--- Kokran (K) <--- Akram (A) <--- Tokhada (T) <--- Paranda (P)
From this arrangement, Bopri is the leftmost town, meaning it is the farthest west.
Correct Option: D) Bopri is the farthest west town as per the established relative positions.
This is an analogy question where you need to identify the relationship between the first pair of words and then apply the same relationship to the second pair to find the missing word.
D) Ore — Metal is extracted from ore. Ore is a naturally occurring solid material from which a metal or valuable mineral can be profitably extracted. For example, iron is extracted from iron ore, aluminum from bauxite ore, etc. This relationship perfectly matches the "product : source" analogy established by "Oil : Seed".
To find the time taken for two trains running in the same direction to cross each other, we need to calculate their relative speed and the total distance they need to cover. The total distance is the sum of their lengths, and the relative speed is the difference between their individual speeds when moving in the same direction.
When two trains cross each other, the total distance covered is the sum of their lengths.
\[ \text{Total Distance} (D) = L_1 + L_2 = 200 \text{ m} + 150 \text{ m} = 350 \text{ m} \]Since the trains are running in the same direction, their relative speed is the difference between their individual speeds. We assume the faster train is catching up to and crossing the slower train.
\[ \text{Relative Speed} (S_{\text{rel}}) = S_2 - S_1 = 45 \text{ km/h} - 40 \text{ km/h} = 5 \text{ km/h} \]To convert km/h to m/s, multiply by \(\frac{5}{18}\).
\[ S_{\text{rel}} = 5 \times \frac{5}{18} \text{ m/s} = \frac{25}{18} \text{ m/s} \]Time is calculated by dividing the total distance by the relative speed.
\[ \text{Time} (T) = \frac{\text{Total Distance}}{\text{Relative Speed}} = \frac{350 \text{ m}}{\frac{25}{18} \text{ m/s}} \]\[ T = 350 \times \frac{18}{25} \text{ s} \]\[ T = 14 \times 18 \text{ s} \]\[ T = 252 \text{ s} \]C) 252 seconds
To hear a distinct echo, the human ear must be able to distinguish between the original sound and the reflected sound. This requires a minimum time interval, which is related to the persistence of hearing.
B) 0.1 second — This is the established minimum time interval required for the human ear to distinguish between two separate sounds, which is essential for perceiving a distinct echo.
The twinkling of stars is a common atmospheric phenomenon. This question asks for the underlying physical principle responsible for it. We need to analyze how light from distant stars interacts with Earth's atmosphere.
A) Atmospheric refraction is the correct explanation. The continuous bending of starlight as it passes through the turbulent and varying refractive index layers of the Earth's atmosphere causes the apparent brightness and position of the star to fluctuate, leading to the twinkling effect.
The question asks to identify the country that awarded Shri. Ram Nath Kovind, the former President of India, with its highest civilian award, the "Grand Order of the King of Tomislav". This is a factual question related to international relations and awards received by Indian dignitaries.
Correct Option: B) Croatia is the country that awarded Shri. Ram Nath Kovind with its highest civilian award, the "Grand Order of the King of Tomislav". He received this award in March 2019 during his visit to the country.
Efficiencies of P, Q, R and S in doing a job are in the ratio 2 : 3 : 5 : 4. The wages paid for the job are Rs.4,200. Who is paid the largest amount and how much is paid?
The problem states that the wages paid for a job are distributed among P, Q, R, and S based on their efficiencies. The wages are directly proportional to their efficiencies. We need to find out who gets the largest amount and what that amount is.
Understand the relationship between efficiency and wages: In such problems, the wages are distributed in the ratio of the efficiencies of the individuals. This means a person with higher efficiency will get a larger share of the wages.
Identify the given ratio of efficiencies: The efficiencies of P, Q, R, and S are in the ratio \(2 : 3 : 5 : 4\).
Determine who has the highest efficiency: By comparing the numbers in the ratio, \(5\) is the largest number, which corresponds to R. Therefore, R has the highest efficiency and will be paid the largest amount.
Calculate the total ratio units: Sum the individual ratio units to find the total number of parts the wages are divided into:
\[ \text{Total ratio units} = 2 + 3 + 5 + 4 = 14 \]Calculate the share of wages for R: The total wages paid are Rs. 4,200. R's share will be R's efficiency ratio divided by the total ratio units, multiplied by the total wages.
\[ \text{R's share} = \left( \frac{\text{R's efficiency ratio}}{\text{Total ratio units}} \right) \times \text{Total wages} \]\[ \text{R's share} = \frac{5}{14} \times 4200 \]\[ \text{R's share} = 5 \times \frac{4200}{14} \]\[ \text{R's share} = 5 \times 300 \]\[ \text{R's share} = 1500 \]Conclusion: R is paid the largest amount, which is Rs. 1,500.
Correct Option: A) R, Rs.1,500