This is an analogy-based reasoning question. We need to identify the relationship between the first pair of words ('Horse' and 'Cart') and then find a word that shares the same relationship with 'Car'.
Correct Option: A) Engine
The question asks to identify the Chief Minister of Uttarakhand who was sworn in during March 2021. This requires knowledge of recent political appointments in Indian states.
Correct Option: B) Tirath Singh Rawat was sworn in as the Chief Minister of Uttarakhand on March 10, 2021, succeeding Trivendra Singh Rawat.
To find the time taken for the train to pass the man, we need to calculate the relative speed between the train and the man. Since they are moving in opposite directions, their speeds add up. Then, we use the formula \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\), where the distance is the length of the train.
Since the train and the man are moving in opposite directions, their relative speed is the sum of their individual speeds.
\[ \text{Relative Speed} = S_T + S_M \]\[ \text{Relative Speed} = 64 \text{ km/h} + 8 \text{ km/h} \]\[ \text{Relative Speed} = 72 \text{ km/h} \]The distance is in meters, so we need to convert the speed from km/h to m/s. To convert km/h to m/s, multiply by \(\frac{5}{18}\).
\[ \text{Relative Speed (m/s)} = 72 \times \frac{5}{18} \]\[ \text{Relative Speed (m/s)} = 4 \times 5 \]\[ \text{Relative Speed (m/s)} = 20 \text{ m/s} \]The time taken for the train to pass the man is the length of the train divided by the relative speed.
\[ \text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} \]\[ \text{Time} = \frac{100 \text{ m}}{20 \text{ m/s}} \]\[ \text{Time} = 5 \text{ seconds} \]Correct Option: D) 5 sec
The question asks to identify the dynasty to which Pulakeshin II, a prominent ruler, belonged. This requires knowledge of ancient Indian history and the major dynasties and their significant rulers.
Correct Option: D) Pulakeshin II was the most illustrious ruler of the Chalukya dynasty, specifically the Badami Chalukyas. His reign was marked by military victories and cultural advancements.
The problem asks for the ratio of the areas of two parallelograms that share an equal base and lie between the same parallel lines. We need to recall the formula for the area of a parallelogram and understand the implications of the given conditions.
Recall the formula for the area of a parallelogram: The area of a parallelogram is given by the product of its base and its corresponding height.
\[ \text{Area} = \text{Base} \times \text{Height} \]
Analyze the given conditions:
Calculate the area for the first parallelogram: Let the first parallelogram be \(P_1\). Its base is \(b\) and its height is \(h\).
\[ \text{Area}(P_1) = b \times h \]
Calculate the area for the second parallelogram: Let the second parallelogram be \(P_2\). Its base is also \(b\) (since they are on equal bases) and its height is also \(h\) (since they are between the same parallels).
\[ \text{Area}(P_2) = b \times h \]
Determine the ratio of their areas: Now, we find the ratio of the area of the first parallelogram to the area of the second parallelogram.
\[ \frac{\text{Area}(P_1)}{\text{Area}(P_2)} = \frac{b \times h}{b \times h} \]
Since \(b \neq 0\) and \(h \neq 0\), we can cancel out \(b\) and \(h\) from the numerator and denominator.
\[ \frac{\text{Area}(P_1)}{\text{Area}(P_2)} = \frac{1}{1} \]
So, the ratio of their areas is \(1:1\).
Correct Option: B) 1 : 1
This is a syllogism problem. We need to evaluate the given statements and determine which of the conclusions logically follow. We can use Venn diagrams to visualize the relationships described in the statements.
Correct Option: A) Only conclusion I follows.
We are given relationships between four numbers and their mean. We will express all numbers in terms of the first number, then use the formula for the mean to find the value of the first number. Once the first number is known, we can identify the smallest among the four.
Let the first number be \(x\).
According to the problem, the second number is thrice of the first. So, the second number is \(3x\).
The third number is four times the second. So, the third number is \(4 \times (3x) = 12x\).
The fourth number is five times the third. So, the fourth number is \(5 \times (12x) = 60x\).
The four numbers are \(x\), \(3x\), \(12x\), and \(60x\).
The mean of these four numbers is given as 1900. The formula for the mean is:
\[ \text{Mean} = \frac{\text{Sum of numbers}}{\text{Count of numbers}} \]Substitute the numbers and the given mean into the formula:
\[ 1900 = \frac{x + 3x + 12x + 60x}{4} \]Simplify the sum in the numerator:
\[ 1900 = \frac{76x}{4} \]Further simplify the right side:
\[ 1900 = 19x \]Solve for \(x\):
\[ x = \frac{1900}{19} \]\[ x = 100 \]So, the first number is 100.
Now, let's find all four numbers:
Comparing these four numbers (100, 300, 1200, 6000), the smallest number is 100.
Correct Option: D) 100
The question asks about the term duration for a Member of Parliament (MP) in the Rajya Sabha. This is a fundamental concept in the Indian parliamentary system, specifically related to the structure and functioning of the Upper House.
D) Six years is the correct duration for the term of a Rajya Sabha MP, as stipulated by the Representation of the People Act, 1951, and based on the constitutional provisions for the Rajya Sabha.
To determine the percentage increase in the surface area of a sphere when its radius is tripled, we will first recall the formula for the surface area of a sphere. Then, we will calculate the new surface area with the tripled radius and compare it to the original surface area to find the percentage increase.
Original Surface Area:
Let the original radius of the sphere be \(r\). The formula for the surface area of a sphere is:
\[ A_1 = 4 \pi r^2 \]New Radius:
The radius is tripled, so the new radius \(r'\) will be:
\[ r' = 3r \]New Surface Area:
Substitute the new radius into the surface area formula:
\[ A_2 = 4 \pi (r')^2 \]\[ A_2 = 4 \pi (3r)^2 \]\[ A_2 = 4 \pi (9r^2) \]\[ A_2 = 36 \pi r^2 \]Increase in Surface Area:
The increase in surface area is the difference between the new and original surface areas:
\[ \text{Increase} = A_2 - A_1 \]\[ \text{Increase} = 36 \pi r^2 - 4 \pi r^2 \]\[ \text{Increase} = 32 \pi r^2 \]Percentage Increase:
The percentage increase is calculated as \[\left( \frac{\text{Increase}}{\text{Original Surface Area}} \right) \times 100\%\]:
\[ \text{Percentage Increase} = \frac{32 \pi r^2}{4 \pi r^2} \times 100\% \]\[ \text{Percentage Increase} = 8 \times 100\% \]\[ \text{Percentage Increase} = 800\% \]Correct Option: A) 800%
The question asks for the conversion of 1 Gigabyte (GB) to Megabytes (MB) in binary. In computing, memory and storage capacities are typically measured in powers of 2, not powers of 10. We need to recall the standard binary prefixes used in computer science.
D) 1024 — As calculated, 1 Gigabyte is equal to \(2^{10}\) Megabytes, and \(2^{10}\) is 1024. This is the standard binary conversion used in computer science for memory and storage.