If the mean of the following data is k, then find the value of k.
| Value | 3 | 2 | k | 4 | 5 |
| Frequency | k | 2k | 3k | 4k | 5k |
To find the value of 'k', we will use the formula for the mean of grouped data. The mean is calculated as the sum of the product of each value and its frequency, divided by the sum of the frequencies.
First, let's list the given data:
| Value (\(x_i\)) | Frequency (\(f_i\)) |
|---|---|
| 3 | k |
| 2 | 2k |
| k | 3k |
| 4 | 4k |
| 5 | 5k |
Calculate the sum of the products of each value and its frequency (\(\sum f_i x_i\)):
\[ \sum f_i x_i = (3 \times k) + (2 \times 2k) + (k \times 3k) + (4 \times 4k) + (5 \times 5k) \]\[ \sum f_i x_i = 3k + 4k + 3k^2 + 16k + 25k \]\[ \sum f_i x_i = 3k^2 + (3+4+16+25)k \]\[ \sum f_i x_i = 3k^2 + 48k \]Calculate the sum of the frequencies (\(\sum f_i\)):
\[ \sum f_i = k + 2k + 3k + 4k + 5k \]\[ \sum f_i = (1+2+3+4+5)k \]\[ \sum f_i = 15k \]The mean of the data is given as 'k'. The formula for the mean (\(\bar{x}\)) is:
\[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \]Substitute the calculated sums and the given mean into the formula:
\[ k = \frac{3k^2 + 48k}{15k} \]Since 'k' is a frequency, it cannot be zero. Therefore, we can divide both the numerator and the denominator by 'k' (assuming \(k \neq 0\)):
\[ k = \frac{k(3k + 48)}{15k} \]\[ k = \frac{3k + 48}{15} \]Now, solve for 'k':
\[ 15k = 3k + 48 \]\[ 15k - 3k = 48 \]\[ 12k = 48 \]\[ k = \frac{48}{12} \]\[ k = 4 \]A) 4 is the correct value for k, as derived from the mean formula.
The question asks to identify the option that shares the same relationship as the given pair: "Actor = Theatre". We need to determine the nature of the relationship between "Actor" and "Theatre" and then find an option that replicates this relationship.
Correct Option: D) Swimmer = Swimming pool. This option establishes a relationship where the first word (Swimmer) is a person who performs their characteristic activity in the place denoted by the second word (Swimming pool).
The problem requires us to determine the number of students shorter than D based on a series of height comparisons. We will establish the relative heights of all six students (A, B, C, D, E, F) by converting the given statements into inequalities and then arrange them in decreasing order of height.
C) Three — Based on the height comparisons, A, E, and F are all shorter than D. This makes a total of three students.
To find what percentage one number is of another, we can use the formula: \[\text{Percentage} = \left(\frac{\text{Part}}{\text{Whole}}\right) \times 100\%\]. In this case, the "part" is 0.06 and the "whole" is 25.
The question asks "What percent of 25 is the number 0.06?". Here, 0.06 is the 'part' and 25 is the 'whole'.
The fraction representing the relationship is \[\frac{\text{Part}}{\text{Whole}} = \frac{0.06}{25}\].
Divide 0.06 by 25:
\[ \frac{0.06}{25} = 0.0024 \]To convert a decimal to a percentage, multiply by 100%:
\[ 0.0024 \times 100\% = 0.24\% \]Thus, 0.06 is 0.24% of 25.
A) 0.24% is the correct answer, as calculated by dividing 0.06 by 25 and then multiplying by 100.
The question asks about the president of the Karachi session of the Indian National Congress in 1931. This is a factual question related to Indian history, specifically the Indian National Congress sessions. Recalling the key figures and decisions of significant Congress sessions is crucial for competitive exams.
Correct Option: D) Sardar Patel
Sardar Vallabhbhai Patel presided over the historic Karachi session of the Indian National Congress in 1931. This session is notable for passing resolutions on Fundamental Rights and the National Economic Programme, which laid down the basic political, social, and economic programs of the Congress.
The question asks to identify which option is NOT a navigation satellite system. We need to evaluate each option to determine if it represents a global or regional navigation satellite system.
Based on the evaluation, MEO is an orbital classification, not a navigation satellite system.
Correct Option: C) MEO is a type of Earth orbit (Medium Earth Orbit) and not a navigation satellite system itself. Many navigation satellites, including those for GPS, Galileo, and BeiDou, operate in MEO, but MEO is not a system name.
The question asks to identify the Indian independence movement that was initiated by the Dandi March. This requires knowledge of the timeline and key events associated with major Gandhian movements.
Correct Option: A) Civil Disobedience Movement. The Dandi March (Salt Satyagraha) was the direct trigger and the beginning of the Civil Disobedience Movement in 1930. Mahatma Gandhi led this march to protest the British salt tax, which then expanded into a nationwide movement of civil disobedience against various British laws.
The question asks to identify the international institution that supported the Neeranchal National Watershed Project. This requires knowledge of major development projects in India and their funding/supporting agencies.
Correct Option: A) World Bank
The question asks to identify the type of server responsible for converting IP addresses to domain names. This process is known as reverse DNS lookup. We need to evaluate each option based on its primary function in networking.
D) DNS — DNS (Domain Name System) servers are responsible for translating domain names into IP addresses (forward lookup) and also for translating IP addresses back into domain names (reverse lookup). This is a fundamental service for the internet's operation.
The question asks to identify the Indian center specifically used for testing missiles. This requires knowledge of India's defense and space infrastructure.
Correct Option: C) Wheeler Island
Wheeler Island, located off the coast of Odisha, India, is the primary missile testing facility for India's integrated test range (ITR) program. It is now officially known as Dr. Abdul Kalam Island, named after the former President of India and renowned scientist, Dr. A.P.J. Abdul Kalam, who played a pivotal role in India's missile development program.