To find the median of a given set of data, the first step is to arrange the data in ascending or descending order. Once ordered, the median is the middle value if the number of data points is odd. If the number of data points is even, the median is the average of the two middle values.
List the given data:
\(15, 70, 34, 37, 25, 36, 74, 36, 54, 78, 33\)
Count the number of data points (n):
There are \(11\) data points.
Arrange the data in ascending order:
\(15, 25, 33, 34, 36, 36, 37, 54, 70, 74, 78\)
Determine the position of the median:
Since the number of data points \(n = 11\) is odd, the median is the \(\left(\frac{n+1}{2}\right)^{\text{th}}\) term.
\[ \text{Median position} = \frac{11+1}{2} = \frac{12}{2} = 6^{\text{th}} \text{ term} \]
Identify the median value:
The \(6^{\text{th}}\) term in the ordered list is \(36\).
Correct Option: B) 36 is the median of the given data set.
To find the average of the first 17 multiples of 13, we can use the property of arithmetic progressions. The multiples of a number form an arithmetic progression. For an arithmetic progression, the average is simply the average of the first and the last term.
Alternatively, we can also think of it as finding the average of the first 17 natural numbers and then multiplying by 13. The average of the first 'n' natural numbers is \((n+1)/2\). So, the average of the first 17 natural numbers is \((17+1)/2 = 18/2 = 9\). Therefore, the average of the first 17 multiples of 13 is \(13 \times 9 = 117\).
C) 117
The problem asks us to find the value of \( (a + b)^{2} \) given two equations: \( a + b = 50 \) and \( a - b = 21 \). We can directly use the given value of \( a + b \) to calculate the required expression.
We are given two equations:
We need to find the value of \( (a + b)^{2} \).
From Equation 1, we already know the value of \( a + b \) which is 50.
Substitute \( a + b = 50 \) into the expression \( (a + b)^{2} \):
\[ (a + b)^{2} = (50)^{2} \]\[ (50)^{2} = 50 \times 50 \]\[ (50)^{2} = 2500 \]The value of \( (a + b)^{2} \) is 2500.
B) 2500
This problem requires us to calculate the number of molecules in a given volume of gas at Standard Temperature and Pressure (STP). We will use the molar volume concept at STP and Avogadro's number.
Recall the molar volume at STP: At STP (0 °C or 273.15 K and 1 atm pressure), 1 mole of any ideal gas occupies a volume of 22.4 liters.
Recall Avogadro's number: One mole of any substance contains Avogadro's number of particles (atoms, molecules, ions, etc.), which is approximately \( 6.022 \times 10^{23} \).
Calculate the number of moles of O₂ gas: We are given 5.6 L of O₂ gas at STP. We can find the number of moles using the molar volume:
\[ \text{Number of moles} = \frac{\text{Given volume}}{\text{Molar volume at STP}} \] \[ \text{Number of moles} = \frac{5.6 \text{ L}}{22.4 \text{ L/mol}} \] \[ \text{Number of moles} = 0.25 \text{ mol} \]
Calculate the number of molecules: Now that we have the number of moles, we can multiply it by Avogadro's number to find the total number of molecules:
\[ \text{Number of molecules} = \text{Number of moles} \times \text{Avogadro's number} \] \[ \text{Number of molecules} = 0.25 \text{ mol} \times 6.022 \times 10^{23} \text{ molecules/mol} \] \[ \text{Number of molecules} = 1.5055 \times 10^{23} \text{ molecules} \]
B) \( 1.5055 \times 10^{23} \) is the correct number of molecules present in 5.6 L of O₂ gas at STP.
A) \( 2.24 \times 10^{23} \) — This value is incorrect. It might result from an incorrect calculation or a misunderstanding of the relationship between volume and moles.
C) \( 6.022 \times 10^{23} \) — This is Avogadro's number, which represents the number of molecules in 1 mole (or 22.4 L at STP) of gas, not 5.6 L.
D) \( 3.011 \times 10^{23} \) — This value is half of Avogadro's number, corresponding to 0.5 moles (or 11.2 L at STP) of gas, not 5.6 L.
The question asks about the government's shareholding threshold in a Public Sector Undertaking (PSU) for it to be considered privatised. Privatisation implies a transfer of ownership and control from the government to the private sector. This typically means the government loses its majority stake.
Correct Option: D) Less than 51%
If the government's shareholding falls to less than 51%, it no longer holds a majority stake, and therefore, the control of the Public Sector Undertaking (PSU) effectively shifts to the private sector. This is the key criterion for an enterprise to be considered privatised, even if the government retains a minority share.
The question asks to identify the flagship program launched in 2015 to improve employability through skill training, aligning the workforce with economic needs. This requires knowledge of major government initiatives related to skill development in India.
Correct Option: B) Skill India Mission
The Skill India Mission, also known as the National Skill Development Mission, was launched by the Government of India on July 15, 2015, on the occasion of World Youth Skills Day. Its primary objective is to train over 40 crore (400 million) people in India in various skills by 2022 and to improve their employability. It encompasses several sub-schemes and initiatives, making it the flagship program for skill development.
The question asks for the full form of the acronym PCB in the context of electronics. This is a common acronym in the field, and knowing its full form is fundamental for anyone studying or working with electronic components.
Correct Option: D) Printed Circuit Board is the correct full form of PCB in electronics. It refers to the board used to mechanically support and electrically connect electronic components using conductive pathways, tracks, or signal traces etched from copper sheets laminated onto a non-conductive substrate.
The problem asks us to find the missing letter-clusters, # and %, based on a pattern observed in the given analogy. We need to determine the relationship between the letters in a known pair and apply the same relationship to find the unknown clusters.
Correct Option: D) # = RVU, % = GKJ
The pattern is that each letter in the first cluster is transformed into the corresponding letter in the second cluster by subtracting 3 from its alphabetical position.
The question asks for the range of the Hansa NG trainer aircraft. This is a factual recall question related to current affairs and defense technology. The Hansa NG (New Generation) is an Indian-developed aircraft, and its specifications, including its range, are publicly available information.
Correct Option: D) 620 nautical miles
The Hansa NG (New Generation) is an indigenous two-seater flying trainer aircraft designed and developed by CSIR-National Aerospace Laboratories (NAL). It successfully completed its maiden flight in September 2021. Its specifications include a maximum speed of 200 km/h and a range of 620 nautical miles (approximately 1150 km).
The question asks to identify the foreign traveller who mentioned the Ajanta caves in their accounts. This requires knowledge of prominent foreign travellers to India and the periods they visited, as well as the historical context of the Ajanta caves.
Correct Option: B) Hiuen Tsang was a Chinese Buddhist pilgrim who visited India in the 7th century CE. His travelogue, "Si-Yu-Ki" (Records of the Western Regions), provides valuable information about India during that period. He is known to have visited and described the Ajanta caves, which were active Buddhist monastic sites during his time. His detailed descriptions match the location and characteristics of the Ajanta caves.