If \( \text{mean} = \frac{3\,\text{median} - \text{mode}}{p} \), then the value of p is:
This question tests knowledge of the empirical relationship between mean, median, and mode for a moderately skewed distribution. The relationship is given by the formula: \( \text{Mode} = 3 \text{Median} - 2 \text{Mean} \). We need to rearrange this formula to match the given expression and find the value of \( p \).
Recall the empirical relationship between mean, median, and mode for a moderately skewed distribution:
\[ \text{Mode} = 3 \text{Median} - 2 \text{Mean} \]Rearrange the formula to isolate the term involving 'Mean' on one side. Move \( 2 \text{Mean} \) to the left side and \( \text{Mode} \) to the right side:
\[ 2 \text{Mean} = 3 \text{Median} - \text{Mode} \]Divide both sides by 2 to express 'Mean' in the desired form:
\[ \text{Mean} = \frac{3 \text{Median} - \text{Mode}}{2} \]Compare this derived formula with the given expression in the question:
\[ \text{Mean} = \frac{3 \text{Median} - \text{Mode}}{p} \]By comparing the two equations, we can clearly see that the value of \( p \) is 2.
B) The value of \( p \) is 2, as derived from the empirical relationship between mean, median, and mode.
To solve this problem, we first need to find the Highest Common Factor (HCF) of 51 and 85. Once we have the HCF, we will equate it to the given expression \(51m - 85\) and solve for the value of \(m\).
Find the HCF of 51 and 85:
We can use the prime factorization method or the Euclidean algorithm.
Prime Factorization Method:
The common prime factor is \(17\). Therefore, the HCF of 51 and 85 is 17.
Euclidean Algorithm:
The last non-zero remainder is 17. So, HCF(51, 85) = 17.
Equate the HCF to the given expression:
We are given that the HCF is expressed in the form \(51m - 85\). So, we set up the equation:
\[ 51m - 85 = 17 \]Solve for \(m\):
Add 85 to both sides of the equation:
\[ 51m = 17 + 85 \] \[ 51m = 102 \]Divide both sides by 51:
\[ m = \frac{102}{51} \] \[ m = 2 \]D) The value of \(m\) is 2, which matches option D.
To find the time taken for a train to cross a bridge, we need to calculate the total distance the train travels, which is the sum of its own length and the length of the bridge. Then, we use the formula \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\), ensuring all units are consistent.
When a train crosses a bridge, the total distance it travels is its own length plus the length of the bridge.
\[ \text{Total Distance} (D) = L_T + L_B \] \[ D = 400 \text{ m} + 600 \text{ m} \] \[ D = 1000 \text{ m} \]
Since the distance is in meters, it's convenient to convert the speed from km/h to m/s.
\[ S = 60 \text{ km/h} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} \] \[ S = 60 \times \frac{5}{18} \text{ m/s} \] \[ S = \frac{10 \times 5}{3} \text{ m/s} \] \[ S = \frac{50}{3} \text{ m/s} \]
Using the formula \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\).
\[ \text{Time} (T) = \frac{D}{S} \] \[ T = \frac{1000 \text{ m}}{\frac{50}{3} \text{ m/s}} \] \[ T = \frac{1000 \times 3}{50} \text{ s} \] \[ T = 20 \times 3 \text{ s} \] \[ T = 60 \text{ s} \]
Since 1 minute = 60 seconds:
\[ T = 60 \text{ s} = 1 \text{ minute} \]
Correct Option: A) 1 min
The question asks to identify the winner of the Orange Cap for the most runs scored in the Indian Premier League (IPL) 2020 season. This requires knowledge of IPL records and awards.
In IPL 2020, K L Rahul, playing for Kings XI Punjab (now Punjab Kings), scored the most runs. He accumulated a total of 670 runs in 14 matches, securing the Orange Cap.
A) K L Rahul — K L Rahul was indeed the highest run-scorer in IPL 2020 with 670 runs, thus winning the Orange Cap.
(\(\sqrt{3}\) + \(\sqrt{11}\))² is a/an:
irrational number
natural number
rational number
whole number
To determine the nature of the given expression \((\sqrt{3} + \sqrt{11}))^2\), we will expand it using the algebraic identity \((a+b)^2 = a^2 + b^2 + 2ab\). Then, we will simplify the result and classify it as a natural, whole, rational, or irrational number.
The given expression is \((\sqrt{3} + \sqrt{11}))^2\). We use the identity \((a+b)^2 = a^2 + b^2 + 2ab\), where \(a = \sqrt{3}\) and \(b = \sqrt{11}\).
\[ (\sqrt{3} + \sqrt{11}))^2 = (\sqrt{3})^2 + (\sqrt{11})^2 + 2(\sqrt{3})(\sqrt{11}) \]Calculate each term:
Substitute these values back into the expanded expression:
\[ (\sqrt{3} + \sqrt{11}))^2 = 3 + 11 + 2\sqrt{33} \]Add the rational numbers:
\[ 3 + 11 = 14 \]The simplified expression is:
\[ 14 + 2\sqrt{33} \]We know that \(\sqrt{33}\) is an irrational number because 33 is not a perfect square. The product of a non-zero rational number (2) and an irrational number (\(\sqrt{33}\)) is irrational. The sum of a rational number (14) and an irrational number (\(2\sqrt{33}\)) is always irrational.
Therefore, \(14 + 2\sqrt{33}\) is an irrational number.
Correct Option: A) irrational number. The simplified expression \(14 + 2\sqrt{33}\) contains the term \(2\sqrt{33}\), which is irrational. The sum of a rational number (14) and an irrational number (\(2\sqrt{33}\)) is always irrational.
The question asks to identify the President of Sri Lanka as of October 2020. This requires knowledge of contemporary political leaders in Sri Lanka.
Correct Option: C) Gotabaya Rajapaksa was the President of Sri Lanka from November 2019 to July 2022. Thus, in October 2020, he held the presidential office.
The question asks about the origin of the Gulf Stream, a major ocean current. To answer this, one needs to recall the geographical starting point of this specific current.
Correct Option: B) The Gulf Stream emanates from the Gulf of Mexico. This is a well-established geographical fact regarding ocean currents.
Which constitutional amendment gave the Directive Principles precedence over Fundamental Rights?
The question asks about a specific constitutional amendment that granted Directive Principles of State Policy (DPSP) precedence over Fundamental Rights. This is a key event in Indian constitutional history, particularly related to the balance between individual liberties and socio-economic goals. We need to identify the amendment responsible for this change.
The question asks to identify the program related to road transport development in India. We need to evaluate each option to determine its primary focus.
Correct Option: C) Golden Quadrilateral is a major highway network project directly associated with the development of road transport in India.
This problem involves calculating the time period for a sum of money to grow to a certain amount under compound interest. We will use the compound interest formula to solve for the number of years.
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]We need to find what power of 1.15 equals 1.3225.
Thus, \(n = 2\) years.
Correct Option: C) 2