This problem requires the application of an empirical relationship between the mean, median, and mode for a moderately skewed distribution. The empirical formula states that the difference between the mode and the mean is approximately three times the difference between the mean and the median.
Identify the given values:
Mean \( = 36 \)
Mode \( = 63 \)
Recall the empirical relation:
For a moderately skewed distribution, the empirical relationship between mean, median, and mode is given by:
Substitute the given values into the formula:
We have Mode \( = 63 \) and Mean \( = 36 \). Let Median \( = M \).
So, the equation becomes:
Simplify the equation:
\[ 63 = 3M - 72 \]Isolate the term with Median:
Add 72 to both sides of the equation:
Solve for Median (M):
Divide both sides by 3:
C) The median of the data set is 45.
The question asks to identify the part of the Indian Constitution that deals with Panchayati Raj Institutions. To answer this, one needs to recall the various parts of the Constitution and their respective subjects.
Correct Option: B) Part-IX of the Constitution of India is related to Panchayati Raj Institutions. This part was added by the 73rd Constitutional Amendment Act of 1992, which aimed to provide constitutional status to Panchayati Raj and ensure their effective functioning as institutions of local self-government.
The question asks about the fundamental characteristic of an isoquant in economics. An isoquant is a concept used in microeconomics to represent different combinations of inputs that yield the same level of output. Understanding this definition is key to identifying the constant factor along such a curve.
B) Level of output. An isoquant is a locus of points representing all possible combinations of two inputs (like labor and capital) that yield the same maximum level of output. Therefore, the level of output remains constant along an isoquant.
Study the given arrangement carefully and answer the question that follows.
(Left) £ & 6 @ 4 ₱ 2 4 % 1 3 ^ 3 ¥ @ 8 8 % 7 (Right)
How many such symbols are there in the above arrangement each of which is immediately followed by a number and immediately preceded by a symbol?
The problem asks us to identify symbols that fit a specific pattern: they must be immediately preceded by another symbol and immediately followed by a number. We need to scan the given arrangement from left to right and check each symbol against this condition.
(Left) £ & 6 @ 4 ₱ 2 4 % 1 3 ^ 3 ¥ @ 8 8 % 7 (Right)A) There are 2 such symbols in the arrangement that are immediately followed by a number and immediately preceded by a symbol. These are the first '&' (in '£ & 6') and the second '@' (in '¥ @ 8').
The problem asks us to identify a set of three numbers that follow the same logical relationship as the given sets: (7, 150, 23) and (14, 100, 6). We need to find a pattern involving the first, second, and third numbers in each set.
A) (15, 140, 13)
First number \(A=15\), Third number \(C=13\).
Sum: \(15 + 13 = 28\).
Apply the pattern: \(28 \times 5 = 140\).
The second number in the option is 140. This matches the calculated value. Therefore, option A follows the same relationship.
What is the nature of the roots of \( \sqrt{3} \)x² + 6x − \( \sqrt{3} \) = 0?
The roots are real and more than 2.
The roots are real and equal.
There are no real roots.
The roots are real and distinct.
To determine the nature of the roots of a quadratic equation of the form \(ax^2 + bx + c = 0\), we use the discriminant, denoted by \(\Delta\) or \(D\). The discriminant is given by the formula \(D = b^2 - 4ac\). The nature of the roots depends on the value of \(D\):
The given quadratic equation is \(\sqrt{3}x^2 + 6x - \sqrt{3} = 0\).
Comparing this with the standard form \(ax^2 + bx + c = 0\), we have:
Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula \(D = b^2 - 4ac\):
\[ D = (6)^2 - 4(\sqrt{3})(-\sqrt{3}) \]\[ D = 36 - 4(-3) \]\[ D = 36 + 12 \]\[ D = 48 \]Since \(D = 48\), which is greater than 0 (\(D > 0\)), the roots of the quadratic equation are real and distinct (unequal).
D) The roots are real and distinct. This is correct because the discriminant \(D = 48\), which is positive, indicating real and distinct roots.
The problem provides the ratio of boys to girls and the actual number of girls. We need to find the actual number of boys. We can achieve this by setting up a proportion or by finding the value of one unit in the ratio.
C) 147 is the correct number of boys in the school.
The roots of the quadratic equation 2x² − x − 3 = 0 are:
1, −\( \frac{3}{2} \)
−1, \( \frac{3}{2} \)
1, \( \frac{3}{2} \)
−1, −\( \frac{3}{2} \)
To find the roots of a quadratic equation of the form \(ax^2 + bx + c = 0\), we can use either factorization (splitting the middle term) or the quadratic formula. In this case, factorization is a straightforward method.
Correct Option: B) \(-1, \frac{3}{2}\)
The question asks to identify the political party that was recognized as a national party in 2016 by the Election Commission and has "Jora Ghas Phool" (Grass with two flowers) as its election symbol. This requires knowledge of Indian political parties, their symbols, and their recognition status.
C) All India Trinamool Congress (AITC) - The All India Trinamool Congress (AITC) uses "Jora Ghas Phool" (Grass with two flowers) as its election symbol and was officially recognized as a national political party by the Election Commission of India in 2016.
The question asks for the target year for achieving 'Housing for All' under the Pradhan Mantri Awaas Yojana–Gramin (PMAY-G) scheme. This requires knowledge of the scheme's objectives and timeline as set by the Indian government.
D) 2022 — The Pradhan Mantri Awaas Yojana–Gramin (PMAY-G) was launched with the explicit aim of achieving "Housing for All" by the year 2022. This target year is a well-known fact associated with the scheme.