This question tests the understanding of Archimedes' principle and the conditions for an object to float or sink. The key is to compare the buoyant force acting on an object with its weight.
Correct Option: C) If the buoyant force is less than the weight of the object, then the object floats.
This statement is incorrect. If the buoyant force (\(F_B\)) is less than the weight of the object (\(W\)), meaning \(F_B < W\), the net force acting on the object is downwards, causing the object to sink, not float. Therefore, this statement is wrong.
To find the mean (average) of a set of numbers, we need to sum all the numbers and then divide the sum by the total count of the numbers.
Correct Option: A) 3.5 is the correct mean of the given numbers.
The geometric mean (GM) of a set of \(n\) numbers is the \(n\)-th root of their product. We will first find the product of the given numbers and then raise it to the power of \(1/n\) to find the geometric mean.
Identify the numbers: The given numbers are \(x_1 = 7\), \(x_2 = 7^2\), \(x_3 = 7^3\), ..., \(x_n = 7^n\).
Calculate the product of the numbers: Let \(P\) be the product of these \(n\) numbers.
\[ P = 7 \times 7^2 \times 7^3 \times \dots \times 7^n \]Simplify the product using exponent rules: When multiplying powers with the same base, we add the exponents.
\[ P = 7^{(1 + 2 + 3 + \dots + n)} \]Calculate the sum of the exponents: The sum \(1 + 2 + 3 + \dots + n\) is the sum of the first \(n\) natural numbers, which is given by the formula \(\frac{n(n+1)}{2}\).
\[ P = 7^{\frac{n(n+1)}{2}} \]Calculate the geometric mean (GM): The geometric mean of \(n\) numbers is the \(n\)-th root of their product. So, \(GM = P^{\frac{1}{n}}\).
\[ GM = \left(7^{\frac{n(n+1)}{2}}\right)^{\frac{1}{n}} \]Simplify the expression for GM: When raising a power to another power, we multiply the exponents.
\[ GM = 7^{\left(\frac{n(n+1)}{2} \times \frac{1}{n}\right)} \]\[ GM = 7^{\frac{n+1}{2}} \]Correct Option: A) \(7^{\frac{n+1}{2}}\)
The problem asks us to find an integer \(n\) that satisfies a given inequality involving fractions. We need to set up the inequality and then solve for \(n\) by isolating it. Since \(n\) must be an integer, we will look for integer values within the derived range.
The problem states that \( \frac{n}{60} \) should be less than or equal to \( \frac{1}{5} \) but greater than \( \frac{1}{6} \). This can be written as a compound inequality:
\[ \frac{1}{6} < \frac{n}{60} \le \frac{1}{5} \]To isolate \(n\), we need to multiply all parts of the inequality by 60. Since 60 is a positive number, the direction of the inequality signs will not change.
\[ 60 \times \frac{1}{6} < 60 \times \frac{n}{60} \le 60 \times \frac{1}{5} \]Perform the multiplications:
So the inequality becomes:
\[ 10 < n \le 12 \]This inequality means that \(n\) must be strictly greater than 10 and less than or equal to 12. Since \(n\) must be an integer, the possible values for \(n\) are 11 and 12.
We found that \(n\) can be 11 or 12. Let's look at the given options:
Options A and C are 11 and 12, respectively, which are the values we found. Option D states "Both option 1 and 3", which means both 11 and 12.
D) Both option 1 and 3. The integer values for \(n\) that satisfy the given condition are 11 and 12. Option D correctly includes both of these values.
The question asks to identify the act that replaced the Monopolies and Restricted Trade Practices Act (MRTP Act), 1969. This requires knowledge of Indian economic and competition law history.
D) Competition Act — The Competition Act, 2002, was enacted to replace the Monopolies and Restricted Trade Practices Act, 1969. Its objective is to promote and sustain competition in markets, protect the interests of consumers, and ensure freedom of trade carried on by other participants in markets in India.
This problem involves calculating compound growth. The population increases by a fixed percentage each year, similar to compound interest. We can use the compound interest formula or calculate the increase year by year.
Identify the given values:
Use the compound growth formula:
The formula for population after \(n\) years is given by:
Substituting the given values:
\[ A = 30,00,000 \left(1 + \frac{3}{100}\right)^3 \] \[ A = 30,00,000 (1 + 0.03)^3 \] \[ A = 30,00,000 (1.03)^3 \]Calculate \((1.03)^3\):
\[ (1.03)^1 = 1.03 \] \[ (1.03)^2 = 1.03 \times 1.03 = 1.0609 \] \[ (1.03)^3 = 1.0609 \times 1.03 = 1.092727 \]Calculate the final population:
\[ A = 30,00,000 \times 1.092727 \] \[ A = 32,78,181 \]Correct Option: C) 3278181
The problem asks us to identify which assumption(s) are implicit in the given statement. An implicit assumption is something that is taken for granted or presupposed by the statement, even if not explicitly stated. We need to analyze the statement and each assumption to see if the statement logically requires that assumption to be true.
A) Only Assumption II is implicit. As explained above, the statement logically implies that the company has a need for a software engineer to be making such an offer. Assumption I is a condition for the offer, not an assumption made by the statement itself.
The question asks for the specific term that describes the temperature at which a liquid starts to boil under atmospheric pressure. We need to identify the correct physical property from the given options.
Correct Option: C) Boiling point is the temperature at which a liquid's vapor pressure equals the surrounding atmospheric pressure, causing it to change into a gas (boil).
This problem can be solved by setting up a system of two linear equations based on the given information: the total number of questions attempted and the total marks obtained. We will define variables for the number of correct and incorrect answers and then solve for the number of correct answers.
Let \(C\) be the number of correct questions answered.
Let \(W\) be the number of wrong questions answered.
The total number of questions attempted is 98. So, we can write the first equation:
\[ C + W = 98 \quad \text{(Equation 1)} \]For each correct answer, 4 marks are awarded. For each wrong answer, 1 mark is deducted (negative mark). The total marks obtained are 202. So, we can write the second equation:
\[ 4C - 1W = 202 \quad \text{(Equation 2)} \]Now we have a system of two linear equations:
\[ \begin{cases} C + W = 98 \ 4C - W = 202 \end{cases} \]We can solve this system by adding Equation 1 and Equation 2:
\[ (C + W) + (4C - W) = 98 + 202 \]\[ C + 4C + W - W = 300 \]\[ 5C = 300 \]Now, solve for \(C\):
\[ C = \frac{300}{5} \]\[ C = 60 \]So, the number of correct questions answered by Mr. X is 60.
To verify, we can find \(W\) using Equation 1:
\[ W = 98 - C \]\[ W = 98 - 60 \]\[ W = 38 \]Check with Equation 2:
\[ 4C - W = 4(60) - 38 = 240 - 38 = 202 \]The calculated marks match the given total marks, confirming our answer.
D) 60 — The calculation shows that Mr. X answered 60 questions correctly.
The question asks for the ordinal number of Justice Jagdish Singh Khehar's appointment as the Chief Justice of India. To answer this, one needs to recall the sequence of Chief Justices of India.
Correct Option: B) Justice Jagdish Singh Khehar was the 44th Chief Justice of India. He assumed office on January 4, 2017, succeeding Justice T.S. Thakur (43rd CJI).