The question asks to identify the first and oldest intergovernmental organization. This requires knowledge of the history and founding dates of major international organizations.
Correct Option: B) Central Commission for Navigation on the Rhine
To find the rate of compound interest, we will use the compound interest formula, which relates the amount, principal, rate, and time. We will substitute the given values into the formula and solve for the rate.
Correct Option: A) 6%
The question asks to identify the winner of the Men's Individual Title in the Asian Squash Championship in 2019. This is a factual question related to sports current events and general knowledge. To answer this, one needs to recall the results of the specified championship.
B) Saurav Ghosal — Saurav Ghosal won the Men's Individual Title at the Asian Squash Championship in 2019, making him the first Indian male squash player to achieve this feat.
The expression \(\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}\) can be written as:
sec A + cosec A
tan A + cot A
1 + sec A cosec A
1 + sin A cos A
To simplify the given trigonometric expression, we will convert all terms into sine and cosine. Then, we will find a common denominator and combine the fractions. Finally, we will simplify the resulting expression to match one of the given options.
Recall the identities: \(\tan A = \frac{\sin A}{\cos A}\) and \(\cot A = \frac{\cos A}{\sin A}\).
Substitute these into the given expression:
\[ \frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A} = \frac{\frac{\sin A}{\cos A}}{1-\frac{\cos A}{\sin A}}+\frac{\frac{\cos A}{\sin A}}{1-\frac{\sin A}{\cos A}} \]Notice that \((\cos A - \sin A) = -(\sin A - \cos A)\). We can use this to make the denominators similar:
\[ = \frac{\sin^2 A}{\cos A (\sin A - \cos A)} + \frac{\cos^2 A}{-\sin A (\sin A - \cos A)} \]\[ = \frac{\sin^2 A}{\cos A (\sin A - \cos A)} - \frac{\cos^2 A}{\sin A (\sin A - \cos A)} \]Now, find a common denominator, which is \(\sin A \cos A (\sin A - \cos A)\):
\[ = \frac{\sin^2 A \cdot \sin A - \cos^2 A \cdot \cos A}{\sin A \cos A (\sin A - \cos A)} \]\[ = \frac{\sin^3 A - \cos^3 A}{\sin A \cos A (\sin A - \cos A)} \]Recall the formula: \(a^3 - b^3 = (a-b)(a^2+ab+b^2)\).
Here, \(a = \sin A\) and \(b = \cos A\).
\[ = \frac{(\sin A - \cos A)(\sin^2 A + \sin A \cos A + \cos^2 A)}{\sin A \cos A (\sin A - \cos A)} \]Cancel \((\sin A - \cos A)\) from the numerator and denominator (assuming \(\sin A \neq \cos A\)).
Recall the identity: \(\sin^2 A + \cos^2 A = 1\).
\[ = \frac{\sin^2 A + \cos^2 A + \sin A \cos A}{\sin A \cos A} \]\[ = \frac{1 + \sin A \cos A}{\sin A \cos A} \]Recall the identities: \(\frac{1}{\sin A} = \operatorname{cosec} A\) and \(\frac{1}{\cos A} = \sec A\).
\[ = \operatorname{cosec} A \sec A + 1 \]\[ = 1 + \sec A \operatorname{cosec} A \]C) \(1 + \sec A \operatorname{cosec} A\) is the simplified form of the given expression.
The question asks about the specific reason for Bhagat Singh and Batukeshwar Dutt throwing smoke bombs in the Delhi Central Legislative Assembly. This event is a significant moment in India's freedom struggle, and understanding the context of their protest is key.
Correct Option: A) Trade Disputes Bill and Public Safety Bill — Bhagat Singh and Batukeshwar Dutt specifically protested against these two repressive bills, which were being debated and passed in the Central Legislative Assembly. Their act was a symbolic protest to draw attention to these legislative measures.
On simplification, \(\frac{2^{10}-3^{10}}{5^{10}-6^{10}}\) is:
Cannot be defined
a negative rational number
neither a positive nor a negative rational number
a positive rational number
To simplify the given expression, we need to evaluate the numerator and the denominator separately. We will determine the sign of each part and then use these signs to find the sign of the overall fraction.
Analyze the numerator:
The numerator is \(2^{10} - 3^{10}\).
We know that \(2^{10}\) is a positive number. Its value is \(1024\).
We know that \(3^{10}\) is also a positive number. Its value is \(59049\).
Since \(3^{10} > 2^{10}\), the difference \(2^{10} - 3^{10}\) will be a negative number.
\[ 2^{10} - 3^{10} = 1024 - 59049 = -58025 \]Analyze the denominator:
The denominator is \(5^{10} - 6^{10}\).
We know that \(5^{10}\) is a positive number.
We know that \(6^{10}\) is also a positive number.
Since \(6^{10} > 5^{10}\), the difference \(5^{10} - 6^{10}\) will also be a negative number.
Determine the sign of the fraction:
The given expression is a fraction where the numerator is a negative number and the denominator is also a negative number.
When a negative number is divided by a negative number, the result is a positive number.
\[ \frac{\text{Negative Number}}{\text{Negative Number}} = \text{Positive Number} \]Conclusion:
The simplified expression will be a positive rational number.
D) a positive rational number. As determined in the step-by-step analysis, both the numerator and the denominator are negative, leading to a positive result.
The question asks for the SI unit of resistance. We need to recall the standard units for various electrical quantities to identify the correct answer.
B) Ohm — The Ohm (\(\Omega\)) is the internationally recognized SI unit for electrical resistance. It is defined as one volt per ampere (\(1\ \Omega = 1\ \text{V/A}\)).
The problem asks us to identify the pattern in the given series of numbers and then find the missing number. We should look for mathematical relationships between consecutive terms or a common property shared by all terms.
Correct Option: A) 16 is the correct answer because it follows the pattern of decreasing even numbers squared.
This question asks about the number of parts in the Constitution of India as of October 2020. To answer this, one needs to recall the original structure of the Constitution and how it has evolved through amendments.
Correct Option: D) 25. As of October 2020, due to various amendments, the Constitution of India has 25 Parts. The original 22 parts, with the addition of Part IVA, Part IX, Part IXA, Part IXB, and Part XIVA, and the repeal of Part VII, result in a total of 25 parts.
The problem asks us to identify the number that is different from the rest in a given set of numbers. This type of problem usually involves finding a common mathematical property (like being a perfect square, perfect cube, prime number, etc.) shared by most of the numbers, while one number does not share that property.
Let's list the properties:
Notice that 512, 729, and 343 are all perfect cubes:
However, 256 is a perfect square (\(16^2\)), but it is not a perfect cube of an integer.
A) 256 — This number is a perfect square (\(16^2\)) but not a perfect cube of an integer. The other three numbers are perfect cubes.