The question asks to identify the longest canal in the world. This requires knowledge of major canals globally and their respective lengths.
Correct Option: A) Beijing-Hangzhou Grand Canal
The Beijing-Hangzhou Grand Canal, also known as the Grand Canal of China, is the longest canal or artificial river in the world. It stretches approximately 1,776 kilometers (1,104 miles) and connects Beijing in the north with Hangzhou in the south. It is a UNESCO World Heritage Site and has a history of over 2,500 years.
The problem involves compound interest. We are given the principal amount, the amount after the first year, and we need to find the annual rate of interest. For the first year, compound interest is the same as simple interest.
The interest earned is the difference between the amount after the first year and the principal.
\[ \text{Interest} = A_1 - P \\ \text{Interest} = \text{₹}5,600 - \text{₹}5,000 \\ \text{Interest} = \text{₹}600 \]For the first year, compound interest is calculated similarly to simple interest. The formula for simple interest is:
\[ \text{Interest} = \frac{P \times R \times T}{100} \]where \(R\) is the rate of interest per annum.The rate of interest per annum is 12%.
Correct Option: C) 12%
The problem asks us to find the sum of an alternating series: \(1 - 1 + 1 - 1 + \dots\), repeated 101 times. We need to observe the pattern of the sum as we add more terms.
The given series is \(1 - 1 + 1 - 1 + 1 - 1 + \dots\).
We can observe a clear pattern:
The series is repeated 101 times. Since 101 is an odd number, the sum of the series will be 1.
We can also group the terms:
\[ (1 - 1) + (1 - 1) + \dots \]Since there are 101 terms, we can form 50 pairs of \((1 - 1)\) and one remaining term:
\[ (1 - 1) + (1 - 1) + \dots + (1 - 1) + 1 \]There are 50 such pairs, and each pair sums to 0:
\[ \underbrace{(1 - 1)}_{0} + \underbrace{(1 - 1)}_{0} + \dots + \underbrace{(1 - 1)}_{0} + 1 \]So, the total sum is \(50 \times 0 + 1 = 0 + 1 = 1\).
A) 1. As determined by analyzing the pattern for an odd number of terms, the sum of the series is 1.
What is the equation for Newton's second law of motion?
\( F = mc^{2} \)
F = AP
F = ma
F = \( \frac{1}{2} \)mv²
The question asks for the correct equation representing Newton's second law of motion. We need to recall the fundamental laws of motion formulated by Sir Isaac Newton.
Correct Option: C) F = ma
The question asks to identify the tribe that celebrates the Bhagoriya festival. This requires knowledge of the cultural practices and festivals associated with various tribal communities in India.
Correct Option: D) Bhil
The Bhil tribe is well-known for celebrating the Bhagoriya festival. This festival is a traditional fair held before the Holi festival, often associated with finding life partners, and is a vibrant cultural event for the community.
The question asks to identify which of the given options is NOT a legal instrument of the United Nations for the protection of the climate system. This requires understanding the nature and role of each option provided.
Correct Option: B) United Nations International Panel on Climate Change
This question tests general knowledge about major international sporting events. Specifically, it asks for the host country of a recent ICC Men's Cricket World Cup.
Correct Option: A) United Kingdom
This problem involves simple interest. We need to first find the rate of interest using the given information that the sum doubles in 10 years. Then, we use this rate to find the time it takes for the sum to triple.
The formula for simple interest is:
\[ I = \frac{P \times R \times T}{100} \]Where:
The Amount (\(A\)) after \(T\) years is \(A = P + I\).
Given that the sum doubles in 10 years, this means the Amount (\(A\)) becomes \(2P\). The interest earned (\(I\)) is \(A - P = 2P - P = P\).
We have \(I = P\), \(T = 10\) years. Substitute these into the simple interest formula:
\[ P = \frac{P \times R \times 10}{100} \]Divide both sides by \(P\) (assuming \(P \neq 0\)):
\[ 1 = \frac{R \times 10}{100} \] \[ 1 = \frac{R}{10} \]So, the rate of interest \(R\) is:
\[ R = 10\% \text{ per annum} \]If the sum triples, the Amount (\(A\)) becomes \(3P\). The interest earned (\(I\)) is \(A - P = 3P - P = 2P\).
We now have \(I = 2P\), \(R = 10\%\). We need to find \(T\). Substitute these into the simple interest formula:
\[ 2P = \frac{P \times 10 \times T}{100} \]Divide both sides by \(P\):
\[ 2 = \frac{10 \times T}{100} \] \[ 2 = \frac{T}{10} \]So, the time \(T\) is:
\[ T = 2 \times 10 = 20 \text{ years} \]Correct Option: A) 20
To answer this question, we need to recall the chronological order of significant events in the Indian independence movement, specifically focusing on the Gandhi-Irwin Pact and the Round Table Conferences.
Correct Option: C) Second Round Table Conference
The problem asks us to simplify the expression \(\sin(\alpha - \beta)\) given the condition \(\cos(\alpha + \beta) = 0\). We will use trigonometric identities to find the relationship between \(\alpha\) and \(\beta\) from the given condition and then substitute it into the expression to be simplified.
Analyze the given condition:
We are given that \(\cos(\alpha + \beta) = 0\).
We know that the cosine function is zero for angles of the form \((2n+1)\frac{\pi}{2}\), where \(n\) is an integer. The principal value for which \(\cos \theta = 0\) is \(\theta = \frac{\pi}{2}\).
So, we can write:
\[ \alpha + \beta = \frac{\pi}{2} \]Express one variable in terms of the other:
From the above equation, we can express \(\alpha\) in terms of \(\beta\):
Substitute this into the expression to be simplified:
We need to simplify \(\sin(\alpha - \beta)\). Substitute the expression for \(\alpha\) into this:
Simplify the expression:
Combine the terms inside the sine function:
Apply the complementary angle identity:
We know the trigonometric identity \(\sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta\).
Using this identity, with \(\theta = 2\beta\), we get:
\[ \sin\left(\frac{\pi}{2} - 2\beta\right) = \cos(2\beta) \]D) \(\cos 2\beta\) is the simplified form of \(\sin(\alpha - \beta)\) given the condition \(\cos(\alpha + \beta) = 0\).