To find the median of a set of values, we first need to arrange the values in ascending or descending order. Once ordered, the median is the middle value if the number of values is odd. If the number of values is even, the median is the average of the two middle values.
A) 6 is the median of the given set of values.
Find the value of sin(\(\frac{5\pi}{3}\)).
\(\frac{\sqrt{3}}{2}\)
\(\frac{1}{2}\)
\(-\frac{\sqrt{3}}{2}\)
\(\frac{1}{\sqrt{2}}\)
To find the value of \(\sin\left(\frac{5\pi}{3}\right)\), we need to determine the quadrant in which the angle \(\frac{5\pi}{3}\) lies and then use the reference angle to find the sine value, considering the sign in that quadrant.
We know that \(\pi \text{ radians} = 180^{\circ}\).
\[ \frac{5\pi}{3} = \frac{5 \times 180^{\circ}}{3} = 5 \times 60^{\circ} = 300^{\circ} \]An angle of \(300^{\circ}\) lies in the fourth quadrant, as it is between \(270^{\circ}\) and \(360^{\circ}\).
For an angle \(\theta\) in the fourth quadrant, the reference angle \(\theta'\) is given by \(360^{\circ} - \theta\) or \(2\pi - \theta\).
\[ \theta' = 360^{\circ} - 300^{\circ} = 60^{\circ} \]Or in radians:\[ \theta' = 2\pi - \frac{5\pi}{3} = \frac{6\pi - 5\pi}{3} = \frac{\pi}{3} \]In the fourth quadrant, the sine function (y-coordinate) is negative.
Therefore, \[\sin\left(\frac{5\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right)\].
We know that \[\sin\left(\frac{\pi}{3}\right) = \sin(60^{\circ}) = \frac{\sqrt{3}}{2}\].
So, \[\sin\left(\frac{5\pi}{3}\right) = -\frac{\sqrt{3}}{2}\].
Correct Option: C) \(-\frac{\sqrt{3}}{2}\)
The question asks for the full form of 'CPI' in the context of inflation. This is a direct knowledge-based question related to economic terminology commonly used in competitive exams.
B) Consumer Price Index
The Consumer Price Index (CPI) is a measure that examines the weighted average of prices of a basket of consumer goods and services, such as transportation, food, and medical care. It is calculated by taking price changes for each item in the predetermined basket of goods and averaging them. Changes in the CPI are used to assess price changes associated with the cost of living; it is one of the most frequently used statistics for identifying periods of inflation or deflation.
If \(\sqrt{5^n}\) = 5, then n = ?
The problem requires us to find the value of 'n' in the given equation involving a square root and an exponent. We will use the properties of exponents and square roots to solve for 'n'.
The given equation is:
\[ \sqrt{5^n} = 5 \]Recall that the square root of a number can be expressed as that number raised to the power of \(\frac{1}{2}\). So, \(\sqrt{x} = x^{\frac{1}{2}}\).
Applying this to the left side of the equation:
\[ \left(5^n\right)^{\frac{1}{2}} = 5 \]Using the exponent rule \(\left(a^m\right)^p = a^{m \cdot p}\), we can simplify the left side:
\[ 5^{n \cdot \frac{1}{2}} = 5 \]\[ 5^{\frac{n}{2}} = 5 \]We know that any number raised to the power of 1 is the number itself. So, \(5 = 5^1\).
Substitute this into the equation:
\[ 5^{\frac{n}{2}} = 5^1 \]If the bases are equal, then the exponents must also be equal. Therefore:
\[ \frac{n}{2} = 1 \]Solve for 'n' by multiplying both sides by 2:
\[ n = 1 \cdot 2 \]\[ n = 2 \]C) 2 is the correct answer. When \(n=2\), the equation becomes \(\sqrt{5^2} = \sqrt{25} = 5\), which satisfies the given condition.
The question asks for the city where Emperor Akbar's tomb is located. This is a factual question related to Indian history and geography, specifically concerning the burial sites of prominent historical figures.
Correct Option: D) Agra
Emperor Akbar's tomb is located in Sikandra, which is a suburb of Agra, Uttar Pradesh. Agra was a prominent capital during the Mughal Empire and is home to several important historical monuments, including the Taj Mahal and Agra Fort.
To find the value of the smaller number, we will use the given ratio and sum. We can represent the two numbers using a common multiplier based on their ratio. Then, we will set up an equation using their sum to find the value of this multiplier, and finally calculate the smaller number.
Let the two numbers be \(5x\) and \(3x\), since their ratio is \(5:3\).
The sum of the two numbers is given as 560. So, we can write the equation:
\[ 5x + 3x = 560 \]Combine the terms on the left side of the equation:
\[ 8x = 560 \]Solve for \(x\) by dividing both sides by 8:
\[ x = \frac{560}{8} \]\[ x = 70 \]Now that we have the value of \(x\), we can find the two numbers:
The question asks for the value of the smaller of the two numbers. Comparing 350 and 210, the smaller number is 210.
Correct Option: D) 210 is the smaller of the two numbers, calculated as \(3x\).
The question asks about the conference where the World Bank and the International Monetary Fund (IMF) were established. This is a direct factual recall question related to international economic history.
Correct Option: B) Bretton Woods Conference
The Bretton Woods Conference, officially known as the United Nations Monetary and Financial Conference, was held in July 1944 in Bretton Woods, New Hampshire, USA. Representatives from 44 Allied nations gathered to regulate the international monetary and financial order after the conclusion of World War II. This conference led to the creation of the International Monetary Fund (IMF) and the International Bank for Reconstruction and Development (IBRD), which is the original institution of the World Bank Group. These institutions are often referred to as the "Bretton Woods Institutions."
The question asks for the launch date of the Direct Benefit Transfer (DBT) scheme by the Government of India. This is a factual question requiring knowledge of key government initiatives and their timelines.
Correct Option: A) 1 January 2013 is the correct launch date for the Direct Benefit Transfer (DBT) scheme in India. It was introduced to streamline the delivery of government subsidies and benefits.
The question asks to identify the author of the famous Tamil epic 'Manimekalai'. This requires knowledge of classical Tamil literature and its prominent poets.
D) Sattanar: Sattanar (also known as Chithalai Chathanar) is widely recognized as the author of the Tamil epic 'Manimekalai'. This epic is a sequel to 'Silappatikaram' and focuses on the life of Manimekalai, the daughter of Kovalan and Madhavi.
The problem involves decoding a pattern in a given word transformation and applying the same pattern to a new word. We need to find the positional difference between the letters of the original word 'FLOW' and its coded form 'DJMU'. Once the pattern is identified, it will be applied to the word 'WING'.
The pattern observed is that each letter in the original word is shifted backward by 2 positions in the alphabet to get its coded letter.
The coded word for 'WING' is 'UGLE'.
Correct Option: A) UGLE