The question asks to identify the category for which British Prime Minister Winston Churchill received a Nobel Prize. This requires knowledge of Nobel laureates and their respective fields.
Correct Option: B) Literature
Winston Churchill was awarded the Nobel Prize in Literature in 1953. The Nobel Committee recognized his historical and biographical writings, as well as his powerful speeches.
The question asks to identify the river with the largest volume of water in the world. This requires knowledge of major world rivers and their characteristics, specifically their discharge volume.
Comparing these values, the Amazon River clearly has the largest volume.
B) Amazon — The Amazon River, located in South America, is renowned for having the largest discharge volume of any river in the world. It carries more water than the next seven largest rivers combined, accounting for approximately 20% of the global riverine discharge into oceans. Its vast drainage basin and high rainfall contribute to its enormous volume.
The question asks to identify the first Chinese citizen to win the Nobel Peace Prize. This requires knowledge of Nobel laureates and their nationalities.
Correct Option: D) Liu Xiaobo
The question asks about the substance that causes silver to turn black. This phenomenon is known as tarnishing, which is a chemical reaction of silver with certain elements or compounds present in the atmosphere.
Correct Option: D) Sulphur
The question asks for an alternative name for Urdu. We need to identify which of the given options is a known historical or common name for the Urdu language.
Correct Option: D) Lashkari Zaban
"Lashkari Zaban" directly translates to "language of the army" or "camp language." This term was historically used to refer to Urdu, reflecting its development in military camps and bazaars where various linguistic groups (Persian, Arabic, Turkish, and local Indian dialects) interacted and contributed to its formation. The word "Urdu" itself is derived from the Turkic word "ordu" (camp/army).
The question asks to identify the pair of a country and its capital that is NOT correct. This requires knowledge of world geography, specifically the capitals of various countries.
C) Kazakhstan – Bishkek
The question asks to identify the head of the committee responsible for drafting the National Education Policy (NEP) 2019. This requires knowledge of key figures involved in major government policy formulations, particularly in the education sector.
A) K Kasturirangan was indeed the head of the committee that drafted the National Education Policy (NEP) 2019. He is a distinguished scientist and former chairman of the Indian Space Research Organisation (ISRO).
The greatest value of sin x + cos x is:
\( \frac{1}{4} \)
1
\(\sqrt{2}\)
\(\frac{1}{\sqrt{2}}\)
To find the greatest value of the expression \(\sin x + \cos x\), we can use a trigonometric identity that transforms the sum of sine and cosine into a single sine function. This allows us to easily determine the maximum value based on the known range of the sine function.
Let the given expression be \(E = \sin x + \cos x\).
We can rewrite this expression using the identity \(a \sin x + b \cos x = R \sin(x + \alpha)\), where \(R = \sqrt{a^2 + b^2}\) and \(\cos \alpha = \frac{a}{R}\), \(\sin \alpha = \frac{b}{R}\).
In our case, \(a = 1\) and \(b = 1\). So, calculate \(R\):
\[ R = \sqrt{1^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \]Now, we can write the expression as:
\[ \sin x + \cos x = \sqrt{2} \left( \frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x \right) \]We know that \[\cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}\] and \[\sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}\]. Substitute these values:
\[ \sin x + \cos x = \sqrt{2} \left( \cos \frac{\pi}{4} \sin x + \sin \frac{\pi}{4} \cos x \right) \]Using the trigonometric identity \(\sin(A+B) = \sin A \cos B + \cos A \sin B\), we get:
\[ \sin x + \cos x = \sqrt{2} \sin \left( x + \frac{\pi}{4} \right) \]The maximum value of \(\sin \theta\) is 1. Therefore, the maximum value of \(\sin \left( x + \frac{\pi}{4} \right)\) is 1.
The greatest value of the expression \(\sin x + \cos x\) will be:
\[ \text{Greatest Value} = \sqrt{2} \times 1 = \sqrt{2} \]Correct Option: C) \(\sqrt{2}\)
For a moderately asymmetrical distribution, there is an empirical relationship between the mean, median, and mode. This relationship is given by Karl Pearson's empirical formula:
\[ \text{Mode} \approx 3 \times \text{Median} - 2 \times \text{Mean} \]We are given the values of the mode and the mean, and we need to find the median. We can rearrange the formula to solve for the median.
Correct Option: C) The median of the distribution is 14.3. This is calculated using Karl Pearson's empirical formula for moderately asymmetrical distributions, which relates the mean, median, and mode.
To find out how many times the bells toll together, we first need to determine the least common multiple (LCM) of their individual tolling intervals. This LCM will represent the time after which all bells will toll together again. Then, we will calculate how many such intervals occur within the given time frame of 10 minutes.
D) 6 times — The bells toll together at 0 seconds (when they commence), and then every 120 seconds. In 600 seconds (10 minutes), they will toll together at 0s, 120s, 240s, 360s, 480s, and 600s, which is a total of 6 times.