The question asks to identify the river that flows through London. This is a general knowledge question related to world geography.
Correct Option: D) Thames — The River Thames is a well-known river in southern England, and it flows directly through the heart of London, playing a significant role in the city's history and development.
The problem asks us to identify a specific element in a given alphanumeric arrangement after a certain modification. The modification involves dropping all numbers from the arrangement. Then, we need to find the element that is ninth to the left of 'S' in the modified arrangement.
Therefore, 'Q' is the ninth element to the left of the second 'S' in the modified arrangement.
C) Q
The question asks to identify a critically endangered bird species from the given options. This requires knowledge of the conservation status of common Indian bird species as classified by organizations like the IUCN (International Union for Conservation of Nature).
The Great Indian Bustard (Ardeotis nigriceps) is well-known for its precarious conservation status. It is one of the heaviest flying birds and is native to the Indian subcontinent. Its population has drastically declined due to habitat loss, hunting, and other anthropogenic pressures, leading to its classification as Critically Endangered by the IUCN.
Correct Option: C) Great Indian Bustard
The question asks about the location of the "last Earth Summit." This requires knowledge of major international environmental conferences, specifically those referred to as Earth Summits, and their chronological order.
Correct Option: B) Rio de Janeiro. The last major conference widely recognized as an "Earth Summit" or its direct successor was the United Nations Conference on Sustainable Development (Rio+20) held in Rio de Janeiro, Brazil, in 2012. The original Earth Summit was also held in Rio de Janeiro in 1992.
This problem can be solved by visualizing the situation as a right-angled triangle. We can form a right-angled triangle by considering the difference in heights of the poles as one leg, the distance between their feet as the other leg, and the distance between their tops as the hypotenuse. We will then apply the Pythagorean theorem.
Let the difference in heights be \(h\).
\[ h = P_2 - P_1 = 30 \text{ m} - 15 \text{ m} = 15 \text{ m} \]Imagine a horizontal line drawn from the top of the shorter pole to the taller pole. This line will be parallel to the ground and have a length equal to the distance between the feet of the poles (36 m). The vertical distance from the top of the shorter pole to the top of the taller pole is the difference in their heights (15 m). The distance between their tops will be the hypotenuse of the right-angled triangle formed by these two lengths.
Let the distance between their tops be \(x\). According to the Pythagorean theorem, for a right-angled triangle with legs \(a\) and \(b\) and hypotenuse \(c\), we have \(a^2 + b^2 = c^2\).
In our case, \(a = h = 15 \text{ m}\) and \(b = D = 36 \text{ m}\). We need to find \(c = x\).
\[ x^2 = h^2 + D^2 \]\[ x^2 = (15)^2 + (36)^2 \]\[ x^2 = 225 + 1296 \]\[ x^2 = 1521 \]B) 39 m is the correct distance between the tops of the poles, calculated using the Pythagorean theorem.
To find the number of integers divisible by 5 between two given numbers, we first identify the largest three-digit number and the largest four-digit number. Then, we find the smallest number greater than the largest three-digit number that is divisible by 5, and the largest number less than or equal to the largest four-digit number that is divisible by 5. Finally, we use the formula for the number of terms in an arithmetic progression to count them.
Therefore, there are 1800 numbers between the largest three-digit number and the largest four-digit number that are divisible by 5.
Correct Option: C) 1800
This problem can be solved using the concept of similar triangles. When the sun casts shadows, the angle of elevation of the sun is the same for all objects at a given time and location. This means that the triangles formed by an object, its shadow, and the line from the top of the object to the end of its shadow are similar. For similar triangles, the ratio of corresponding sides is equal.
Since the angle of elevation of the sun is the same for both the pole and the tree, the triangles formed by each object and its shadow are similar. Therefore, the ratio of height to shadow length will be the same for both the pole and the tree.
\[ \frac{H_{\text{pole}}}{S_{\text{pole}}} = \frac{H_{\text{tree}}}{S_{\text{tree}}} \]Multiply both sides by 28 m:
\[ H_{\text{pole}} = \frac{8}{14} \times 28 \]\[ H_{\text{pole}} = 8 \times \frac{28}{14} \]\[ H_{\text{pole}} = 8 \times 2 \]\[ H_{\text{pole}} = 16 \text{ m} \]Thus, the height of the electric pole is 16 m.
A) 16 m is the correct height of the pole, calculated using the principle of similar triangles.
When four numbers are in proportion, it means that the ratio of the first two numbers is equal to the ratio of the last two numbers. For numbers \(a, b, c, d\) to be in proportion, the relationship is \(a:b :: c:d\), which can be written as \[\frac{a}{b} = \frac{c}{d}\]. This implies that the product of the extremes (first and fourth terms) is equal to the product of the means (second and third terms), i.e., \(ad = bc\).
Identify the given numbers: The given numbers are 9, x, x, 49.
Apply the condition for proportion: Since the numbers 9, x, x, 49 are in proportion, we can write the relationship as:
\[ 9 : x :: x : 49 \]Convert the proportion to an equation: This proportion can be written as a fraction equality:
\[ \frac{9}{x} = \frac{x}{49} \]Cross-multiply: To solve for \(x\), cross-multiply the terms:
\[ 9 \times 49 = x \times x \]\[ 441 = x^2 \]Solve for \(x\): Take the square root of both sides to find the value of \(x\):
\[ x = \sqrt{441} \]\[ x = 21 \]Consider the context: In most proportion problems in competitive exams, we consider the positive root unless specified otherwise. Thus, \(x = 21\).
Correct Option: D) 21
If x = \(\frac{\sqrt{3}+1}{\sqrt{3}-1}\) and y = \(\frac{\sqrt{3}-1}{\sqrt{3}+1}\), then find the value of \( x^{2} + y^{2} \).
To find the value of \(x^2 + y^2\), we first need to simplify the expressions for \(x\) and \(y\). This can be done by rationalizing the denominators. Once \(x\) and \(y\) are simplified, we can calculate \(x^2\) and \(y^2\) and then sum them up. Alternatively, we can notice the relationship between \(x\) and \(y\) and use the identity \((x+y)^2 = x^2 + y^2 + 2xy\).
Given \(x = \frac{\sqrt{3}+1}{\sqrt{3}-1}\). To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{3}+1\).
\[ x = \frac{\sqrt{3}+1}{\sqrt{3}-1} \times \frac{\sqrt{3}+1}{\sqrt{3}+1} \]\[ x = \frac{(\sqrt{3}+1)^2}{(\sqrt{3})^2 - (1)^2} \]\[ x = \frac{(\sqrt{3})^2 + 2(\sqrt{3})(1) + (1)^2}{3 - 1} \]\[ x = \frac{3 + 2\sqrt{3} + 1}{2} \]\[ x = \frac{4 + 2\sqrt{3}}{2} \]\[ x = 2 + \sqrt{3} \]Given \(y = \frac{\sqrt{3}-1}{\sqrt{3}+1}\). Notice that \(y\) is the reciprocal of \(x\). So, \(y = \frac{1}{x}\).
We can also simplify it by rationalizing the denominator:
\[ y = \frac{\sqrt{3}-1}{\sqrt{3}+1} \times \frac{\sqrt{3}-1}{\sqrt{3}-1} \]\[ y = \frac{(\sqrt{3}-1)^2}{(\sqrt{3})^2 - (1)^2} \]\[ y = \frac{(\sqrt{3})^2 - 2(\sqrt{3})(1) + (1)^2}{3 - 1} \]\[ y = \frac{3 - 2\sqrt{3} + 1}{2} \]\[ y = \frac{4 - 2\sqrt{3}}{2} \]\[ y = 2 - \sqrt{3} \]Now that we have \(x = 2 + \sqrt{3}\), we can find \(x^2\).
\[ x^2 = (2 + \sqrt{3})^2 \]\[ x^2 = (2)^2 + 2(2)(\sqrt{3}) + (\sqrt{3})^2 \]\[ x^2 = 4 + 4\sqrt{3} + 3 \]\[ x^2 = 7 + 4\sqrt{3} \]Now that we have \(y = 2 - \sqrt{3}\), we can find \(y^2\).
\[ y^2 = (2 - \sqrt{3})^2 \]\[ y^2 = (2)^2 - 2(2)(\sqrt{3}) + (\sqrt{3})^2 \]\[ y^2 = 4 - 4\sqrt{3} + 3 \]\[ y^2 = 7 - 4\sqrt{3} \]Add the values of \(x^2\) and \(y^2\).
\[ x^2 + y^2 = (7 + 4\sqrt{3}) + (7 - 4\sqrt{3}) \]\[ x^2 + y^2 = 7 + 4\sqrt{3} + 7 - 4\sqrt{3} \]\[ x^2 + y^2 = 7 + 7 \]\[ x^2 + y^2 = 14 \]Alternative Method:
We know that \(x = 2 + \sqrt{3}\) and \(y = 2 - \sqrt{3}\).
Calculate \(x+y\):
\[ x+y = (2 + \sqrt{3}) + (2 - \sqrt{3}) = 4 \]Calculate \(xy\):
\[ xy = (2 + \sqrt{3})(2 - \sqrt{3}) = (2)^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]Using the identity \(x^2 + y^2 = (x+y)^2 - 2xy\):
\[ x^2 + y^2 = (4)^2 - 2(1) \]\[ x^2 + y^2 = 16 - 2 \]\[ x^2 + y^2 = 14 \]C) The calculated value of \(x^2 + y^2\) is 14.
The question asks to identify the freedom fighter who started the newspaper Al-Hilal. This requires knowledge of prominent Indian freedom fighters and their contributions, specifically in the field of journalism and public awareness during the independence movement.
B) Maulana Abul Kalam Azad
Maulana Abul Kalam Azad was a prominent Indian independence activist, Islamic theologian, writer, and a senior leader of the Indian National Congress. He launched the Urdu weekly newspaper Al-Hilal in 1912. The newspaper played a crucial role in propagating nationalist ideas and critiquing British policies, advocating for Hindu-Muslim unity and the cause of Indian independence. Its strong anti-British stance led to its suppression by the government in 1914.