The perimeter of an equilateral triangle is 36\(\sqrt{3}\) m. Find the height (in metres).
To find the height of an equilateral triangle, we first need to determine the length of its side. The perimeter of an equilateral triangle is given by three times its side length. Once the side length is known, we can use the formula for the height of an equilateral triangle, which is \( \frac{\sqrt{3}}{2} \times \text{side} \).
Let the side length of the equilateral triangle be \(a\) metres.
The perimeter of an equilateral triangle is given by the formula:
\[ \text{Perimeter} = 3a \]
We are given that the perimeter is \(36\sqrt{3}\) m. So,
\[ 3a = 36\sqrt{3} \]
Divide both sides by 3 to find \(a\):
\[ a = \frac{36\sqrt{3}}{3} \]
\[ a = 12\sqrt{3} \text{ m} \]
The formula for the height \((h)\) of an equilateral triangle with side length \(a\) is:
\[ h = \frac{\sqrt{3}}{2} a \]
Substitute the value of \(a = 12\sqrt{3}\) m into the height formula:
\[ h = \frac{\sqrt{3}}{2} \times (12\sqrt{3}) \]
\[ h = \frac{12 \times (\sqrt{3} \times \sqrt{3})}{2} \]
Since \( \sqrt{3} \times \sqrt{3} = 3 \):
\[ h = \frac{12 \times 3}{2} \]
\[ h = \frac{36}{2} \]
\[ h = 18 \text{ m} \]
Correct Option: D) 18
The question asks about the number of founding member countries of the United Nations Organization (UNO) when it was established in 1945. This requires recalling a specific historical fact about the UN's formation.
Correct Option: B) The United Nations was founded by 51 member countries in 1945. Fifty nations signed the UN Charter at the San Francisco Conference, and Poland, which was not represented at the conference, signed it later, making it the 51st founding member.
The question asks to identify the architect responsible for designing the India Gate in New Delhi. This requires knowledge of historical architecture and prominent figures involved in the planning of New Delhi.
Correct Option: D) Edwin Lutyens
Sir Edwin Lutyens was the principal architect responsible for designing the India Gate, which is a war memorial located in New Delhi. He was also a key figure in planning and designing much of New Delhi during the British Raj, including the Viceroy's House (now Rashtrapati Bhavan).
The question asks about the causative agent of Bird Flu. This requires knowledge of common diseases and their respective pathogens or vectors.
Correct Option: A) H5N1 virus. The H5N1 virus is a highly pathogenic avian influenza (HPAI) virus that causes a severe respiratory disease in birds, commonly known as Bird Flu. It can also, on rare occasions, infect humans, leading to severe illness and high mortality rates.
The question asks to identify the Indian Nationalist leader who led the Bardoli Satyagraha. This requires knowledge of key figures and events in the Indian independence movement, specifically peasant agitations.
A) Vallabhbhai Patel spearheaded the Bardoli Satyagraha in 1928. He successfully organized the peasants against the unjust increase in land revenue, leading to the government eventually conceding to their demands and reducing the tax hike.
To solve this blood relation problem, we need to break down the statement made by Rajat step-by-step, starting from the end of the sentence and working backward to establish the relationship between Rajat and the person in the photograph.
Correct Option: D) Son
This problem involves a man traveling a total distance in a given time, partly by foot and partly by bicycle, each at a different speed. We need to find the distance traveled on foot. We can set up a system of two linear equations based on the total distance and total time, and then solve for the unknown distances or times.
Let \(d_f\) be the distance traveled on foot (in km) and \(d_b\) be the distance traveled on bicycle (in km).
The total distance traveled is 28 km. So, we have the equation:
\[ d_f + d_b = 28 \quad \text{(Equation 1)} \]Let \(t_f\) be the time spent traveling on foot (in hours) and \(t_b\) be the time spent traveling on bicycle (in hours).
The total time taken is 8 hours. So, we have the equation:
\[ t_f + t_b = 8 \quad \text{(Equation 2)} \]We know the relationship between distance, speed, and time: \(\text{Distance} = \text{Speed} \times \text{Time}\), which implies \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\).
For travel on foot:
Speed on foot \(= 2 \text{ km/h}\)
\[ t_f = \frac{d_f}{2} \quad \text{(Equation 3)} \]For travel on bicycle:
Speed on bicycle \(= 6 \text{ km/h}\)
\[ t_b = \frac{d_b}{6} \quad \text{(Equation 4)} \]Substitute Equation 3 and Equation 4 into Equation 2:
\[ \frac{d_f}{2} + \frac{d_b}{6} = 8 \]To eliminate the denominators, multiply the entire equation by the least common multiple of 2 and 6, which is 6:
\[ 6 \left( \frac{d_f}{2} \right) + 6 \left( \frac{d_b}{6} \right) = 6 \times 8 \] \[ 3d_f + d_b = 48 \quad \text{(Equation 5)} \]Now we have a system of two linear equations with two variables \(d_f\) and \(d_b\):
Equation 1: \(d_f + d_b = 28\)
Equation 5: \(3d_f + d_b = 48\)
Subtract Equation 1 from Equation 5 to solve for \(d_f\):
\[ (3d_f + d_b) - (d_f + d_b) = 48 - 28 \] \[ 3d_f - d_f + d_b - d_b = 20 \] \[ 2d_f = 20 \] \[ d_f = \frac{20}{2} \] \[ d_f = 10 \text{ km} \]The distance traveled on foot is 10 km.
C) 10
The distance traveled on foot is 10 km, which matches option C.
The question asks about the number of major banks nationalised by the Indian government in 1969. This is a factual question related to the history of banking in India.
Correct Option: B) 14 major private commercial banks were nationalised by the Indian government in 1969. This was a landmark decision aimed at greater government control over the financial sector to achieve socio-economic objectives.
This problem requires us to establish the relative heights of different trees based on the given comparative statements. We will use inequalities to represent the height relationships and then combine them to find the tallest tree.
Correct Option: D) Palm tree — The combined height comparison clearly shows that the palm tree is taller than all other trees mentioned.
The question asks to identify which substance is NOT used as a moderator in nuclear reactors. A moderator's primary function is to slow down fast neutrons produced during nuclear fission to thermal energies, making them more likely to cause further fission reactions. We need to evaluate each option based on its properties as a moderator.
C) Boron is not used as a moderator in nuclear reactors. Instead, it is used as a neutron absorber to control the chain reaction because of its high neutron absorption cross-section.