If sinA =\(\frac{4}{\sqrt{17}}\) then tan 2A = ? (0° < A < 90°)
-\( \frac{8}{15} \)
-\( \frac{21}{15} \)
-\( \frac{7}{15} \)
-\( \frac{4}{15} \)
To find \(\tan 2A\), we first need to find \(\tan A\). We are given \(\sin A\). We can use the Pythagorean identity \(\sin^2 A + \cos^2 A = 1\) to find \(\cos A\), and then calculate \(\tan A = \frac{\sin A}{\cos A}\). Finally, we will use the double angle formula for tangent, which is \(\tan 2A = \frac{2 \tan A}{1 - \tan^2 A}\).
Find \(\cos A\):
Given \(\sin A = \frac{4}{\sqrt{17}}\). Since \(0^\circ < A < 90^\circ\), angle \(A\) is in the first quadrant, so \(\sin A\) and \(\cos A\) are both positive.
Find \(\tan A\):
\[ \tan A = \frac{\sin A}{\cos A} \]\[ \tan A = \frac{\frac{4}{\sqrt{17}}}{\frac{1}{\sqrt{17}}} \]\[ \tan A = 4 \]Find \(\tan 2A\):
Use the double angle formula for tangent:
Correct Option: A) \(-\frac{8}{15}\)
The question asks to identify the agency responsible for constructing and maintaining the Leh-Manali Highway. This requires knowledge of the specific responsibilities of various highway and road construction agencies in India.
Correct Option: C) The Border Road Organisation
The Border Roads Organisation (BRO) is a statutory body under the Ministry of Defence, Government of India. It is responsible for road construction and maintenance in border areas and strategically important regions of India, including the challenging terrains of the Himalayas. The Leh-Manali Highway is a prime example of such a strategic road, known for its high-altitude passes and extreme weather conditions, making it a crucial project for the BRO.
The question asks to identify the author of the Sanskrit epic Mahabharata. This requires knowledge of ancient Indian literature and its traditional authors.
Correct Option: C) Maharishi Veda Vyasa is traditionally credited as the author of the Mahabharata. He is also known as Vyasa or Krishna Dvaipayana.
The question asks about the maximum number of seats allowed in a State Legislative Assembly as per Article 170 of the Indian Constitution. This requires knowledge of the specific constitutional provision regarding the composition of State Legislative Assemblies.
Correct Option: D) 500. Article 170(1) of the Indian Constitution clearly states that the Legislative Assembly of a State shall consist of not more than five hundred members.
To find the difference between two numbers given their ratio and LCM, we first represent the numbers using a common factor. Then, we use the relationship between the numbers, their LCM, and their HCF (Highest Common Factor) to find the value of this common factor. Once the numbers are determined, their difference can be calculated.
Represent the numbers:
Let the two numbers be \(3x\) and \(5x\), since their ratio is \(3:5\). Here, \(x\) is their Highest Common Factor (HCF).
Understand the relationship between numbers, HCF, and LCM:
For any two numbers \(A\) and \(B\), the product of the numbers is equal to the product of their HCF and LCM. That is, \(A \times B = \text{HCF}(A, B) \times \text{LCM}(A, B)\).
Alternatively, for numbers \(ax\) and \(bx\) where \(x\) is the HCF, their LCM is given by \(abx\).
Calculate the LCM in terms of \(x\):
The numbers are \(3x\) and \(5x\). Their HCF is \(x\).
The LCM of \(3x\) and \(5x\) is \(3 \times 5 \times x = 15x\).
Equate the calculated LCM with the given LCM:
We are given that the LCM is 105.
So, \(15x = 105\).
Solve for \(x\):
\[ x = \frac{105}{15} \]
\[ x = 7 \]
Find the two numbers:
The first number is \(3x = 3 \times 7 = 21\).
The second number is \(5x = 5 \times 7 = 35\).
Calculate the difference between the numbers:
Difference \( = 35 - 21 = 14\).
Correct Option: C) 14 is the difference between the two numbers.
The question asks to identify the leader who organized the "Red Shirts" movement. This requires knowledge of key figures and movements during the Indian independence struggle.
The "Red Shirts" movement, or Khudai Khidmatgar (Servants of God), was a non-violent Pashtun independence movement in the North-West Frontier Province (NWFP) of British India. It was led by Khan Abdul Gaffar Khan, who was a staunch follower of Mahatma Gandhi's principles of non-violence. The members wore red uniforms, which gave the movement its popular name.
B) Khan Abdul Gaffar Khan — He was the founder and leader of the Khudai Khidmatgar movement, popularly known as the "Red Shirts" movement. He was also known as "Frontier Gandhi" due to his commitment to non-violence.
This problem involves work and wages, where the earning capacity of men and women needs to be determined first. We will set up equations based on the given information to find the individual daily earning rates of a man and a woman. Once these rates are known, we can calculate the total daily earning of 1 man and 1 woman and then find the time required to earn a specific amount.
Let \(M\) be the daily earning of one man and \(W\) be the daily earning of one woman.
From the first statement: 2 men and 3 women earn ₹49 in 7 days. This means their daily earning is \(\frac{49}{7} = \text{₹}7\).
So, we have the equation:
\[ 2M + 3W = 7 \quad \text{(Equation 1)} \]From the second statement: 3 men and 6 women earn ₹96 in 8 days. This means their daily earning is \(\frac{96}{8} = \text{₹}12\).
So, we have the equation:
\[ 3M + 6W = 12 \quad \text{(Equation 2)} \]Now, we solve these two linear equations to find \(M\) and \(W\).
Multiply Equation 1 by 2:
\[ 2 \times (2M + 3W) = 2 \times 7 \]\[ 4M + 6W = 14 \quad \text{(Equation 3)} \]Subtract Equation 2 from Equation 3:
\[ (4M + 6W) - (3M + 6W) = 14 - 12 \]\[ M = 2 \]Substitute \(M = 2\) into Equation 1:
\[ 2(2) + 3W = 7 \]\[ 4 + 3W = 7 \]\[ 3W = 7 - 4 \]\[ 3W = 3 \]\[ W = 1 \]So, a man earns ₹2 per day, and a woman earns ₹1 per day.
Now, we need to find the time it takes for 1 man and 1 woman to earn ₹27.
The daily earning of 1 man and 1 woman is \(1M + 1W = 1(2) + 1(1) = 2 + 1 = \text{₹}3\).
To earn ₹27, the number of days required is:
\[ \text{Days} = \frac{\text{Total amount to be earned}}{\text{Daily earning}} = \frac{27}{3} = 9 \text{ days} \]A) 9 days — As calculated, 1 man and 1 woman will earn ₹27 in 9 days.
To calculate the mean (average) of a set of numbers, we need to sum all the numbers in the set and then divide the sum by the total count of numbers in that set.
Correct Option: A) 15.38 is the calculated mean of the given data.
If with simple interest, an amount doubles in 5 years, how many times of the original amount, will it be after 8 years with same rate of simple interest?
2 \( \frac{1}{5} \) times
3 \( \frac{2}{5} \) times
2 \( \frac{3}{5} \) times
3 \( \frac{3}{5} \) times
The problem involves simple interest. We are given that an amount doubles in 5 years. We need to find out how many times the original amount it will be after 8 years with the same simple interest rate. We will first calculate the rate of interest using the given information and then use this rate to find the total amount after 8 years.
Define variables:
The formula for simple interest is: \[ SI = \frac{P \times R \times T}{100} \]
The formula for the total amount is: \[ A = P + SI \]
Use the first condition to find the rate (R):
Given that the amount doubles in 5 years. This means \(A = 2P\) when \(T = 5\) years.
Since \(A = P + SI\), we have \(2P = P + SI\). This implies \(SI = P\).
Now, substitute \(SI = P\) and \(T = 5\) into the simple interest formula:
\[ P = \frac{P \times R \times 5}{100} \]To find \(R\), we can cancel \(P\) from both sides (assuming \(P \neq 0\)):
\[ 1 = \frac{R \times 5}{100} \]Multiply both sides by 100:
\[ 100 = 5R \]Divide by 5:
\[ R = \frac{100}{5} = 20\% \]Calculate the amount after 8 years:
Now we need to find the total amount after \(T = 8\) years with the same rate \(R = 20\%\).
First, calculate the simple interest for 8 years:
\[ SI = \frac{P \times R \times T}{100} = \frac{P \times 20 \times 8}{100} \]Simplify the expression:
\[ SI = \frac{P \times 160}{100} = \frac{160P}{100} = \frac{16}{10}P = \frac{8}{5}P \]Calculate the total amount:
The total amount \(A\) after 8 years will be:
\[ A = P + SI = P + \frac{8}{5}P \]To add these, find a common denominator:
\[ A = \frac{5P}{5} + \frac{8P}{5} = \frac{5P + 8P}{5} = \frac{13P}{5} \]Express the amount as a multiple of the original amount:
The amount is \(\frac{13}{5}\) times the original amount \(P\).
Convert the improper fraction to a mixed number:
\[ \frac{13}{5} = 2 \frac{3}{5} \]C) 2 \( \frac{3}{5} \) times
As calculated, after 8 years, the amount will be \(2 \frac{3}{5}\) times the original principal amount.
The question asks to identify the software most suitable for creating presentations in a classroom setting. We need to evaluate the primary function of each given Microsoft Office application.
Correct Option: C) Microsoft PowerPoint is a presentation software that allows users to create slide shows comprising text, graphics, audio, and video. It is specifically designed for delivering information visually and interactively to an audience, making it ideal for classroom teaching.