If sec θ + tan θ = 2, then sec θ is equal to:
\( \frac{5}{4} \)
\( \frac{1}{4} \)
\( \frac{1}{2} \)
\( \frac{3}{4} \)
The problem asks us to find the value of \(\sec \theta\) given the equation \(\sec \theta + \tan \theta = 2\). We can use the fundamental trigonometric identity \(\sec^2 \theta - \tan^2 \theta = 1\) to solve this problem. This identity can be factored as a difference of squares, which will allow us to find another relationship between \(\sec \theta\) and \(\tan \theta\).
We are given the equation:
\[ \sec \theta + \tan \theta = 2 \quad \text{(Equation 1)} \]Recall the fundamental trigonometric identity:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]Factor the left side of the identity using the difference of squares formula \(a^2 - b^2 = (a-b)(a+b)\):
\[ (\sec \theta - \tan \theta)(\sec \theta + \tan \theta) = 1 \]Substitute the value from Equation 1 into the factored identity:
\[ (\sec \theta - \tan \theta)(2) = 1 \]Solve for \((\sec \theta - \tan \theta)\):
\[ \sec \theta - \tan \theta = \frac{1}{2} \quad \text{(Equation 2)} \]Now we have a system of two linear equations with two variables (\(\sec \theta\) and \(\tan \theta\)):
Add Equation 1 and Equation 2:
\[ (\sec \theta + \tan \theta) + (\sec \theta - \tan \theta) = 2 + \frac{1}{2} \]Simplify the equation:
\[ 2 \sec \theta = \frac{4}{2} + \frac{1}{2} \] \[ 2 \sec \theta = \frac{5}{2} \]Solve for \(\sec \theta\):
\[ \sec \theta = \frac{5}{2} \times \frac{1}{2} \] \[ \sec \theta = \frac{5}{4} \]Correct Option: A) \( \frac{5}{4} \)
The question asks to identify the paramilitary force associated with the Nizam of Hyderabad. This requires knowledge of the historical context of the Hyderabad State, particularly during the period leading up to its integration into India.
During the period leading up to Operation Polo (the Indian military action to integrate Hyderabad in 1948), the Nizam of Hyderabad, Mir Osman Ali Khan, relied on a powerful and often violent paramilitary organization to maintain his rule and resist integration with India. This organization was known as the 'Razakars'. They were an armed militia formed by Qasim Razvi and were instrumental in perpetrating violence against the Hindu population and those who supported accession to India.
Correct Option: D) Razakars was a paramilitary force of the Nizam of Hyderabad, notorious for their actions during the period leading up to Hyderabad's integration into India.
The question asks to identify the winner of the Purple Cap in the IPL 2020 (IPL-13) season. The Purple Cap is awarded to the bowler who takes the maximum number of wickets in a single season of the Indian Premier League.
Correct Option: A) Kagiso Rabada was the recipient of the Purple Cap in the IPL 2020 season (IPL-13) for taking 30 wickets, the highest in that tournament.
To find the ratio of the total number of girls to the total number of boys, we first need to determine the number of girls. We can do this by subtracting the number of boys from the total number of students. Once we have the number of girls and boys, we can form the ratio and simplify it to its lowest terms.
The total number of students is the sum of the number of boys and the number of girls.
\[ \text{Number of girls} = \text{Total number of students} - \text{Number of boys} \]\[ \text{Number of girls} = 240 - 135 \]\[ \text{Number of girls} = 105 \]The question asks for the ratio of the total number of girls to the total number of boys.
\[ \text{Ratio} = \text{Number of girls} : \text{Number of boys} \]\[ \text{Ratio} = 105 : 135 \]To simplify the ratio, find the greatest common divisor (GCD) of 105 and 135. Both numbers are divisible by 5.
\[ 105 \div 5 = 21 \]\[ 135 \div 5 = 27 \]\[ \text{The ratio becomes } 21 : 27 \]Both 21 and 27 are divisible by 3.
\[ 21 \div 3 = 7 \]\[ 27 \div 3 = 9 \]\[ \text{The simplified ratio is } 7 : 9 \]C) 7 : 9 — This is the correct ratio of the number of girls to the number of boys after simplification.
The question asks to identify the state where the Mathuri folk dance is practiced. This requires knowledge of Indian folk dances and their associated regions.
Correct Option: C) Telangana
The Mathuri folk dance is a significant cultural expression of the Mathuri community residing in the Adilabad district of Telangana. It is performed by both men and women, often depicting themes related to daily life, festivals, and religious rituals. The dance is characterized by its rhythmic movements and traditional attire.
We are given the sum and difference of two numbers. We need to find two times their product. We can use algebraic equations to find the individual numbers first, then their product, and finally two times the product. Alternatively, we can use an algebraic identity to directly find the product.
Let the two numbers be \(x\) and \(y\).
According to the problem statement, the sum of the two numbers is 24:
\[ x + y = 24 \quad \text{(Equation 1)} \]The difference between the two numbers is 10:
\[ x - y = 10 \quad \text{(Equation 2)} \]To find the values of \(x\) and \(y\), we can add Equation 1 and Equation 2:
\[ (x + y) + (x - y) = 24 + 10 \] \[ 2x = 34 \] \[ x = \frac{34}{2} \] \[ x = 17 \]Substitute the value of \(x\) into Equation 1:
\[ 17 + y = 24 \] \[ y = 24 - 17 \] \[ y = 7 \]So, the two numbers are 17 and 7.
Now, we need to find the product of the numbers, \(xy\):
\[ xy = 17 \times 7 \] \[ xy = 119 \]Finally, we need to find two times the product of the numbers:
\[ 2xy = 2 \times 119 \] \[ 2xy = 238 \]We know the identity: \((x+y)^2 - (x-y)^2 = 4xy\)
Given \(x+y = 24\) and \(x-y = 10\).
Substitute these values into the identity:
\[ (24)^2 - (10)^2 = 4xy \] \[ 576 - 100 = 4xy \] \[ 476 = 4xy \]We need to find \(2xy\). We can divide both sides by 2:
\[ \frac{476}{2} = \frac{4xy}{2} \] \[ 238 = 2xy \]Correct Option: C) 238
The question asks to identify the first woman pilot of the Indian Air Force (IAF) who qualified to undertake combat missions on a fighter jet. This requires knowledge of significant milestones achieved by women in the Indian armed forces.
B) Bhawana Kanth is the correct answer. She became the first woman fighter pilot of the IAF to qualify for combat missions in May 2019.
Find the value of 2.\(\overline{43}\) - 3.\(\overline{04}\) + 4.\(\overline{44}\)
To solve this problem, we need to convert the given recurring decimals into fractions. Once converted, we can perform the addition and subtraction operations. Finally, we will convert the resulting fraction back into a decimal form, preferably a recurring decimal, to match the options.
The expression is \(2.\overline{43} - 3.\overline{04} + 4.\overline{44}\). Substituting the fractional forms:
\[\frac{241}{99} - \frac{301}{99} + \frac{440}{99}\]Since all fractions have the same denominator, we can combine the numerators:
\[\frac{241 - 301 + 440}{99}\] \[\frac{-60 + 440}{99}\] \[\frac{380}{99}\]To convert \(\frac{380}{99}\) to a decimal, divide 380 by 99:
\[380 \div 99 = 3 \text{ with a remainder of } 83\]So, \[\frac{380}{99} = 3 + \frac{83}{99}\].
The fraction \(\frac{83}{99}\) represents a recurring decimal where the digits 83 repeat. Thus, \(\frac{83}{99} = 0.\overline{83}\).
Therefore, \(3 + 0.\overline{83} = 3.\overline{83}\).
D) \(3.\overline{83}\) is the correct answer, as derived from the calculations.
The question asks to identify the winner of a specific award in a particular category and year. This requires knowledge of current affairs and notable achievements in the field of photography, specifically wildlife photography.
In 2018, the Wildlife Photographer of the Year award in the "10 years and under" category was won by Arshdeep Singh from India for his photograph titled "Pipe Owl".
Correct Option: B) Arshdeep Singh
The problem requires us to find the ratio \(x:y\) given a relationship between 30% of \(x\) and five-sixth of \(y\). We will translate the given statement into an algebraic equation and then rearrange it to express \(x/y\) as a fraction, which can then be written as a ratio.
"30% of \(x\)" can be written as \(\frac{30}{100} \times x\).
"Five-sixth of \(y\)" can be written as \(\frac{5}{6} \times y\).
The statement "30% of \(x\) is equal to five-sixth of \(y\)" translates to:
\[ \frac{30}{100} x = \frac{5}{6} y \]Simplify the fraction on the left side:
\[ \frac{3}{10} x = \frac{5}{6} y \]To find \(x:y\), we need to express \(\frac{x}{y}\). Divide both sides by \(y\) and multiply both sides by \(\frac{10}{3}\) (the reciprocal of \(\frac{3}{10}\)):
\[ \frac{x}{y} = \frac{5}{6} \times \frac{10}{3} \]Both 50 and 18 are divisible by 2:
\[ \frac{x}{y} = \frac{50 \div 2}{18 \div 2} \]\[ \frac{x}{y} = \frac{25}{9} \]The ratio \(x:y\) is \(25:9\).
Correct Option: A) 25 : 9