This is a logical reasoning question based on a given statement and conclusions. We need to determine which conclusion(s) logically follow directly from the information provided in the statement, without introducing outside knowledge or assumptions.
C) Only conclusion I follows. The statement explicitly states that Mukesh Ambani is the richest person in Asia. Since India is a part of Asia, it is a direct and undeniable logical inference that he must also be the richest person in India. If he were not the richest person in India, he could not be the richest person in Asia.
The problem involves a tangent to a circle. We will use the property that the radius drawn to the point of tangency is perpendicular to the tangent. This forms a right-angled triangle, allowing us to use the Pythagorean theorem to find the required distance.
Let the circle have its center at O and radius \(r\). We are given that \(r = 8 \, \text{cm}\).
Let P be an external point from which a tangent is drawn to the circle. Let A be the point where the tangent touches the circle. We are given that the length of the tangent \(AP = 15 \, \text{cm}\).
According to the property of tangents, the radius drawn to the point of tangency is perpendicular to the tangent. Therefore, the angle \(\angle OAP\) is a right angle (\(90^\circ\)).
This means that triangle \(\triangle OAP\) is a right-angled triangle, with the right angle at A.
In \(\triangle OAP\), the sides are \(OA\) (radius), \(AP\) (tangent), and \(OP\) (hypotenuse, which is the distance of P from the center O).
We can apply the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
\[ OP^2 = OA^2 + AP^2 \]Substitute the given values: \(OA = r = 8 \, \text{cm}\) and \(AP = 15 \, \text{cm}\).
\[ OP^2 = (8 \, \text{cm})^2 + (15 \, \text{cm})^2 \]\[ OP^2 = 64 \, \text{cm}^2 + 225 \, \text{cm}^2 \]\[ OP^2 = 289 \, \text{cm}^2 \]To find \(OP\), take the square root of both sides:
\[ OP = \sqrt{289 \, \text{cm}^2} \]\[ OP = 17 \, \text{cm} \]B) 17 cm is the correct answer. The distance of P from the center of the circle is 17 cm.
If \(\frac{\sin\theta+\cos\theta}{\sin\theta-\cos\theta}\) = \(\frac{\sqrt{3}-1}{\sqrt{3}+1}\), then the angle θ is.
The problem asks us to find the value of \(\theta\) given a trigonometric equation. We can solve this by using the componendo and dividendo rule, or by simplifying the given expression and comparing it with known trigonometric values.
Given the equation:
\[ \frac{\sin\theta+\cos\theta}{\sin\theta-\cos\theta} = \frac{\sqrt{3}-1}{\sqrt{3}+1} \]Apply the componendo and dividendo rule. If \[\frac{a}{b} = \frac{c}{d}\], then \[\frac{a+b}{a-b} = \frac{c+d}{c-d}\].
Here, let \(a = \sin\theta+\cos\theta\), \(b = \sin\theta-\cos\theta\), \(c = \sqrt{3}-1\), and \(d = \sqrt{3}+1\).
So, we have:
\[ \frac{(\sin\theta+\cos\theta) + (\sin\theta-\cos\theta)}{(\sin\theta+\cos\theta) - (\sin\theta-\cos\theta)} = \frac{(\sqrt{3}-1) + (\sqrt{3}+1)}{(\sqrt{3}-1) - (\sqrt{3}+1)} \]Simplify both sides of the equation:
Left Hand Side (LHS):
\[ \frac{\sin\theta+\cos\theta+\sin\theta-\cos\theta}{\sin\theta+\cos\theta-\sin\theta+\cos\theta} = \frac{2\sin\theta}{2\cos\theta} = \tan\theta \]Right Hand Side (RHS):
\[ \frac{\sqrt{3}-1+\sqrt{3}+1}{\sqrt{3}-1-\sqrt{3}-1} = \frac{2\sqrt{3}}{-2} = -\sqrt{3} \]Equating LHS and RHS, we get:
\[ \tan\theta = -\sqrt{3} \]We know that \(\tan(60^\circ) = \sqrt{3}\). Since \(\tan\theta\) is negative, \(\theta\) must lie in the second or fourth quadrant.
In the second quadrant, \(\tan(180^\circ - x) = -\tan x\). So, \(\tan(180^\circ - 60^\circ) = \tan(120^\circ) = -\sqrt{3}\).
In the fourth quadrant, \(\tan(360^\circ - x) = -\tan x\). So, \(\tan(360^\circ - 60^\circ) = \tan(300^\circ) = -\sqrt{3}\).
Comparing with the given options, \(300^\circ\) is one of the possible values for \(\theta\).
Correct Option: C) 300°
To identify an ester, we need to recognize its characteristic functional group. An ester is a derivative of a carboxylic acid where the hydrogen of the hydroxyl group is replaced by an alkyl or aryl group. The general formula for an ester is \(RCOOR'\), where \(R\) and \(R'\) are alkyl or aryl groups.
C) CH₃COOC₂H₅ is ethyl acetate, which is an ester. The functional group \(-COOC-\) is characteristic of an ester. It is formed from acetic acid (\(CH_3COOH\)) and ethanol (\(C_2H_5OH\)).
The question asks for the real name of the renowned Hindi literary writer Munshi Premchand. This is a factual question related to Indian literature and general knowledge.
Correct Option: D) Dhanpat Rai
Munshi Premchand, one of the most celebrated writers in Hindi and Urdu literature, was born as Dhanpat Rai Srivastava. He initially started writing under the pen name 'Nawab Rai' but later switched to 'Premchand' after his collection of short stories, 'Soz-e-Watan', was confiscated by the British government. 'Munshi' was an honorific title bestowed upon him.
To find the Highest Common Factor (HCF) of two numbers, we need to identify the common prime factors and multiply them. The problem states that \(a, b, m, r\) are odd primes. We are given the expressions for \(P\) and \(Q\).
Write down the given expressions for \(P\) and \(Q\):
\[ P = a \times m \times r \] \[ Q = b \times m \times 2 \times r \]Identify the factors of \(P\): The factors of \(P\) are \(a, m, r\).
Identify the factors of \(Q\): The factors of \(Q\) are \(b, m, 2, r\).
Identify the common factors between \(P\) and \(Q\). We look for factors that appear in both expressions.
Common factors are \(m\) and \(r\).
The HCF is the product of all common prime factors. In this case, the common prime factors are \(m\) and \(r\).
\[ \text{HCF}(P, Q) = m \times r = mr \]D) The common factors of \(P\) and \(Q\) are \(m\) and \(r\). Therefore, the HCF is \(m \times r = mr\).
This is a blood relation problem. We need to trace the relationships described in the statement from Ram's perspective to determine Sohan's relationship with Ram's mother.
Correct Option: B) Brother
To find the percentage of students who are NOT scholarship holders, we first determine the proportion of boys and girls in the school. Then, we calculate the percentage of non-scholarship holders among boys and girls separately. Finally, we combine these to find the overall percentage.
Assume a total number of students:
Given the ratio of boys to girls is \(3:2\), let the number of boys be \(3x\) and the number of girls be \(2x\). For simplicity, we can assume a total number of students that is a multiple of \(3+2=5\). Let's assume the total number of students is \(100\).
If total students = \(100\), then:
Calculate scholarship holders among boys:
\(20\%\) of boys are scholarship holders. So, scholarship holder boys \( = 20\% \text{ of } 60 \)
\[ = \frac{20}{100} \times 60 = 12 \]Calculate non-scholarship holders among boys:
Non-scholarship holder boys \( = \text{Total boys} - \text{Scholarship holder boys} \)
\[ = 60 - 12 = 48 \]Calculate scholarship holders among girls:
\(25\%\) of girls are scholarship holders. So, scholarship holder girls \( = 25\% \text{ of } 40 \)
\[ = \frac{25}{100} \times 40 = 10 \]Calculate non-scholarship holders among girls:
Non-scholarship holder girls \( = \text{Total girls} - \text{Scholarship holder girls} \)
\[ = 40 - 10 = 30 \]Calculate total non-scholarship holders:
Total non-scholarship holders \( = \text{Non-scholarship holder boys} + \text{Non-scholarship holder girls} \)
\[ = 48 + 30 = 78 \]Calculate the percentage of non-scholarship holders:
Percentage of non-scholarship holders \[= \frac{\text{Total non-scholarship holders}}{\text{Total students}} \times 100\%\]
\[ = \frac{78}{100} \times 100\% = 78\% \]B) 78% is the correct percentage of students who are NOT scholarship holders.
Find:
The problem asks us to find the value of a square root expression involving sums of consecutive odd numbers. The key to solving this is to recognize the pattern of the sum of the first \(n\) odd numbers.
D) 60 is the correct answer.
Who won the 65th Filmfare Awards, 2020 for Best Actor in a leading role (male)?
Ishaan Khattar
Gajraj Rao
Ayushmann Khurrana
Ranveer Singh
The question asks to identify the winner of the Best Actor in a Leading Role (Male) award at the 65th Filmfare Awards in 2020. This requires knowledge of the Filmfare Awards results for that specific year.
The 65th Filmfare Awards ceremony was held on February 15, 2020, in Guwahati, Assam. Ranveer Singh won the award for Best Actor in a Leading Role (Male) for his performance in the film Gully Boy.
Correct Option: D) Ranveer Singh
Ranveer Singh won the Best Actor in a Leading Role (Male) award at the 65th Filmfare Awards for his role as Murad Ahmed in the film Gully Boy.