This is a question on logical deduction. We need to analyze the given statement and determine which conclusion logically and necessarily follows from it, without introducing any outside information or assumptions.
Correct Option: A) Conclusion 3: No school is without students.
This conclusion is a direct logical equivalent of the given statement. If "Every school has students," it inherently means that there is no school that lacks students.
The problem is a coding-decoding question. We need to identify the pattern or rule used to transform "MODERN" into "QSHIVR". Once the rule is found, we apply the reverse of that rule to "WMIRG" to find the original word.
The pattern is that each letter in the original word is shifted forward by 4 positions in the English alphabet to get the coded letter.
Correct Option: D) SIENC
To arrange words in dictionary order, we compare them letter by letter from left to right. If the first letters are the same, we move to the second letter, and so on. If one word is a prefix of another (e.g., "cat" and "catalog"), the shorter word comes first.
The correct order is: Practical → Practise → Praise → Prank → Prayer.
Correct Option: A) Practical → Practise → Praise → Prank → Prayer
The goal is to find the relationship between the numbers in the given set (4, 20, 28) and then apply the same relationship to the given options to find the correct one.
The relationship is that if the first number is \(x\), the second number is \(5x\), and the third number is \(7x\).
Correct Option: C) 12, 60, 84
The value of \( \frac{32}{5} \) × 92 + \( \frac{3}{5} \) × 2 is:
To solve this problem, we need to perform the multiplication operations first, following the order of operations (BODMAS/PEMDAS), and then add the results. We will multiply the fractions by the whole numbers and then sum the two terms.
B) 590 is the correct answer. The sum of the two products \( \frac{32}{5} \times 92 \) and \( \frac{3}{5} \times 2 \) is 590.
The question asks to identify the first post-communist EU member state to legalize same-sex marriage in 2022. This requires knowledge of recent social legislation in European countries, specifically those with a post-communist history that are also members of the European Union.
Correct Option: C) Slovenia became the first post-communist member of the EU to legalize same-sex marriage in 2022. The country's Constitutional Court ruled in July 2022 that the ban on same-sex marriage and adoption was unconstitutional, and the parliament subsequently passed legislation to implement this decision.
The question asks to identify the highest law officer of the Government of India. This requires knowledge of the constitutional and legal hierarchy of officials in India.
Correct Option: D) Attorney General for India
The Attorney General for India is the highest law officer of the Government of India. Article 76 of the Indian Constitution provides for the appointment of the Attorney General by the President. The Attorney General is the chief legal advisor to the Government of India and represents the Union Government in all its legal matters before the Supreme Court and other courts.
The value of cos 75° + sin 15° is equal to:
\(\frac{\sqrt{3}-1}{\sqrt{2}}\)
\(\frac{\sqrt{3}}{\sqrt{2}}\)
\(\frac{\sqrt{2}}{\sqrt{3}}\)
\(\frac{\sqrt{3}+1}{\sqrt{2}}\)
To find the value of \(\cos 75^\circ + \sin 15^\circ\), we can use trigonometric identities. Specifically, we can express both terms in terms of known angles (like \(45^\circ\) and \(30^\circ\)) or use the complementary angle identity \(\sin \theta = \cos (90^\circ - \theta)\).
We know that \(\sin \theta = \cos (90^\circ - \theta)\).
So, \(\sin 15^\circ = \cos (90^\circ - 15^\circ) = \cos 75^\circ\).
Therefore, the expression becomes:
\[ \cos 75^\circ + \sin 15^\circ = \cos 75^\circ + \cos 75^\circ = 2 \cos 75^\circ \]We can write \(75^\circ\) as \(45^\circ + 30^\circ\).
Using the cosine addition formula \(\cos (A+B) = \cos A \cos B - \sin A \sin B\):
\[ \cos 75^\circ = \cos (45^\circ + 30^\circ) = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ \]We know the standard trigonometric values:
Substitute these values into the expression for \(\cos 75^\circ\):
\[ \cos 75^\circ = \left(\frac{1}{\sqrt{2}}\right) \left(\frac{\sqrt{3}}{2}\right) - \left(\frac{1}{\sqrt{2}}\right) \left(\frac{1}{2}\right) \]\[ \cos 75^\circ = \frac{\sqrt{3}}{2\sqrt{2}} - \frac{1}{2\sqrt{2}} \]\[ \cos 75^\circ = \frac{\sqrt{3}-1}{2\sqrt{2}} \]Now, substitute the value of \(\cos 75^\circ\) back into \(2 \cos 75^\circ\):
\[ 2 \cos 75^\circ = 2 \left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right) \]\[ 2 \cos 75^\circ = \frac{\sqrt{3}-1}{\sqrt{2}} \]Correct Option: A) \(\frac{\sqrt{3}-1}{\sqrt{2}}\) is the correct value obtained from the calculation.
What is the value of the following expression?
\(\frac{\sqrt{7}-\sqrt{3}}{\sqrt{7}+\sqrt{3}}+\frac{\sqrt{7}+\sqrt{3}}{\sqrt{7}-\sqrt{3}}+\frac{\sqrt{3}+1}{\sqrt{3}-1}+\frac{\sqrt{3}-1}{\sqrt{3}+1}\)
The problem requires simplifying a complex expression involving square roots. We will rationalize each fraction by multiplying the numerator and denominator by the conjugate of the denominator. We will use the algebraic identity \((a-b)(a+b) = a^2 - b^2\).
Let's simplify the first two terms: \[\frac{\sqrt{7}-\sqrt{3}}{\sqrt{7}+\sqrt{3}}+\frac{\sqrt{7}+\sqrt{3}}{\sqrt{7}-\sqrt{3}}\].
First term: \(\frac{\sqrt{7}-\sqrt{3}}{\sqrt{7}+\sqrt{3}}\)
\[ \frac{\sqrt{7}-\sqrt{3}}{\sqrt{7}+\sqrt{3}} = \frac{(\sqrt{7}-\sqrt{3})(\sqrt{7}-\sqrt{3})}{(\sqrt{7}+\sqrt{3})(\sqrt{7}-\sqrt{3})} = \frac{(\sqrt{7}-\sqrt{3})^2}{(\sqrt{7})^2 - (\sqrt{3})^2} = \frac{7 - 2\sqrt{21} + 3}{7 - 3} = \frac{10 - 2\sqrt{21}}{4} = \frac{5 - \sqrt{21}}{2} \]Second term: \(\frac{\sqrt{7}+\sqrt{3}}{\sqrt{7}-\sqrt{3}}\)
\[ \frac{\sqrt{7}+\sqrt{3}}{\sqrt{7}-\sqrt{3}} = \frac{(\sqrt{7}+\sqrt{3})(\sqrt{7}+\sqrt{3})}{(\sqrt{7}-\sqrt{3})(\sqrt{7}+\sqrt{3})} = \frac{(\sqrt{7}+\sqrt{3})^2}{(\sqrt{7})^2 - (\sqrt{3})^2} = \frac{7 + 2\sqrt{21} + 3}{7 - 3} = \frac{10 + 2\sqrt{21}}{4} = \frac{5 + \sqrt{21}}{2} \]Sum of the first two terms:
\[ \frac{5 - \sqrt{21}}{2} + \frac{5 + \sqrt{21}}{2} = \frac{5 - \sqrt{21} + 5 + \sqrt{21}}{2} = \frac{10}{2} = 5 \]Next, let's simplify the last two terms: \[\frac{\sqrt{3}+1}{\sqrt{3}-1}+\frac{\sqrt{3}-1}{\sqrt{3}+1}\].
Third term: \(\frac{\sqrt{3}+1}{\sqrt{3}-1}\)
\[ \frac{\sqrt{3}+1}{\sqrt{3}-1} = \frac{(\sqrt{3}+1)(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)} = \frac{(\sqrt{3}+1)^2}{(\sqrt{3})^2 - (1)^2} = \frac{3 + 2\sqrt{3} + 1}{3 - 1} = \frac{4 + 2\sqrt{3}}{2} = 2 + \sqrt{3} \]Fourth term: \(\frac{\sqrt{3}-1}{\sqrt{3}+1}\)
\[ \frac{\sqrt{3}-1}{\sqrt{3}+1} = \frac{(\sqrt{3}-1)(\sqrt{3}-1)}{(\sqrt{3}+1)(\sqrt{3}-1)} = \frac{(\sqrt{3}-1)^2}{(\sqrt{3})^2 - (1)^2} = \frac{3 - 2\sqrt{3} + 1}{3 - 1} = \frac{4 - 2\sqrt{3}}{2} = 2 - \sqrt{3} \]Sum of the last two terms:
\[ (2 + \sqrt{3}) + (2 - \sqrt{3}) = 2 + \sqrt{3} + 2 - \sqrt{3} = 4 \]Finally, add the results from step 1 and step 2:
\[ 5 + 4 = 9 \]B) The value of the expression is 9.
The question asks for the name of the specific scheme launched by the Government of India to achieve a clean and open defecation-free India. This requires knowledge of major government initiatives related to sanitation.
Correct Option: B) Swachh Bharat Abhiyan is the correct name of the scheme launched by the Government of India with the goal of achieving a clean and open defecation-free India. It was launched on October 2, 2014, by Prime Minister Narendra Modi.