This is a logical reasoning question based on statements and conclusions. We need to determine which conclusions logically follow from the given statement, assuming the statement is absolutely true, even if it contradicts common knowledge. The key is to stick strictly to the information provided in the statement and not introduce outside information.
A) Neither conclusion I nor II follows. This is correct because, as analyzed above, both conclusions are not logically derivable from the given statement. Conclusion I makes an unwarranted assumption of exclusivity, and Conclusion II directly contradicts the statement.
To find the square root of a decimal number, we can either use the long division method or convert the decimal to a fraction and then find its square root. Alternatively, for multiple-choice questions, we can square the given options to see which one results in the original number.
The given number is \(182.25\). We can write this as a fraction:
\[ 182.25 = \frac{18225}{100} \]We need to find \(\sqrt{\frac{18225}{100}}\), which is equal to \(\frac{\sqrt{18225}}{\sqrt{100}}\).
Let's try squaring the options provided, as this is often the quickest method for MCQs.
We can multiply \(135 \times 135\):
\[ 135 \times 135 = 18225 \]Since there are two decimal places in total (\(13.5\) has one, and \(13.5\) has one), the result will have two decimal places.
\[ (13.5)^2 = 182.25 \]This matches the given number.
A) \(13.5\) is the correct answer because \((13.5)^2 = 182.25\).
The question asks to identify the British Viceroy who introduced the Vernacular Press Act in India. This requires knowledge of significant historical acts and the Viceroys associated with them during British rule.
Correct Option: B) Lord Lytton introduced the Vernacular Press Act in 1878. This act allowed the government to confiscate the printing press and other assets of any vernacular newspaper that published "seditious" material, effectively curbing the freedom of the press in India.
The problem presents a letter series, and we need to find the pattern to determine the next letter. We will analyze the position of each letter in the English alphabet and identify the difference between consecutive letters.
C) V is the correct answer because the pattern of differences between consecutive letters' alphabetical positions is an increasing sequence of natural numbers (1, 2, 3, 4, 5, ...). The next difference is 6, and adding 6 to the position of P (16) gives 22, which corresponds to the letter V.
The question asks to identify the Satyagraha movement during which Vallabhbhai Patel earned the title "Sardar". We need to recall the major Satyagraha movements led by or involving Vallabhbhai Patel and the historical context of each.
Correct Option: C) Bardoli — Vallabhbhai Patel was given the title "Sardar" by the women of Bardoli during the Bardoli Satyagraha in 1928 for his successful leadership of the peasant movement against increased land revenue.
Who was the President of Indian National Congress on 15 August 1947?
Sardar Patel
Pandit J L Nehru
J.B. Kripalani
Rajendra Prasad
The question asks to identify the President of the Indian National Congress on India's Independence Day, August 15, 1947. This requires knowledge of the history of the Indian National Congress and its leadership during the crucial period of India's independence.
C) J.B. Kripalani was the President of the Indian National Congress on August 15, 1947. He was elected President at the Meerut session in November 1946 and remained in office until November 1947, thus overseeing the period of India's independence.
The question asks to identify the country where the historical site known as the 'Forbidden City' is located. This requires general knowledge of world geography and famous landmarks.
Correct Option: B) China is the correct answer because the Forbidden City is a historical imperial palace located in Beijing, China.
The question asks to identify the Englishman who taught Philosophy at St. Stephen's College and was later known as 'Deenbandhu'. This requires knowledge of prominent historical figures associated with India during the British Raj and their contributions or titles.
Correct Option: D) C F Andrews was an English missionary and social reformer who taught Philosophy at St. Stephen's College, Delhi. He was a close associate of Mahatma Gandhi and Rabindranath Tagore and was affectionately called 'Deenbandhu' by Mahatma Gandhi for his deep empathy and work for the poor and oppressed in India.
If x + \( \frac{1}{x} \) = 2 cos θ, then what is the value of x² + \( \frac{1}{x^2} \)?
The problem asks us to find the value of \(x^2 + \frac{1}{x^2}\) given the equation \(x + \frac{1}{x} = 2 \cos \theta\). We can solve this by squaring the given equation and then using trigonometric identities.
Start with the given equation:
\[ x + \frac{1}{x} = 2 \cos \theta \]Square both sides of the equation:
\[ \left(x + \frac{1}{x}\right)^2 = (2 \cos \theta)^2 \]Expand the left side using the identity \((a+b)^2 = a^2 + b^2 + 2ab\):
\[ x^2 + \left(\frac{1}{x}\right)^2 + 2 \cdot x \cdot \frac{1}{x} = 4 \cos^2 \theta \]\[ x^2 + \frac{1}{x^2} + 2 = 4 \cos^2 \theta \]Isolate \(x^2 + \frac{1}{x^2}\) by subtracting 2 from both sides:
\[ x^2 + \frac{1}{x^2} = 4 \cos^2 \theta - 2 \]Factor out 2 from the right side:
\[ x^2 + \frac{1}{x^2} = 2(2 \cos^2 \theta - 1) \]Recall the double angle identity for cosine: \(\cos 2\theta = 2 \cos^2 \theta - 1\). Substitute this into the equation:
\[ x^2 + \frac{1}{x^2} = 2 \cos 2\theta \]Thus, the value of \(x^2 + \frac{1}{x^2}\) is \(2 \cos 2\theta\).
Correct Option: B) 2 cos 2θ
In tossing three coins at a time, the probability of getting at least one heads is:
\( \frac{7}{8} \)
\( \frac{1}{2} \)
\( \frac{1}{8} \)
\( \frac{3}{8} \)
To find the probability of getting at least one head when tossing three coins, we can either list all possible outcomes and count the favorable ones, or use the complementary probability rule. The complementary rule states that the probability of an event happening is 1 minus the probability of the event not happening.
Determine the total number of possible outcomes:
When tossing one coin, there are 2 possible outcomes (Head or Tail).
When tossing three coins, the total number of outcomes is \(2 \times 2 \times 2 = 2^3 = 8\).
List all possible outcomes:
Let H denote Head and T denote Tail. The 8 possible outcomes are:
(H, H, H)
(H, H, T)
(H, T, H)
(T, H, H)
(H, T, T)
(T, H, T)
(T, T, H)
(T, T, T)
Identify the event "at least one head":
This means we are looking for outcomes that have one head, two heads, or three heads. It is easier to identify the complementary event, which is "no heads" (i.e., all tails).
Identify the outcome(s) for "no heads":
The only outcome with no heads is (T, T, T).
Calculate the probability of "no heads":
Number of outcomes with no heads = 1
Total number of outcomes = 8
Probability of no heads \[P(\text{no heads}) = \frac{\text{Number of outcomes with no heads}}{\text{Total number of outcomes}} = \frac{1}{8}\]
Calculate the probability of "at least one head" using the complementary rule:
\(P(\text{at least one head}) = 1 - P(\text{no heads})\)
\[ P(\text{at least one head}) = 1 - \frac{1}{8} = \frac{8}{8} - \frac{1}{8} = \frac{7}{8} \]
A) \( \frac{7}{8} \) is the correct probability of getting at least one head when tossing three coins.