In the triangle ABC, a = 25, b = 45 and c = 30. The value of cos A is:
\( \frac{21}{27} \)
\( \frac{25}{27} \)
\( \frac{20}{27} \)
\( \frac{23}{27} \)
To find the value of \(\cos A\) in a triangle when all three side lengths are given, we use the Law of Cosines. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.
For a triangle with sides \(a\), \(b\), \(c\) and angles \(A\), \(B\), \(C\) opposite to these sides respectively, the Law of Cosines states:
\[ a^2 = b^2 + c^2 - 2bc \cos A \]\[ b^2 = a^2 + c^2 - 2ac \cos B \]\[ c^2 = a^2 + b^2 - 2ab \cos C \]From the first equation, we can isolate \(\cos A\):
\[ 2bc \cos A = b^2 + c^2 - a^2 \]\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]We are given \(a = 25\), \(b = 45\), and \(c = 30\).
\[ \cos A = \frac{(45)^2 + (30)^2 - (25)^2}{2 \times 45 \times 30} \]Numerator:
\[ 2025 + 900 - 625 = 2925 - 625 = 2300 \]Denominator:
\[ 2 \times 45 \times 30 = 90 \times 30 = 2700 \]We can cancel out the two zeros from the numerator and denominator:
\[ \cos A = \frac{23}{27} \]D) \( \frac{23}{27} \) is the calculated value of \(\cos A\) after applying the Law of Cosines and simplifying the expression.
To find the value of the given expression, we can either calculate each cubic term separately and then subtract, or use the algebraic identity \(a^3 - b^3 = (a-b)(a^2+ab+b^2)\). Given the numbers, direct calculation of cubes might be simpler and less prone to calculation errors with decimals.
\[ (0.2)^3 = 0.2 \times 0.2 \times 0.2 \]
First, \(0.2 \times 0.2 = 0.04\).
Then, \(0.04 \times 0.2 = 0.008\).
So, \( (0.2)^3 = 0.008 \).
\[ (0.02)^3 = 0.02 \times 0.02 \times 0.02 \]
First, \(0.02 \times 0.02 = 0.0004\).
Then, \(0.0004 \times 0.02 = 0.000008\).
So, \( (0.02)^3 = 0.000008 \).
We need to calculate \( (0.2)^3 - (0.02)^3 = 0.008 - 0.000008 \).
To subtract, it's helpful to align the decimal points and add trailing zeros to \(0.008\):
\[ \begin{array}{r} 0.008000 \\ - 0.000008 \\ \hline 0.007992 \end{array} \]
Thus, \( (0.2)^3 - (0.02)^3 = 0.007992 \).
C) 0.007992 is the correct result obtained from the calculation.
The question asks about the origin of the principles of 'Liberty, Equality & Fraternity' in the Indian Constitution. This requires knowledge of the sources of various constitutional provisions.
Correct Option: B) France — The ideals of Liberty, Equality, and Fraternity, which are enshrined in the Preamble of the Indian Constitution, were inspired by the French Revolution and subsequently adopted from the French Constitution.
The question asks to identify the digestive organ that resembles the English alphabet 'J' in shape. This requires knowledge of the basic anatomy of the human digestive system.
C) Stomach. The stomach is a muscular, J-shaped organ located on the left side of the upper abdomen. It receives food from the esophagus and mixes it with digestive juices before passing it to the small intestine. Its distinctive shape is widely recognized as resembling the letter 'J'.
A dice is cast twice, and the sum of the appearing numbers is 10. Then the probability that the number 5 has appeared at least once is:
\( \frac{1}{4} \)
\( \frac{1}{3} \)
\( \frac{1}{2} \)
\( \frac{2}{3} \)
The question asks to identify which of the given museums is located in Kerala. This requires knowledge of the geographical locations of prominent museums in India.
Based on this evaluation, the Napier Museum is the only one situated in Kerala.
D) Napier Museum is located in Thiruvananthapuram, Kerala. It is one of the oldest museums in India and houses a rare collection of archaeological and historic artifacts, bronze idols, ancient ornaments, a temple chariot, and ivory carvings.
The question asks about the geographical origin of the human species. This is a topic covered in human evolution and anthropology, specifically the "Out of Africa" theory, which is widely accepted in the scientific community.
Correct Option: D) Africa — Extensive scientific evidence, including genetic studies and fossil discoveries, points to Africa as the continent where the human species (Homo sapiens) first evolved before migrating to other parts of the world. This is known as the "Out of Africa" theory.
This question tests knowledge about the regional names given to tropical cyclones based on where they originate. Tropical cyclones are intense low-pressure systems that form over warm tropical oceans. Their names vary geographically.
Correct Option: D) Hurricane
We need to evaluate each conclusion based on the given statement. We should determine if the conclusion is a direct logical consequence or a reasonable inference from the statement, without introducing outside information or assumptions.
Correct Option: B) Only conclusion I follows.
The question asks to identify the personality known as the 'Martin Luther of India'. This title is bestowed upon individuals who initiated significant reform movements, similar to Martin Luther's role in the Protestant Reformation. We need to recall prominent Indian social and religious reformers and their contributions.
Correct Option: D) Dayanand Saraswati is known as the 'Martin Luther of India' due to his significant role in reforming Hinduism, particularly through the establishment of the Arya Samaj. He advocated for a return to the pristine teachings of the Vedas, rejecting idol worship, the caste system, and other social evils prevalent in his time. His call for "Go back to the Vedas" was a revolutionary step against prevailing religious dogmas, much like Martin Luther's challenge to the Catholic Church.