The given series is a combination of sub-series. We need to identify the pattern by grouping the numbers. Observe that the series can be broken down into groups of three numbers, where each group follows a specific mathematical relationship.
C) 64 — The series is composed of groups of three numbers. Each group consists of a base number \(n\), followed by \(n^2\) and \(n^3\). The base numbers for these groups are \(1, 2, 3, 4, \ldots\). The last complete group is \(3, 9, 27\) (i.e., \(3^1, 3^2, 3^3\)). The next group starts with \(4\), so the numbers are \(4^1, 4^2, 4^3\). The given numbers are \(4, 16\), which are \(4^1\) and \(4^2\). The missing number is \(4^3 = 64\).
The question asks to identify the book authored by Maulana Abul Kalam Azad from the given options. This requires knowledge of prominent authors and their famous works, especially in the context of Indian history and literature.
A) India Wins Freedom: This is the autobiography of Maulana Abul Kalam Azad. It provides a detailed account of India's independence movement from his perspective as a key leader.
The descending order of the fractions \( \frac{2}{3} \), \( \frac{1}{6} \), \( \frac{1}{5} \), \( \frac{3}{7} \) is:
\( \frac{3}{7} \), \( \frac{2}{3} \), \( \frac{1}{6} \), \( \frac{1}{5} \)
\( \frac{3}{7} \), \( \frac{1}{6} \), \( \frac{1}{5} \), \( \frac{2}{3} \)
\( \frac{2}{3} \), \( \frac{3}{7} \), \( \frac{1}{5} \), \( \frac{1}{6} \)
\( \frac{1}{6} \),\( \frac{1}{5} \), \( \frac{3}{7} \), \( \frac{2}{3} \)
To arrange fractions in descending order, we need to compare their values. A common method is to convert all fractions to equivalent fractions with a common denominator, or convert them to decimal form. Converting to decimals is often quicker for competitive exams.
Thus, the descending order is \( \frac{2}{3} \), \( \frac{3}{7} \), \( \frac{1}{5} \), \( \frac{1}{6} \).
Correct Option: C) \( \frac{2}{3} \), \( \frac{3}{7} \), \( \frac{1}{5} \), \( \frac{1}{6} \)
The question asks to identify the incorrect statement among the given options regarding the solar system and its planets. We need to evaluate each statement based on established astronomical facts.
Based on this evaluation, statement A is the only incorrect statement.
Correct Option: A) Solar system is 2.3 billion years old.
The question asks to identify the judge who presided over Mahatma Gandhi's trial following the Non-Cooperation Movement. This is a specific historical fact related to India's freedom struggle.
B) Justice CN Broomfield — Justice C.N. Broomfield was indeed the Sessions Judge of Ahmedabad who presided over the trial of Mahatma Gandhi and Shankarlal Banker on charges of sedition (under Section 124A of the Indian Penal Code) in March 1922. This trial is famous for Gandhi's statement where he pleaded guilty and accepted the punishment.
The question asks for the Indian state where Golconda Fort is located. This is a factual question related to Indian geography and history, specifically about famous historical sites.
Correct Option: D) Telangana
The question asks to identify an anti-tank guided missile from the given options. This requires knowledge of various missile systems developed by India, particularly under the Integrated Guided Missile Development Programme (IGMDP).
D) Nag is an anti-tank guided missile (ATGM) developed by India. It is designed to engage and destroy modern main battle tanks and other armored vehicles.
When two objects move in opposite directions on a circular track, their relative speed is the sum of their individual speeds. They will meet when the total distance covered by them collectively equals the length of the track.
Correct Option: A) They will meet after 6 minutes.
The question asks to identify the athlete who has won the maximum number of medals in the Olympics. This requires knowledge of Olympic records, specifically medal counts of prominent Olympians.
Correct Option: A) Michael Phelps has won the maximum number of medals in the Olympics with a total of 28 medals.
Simplify the following.
To simplify the given trigonometric expression, we will use complementary angle identities. Specifically, we know that \(\tan(90^\circ - \theta) = \cot \theta\), \(\cot(90^\circ - \theta) = \tan \theta\), \(\sec(90^\circ - \theta) = \text{cosec } \theta\), and \(\text{cosec}(90^\circ - \theta) = \sec \theta\). We will convert the angles to a common reference (e.g., \(40^\circ\)) and then use the reciprocal identities \(\text{cosec } \theta = 1/\sin \theta\), \(\sec \theta = 1/\cos \theta\), \(\tan \theta = \sin \theta / \cos \theta\), and \(\cot \theta = \cos \theta / \sin \theta\).
Consider the first term of the expression: \[\left(\frac{\tan40^\circ}{\text{cosec } 50^\circ}\right)^2\].
We know that \(50^\circ = 90^\circ - 40^\circ\). Using the complementary angle identity, \(\text{cosec } 50^\circ = \text{cosec }(90^\circ - 40^\circ) = \sec 40^\circ\).
Substitute this into the first term:
\[ \left(\frac{\tan40^\circ}{\sec40^\circ}\right)^2 \]Now, express \(\tan40^\circ\) and \(\sec40^\circ\) in terms of sine and cosine:
Substitute these into the expression:
\[ \left(\frac{\frac{\sin40^\circ}{\cos40^\circ}}{\frac{1}{\cos40^\circ}}\right)^2 = \left(\frac{\sin40^\circ}{\cos40^\circ} \times \cos40^\circ\right)^2 = (\sin40^\circ)^2 = \sin^2 40^\circ \]Now, consider the second term of the expression: \(\left(\frac{\cot40^\circ}{\sec50^\circ}\right)^2\).
We know that \(50^\circ = 90^\circ - 40^\circ\). Using the complementary angle identity, \(\sec 50^\circ = \sec (90^\circ - 40^\circ) = \text{cosec } 40^\circ\).
Substitute this into the second term:
\[ \left(\frac{\cot40^\circ}{\text{cosec } 40^\circ}\right)^2 \]Now, express \(\cot40^\circ\) and \(\text{cosec } 40^\circ\) in terms of sine and cosine:
Substitute these into the expression:
\[ \left(\frac{\frac{\cos40^\circ}{\sin40^\circ}}{\frac{1}{\sin40^\circ}}\right)^2 = \left(\frac{\cos40^\circ}{\sin40^\circ} \times \sin40^\circ\right)^2 = (\cos40^\circ)^2 = \cos^2 40^\circ \]Add the simplified first and second terms:
\[ \sin^2 40^\circ + \cos^2 40^\circ \]Using the fundamental trigonometric identity \(\sin^2 \theta + \cos^2 \theta = 1\), we get:
\[ \sin^2 40^\circ + \cos^2 40^\circ = 1 \]Correct Option: D) 1