The question asks to identify the assumption that can be drawn from the statement: "Doctors who charge high consultation fees are good". This is a statement of perceived correlation, and we need to find the underlying belief that connects high fees with goodness.
Correct Option: C) A doctor's proficiency is directly related to consultation fees.
This is a syllogism problem, which tests logical deduction. We are given two premises and need to determine which conclusion logically follows from them. We can visualize this using Venn diagrams or simply follow the rules of deductive reasoning.
D) Socrates is a rationalist. This conclusion directly follows from combining the two given premises. If all members of group A are also members of group B, and an individual X is a member of group A, then individual X must also be a member of group B.
The question asks to identify the plant from which Taxol, an important anticancer drug, is extracted. This requires knowledge of common medicinal plants and their derived compounds.
Correct Option: D) Yew. Taxol (Paclitaxel) is a natural product extracted primarily from the bark of various species of yew trees, particularly the Pacific yew (Taxus brevifolia) and Himalayan yew (Taxus wallichiana). It is a crucial drug in cancer chemotherapy.
To find the value of \(\tan 15^\circ + \cot 15^\circ\), we can use the trigonometric identities for \(\tan 15^\circ\) and \(\cot 15^\circ\). Alternatively, we can express \(\cot 15^\circ\) as \(\frac{1}{\tan 15^\circ}\) and simplify the expression. We can also use the identity \[\tan \theta + \cot \theta = \frac{1}{\sin \theta \cos \theta}\] or \(\frac{2}{\sin 2\theta}\).
Method 1: Using known values of \(\tan 15^\circ\) and \(\cot 15^\circ\)
We know that:
\[ \tan 15^\circ = \tan (45^\circ - 30^\circ) = \frac{\tan 45^\circ - \tan 30^\circ}{1 + \tan 45^\circ \tan 30^\circ} \] \[ = \frac{1 - \frac{1}{\sqrt{3}}}{1 + 1 \cdot \frac{1}{\sqrt{3}}} = \frac{\frac{\sqrt{3} - 1}{\sqrt{3}}}{\frac{\sqrt{3} + 1}{\sqrt{3}}} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \]To rationalize the denominator, multiply the numerator and denominator by \((\sqrt{3} - 1)\):
\[ \tan 15^\circ = \frac{(\sqrt{3} - 1)(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} = \frac{(\sqrt{3})^2 - 2\sqrt{3} + 1^2}{(\sqrt{3})^2 - 1^2} = \frac{3 - 2\sqrt{3} + 1}{3 - 1} = \frac{4 - 2\sqrt{3}}{2} = 2 - \sqrt{3} \]Now, for \(\cot 15^\circ\):
\[ \cot 15^\circ = \frac{1}{\tan 15^\circ} = \frac{1}{2 - \sqrt{3}} \]To rationalize the denominator, multiply the numerator and denominator by \((2 + \sqrt{3})\):
\[ \cot 15^\circ = \frac{1 \cdot (2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})} = \frac{2 + \sqrt{3}}{2^2 - (\sqrt{3})^2} = \frac{2 + \sqrt{3}}{4 - 3} = \frac{2 + \sqrt{3}}{1} = 2 + \sqrt{3} \]Now, add the values:
\[ \tan 15^\circ + \cot 15^\circ = (2 - \sqrt{3}) + (2 + \sqrt{3}) = 2 - \sqrt{3} + 2 + \sqrt{3} = 4 \]Method 2: Using trigonometric identities
We know that \(\cot \theta = \frac{1}{\tan \theta}\). So, the expression becomes:
\[ \tan 15^\circ + \cot 15^\circ = \tan 15^\circ + \frac{1}{\tan 15^\circ} \] \[ = \frac{\tan^2 15^\circ + 1}{\tan 15^\circ} \]We also know that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). So,
\[ \tan 15^\circ + \cot 15^\circ = \frac{\sin 15^\circ}{\cos 15^\circ} + \frac{\cos 15^\circ}{\sin 15^\circ} \]Find a common denominator:
\[ = \frac{\sin^2 15^\circ + \cos^2 15^\circ}{\sin 15^\circ \cos 15^\circ} \]Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\):
\[ = \frac{1}{\sin 15^\circ \cos 15^\circ} \]We know the double angle identity \(\sin 2\theta = 2 \sin \theta \cos \theta\). So, \(\sin \theta \cos \theta = \frac{\sin 2\theta}{2}\).
Substitute this into the expression:
\[ = \frac{1}{\frac{\sin (2 \cdot 15^\circ)}{2}} = \frac{2}{\sin 30^\circ} \]We know that \(\sin 30^\circ = \frac{1}{2}\).
\[ = \frac{2}{\frac{1}{2}} = 2 \cdot 2 = 4 \]The question asks for the year when the first passenger train service was inaugurated in India. This is a factual question related to Indian history, specifically the history of Indian Railways. We need to recall the exact year of this significant event.
Correct Option: C) 1853
The first passenger train in India ran on April 16, 1853, from Bori Bunder in Mumbai to Thane, covering a distance of 34 km. This event marked the birth of Indian Railways.
The question asks about the historical reason behind the construction of the Buland Darwaza by Emperor Akbar. This requires knowledge of significant events during Akbar's reign and the monuments associated with them.
Correct Option: D) Gujarat — The Buland Darwaza was erected by Akbar in 1575 to commemorate his decisive victory over the Sultanate of Gujarat in 1573. This was a significant achievement that expanded the Mughal Empire's reach and economic power.
The question asks about the number of operational nuclear power reactors in India as of November 2020. This requires factual knowledge regarding India's nuclear power program at that specific time.
Correct Option: C) 22
The question asks for the completion year of the Jama Masjid in Delhi. This requires knowledge of historical facts related to Mughal architecture, specifically the reign of Emperor Shah Jahan.
Correct Option: D) 1656
The construction of the Jama Masjid in Delhi was completed in the year 1656. It was built under the patronage of Mughal Emperor Shah Jahan and remains a significant example of Mughal architecture.
The question asks us to find the relationship between the first two terms and apply the same relationship to the third term to find the missing fourth term. This is a classic analogy question that tests logical reasoning and general knowledge.
C) Education — A hospital is associated with health, and similarly, a school is associated with education. Education is the primary function and output of a school.
The problem presents a series of numbers, and we need to find the pattern to determine the missing number. We will analyze the differences between consecutive terms or look for a relationship with squares or cubes of numbers.
The differences are increasing by 2 each time (\(26, 28, \dots\)). This suggests the next difference should be \(28 + 2 = 30\).
The next term in the series will be the last given term plus the next difference:
\[ 210 + 30 = 240 \]Following this pattern, the next number should be the product of \(15 \times 16\).
A) 240 — Both methods (differences and products of consecutive integers) consistently lead to 240 as the next number in the series.