The question asks to identify the Governor-General responsible for introducing the Permanent Settlement of Bengal in 1793. This requires knowledge of key historical figures and their administrative reforms during British rule in India.
Correct Option: B) Charles Cornwallis was the Governor-General of Bengal who introduced the Permanent Settlement in 1793. This land revenue system made zamindars the owners of the land and fixed the revenue demand.
The question asks about the number of Fundamental Duties in the Indian Constitution as of December 2020. To answer this, we need to recall the constitutional provisions related to Fundamental Duties and their amendments.
C) 11 is the correct answer because the Constitution of India originally had 10 Fundamental Duties, and one more was added by the 86th Amendment Act of 2002, bringing the total to 11.
This problem involves work and time, where we need to find the individual time taken by B to complete a task. We are given the combined work rates of A and B, and B and C, along with a relationship between the efficiencies of A and C. We will use the concept that work rate is the reciprocal of the time taken to complete a task.
Let the work done by A, B, and C in one day be \(W_A\), \(W_B\), and \(W_C\) respectively.
From the first statement, A and B together can finish a task in 12 days. This means their combined one-day work is:
\[ W_A + W_B = \frac{1}{12} \quad \text{(Equation 1)} \]From the second statement, B and C together can finish it in 18 days. Their combined one-day work is:
\[ W_B + W_C = \frac{1}{18} \quad \text{(Equation 2)} \]From the third statement, A is three times more efficient than C. This means A's work rate is 3 times C's work rate:
\[ W_A = 3W_C \quad \text{(Equation 3)} \]Substitute Equation 3 into Equation 1:
\[ 3W_C + W_B = \frac{1}{12} \quad \text{(Equation 4)} \]Now we have a system of two equations (Equation 2 and Equation 4) with two variables (\(W_B\) and \(W_C\)):
\[ W_B + W_C = \frac{1}{18} \]\[ W_B + 3W_C = \frac{1}{12} \]Subtract Equation 2 from Equation 4:
\[ (W_B + 3W_C) - (W_B + W_C) = \frac{1}{12} - \frac{1}{18} \]\[ 2W_C = \frac{3 - 2}{36} \]\[ 2W_C = \frac{1}{36} \]\[ W_C = \frac{1}{72} \]Now that we have \(W_C\), we can find \(W_B\) using Equation 2:
\[ W_B + \frac{1}{72} = \frac{1}{18} \]\[ W_B = \frac{1}{18} - \frac{1}{72} \]\[ W_B = \frac{4 - 1}{72} \]\[ W_B = \frac{3}{72} \]\[ W_B = \frac{1}{24} \]The work done by B in one day is \(\frac{1}{24}\) of the task. Therefore, B alone would finish the task in 24 days.
C) 24 days
If cos x - 3sin x = \(\sqrt{3}\) sin x, then the value of tan x is:
\(\frac{3-\sqrt{3}}{6}\)
3 - \(\sqrt{3}\)
3 + \(\sqrt{3}\)
\(\sqrt{3}\)
The problem asks us to find the value of \(\tan x\) given an equation involving \(\cos x\) and \(\sin x\). The strategy is to rearrange the given equation to isolate terms involving \(\sin x\) and \(\cos x\) and then divide by \(\cos x\) to express the equation in terms of \(\tan x\).
Start with the given equation:
\[ \cos x - 3\sin x = \sqrt{3}\sin x \]Move all terms involving \(\sin x\) to one side of the equation:
\[ \cos x = \sqrt{3}\sin x + 3\sin x \]Factor out \(\sin x\) from the terms on the right side:
\[ \cos x = (\sqrt{3} + 3)\sin x \]To find \(\tan x\), which is \(\frac{\sin x}{\cos x}\), we need to divide both sides by \(\cos x\) (assuming \(\cos x \neq 0\)) and by \(( \sqrt{3} + 3 )\):
\[ \frac{\sin x}{\cos x} = \frac{1}{\sqrt{3} + 3} \]So, \(\tan x = \frac{1}{3 + \sqrt{3}}\). To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator, which is \((3 - \sqrt{3})\):
\[ \tan x = \frac{1}{3 + \sqrt{3}} \times \frac{3 - \sqrt{3}}{3 - \sqrt{3}} \]Perform the multiplication. Recall that \((a+b)(a-b) = a^2 - b^2\):
\[ \tan x = \frac{3 - \sqrt{3}}{3^2 - (\sqrt{3})^2} \]\[ \tan x = \frac{3 - \sqrt{3}}{9 - 3} \]\[ \tan x = \frac{3 - \sqrt{3}}{6} \]A) The calculated value of \(\tan x\) is \(\frac{3 - \sqrt{3}}{6}\), which matches option A.
The question asks to identify the type of soil best suited for cotton cultivation. This requires knowledge of the characteristics of different soil types and their suitability for specific crops.
Correct Option: B) Black soil
The problem requires us to deduce the professions and games played by four individuals (P, Q, R, S) based on a set of given clues. We need to identify who plays carrom and is also a swimmer by systematically building a table or list of attributes for each person.
| Person | Profession | Games | Other Activities |
|---|---|---|---|
| P | Teacher | Badminton, ? | Swimmer |
| Q | Teacher | ?, ? | Swimmer |
| R | ? | ?, ? | ? |
| S | Lawyer | Carrom, ? | Swimmer |
A) S is the correct answer. Based on the information, "S plays carrom and is a lawyer." Also, "The lawyer and both the teachers are swimmers," which means S (the lawyer) is a swimmer. Therefore, S plays carrom and is a swimmer.
This is a blood relations question. We need to deduce the relationship between S and L by constructing a family tree based on the given statements. We will use symbols to represent gender and relationships to build the tree step-by-step.
B) Sister's Daughter — Based on our step-by-step deduction, S is the daughter of M, and M is the sister of L. Therefore, S is L's sister's daughter.
The question asks to identify a copper ore from the given options. This requires knowledge of common ores and the metals they are associated with.
Correct Option: C) Malachite
Malachite is a copper carbonate hydroxide mineral with the chemical formula \( \text{Cu}_2(\text{CO}_3)(\text{OH})_2 \). It is a well-known ore of copper, characterized by its distinctive green color.
The question asks us to identify the cause-and-effect relationship between two events: Event A (Meghna receiving a prize and medal) and Event B (Meghna winning a race competition). We need to determine which event is the cause and which is the effect, and if the relationship is immediate and principal.
Correct Option: B) Event B is the principal and immediate cause and A is its effect.
The question asks for the common name of E300. This refers to the E-number system used to identify food additives. E-numbers are codes for substances allowed to be used as food additives within the European Union and other countries. Each E-number corresponds to a specific chemical compound or group of compounds.
Correct Option: D) Vitamin C is the common name for E300, which is L-ascorbic acid. It is a well-known antioxidant and essential nutrient.