The question asks about the authority responsible for creating local self-governing bodies like municipalities, municipal corporations, and panchayats. This falls under the constitutional framework of local self-governance in India.
C) State Government is the correct answer because the creation and governance of local bodies (Panchayats and Municipalities) are subjects primarily dealt with by state legislatures and state governments under the Indian Constitution.
The question asks for the state where Ranikhet hill station is located. This is a direct factual question related to Indian geography, specifically the location of prominent tourist places or geographical features.
Correct Option: D) Ranikhet is a well-known hill station situated in the Almora district of the Indian state of Uttarakhand. It is famous for its natural beauty, ancient temples, and orchards.
The question asks for the launch period of the First Five-Year Plan in India. This is a factual question related to India's economic planning history. Recalling the timeline of India's Five-Year Plans is key to answering this question.
Correct Option: B) 1951-1956
The First Five-Year Plan in India was indeed launched in 1951 and covered the period from 1951 to 1956. It was based on the Harrod-Domar model and prioritized agricultural development.
The question asks for the literal meaning of 'Namo Arihantanam' from the Navakara Mantra, a fundamental prayer in Jainism. Understanding the components of this phrase and their significance in Jain philosophy is key to identifying the correct option.
B) I bow down to all souls who have escaped all their ghati karmas. This accurately reflects the meaning of 'Arihant' in Jain philosophy, referring to those who have destroyed the four destructive (ghati) karmas and attained perfect knowledge.
This is a work and time problem. We need to find the work equivalency between men and women. We can set up equations based on the given information, assuming a constant rate of work for each man and woman, and then solve for the ratio of their work rates.
Let \(M\) be the amount of work one man can do in one day, and \(W\) be the amount of work one woman can do in one day.
From the first statement, "8 men and 13 women can complete a piece of work in 12 days", the total work done by them in one day is \(\frac{1}{12}\) of the total work. So, we can write the equation:
\[ 8M + 13W = \frac{1}{12} \quad \ldots(1) \]From the second statement, "10 men and 14 women can complete the same work in 10 days", the total work done by them in one day is \(\frac{1}{10}\) of the total work. So, we can write the equation:
\[ 10M + 14W = \frac{1}{10} \quad \ldots(2) \]Now we have a system of two linear equations with two variables. We need to solve for \(M\) and \(W\) or their ratio. Let's eliminate one variable. Multiply equation (1) by 10 and equation (2) by 8 to make the coefficients of \(M\) equal:
Multiplying (1) by 10:
\[ 10(8M + 13W) = 10 \times \frac{1}{12} \]\[ 80M + 130W = \frac{10}{12} = \frac{5}{6} \quad \ldots(3) \]Multiplying (2) by 8:
\[ 8(10M + 14W) = 8 \times \frac{1}{10} \]\[ 80M + 112W = \frac{8}{10} = \frac{4}{5} \quad \ldots(4) \]Subtract equation (4) from equation (3):
\[ (80M + 130W) - (80M + 112W) = \frac{5}{6} - \frac{4}{5} \]\[ 18W = \frac{25 - 24}{30} \]\[ 18W = \frac{1}{30} \]\[ W = \frac{1}{30 \times 18} = \frac{1}{540} \]Now substitute the value of \(W\) into equation (1):
\[ 8M + 13 \left(\frac{1}{540}\right) = \frac{1}{12} \]\[ 8M + \frac{13}{540} = \frac{1}{12} \]\[ 8M = \frac{1}{12} - \frac{13}{540} \]\[ 8M = \frac{45}{540} - \frac{13}{540} \]\[ 8M = \frac{32}{540} \]\[ M = \frac{32}{540 \times 8} = \frac{4}{540} = \frac{1}{135} \]We need to find the work of how many women is equivalent to that of one man. This means we need to find the value of \(x\) such that \(1M = xW\). We have \(M = \frac{1}{135}\) and \(W = \frac{1}{540}\). So,
\[ \frac{1}{135} = x \times \frac{1}{540} \]\[ x = \frac{540}{135} \]\[ x = 4 \]Therefore, the work of 4 women is equivalent to that of one man.
Correct Option: C) 4
This problem involves setting up equations based on the given information about the age difference and the ratio of ages. We will use a variable to represent a common factor in the ratio and then solve for it using the age difference.
Let the present age of Ramu be \(R\) and the present age of Lakshman be \(L\).
Given that their present ages are in the ratio 6 : 5, we can write:
\[ \frac{R}{L} = \frac{6}{5} \]This means we can express their ages using a common multiplier, say \(x\):
We are given that Ramu is older than Lakshman by 4 years. This can be written as:
\[ R - L = 4 \]Substitute \(R = 6x\) and \(L = 5x\) into the difference equation:
\[ 6x - 5x = 4 \]Simplifying the equation:
\[ x = 4 \]Lakshman's age is \(L = 5x\). Substitute the value of \(x\):
\[ L = 5 \times 4 \] \[ L = 20 \text{ years} \]Correct Option: D) 20 years
The roots of the quadratic equation x² − 4x − 1 = 0 are:
2 ± \(\sqrt{3}\)
2 ± \(\sqrt{5}\)
−2 ± \(\sqrt{3}\)
−2 ± \(\sqrt{5}\)
To find the roots of a quadratic equation of the form \(ax^2 + bx + c = 0\), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]B) The roots of the equation \(x^2 - 4x - 1 = 0\) are \(2 \pm \sqrt{5}\), which matches option B.
To solve the given linear equation, we need to simplify it by distributing the multiplication, combining like terms, and then isolating the variable \(x\).
\[ (5x^2 - 5) + 5(-x^2 + 2x - 7) = 0 \]
Multiply each term inside the second parenthesis by \(5\).
\[ 5x^2 - 5 + 5(-x^2) + 5(2x) + 5(-7) = 0 \]
\[ 5x^2 - 5 - 5x^2 + 10x - 35 = 0 \]
Group the \(x^2\) terms, \(x\) terms, and constant terms together.
\[ (5x^2 - 5x^2) + 10x + (-5 - 35) = 0 \]
\[ 0x^2 + 10x - 40 = 0 \]
\[ 10x - 40 = 0 \]
Add \(40\) to both sides of the equation.
\[ 10x = 40 \]
Divide both sides by \(10\).
\[ x = \frac{40}{10} \]
\[ x = 4 \]
Correct Option: A) \(x = 4\)
This problem involves tracking a person's movement and direction changes. We will trace Kritika's path step-by-step, noting her current direction after each turn, to determine her final facing direction.
Correct Option: B) North
The question asks to identify the correct statement regarding a precipitation reaction. A precipitation reaction is a type of chemical reaction that occurs in an aqueous solution where two ionic compounds react to form an insoluble product, known as a precipitate.
For example, when silver nitrate solution (\(\text{AgNO}_3\)) is mixed with sodium chloride solution (\(\text{NaCl}\)), a white precipitate of silver chloride (\(\text{AgCl}\)) is formed, which is insoluble in water. The reaction is:
\[ \text{AgNO}_3(\text{aq}) + \text{NaCl}(\text{aq}) \rightarrow \text{AgCl}(\text{s}) \downarrow + \text{NaNO}_3(\text{aq}) \]D) An insoluble salt is formed. This statement accurately describes the defining characteristic of a precipitation reaction. The insoluble product formed is commonly referred to as an insoluble salt (or precipitate).