This problem involves calculating the combined work rate of two taps, one filling and one emptying, to determine the time taken to fill a specific fraction of the tank. We will first find the individual rates, then the net rate, and finally the time required for the desired volume.
Calculate the filling rate of the first tap:
The first tap fills the cistern in 18 hours. So, in one hour, it fills:
\[ \text{Filling Rate} = \frac{1}{18} \text{ of the tank per hour} \]Calculate the emptying rate of the second tap:
The second tap empties the full tank in 19 hours. So, in one hour, it empties:
\[ \text{Emptying Rate} = \frac{1}{19} \text{ of the tank per hour} \]Calculate the net filling rate when both taps are open:
Since one tap is filling and the other is emptying, the net rate is the difference between the filling rate and the emptying rate:
\[ \text{Net Rate} = \text{Filling Rate} - \text{Emptying Rate} \] \[ \text{Net Rate} = \frac{1}{18} - \frac{1}{19} \]To subtract these fractions, find a common denominator, which is \(18 \times 19 = 342\):
\[ \text{Net Rate} = \frac{19}{342} - \frac{18}{342} \] \[ \text{Net Rate} = \frac{1}{342} \text{ of the tank per hour} \]This means that when both taps are open, \(\frac{1}{342}\) of the tank is filled in one hour.
Calculate the time to fill the entire tank:
If \(\frac{1}{342}\) of the tank is filled in 1 hour, then the entire tank will be filled in 342 hours.
\[ \text{Time to fill full tank} = \frac{1}{\text{Net Rate}} = \frac{1}{\frac{1}{342}} = 342 \text{ hours} \]Calculate the time to fill one-third of the tank:
We need to find the time to fill one-third of the tank. This will be one-third of the time it takes to fill the entire tank:
\[ \text{Time to fill } \frac{1}{3} \text{ tank} = \frac{1}{3} \times \text{Time to fill full tank} \] \[ \text{Time to fill } \frac{1}{3} \text{ tank} = \frac{1}{3} \times 342 \] \[ \text{Time to fill } \frac{1}{3} \text{ tank} = 114 \text{ hours} \]As calculated, it will take 114 hours to fill one-third of the tank.
A can complete a piece of work in 14 days and B can complete it in 10 days. If they work on alternate days, starting with B on the first day, then in how many days will the work be finished?
11 \( \frac{2}{3} \)
10 \( \frac{2}{3} \)
11 \( \frac{3}{5} \)
10 \( \frac{3}{5} \)
This problem involves calculating the time taken to complete a work when two individuals work on alternate days. We need to find the individual work rates, combine them for a two-day cycle, and then calculate the total days, considering who starts the work.
A can complete the work in 14 days. So, A's one day's work is:
\[ \text{A's work per day} = \frac{1}{14} \]B can complete the work in 10 days. So, B's one day's work is:
\[ \text{B's work per day} = \frac{1}{10} \]Since they work on alternate days, starting with B, the first day B works, and the second day A works. A cycle consists of 2 days.
Work done in 2 days (1 cycle) = (B's work on day 1) + (A's work on day 2)
\[ \text{Work in 2 days} = \frac{1}{10} + \frac{1}{14} \]To add these fractions, find a common denominator, which is the least common multiple (LCM) of 10 and 14. LCM(10, 14) = 70.
\[ \text{Work in 2 days} = \frac{7}{70} + \frac{5}{70} = \frac{12}{70} = \frac{6}{35} \]Let \( N \) be the number of 2-day cycles. We want to find \( N \) such that \( N \times \frac{6}{35} \) is close to, but less than, 1 (total work).
We can estimate:
\[ N \times \frac{6}{35} < 1 \]\[ N < \frac{35}{6} \]\[ N < 5.83 \]So, we consider 5 cycles. In 5 cycles (which is \( 5 \times 2 = 10 \) days), the work done will be:
\[ \text{Work done in 10 days} = 5 \times \frac{6}{35} = \frac{30}{35} = \frac{6}{7} \]Remaining work = Total work - Work done in 10 days
\[ \text{Remaining work} = 1 - \frac{6}{7} = \frac{1}{7} \]After 10 days (5 cycles), it's the start of a new cycle. Since B started the work, B will work on the 11th day.
B's work per day is \( \frac{1}{10} \).
Time taken by B to complete the remaining \( \frac{1}{7} \) work is:
\[ \text{Time} = \frac{\text{Remaining work}}{\text{B's work per day}} = \frac{1/7}{1/10} = \frac{1}{7} \times 10 = \frac{10}{7} \text{ days} \]Since \( \frac{10}{7} > 1 \), B cannot finish the remaining work in one day. B will work for 1 full day (the 11th day).
Work done by B on the 11th day = \( \frac{1}{10} \).
Remaining work after B works on day 11:
\[ \text{Remaining work} = \frac{1}{7} - \frac{1}{10} = \frac{10 - 7}{70} = \frac{3}{70} \]Now it's A's turn (the 12th day). A's work per day is \( \frac{1}{14} \).
Time taken by A to complete the remaining \( \frac{3}{70} \) work is:
\[ \text{Time} = \frac{\text{Remaining work}}{\text{A's work per day}} = \frac{3/70}{1/14} = \frac{3}{70} \times 14 = \frac{3 \times 14}{70} = \frac{3 \times 1}{5} = \frac{3}{5} \text{ days} \]Total time = 10 days (from 5 cycles) + 1 day (B's work on day 11) + \( \frac{3}{5} \) day (A's work on day 12)
\[ \text{Total time} = 10 + 1 + \frac{3}{5} = 11 \frac{3}{5} \text{ days} \]C) \( 11 \frac{3}{5} \)
The question asks about the first introduction of municipal governance in British India. This requires knowledge of the historical timeline of administrative developments under British rule.
Correct Option: A) 1687 is the correct answer because the Madras Municipal Corporation, the first of its kind in British India, was established in 1687 by a Royal Charter.
The problem requires us to evaluate a given mathematical expression after interchanging certain operators. We need to replace '+' with '−', '−' with '+', '×' with '÷', and '÷' with '×'. After performing these substitutions, we will evaluate the new expression following the order of operations (BODMAS/PEMDAS).
B) \(801\) is the correct answer, as calculated by interchanging the operators and then applying the order of operations.
According to the 73rd Constitutional Amendment Act, 1992, which body functions as the foundation of the Panchayati Raj system, ensuring broad-based community participation?
Gram Sabha
Zila Parishad
Nyaya Panchayat
Gram Panchayat
The question asks to identify the body that serves as the foundation of the Panchayati Raj system, ensuring broad community participation, as per the 73rd Constitutional Amendment Act, 1992. Understanding the structure and purpose of different tiers of Panchayati Raj is crucial here.
Correct Option: A) Gram Sabha. The Gram Sabha is explicitly defined by the 73rd Amendment as the foundation of the Panchayati Raj system. It comprises all adult voters of a village and is designed to ensure direct participation and accountability at the grassroots level.
The color of a flame is determined by the completeness of combustion. We need to understand the conditions that lead to a blue flame in a gas stove.
D) Complete combustion of fuel — A blue flame indicates that the fuel (e.g., LPG or natural gas) is burning completely due to an adequate supply of oxygen. This is the desired and most efficient burning condition for a gas stove.
The problem asks us to find a relationship between two sets of three letters based on their positions in the English alphabet. We need to identify the pattern in the given pairs (JKL to KMO, and PQR to QSU) and then apply the same pattern to XYZ to find the related set of letters.
Correct Option: D) YAC
To calculate the simple interest, we use the standard formula for simple interest: \[\text{Simple Interest} = \frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100}\]. We are given the principal amount, the annual interest rate, and the time period.
Identify the given values:
Apply the simple interest formula:
The formula for simple interest (SI) is:
\[ \text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100} \]Substitute the values into the formula:
\[ \text{SI} = \frac{2,20,000 \times 4 \times 2}{100} \]Calculate the simple interest:
\[ \text{SI} = \frac{17,60,000}{100} \] \[ \text{SI} = 17,600 \]Therefore, Manjit will pay Rs. 17,600 as simple interest after 2 years.
Correct Option: A) 17,600
The problem involves finding the angles of a parallelogram using the property that adjacent angles in a parallelogram are supplementary (sum up to 180°). We will set up an equation based on the given relationship between an angle and its adjacent angle and then solve for the angles.
Let the two adjacent angles of the parallelogram be \(x\) and \(y\). According to the property of a parallelogram, adjacent angles are supplementary.
\[ x + y = 180^\circ \]The problem states that one angle (let's say \(x\)) is 27° less than four-fifth of its adjacent angle (\(y\)). We can write this as an equation:
\[ x = \frac{4}{5}y - 27^\circ \]Now we have a system of two linear equations:
Substitute the expression for \(x\) from equation (b) into equation (a):
\[ \left(\frac{4}{5}y - 27^\circ\right) + y = 180^\circ \]Combine the terms with \(y\):
\[ \frac{4}{5}y + y = 180^\circ + 27^\circ \]\[ \frac{4y + 5y}{5} = 207^\circ \]\[ \frac{9y}{5} = 207^\circ \]Solve for \(y\):
\[ 9y = 207^\circ \times 5 \]\[ 9y = 1035^\circ \]\[ y = \frac{1035^\circ}{9} \]\[ y = 115^\circ \]Now that we have \(y\), substitute its value back into \(x + y = 180^\circ\) to find \(x\):
\[ x + 115^\circ = 180^\circ \]\[ x = 180^\circ - 115^\circ \]\[ x = 65^\circ \]The angles of the parallelogram are \(65^\circ\) and \(115^\circ\). The smallest angle is \(65^\circ\).
D) The smallest angle of the parallelogram is \(65^\circ\).
The problem asks for the value of an algebraic expression \( x^{3} + y^{3} + z^{3} - 3xyz \) given the sum of three numbers \( x+y+z \) and the sum of their squares \( x^2+y^2+z^2 \). We will use a standard algebraic identity that relates these expressions.
We are given the following information:
We need to find the value of \( x^{3} + y^{3} + z^{3} - 3xyz \).
Recall the algebraic identity:
\[ x^{3} + y^{3} + z^{3} - 3xyz = (x+y+z)(x^2+y^2+z^2 - xy - yz - zx) \]To use this identity, we need to find the value of \( xy + yz + zx \). We can find this using another identity:
\[ (x+y+z)^2 = x^2+y^2+z^2 + 2(xy+yz+zx) \]Substitute the given values into the identity from step 4:
\[ (12)^2 = 56 + 2(xy+yz+zx) \]\[ 144 = 56 + 2(xy+yz+zx) \]Solve for \( 2(xy+yz+zx) \):
\[ 2(xy+yz+zx) = 144 - 56 \]\[ 2(xy+yz+zx) = 88 \]Solve for \( xy+yz+zx \):
\[ xy+yz+zx = \frac{88}{2} \]\[ xy+yz+zx = 44 \]Now, substitute the known values into the main identity from step 3:
\[ x^{3} + y^{3} + z^{3} - 3xyz = (x+y+z)(x^2+y^2+z^2 - (xy+yz+zx)) \]\[ x^{3} + y^{3} + z^{3} - 3xyz = (12)(56 - 44) \]Perform the subtraction inside the parenthesis:
\[ x^{3} + y^{3} + z^{3} - 3xyz = (12)(12) \]Perform the multiplication:
\[ x^{3} + y^{3} + z^{3} - 3xyz = 144 \]B) The calculated value is 144, which matches option B.