This problem involves calculating the combined work rate of two pipes, one filling and one emptying, and then determining the time required to fill a specific fraction of the tank. We will find the individual rates, then the net rate, and finally the time to fill three-fourths of the tank.
The first pipe fills the tank in 8 hours. So, in 1 hour, it fills \(\frac{1}{8}\) of the tank.
\[ \text{Filling rate} = \frac{1}{8} \text{ tank/hour} \]The second pipe empties the tank in 16 hours. So, in 1 hour, it empties \(\frac{1}{16}\) of the tank.
\[ \text{Emptying rate} = \frac{1}{16} \text{ tank/hour} \]Since one pipe fills and the other empties, their effects are opposite. The net rate is the filling rate minus the emptying rate.
\[ \text{Net rate} = \text{Filling rate} - \text{Emptying rate} \\ \text{Net rate} = \frac{1}{8} - \frac{1}{16} \\ \text{Net rate} = \frac{2}{16} - \frac{1}{16} \\ \text{Net rate} = \frac{1}{16} \text{ tank/hour} \]This means that when both pipes are open, \(\frac{1}{16}\) of the tank is filled every hour.
We want to find the time \(T\) to fill \(\frac{3}{4}\) of the tank. If the net rate is \(\frac{1}{16}\) tank per hour, then:
\[ \text{Time} = \frac{\text{Fraction of tank to be filled}}{\text{Net rate}} \\ T = \frac{\frac{3}{4}}{\frac{1}{16}} \\ T = \frac{3}{4} \times 16 \\ T = 3 \times 4 \\ T = 12 \text{ hours} \]Correct Option: D) 12
If 22% of a number is equal to \( \frac{4}{9} \) of another number, what is the ratio of the first number to the second number?
The problem states a relationship between two numbers: 22% of the first number is equal to \( \frac{4}{9} \) of the second number. We need to find the ratio of the first number to the second number. We will represent the numbers with variables, set up an equation based on the given information, and then solve for the ratio.
Let the first number be \( A \) and the second number be \( B \).
Translate "22% of a number" into a mathematical expression. 22% can be written as \( \frac{22}{100} \). So, 22% of \( A \) is \( \frac{22}{100} A \).
Translate "\( \frac{4}{9} \) of another number" into a mathematical expression. \( \frac{4}{9} \) of \( B \) is \( \frac{4}{9} B \).
Set up the equation based on the problem statement that these two quantities are equal:
\[ \frac{22}{100} A = \frac{4}{9} B \]We need to find the ratio of the first number to the second number, which is \( \frac{A}{B} \). To do this, rearrange the equation to isolate \( \frac{A}{B} \):
\[ A = \frac{4}{9} B \times \frac{100}{22} \]\[ \frac{A}{B} = \frac{4}{9} \times \frac{100}{22} \]Simplify the expression:
\[ \frac{A}{B} = \frac{4 \times 100}{9 \times 22} \]\[ \frac{A}{B} = \frac{400}{198} \]Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
\[ \frac{A}{B} = \frac{400 \div 2}{198 \div 2} \]\[ \frac{A}{B} = \frac{200}{99} \]Thus, the ratio of the first number to the second number is \( 200 : 99 \).
D) 200 : 99
The question asks to identify the caves that contain rock-cut paintings dating back to the Stone Age era. This requires knowledge of prominent cave sites in India and their historical periods.
Correct Option: B) Bhimbetka caves
The Bhimbetka Rock Shelters, located in Madhya Pradesh, India, are a UNESCO World Heritage Site. They are renowned for their extensive collection of prehistoric cave paintings, which date back to the Paleolithic and Mesolithic periods (Stone Age). These paintings depict scenes of daily life, animals, hunting, and rituals, providing valuable insights into early human existence.
This is a syllogism problem. We need to analyze the given statements using Venn diagrams or logical deduction to determine which conclusions logically follow.
Correct Option: B) Only conclusion (II) follows.
If a2 + b2 = 179 and a × b = 14, then find \(\frac{a-b}{a+b}\), where a > b.
\(\frac{411}{445}\)
\(\sqrt{\frac{171}{217}}\)
\(\sqrt{\frac{151}{207}}\)
\(\frac{240}{281}\)
The problem asks us to find the value of the expression \(\frac{a-b}{a+b}\) given the values of \(a^2 + b^2\) and \(a \times b\). We can use algebraic identities involving \((a+b)^2\) and \((a-b)^2\) to find the values of \(a+b\) and \(a-b\) first.
We are given the following equations:
Recall the algebraic identity for \((a+b)^2\):
\[ (a+b)^2 = a^2 + b^2 + 2ab \]Substitute the given values into the identity for \((a+b)^2\):
\[ (a+b)^2 = 179 + 2(14) \]\[ (a+b)^2 = 179 + 28 \]\[ (a+b)^2 = 207 \]Now, take the square root to find \(a+b\):
\[ a+b = \sqrt{207} \]Recall the algebraic identity for \((a-b)^2\):
\[ (a-b)^2 = a^2 + b^2 - 2ab \]Substitute the given values into the identity for \((a-b)^2\):
\[ (a-b)^2 = 179 - 2(14) \]\[ (a-b)^2 = 179 - 28 \]\[ (a-b)^2 = 151 \]Now, take the square root to find \(a-b\). Since it is given that \(a > b\), \(a-b\) must be positive:
\[ a-b = \sqrt{151} \]Finally, substitute the values of \(a-b\) and \(a+b\) into the expression \(\frac{a-b}{a+b}\):
\[ \frac{a-b}{a+b} = \frac{\sqrt{151}}{\sqrt{207}} \]\[ \frac{a-b}{a+b} = \sqrt{\frac{151}{207}} \]C) The calculated value is \(\sqrt{\frac{151}{207}}\), which matches option C.
The question asks for the standard method to create a new text file within a specific folder in Windows. This involves navigating to the desired location and using the context menu options provided by the operating system.
This sequence of actions is the standard and most direct way to create a new text file in a specific folder in Windows.
Correct Option: A) Open the folder and right-click -> New -> Text Document
The question asks to identify the Windows 11 command-line utility used for managing computer storage components like disks, partitions, volumes, and virtual hard disks. We need to recall the function of each given command.
msconfig (System Configuration) is used to configure how Windows starts up, manage boot options, services, and startup programs. It is not for disk management.regedit (Registry Editor) is used to view and edit the Windows Registry, which stores system-wide settings and configurations. It is not for disk management.gpedit.msc (Local Group Policy Editor) is used to configure various operating system settings and security policies for a local computer. It is not for disk management.diskpart is a command-line utility that provides comprehensive control over disk, partition, and volume management. It can create, delete, format, extend, shrink, and assign drive letters to partitions and volumes, and also manage virtual hard disks. This perfectly matches the question's requirement.Correct Option: D) diskpart
The question asks for the most suitable material for making filaments in electric bulbs. The primary function of a filament in an incandescent bulb is to glow brightly when heated by the passage of electric current. This requires the material to withstand very high temperatures without melting or oxidizing, and to have high resistivity to generate sufficient heat.
B) Tungsten is the most suitable material for making filaments in electric bulbs due to its extremely high melting point (\(3422^\circ \text{C}\)), high resistivity, and low rate of evaporation at high temperatures, allowing it to glow brightly without melting.
Which Indian naval ship visited Port Louis, Mauritius, in March 2025 to participate in the 57th National Day celebrations?
The question asks to identify the Indian naval ship that visited Port Louis, Mauritius, in March 2025 for the 57th National Day celebrations. This requires knowledge of recent naval deployments and diplomatic engagements.
Based on official reports, the Indian Navy's latest stealth guided-missile destroyer, INS Imphal, visited Port Louis, Mauritius, in March 2025 to participate in the 57th National Day celebrations of Mauritius. This visit underscored the strong bilateral relations and maritime cooperation between India and Mauritius.
B) INS Imphal — INS Imphal is a Visakhapatnam-class stealth guided-missmissile destroyer. It was indeed deployed to Port Louis, Mauritius, in March 2025 to participate in the 57th National Day celebrations, highlighting India's commitment to its maritime partners in the Indian Ocean Region.
This is a logical reasoning problem involving seating arrangement or floor arrangement. We need to deduce the exact floor on which each person lives based on the given clues. We will start by listing the floors and then place individuals based on the most restrictive clues, gradually filling in the positions.
List the floors: The building has 6 floors, numbered 1 (lowermost) to 6 (topmost).
Floors: 6, 5, 4, 3, 2, 1
Clue 1: H lives on an odd-numbered floor and immediately above K.
Possible (H, K) pairs:
So, (H, K) can be (5, 4) or (3, 2).
Clue 2: Only three people live between K and G, where K lives below G.
Let's test the possibilities from Clue 1:
Case 1: H is on 5, K is on 4.
If K is on floor 4, and there are three people between K and G, and K is below G, then G must be on floor \(4 + 3 + 1 = 8\). This is not possible as there are only 6 floors.
Therefore, H cannot be on floor 5 and K cannot be on floor 4.
Case 2: H is on 3, K is on 2.
If K is on floor 2, and there are three people between K and G, and K is below G, then G must be on floor \(2 + 3 + 1 = 6\).
This is a valid placement.
So, we have:
Clue 3: J lives on one of the floors below F and on one of the floors above L.
This implies the order F > J > L. The remaining floors are 5, 4, 1.
For F > J > L to be true with these floors, F must be on 5, J on 4, and L on 1.
Final Arrangement:
Answer the question: Who lives on floor numbered 4?
From the final arrangement, J lives on floor 4.
Correct Option: D) J
Based on the step-by-step deduction, J lives on floor number 4.