This problem involves calculating the work rates of pipes. We'll determine the individual filling rates of the two inlet pipes, and the combined filling/emptying rate when all three pipes are open. From these, we can deduce the emptying rate of the third pipe and then calculate the time it takes to empty the cistern alone.
Calculate the filling rate of the first pipe:
Pipe 1 fills the cistern in 98 min. So, its rate is:
Calculate the filling rate of the second pipe:
Pipe 2 fills the cistern in 28 min. So, its rate is:
Calculate the combined rate when all three pipes are open:
Let the emptying pipe be Pipe 3. Let its emptying rate be \(R_3\) cistern/min. When all three pipes are open, the cistern fills in 24.5 min. So, the net filling rate is:
Formulate the equation for the net rate:
The net rate is the sum of the filling rates minus the emptying rate:
Solve for the emptying rate \(R_3\):
First, find a common denominator for the fractions on the left side. The least common multiple of 98 and 28 is 196.
Calculate the time taken by Pipe 3 to empty the cistern:
Since Pipe 3 empties \(1/196\) of the cistern per minute, it will take 196 minutes to empty the entire cistern.
C) 196 min is the time taken by the pipe at the bottom to empty the completely filled cistern.
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 apply the BODMAS/PEMDAS rule to solve the new expression.
Correct Option: A) 529 is the correct result after interchanging the operators and evaluating the expression according to the order of operations.
The question asks for a specific statistic from the Global Debt Report 2025 concerning the maturity of OECD sovereign debt. This requires knowledge of the report's findings, which are often highlighted in economic news and summaries.
Correct Option: B) 45%
To find the value of \( \alpha + \beta - \alpha\beta \), we first need to identify the sum of the roots (\( \alpha + \beta \)) and the product of the roots (\( \alpha\beta \)) for the given quadratic equation. For a standard quadratic equation \( ax^2 + bx + c = 0 \), the sum of the roots is given by \( -b/a \) and the product of the roots is given by \( c/a \).
The given quadratic equation is \( y^2 - 9y + 8 = 0 \).
Comparing this with the standard form \( ay^2 + by + c = 0 \), we have:
The sum of the roots is given by the formula \( -b/a \).
\[ \alpha + \beta = -\frac{(-9)}{1} = 9 \]
The product of the roots is given by the formula \( c/a \).
\[ \alpha\beta = \frac{8}{1} = 8 \]
We need to find the value of \( \alpha + \beta - \alpha\beta \).
Substitute the calculated values:
\[ \alpha + \beta - \alpha\beta = 9 - 8 \]
\[ \alpha + \beta - \alpha\beta = 1 \]
Correct Option: B) 1
If the ratio of two numbers is 18 : 5 and the product of their LCM and their HCF is 360, then the sum of the reciprocals of the LCM and the HCF is:
\( \frac{91}{180} \)
\( \frac{91}{185} \)
\( \frac{91}{204} \)
\( \frac{91}{193} \)
We are given the ratio of two numbers and the product of their LCM and HCF. We need to find the sum of the reciprocals of their LCM and HCF. We will use the fundamental property that for any two positive integers, the product of the numbers is equal to the product of their LCM and HCF. We will represent the numbers using the given ratio and their HCF, then find the LCM and HCF, and finally calculate the required sum.
Let the two numbers be \(a\) and \(b\). The ratio of the two numbers is given as \(18:5\). So, we can write the numbers as \(a = 18x\) and \(b = 5x\), where \(x\) is their HCF (Highest Common Factor).
We know that for any two numbers \(a\) and \(b\), the product of the numbers is equal to the product of their LCM and HCF:
\[ a \times b = \text{LCM} \times \text{HCF} \]We are given that the product of their LCM and HCF is 360. So,
\[ \text{LCM} \times \text{HCF} = 360 \]Substitute the expressions for \(a\) and \(b\) into the product equation:
\[ (18x) \times (5x) = 360 \]\[ 90x^2 = 360 \]Solve for \(x\):
\[ x^2 = \frac{360}{90} \]\[ x^2 = 4 \]\[ x = \sqrt{4} \]\[ x = 2 \]Now we have the HCF, which is \(x = 2\). Let's denote HCF as \(H\). So, \(H = 2\).
Using the product of LCM and HCF, we can find the LCM:
\[ \text{LCM} \times H = 360 \]\[ \text{LCM} \times 2 = 360 \]\[ \text{LCM} = \frac{360}{2} \]\[ \text{LCM} = 180 \]We need to find the sum of the reciprocals of the LCM and the HCF:
\[ \frac{1}{\text{LCM}} + \frac{1}{\text{HCF}} \]Substitute the values of LCM and HCF:
\[ \frac{1}{180} + \frac{1}{2} \]To add these fractions, find a common denominator, which is 180:
\[ \frac{1}{180} + \frac{1 \times 90}{2 \times 90} \]\[ \frac{1}{180} + \frac{90}{180} \]\[ \frac{1 + 90}{180} \]\[ \frac{91}{180} \]A) The calculated sum of the reciprocals of the LCM and HCF is \( \frac{91}{180} \), which matches option A.
This problem involves calculating the original sale amount before successive percentage increases. We can work backward from the present sale amount by dividing by the respective growth factors for each year. Alternatively, we can set up an equation where the original sale amount, multiplied by the growth factors, equals the present sale amount.
A) 1,50,000 — The calculation shows that Gopal's sale two years ago was ₹1,50,000.
The problem asks us to identify the highest and lowest numbers from a given set of three-digit numbers, then extract specific digits from them, and finally perform an addition operation. We need to carefully follow the steps of identifying the numbers, extracting the digits, and then adding them.
The resultant value is 5.
B) 5 is the correct answer because, as calculated, the sum of the second digit of the highest number (3 from 733) and the first digit of the lowest number (2 from 265) is \(3 + 2 = 5\).
The question asks to identify the correct statement about Neptunium. We need to evaluate each option based on the known properties and history of Neptunium.
Correct Option: A) It is a naturally occurring element found in pitchblende. Although Neptunium is primarily produced synthetically, trace amounts of Neptunium-237 and Neptunium-239 are indeed found in uranium ores like pitchblende. These are formed by neutron capture by uranium followed by beta decay.
To solve this problem, we need to first determine the volume of the large cube and the small cubes. Then, we can find out how many small cubes can be cut from the large cube. Finally, we calculate the total surface area of all these small cubes.
Calculate the volume of the large cube:
The side of the large cube is \(L = 10 \text{ cm}\).
The volume of the large cube is \(V_L = L^3\).
\[ V_L = (10 \text{ cm})^3 = 1000 \text{ cm}^3 \]Calculate the volume of one small cube:
The side of a small cube is \(s = 2 \text{ cm}\).
The volume of one small cube is \(V_s = s^3\).
\[ V_s = (2 \text{ cm})^3 = 8 \text{ cm}^3 \]Calculate the number of small cubes:
The number of small cubes (\(N\)) that can be cut from the large cube is the ratio of the volume of the large cube to the volume of one small cube.
\[ N = \frac{V_L}{V_s} = \frac{1000 \text{ cm}^3}{8 \text{ cm}^3} = 125 \]Calculate the surface area of one small cube:
The surface area of a cube with side \(s\) is \(A_s = 6s^2\).
\[ A_s = 6 \times (2 \text{ cm})^2 = 6 \times 4 \text{ cm}^2 = 24 \text{ cm}^2 \]Calculate the total surface area of all the small cubes:
The total surface area (\(A_{\text{total}}\)) is the number of small cubes multiplied by the surface area of one small cube.
\[ A_{\text{total}} = N \times A_s = 125 \times 24 \text{ cm}^2 \] \[ A_{\text{total}} = 3000 \text{ cm}^2 \]A) The total surface area of all the small cubes is \(3000 \text{ cm}^2\).
To simplify the given expression, we must follow the order of operations, often remembered by the acronym PEMDAS/BODMAS. This stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
Let's apply these steps to the given expression: \(118 - 5 \times (10 + 20) + 55\).
Thus, the simplified value of the expression is 23.
Correct Option: B) 23 is the correct simplified value of the expression.