To solve this problem, we need to determine the cost price (CP) of one lemon and then calculate the selling price (SP) required to achieve a 100% profit. Once the SP of one lemon is known, we can find out how many lemons must be sold for ₹1.
A) 3 — As calculated, selling 3 lemons for ₹1 results in a 100% profit.
This problem involves setting up a system of two linear equations based on the given information about the present ages and future ages of a father and his son. We will define variables for their present ages, translate the word problem into algebraic equations, solve the system of equations to find their present ages, and then calculate the difference.
Define variables:
Let the present age of the father be \(F\) years.
Let the present age of the son be \(S\) years.
Formulate the first equation:
"The sum of the present ages of a father and his son is 18 years more than 4 times the present age of the son."
\[ F + S = 4S + 18 \]Simplify this equation:
\[ F = 3S + 18 \quad \text{(Equation 1)} \]Formulate the second equation (after 5 years):
After 5 years:
"4 times the father's age will be 8 years less than 14 times the son's age."
\[ 4(F+5) = 14(S+5) - 8 \]Expand and simplify this equation:
\[ 4F + 20 = 14S + 70 - 8 \] \[ 4F + 20 = 14S + 62 \] \[ 4F = 14S + 42 \]Divide by 2 to simplify:
\[ 2F = 7S + 21 \quad \text{(Equation 2)} \]Solve the system of equations:
Substitute Equation 1 into Equation 2:
Substitute \(F = 3S + 18\) into \(2F = 7S + 21\):
\[ 2(3S + 18) = 7S + 21 \] \[ 6S + 36 = 7S + 21 \]Rearrange to solve for \(S\):
\[ 36 - 21 = 7S - 6S \] \[ S = 15 \]So, the son's present age is 15 years.
Find the father's present age:
Substitute \(S = 15\) into Equation 1:
\[ F = 3(15) + 18 \] \[ F = 45 + 18 \] \[ F = 63 \]So, the father's present age is 63 years.
Calculate the difference in their present ages:
Difference \( = F - S \)
\[ \text{Difference} = 63 - 15 \] \[ \text{Difference} = 48 \]C) The difference between the present ages of the father and the son is 48 years.
The volume of a solid cylinder is 5852 cm³ and its height is 38 cm. What is the total surface area of the solid cylinder? (Round your answer to the nearest integer)
(Use π = \( \frac{22}{7} \))
To find the total surface area of a solid cylinder, we first need to determine its radius. We are given the volume and height of the cylinder. Using the formula for the volume of a cylinder, we can calculate the radius. Once the radius is known, we can use the formula for the total surface area of a cylinder to find the required value.
The formula for the volume of a cylinder is \(V = \pi r^2 h\).
Substitute the given values into the formula:
\[ 5852 = \frac{22}{7} \times r^2 \times 38 \]Rearrange the equation to solve for \(r^2\):
\[ r^2 = \frac{5852 \times 7}{22 \times 38} \] \[ r^2 = \frac{5852 \times 7}{836} \] \[ r^2 = 7 \times 7 \] \[ r^2 = 49 \]Take the square root to find \(r\):
\[ r = \sqrt{49} \] \[ r = 7 \text{ cm} \]The formula for the total surface area (\(TSA\)) of a solid cylinder is \(TSA = 2\pi r (h + r)\).
Substitute the values of \(r\), \(h\), and \(\pi\) into the formula:
\[ TSA = 2 \times \frac{22}{7} \times 7 \times (38 + 7) \] \[ TSA = 2 \times 22 \times (45) \] \[ TSA = 44 \times 45 \] \[ TSA = 1980 \text{ cm}^2 \]The calculated total surface area is \(1980 \text{ cm}^2\), which is already an integer.
Correct Option: B) 1980 cm²
The question asks for the city in India where the Arctic Circle India Forum was held on 3-4 May 2025. This is a factual question related to international events and their venues in India. Candidates need to be aware of significant international conferences and forums hosted in different Indian cities.
D) New Delhi — The Arctic Circle India Forum was indeed held in New Delhi on 3-4 May 2025. New Delhi frequently hosts major international conferences and forums due to its status as the national capital and a hub for international diplomacy.
The problem presents a number series, and we need to find the missing term. We will analyze the differences between consecutive terms to identify a pattern. Once the pattern is established, we can apply it to find the missing number.
C) 104 is the correct option because it fits the established pattern where each term is obtained by subtracting consecutive odd numbers (3, 5, 7, 9, 11, 13) from the previous term.
To simplify the given algebraic expression, we will use the standard algebraic identities:
We will expand each term in the expression using these identities and then combine like terms.
Correct Option: C) \( 54x^{2} + 25y^{2} \)
The question asks to identify which Urban Local Government Body (ULB) was discontinued after the 74th Constitutional Amendment Act of 1992. This requires knowledge of the types of ULBs existing before and after this significant amendment.
Correct Option: C) Notified Area Committee was largely discontinued or its functions were integrated into the new constitutional framework for urban local bodies after the 74th Amendment Act of 1992. The Act standardized the urban local government structure, making this specific type of body less relevant or obsolete in its original form.
The question asks about the primary function of a Notified Area Committee in India. To answer this, one needs to understand the different types of urban local bodies in India and their specific roles as defined in the Indian political system.
D) Manage developing towns that are due to get municipality status. This option correctly identifies the primary function of a Notified Area Committee. NACs are formed for areas that are undergoing rapid development and are in the process of transitioning into full-fledged municipalities, providing them with a basic framework of urban administration.