To find the median of a given set of data, we first need to arrange the data in ascending order. Once arranged, the median is the middle value if the number of data points is odd. If the number of data points is even, the median is the average of the two middle values.
Therefore, the median of the given data is 47.
Correct Option: B) 47
When four numbers are in proportion, it means that the ratio of the first two numbers is equal to the ratio of the last two numbers. We will set up this proportion and solve for the unknown variable \(y\).
Identify the given numbers: 12, 60, 13, and \(y\).
Understand the definition of proportion: If four numbers \(a, b, c, d\) are in proportion, then \[\frac{a}{b} = \frac{c}{d}\].
Apply the definition to the given numbers:
\[ \frac{12}{60} = \frac{13}{y} \]Simplify the fraction on the left side:
\[ \frac{12}{60} = \frac{1}{5} \]Now the equation becomes:
\[ \frac{1}{5} = \frac{13}{y} \]To solve for \(y\), cross-multiply:
\[ 1 \times y = 5 \times 13 \]\[ y = 65 \]Correct Option: B) 65 — The value of \(y\) that satisfies the proportion is 65.
We are given three numbers in a specific ratio and their Least Common Multiple (LCM). We need to find their Highest Common Factor (HCF). We can represent the numbers using a common factor, which will be their HCF. Then, we can express their LCM in terms of this HCF and the given ratio components to solve for the HCF.
Let the three numbers be \(3x\), \(2x\), and \(13x\), where \(x\) is their HCF. This is because if numbers are in a ratio \(a:b:c\), and their HCF is \(x\), then the numbers can be written as \(ax, bx, cx\).
The LCM of three numbers \(ax, bx, cx\) can be found by taking the LCM of the ratio components and multiplying it by their HCF \(x\). So, \(\text{LCM}(3x, 2x, 13x) = x \times \text{LCM}(3, 2, 13)\).
Calculate the LCM of 3, 2, and 13. Since 2, 3, and 13 are all prime numbers, their LCM is their product.
\[ \text{LCM}(3, 2, 13) = 3 \times 2 \times 13 = 6 \times 13 = 78 \]Now, substitute this back into the LCM expression for the three numbers:
\[ \text{LCM}(3x, 2x, 13x) = x \times 78 = 78x \]We are given that the LCM of the three numbers is 4758. So, we can set up the equation:
\[ 78x = 4758 \]Solve for \(x\):
\[ x = \frac{4758}{78} \]Perform the division:
\[ x = 61 \]Since \(x\) represents the HCF of the three numbers, their HCF is 61.
Correct Option: C) 61
To solve this problem, we first need to calculate the total distance of the journey using the initial speed and time. Once the total distance is known, we can determine the new speed required to cover the same distance in the reduced time. Finally, we calculate the increase in speed by finding the difference between the new speed and the original speed.
Calculate the total distance of the journey:
Given:
The formula for distance is \( \text{Distance} = \text{Speed} \times \text{Time} \).
\[ \text{Distance} = S_1 \times T_1 = 66 \text{ km/hr} \times 20 \text{ hours} = 1320 \text{ km} \]Calculate the new speed required:
We need to cover the same distance (1320 km) in a new time (\(T_2\)) = 15 hours.
Let the new speed be \(S_2\).
\[ S_2 = \frac{\text{Distance}}{T_2} = \frac{1320 \text{ km}}{15 \text{ hours}} \] \[ S_2 = 88 \text{ km/hr} \]Calculate the increase in speed:
The increase in speed is the difference between the new speed and the original speed.
\[ \text{Increase in speed} = S_2 - S_1 = 88 \text{ km/hr} - 66 \text{ km/hr} \] \[ \text{Increase in speed} = 22 \text{ km/hr} \]Correct Option: D) 22
When objects of the same metal are melted and recast into new objects, the total volume of the metal remains constant. Therefore, the sum of the volumes of the initial cube and cuboid will be equal to the total volume of all the new cubes formed. We need to calculate the volume of each initial object, sum them up, and then divide this total volume by the volume of one new cube to find the number of new cubes.
Calculate the volume of the initial cube:
Side of the cube, \(a_1 = 5 \text{ cm}\)
Volume of the cube, \(V_1 = a_1^3\)
\[V_1 = (5 \text{ cm})^3 = 125 \text{ cm}^3\]Calculate the volume of the initial cuboid:
Measurements of the cuboid: length \(l = 35 \text{ cm}\), breadth \(b = 15 \text{ cm}\), height \(h = 15 \text{ cm}\)
Volume of the cuboid, \(V_2 = l \times b \times h\)
\[V_2 = 35 \text{ cm} \times 15 \text{ cm} \times 15 \text{ cm}\] \[V_2 = 35 \times 225 \text{ cm}^3 = 7875 \text{ cm}^3\]Calculate the total volume of the metal:
Total volume, \(V_{\text{total}} = V_1 + V_2\)
\[V_{\text{total}} = 125 \text{ cm}^3 + 7875 \text{ cm}^3 = 8000 \text{ cm}^3\]Calculate the volume of one new cube:
Side of the new cube, \(a_2 = 10 \text{ cm}\)
Volume of one new cube, \(V_{\text{new cube}} = a_2^3\)
\[V_{\text{new cube}} = (10 \text{ cm})^3 = 1000 \text{ cm}^3\]Calculate the number of new cubes formed:
Number of cubes, \(N = \frac{V_{\text{total}}}{V_{\text{new cube}}}\)
\[N = \frac{8000 \text{ cm}^3}{1000 \text{ cm}^3} = 8\]Thus, 8 cubes of side 10 cm each can be formed.
D) 8
This question tests basic knowledge of common acronyms used in electronics and electrical engineering. The task is to identify the full form of "PCB" in the context of electric circuits.
Correct Option: A) Printed Circuit Board
A Printed Circuit Board (PCB) is a board used in electronics to mechanically support and electrically connect electronic components using conductive pathways, tracks, or signal traces etched from copper sheets laminated onto a non-conductive substrate. It is the foundation for most modern electronic devices.
The problem requires us to evaluate a given mathematical expression after interchanging specific operators. We need to replace '+' with '−', '−' with '+', '×' with '÷', and '÷' with '×'. After performing these substitutions, we will solve the new expression following the order of operations (BODMAS/PEMDAS).
Correct Option: D) 814
The problem requires us to identify the highest and lowest numbers from a given set of three-digit numbers. Then, we need to extract specific digits from these identified numbers and perform an addition operation as specified.
\[ 8 + 1 = 9 \]
The resultant value is 9.
Correct Option: B) 9
The question asks to identify the specific department at IISc that led a 2025 research project on synthetic image generation for MR applications. This requires specific knowledge about recent research projects and their associated departments at IISc.
C) Department of Design and Manufacturing — The Department of Design and Manufacturing (DDM) at IISc has been at the forefront of interdisciplinary research, including projects that integrate advanced computational techniques like synthetic image generation with medical applications such as Magnetic Resonance (MR) imaging. Their work often spans from fundamental design principles to the development of practical applications in healthcare technology.