We are given the LCM of two numbers and their ratio. We need to find the sum of these numbers. We can represent the numbers using a common multiple and then use the property that for any two numbers \(A\) and \(B\), the product of the numbers is equal to the product of their LCM and HCF (Highest Common Factor).
Let the two numbers be \(A\) and \(B\). Given that their ratio is \(2:3\), we can express them as:
\[ A = 2x \]\[ B = 3x \]where \(x\) is the HCF of the two numbers.
The LCM of \(2x\) and \(3x\) can be found as:
\[ \text{LCM}(2x, 3x) = x \times \text{LCM}(2, 3) \]\[ \text{LCM}(2x, 3x) = x \times 6 \]\[ \text{LCM}(2x, 3x) = 6x \]We are given that the LCM of the two numbers is 84. So, we can set up the equation:
\[ 6x = 84 \]Solve for \(x\):
\[ x = \frac{84}{6} \]\[ x = 14 \]Now that we have the value of \(x\), we can find the two numbers:
\[ A = 2x = 2 \times 14 = 28 \]\[ B = 3x = 3 \times 14 = 42 \]Finally, find the sum of the numbers:
\[ \text{Sum} = A + B = 28 + 42 \]\[ \text{Sum} = 70 \]C) 70
To find the diameter of a sphere given its surface area, we first need to use the formula for the surface area of a sphere to calculate its radius. Once the radius is known, the diameter can be easily found as it is twice the radius.
Recall the formula for the surface area of a sphere:
The surface area (\(A\)) of a sphere with radius \(r\) is given by:
\[ A = 4\pi r^2 \]Substitute the given surface area into the formula:
We are given that the surface area is \(256\pi \text{ m}^2\). So,
\[ 256\pi = 4\pi r^2 \]Solve for the radius (\(r\)):
Divide both sides by \(4\pi\):
\[ \frac{256\pi}{4\pi} = r^2 \]\[ 64 = r^2 \]Calculate the radius:
Take the square root of both sides:
\[ r = \sqrt{64} \]\[ r = 8 \text{ m} \]Calculate the diameter (\(D\)):
The diameter is twice the radius:
\[ D = 2r \]\[ D = 2 \times 8 \]\[ D = 16 \text{ m} \]Correct Option: D) 16 m
To solve this problem, we need to apply two rules sequentially to each letter in the word "ROMANCE":
After transforming all letters, we will identify which letter is repeated in the new group.
Let's process each letter of the word ROMANCE:
The new group of letters formed is U, P, O, B, O, E, F.
Now, let's check for repeated letters in this new group:
The letter 'O' is repeated twice in the new group.
Correct Option: B) O
This problem involves decoding a given expression by substituting symbols with their corresponding mathematical operators and then evaluating the expression using the order of operations (BODMAS/PEMDAS).
The original equation is: \((32 \text{ P } 4) \text{ R } (21 \text{ P } 5) \text{ S } (9 \text{ Q } 3) \text{ P } 6 = ?\)
Substituting the operators, we get:
\[ (32 \times 4) - (21 \times 5) + (9 \div 3) \times 6 = ? \]First, solve the operations inside the brackets:
The equation now becomes:
\[ 128 - 105 + 3 \times 6 = ? \]Next, perform the multiplication:
The equation now becomes:
\[ 128 - 105 + 18 = ? \]Finally, perform subtraction and addition:
Correct Option: B) 41
This problem involves analyzing logical statements and conclusions using Venn diagrams or direct logical reasoning. We need to determine which conclusions necessarily follow from the given statements.
Analyze Statement 1: "No bike is a car."
This means that the set of Bikes and the set of Cars are mutually exclusive. There is no overlap between them.
Diagrammatically: Two separate circles for Bikes and Cars.
Analyze Statement 2: "All cars are planes."
This means that the set of Cars is entirely contained within the set of Planes. Every car is also a plane.
Diagrammatically: The circle for Cars is completely inside the circle for Planes.
Combine the statements and evaluate Conclusion (I): "No plane is a bike."
From statement 2, we know that all cars are planes. From statement 1, no bike is a car. However, this does not mean that no plane is a bike. It is possible that some planes (those that are not cars) could be bikes. For example, if we have a set of Planes, and Cars are a subset of Planes, and Bikes are separate from Cars, Bikes could still overlap with the larger set of Planes (but not with the Cars part of Planes).
Consider a Venn diagram: Draw a large circle for Planes. Inside it, draw a smaller circle for Cars. Draw another circle for Bikes. The Bikes circle cannot overlap with the Cars circle. However, the Bikes circle can overlap with the Planes circle (outside the Cars circle). Therefore, this conclusion does not necessarily follow.
Evaluate Conclusion (II): "All planes are cars."
Statement 2 says "All cars are planes." This is the converse of conclusion (II). The statement "All A are B" does not imply "All B are A." For example, all dogs are animals, but not all animals are dogs. Similarly, all cars are planes, but it is not necessary that all planes are cars. There can be planes that are not cars.
Therefore, this conclusion does not necessarily follow.
Since neither conclusion (I) nor conclusion (II) logically follows from the given statements, the correct option is B.
Correct Option: B) Neither conclusion (I) nor (II) follows
The question asks to identify the Brand Ambassador of the Ministry of Ports, Shipping and Waterways of the Government of India as of 2024. This is a factual question related to current affairs and government appointments.
As of 2024, the Ministry of Ports, Shipping and Waterways appointed Indian shooter Manu Bhaker as its brand ambassador to promote its initiatives, particularly those related to youth engagement and sports.
D) Manu Bhaker was appointed as the Brand Ambassador of the Ministry of Ports, Shipping and Waterways of the Government of India. This appointment was made to promote various initiatives of the ministry, especially focusing on engaging the youth.
This problem involves analyzing statements and conclusions based on the principles of syllogism, often visualized using Venn diagrams. We need to determine which conclusions logically follow from the given statements, treating the statements as absolutely true.
C) Only conclusion (I) follows.
The question asks about a specific recent decision made by the Union Cabinet regarding the Ayushman Bharat PM-JAY scheme on 11 September 2024. To answer this, one needs to be aware of recent policy updates related to this flagship health scheme.
Correct Option: A) To extend benefits to all citizens aged 70 and above. This option accurately reflects the Union Cabinet's decision on 11 September 2024, which expanded the Ayushman Bharat PM-JAY scheme to cover all individuals above 70 years of age, irrespective of their income or socio-economic background, ensuring comprehensive health coverage for the elderly.
This is a Statement and Course of Action question. The goal is to identify which course of action logically follows from the given statement to address the problem described. A course of action should be practical, feasible, and directly address the core issue without creating new, bigger problems or being too extreme.
Correct Option: D) Neither I nor II follows
The problem asks us to calculate the magnitude of the work done by a man lifting a load. Work done against gravity is calculated as the product of the force required to lift the object (which is equal to its weight) and the vertical distance through which it is lifted. The weight of an object is given by the product of its mass and the acceleration due to gravity.
The force required to lift the load against gravity is its weight, which is given by the formula:
\[ F = m \times g \]Work done (\(W\)) when lifting an object against gravity is given by the formula:
\[ W = F \times h \]Therefore, the magnitude of the work done by the man is 25 joules.
B) 25 joules — This is the correct calculation for the work done using the given mass, height, and acceleration due to gravity.