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Concept use:

For a set with n elements, the number of possible subsets is 2n.

To find the total number of relations from A into B

We need to find all possible subsets of the Cartesian product A × B.

Calculation:

 let us find A × B:

A × B = {(7,6), (7,8), (5,6), (5,8)}

Since a relation is any subset of A × B

The Cartesian product A × B has 4 elements.

For a set with n elements, the number of possible subsets is 2ⁿ.

This is because for each element, we have 2 choices: include it or not.

∴ The total number of relations from A into B is: 2⁴ = 16

Hence, The Correct answer is 16 relations.

Calculation:

Step 1: Firstly, multiply both the given equations by some suitable non-zero constants to make the coefficients of any one of the variables (either x or y) numerically equal.

Step 2: After that, add or subtract one equation from the other in such a way that one variable gets eliminated. Now, if you get an equation in one variable, go to

Step 3: Solve the equation in one variable (x or y) to get its value.

Step 4: Substitute this value in any of the given equations to get the value of another variable

Calculation:

x + y = 8 ---(1) and 2x + y = 10 ---(2)

Substract 1^st from 2^nd 

2x - x + y - y = 10 - 8 = 2

x = 2 

Hence, The Correct Answer is option 1.

Concept use:

The limit of a function f(x) as x approaches p is a number L with the following property: given any target distance from L, there is a distance from p within which the values of f(x) remain within the target distance.

Calculation:

\(\lim _{x \rightarrow 0}\left(\frac{x^{2}+2}{x+5}\right)\) = \(\lim _{x \rightarrow 0}\left(\frac{0^{2}+2}{0+5}\right)\) = 2/5 

Hence, The Limit of the Function is 2/5.

Concept -

The distance between two points (x1, y1) and (x2, y2), you can use the distance formula:

 \(d = √{(x_2 - x_1)^2 + (y_2 - y_1)^2 } \)

Explanation -

Given the points P(2, 3) and Q(4, 2), you can substitute these values into the formula:

\( d = √{(3 - 0)^2 + (5 - 1)^2} \)

\(d = √{3^2 + (4)^2} = √ {9+16} = √25 = 5\)

Therefore, the distance between the points P(0, 1) and Q(3, 5) is 5 units.

Calculation:

\(\frac{2 x+3}{5}=\frac{x+7}{3}\)

⇒ 6x + 9 = 5x + 35

⇒ x = 35 - 9 = 26

Hence, The Correct Option is 1

The correct answer is Only I.

Key Points

• Coulomb's Law agrees with Newton's Third Law.
• Newton's Third Law states that for every action, there is an equal and opposite reaction. Similarly, Coulomb's Law states that the force exerted by one charge on another is equal in magnitude and opposite in direction to the force exerted by the second charge on the first.
• The SI unit of permittivity of free space, also known as the electric constant (ε₀), is actually C²/N·m² and not c²/N·m. This unit can be derived from Coulomb's Law, where the force between two point charges is inversely proportional to the permittivity of free space.

 Additional Information

• Permittivity of Free Space
• The permittivity of free space (ε₀) is a fundamental physical constant that describes how electric fields interact with the vacuum. It is crucial in the equations of electrostatics and electromagnetism.
• The correct SI unit of ε₀ is F/m (farads per meter), but in the context of Coulomb's Law, it is often represented as C²/N·m².

The correct answer is option 3.

Key Points

• Potential energy is a type of energy associated with the position or configuration of an object in a conservative force field.
• The SI unit of potential energy is the Joule (J), which is equivalent to kg·m²/s².
• The dimension of potential energy can be derived from its formula: Potential Energy (U) = mgh, where:
• m = mass (kg)
• g = acceleration due to gravity (m/s²)
• h = height (m)
• By analyzing the formula U = mgh, the dimensional formula is:
• m (mass) = M
• g (acceleration due to gravity) = LT⁻²
• h (height) = L
• So, the dimensional formula of potential energy is:
• U = M × (LT⁻²) × L = ML²T⁻²
• Therefore, the dimension of potential energy is correctly represented by ML²T⁻².

 Additional Information

• Option 1 (ML⁻²T⁻²)
• This dimensional formula does not align with the physical quantities involved in potential energy, as it incorrectly represents the dimensions of mass, length, and time.
• Option 2 (MLT⁻²)
• This represents the dimension of force, not potential energy, as force is defined by the product of mass and acceleration (F = ma).
• Option 4 (ML²T⁻¹)
• This dimensional formula is incorrect for potential energy. It does not correctly account for the dimensions of time in the context of energy.

The Correct answer is Gold-leaf electroscope.

Key Points

• The Gold-leaf electroscope is an instrument used for detecting and measuring electric charge.
• It consists of a vertical metal rod with two thin gold leaves attached at the bottom.
• When a charged object comes into contact with the metal rod, the charge gets transferred to the gold leaves.
• Due to like charges repelling, the gold leaves diverge, indicating the presence of electric charge.
• This apparatus is highly sensitive and can detect even small amounts of charge.
• It is widely used in physics experiments and educational demonstrations to show the presence of static electricity.

Additional Information

• Silver-leaf electroscope
• No common apparatus named Silver-leaf electroscope exists in scientific literature or common usage.
• Platinum-leaf electroscope
• Similar to the silver-leaf electroscope, no common apparatus named Platinum-leaf electroscope is known.
• Silver-rod electroscope
• This term is not typically used to describe an electroscope. The most commonly used type is the gold-leaf electroscope due to its sensitivity.

The correct answer is accelerate.

Key Points

• The law of inertia, also known as Newton's first law of motion, states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force.
• When an object is moving down an inclined plane, it experiences a component of the gravitational force acting along the plane.
• This component of gravitational force causes the object to accelerate down the slope.
• The acceleration of the object on the inclined plane is given by the equation a = g sin(θ), where g is the acceleration due to gravity and θ is the angle of the incline.
• The absence of other forces (like friction) would mean the object continuously accelerates as it moves down the plane.

 Additional Information

• Option 1: "is an intermediate situation"
• This option is too vague and does not accurately describe the behavior of objects on an inclined plane according to the law of inertia.
• Option 3: "retard"
• "Retard" means to slow down. However, objects on an inclined plane do not decelerate unless there is a significant opposing force like friction.
• Option 4: "has constant velocity"
• Constant velocity would imply no net force and no acceleration, which contradicts the effect of the gravitational component along the inclined plane.

The correct answer is option 3.

Key Points

• Kepler's laws describe the motion of planets around the Sun.
• There are three laws:
• Law of Orbits: States that planets move in elliptical orbits with the Sun at one focus.
• Law of Areas: States that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
• Law of Periods: States that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.
• The Law of Mass is not one of Kepler’s laws. It does not describe the motion of planets but is related to the mass of objects.

Additional Information

• Law of Orbits
• Each planet's orbit around the Sun is an ellipse with the Sun at one focus.
• Law of Areas
• A line segment between a planet and the Sun sweeps out equal areas in equal times.
• Law of Periods
• The square of a planet's orbital period is proportional to the cube of the average distance from the Sun.