JEE Advanced Mix Test 48

Concept:

The problem involves two balls rolling down ramps of different heights. The first ramp has a height of h, and the second ramp has a height of h. According to the principles of energy conservation and kinematics, the speed of the balls when they reach the bottom will depend on the potential energy they convert into kinetic energy as they roll down the ramp. The potential energy at the top is proportional to the height, and this energy is completely converted into kinetic energy (assuming no friction). The speed of the balls at the bottom of the ramp will be determined by the square root of the height from which they fall.

Calculation:

v_ii = √(2g(2h)) = √(4gh) = 2√(gh)

For the ball on ramp (i), the potential energy at height h is converted into kinetic energy at the bottom. The velocity at the bottom is given by:

v_i = √(2gh)

For the ball on ramp (ii), the potential energy at height h is converted into kinetic energy. The velocity at the bottom is given by:

v_ii = √(2g(h)) = √(2gh)

From this, we can see that Both balls will have the same speed after hitting the ground.

The correct answer is Option 3.

Concept:

In this problem, a horizontal force is applied at the top of a square block. We need to determine the minimum coefficient of friction required to topple the block before it starts to slide. The block will topple when the torque generated by the applied force causes the block to rotate about the pivot point (the bottom edge). If the applied force exceeds the frictional force, the block will start to slide. The balance of forces and torques governs the solution to this problem.

Calculation:

The moment of inertia of the square block about point B (the pivot point) is given by:

I = (1/12) m a² + m (a/2)²

On simplifying, we get:

I = (7/12) m a²

Next, the torque about point B due to the applied force is:

Torque = F × a

The torque about point B due to the Gravitational force is:

mg (a/2)

The force required to topple the block is the one that creates a torque sufficient to overcome the static friction:

F = 0.5 m g

For the block to topple, the frictional force must balance this force, and the minimum coefficient of friction (μ) is given by:

F = μ m g

On comparing the two forces, we get:

μ = 0.5

Conclusion:

The minimum coefficient of friction required to topple the block before it starts sliding is 0.5.

Concept:

In this scenario, you are inside a car that is initially moving at 20 km/h and suddenly decelerates because the driver applies the brakes. According to Newton's first law of motion (inertia), your body will continue to move forward when the car slows down. The car stops, but your body will tend to move forward until you are stopped by the seat or other restraints. This phenomenon happens due to inertia and the rapid deceleration of the car.

Calculation:

The deceleration is given as the rate at which the car stops, and it is very abrupt. As the car decelerates, you are still moving at the initial velocity of 20 km/h.

Your body will not instantly stop with the car and will continue its motion forward until you are restrained by the seat or other forces acting on you (e.g., seatbelt). Hence, you will be pushed forward within the seat.

Correct Answer Explanation:

Option 1: "You will barely notice the change" is incorrect. Due to inertia, you will feel the deceleration of the car, and it will push you forward.

Option 2: "You will be pushed forward" is the correct answer. As the car decelerates abruptly, your body will continue to move forward due to inertia until you are stopped by the seat or restraints.

Option 3: "You will jump vertically" is incorrect. You will not jump vertically as there is no vertical force applied to make you jump.

Option 4: "You will be pushed backward in your seat" is incorrect. Since the car is decelerating, your body will move forward relative to the car, not backward.

Conclusion:

The correct answer is Option 2: You will be pushed forward.

Concept:

The power radiated by a body is given by the Stefan-Boltzmann law:

P = σAT⁴

Where:

• P is the power radiated
• A is the surface area of the sphere
• T is the temperature of the sphere
• \sigma is the Stefan-Boltzmann constant

The surface area of a sphere is given by:

A = 4πr²

The energy radiated is proportional to the surface area and the fourth power of the temperature. Thus, we can write the ratio of the energies radiated by the two spheres as:

∴ The energy radiated by the second sphere is equal to the energy radiated by the first sphere, which is E.

CONCEPT:

Vapour Pressure and Colligative Properties

• Vapour pressure of a solution is the pressure exerted by the vapour when the solution is in a closed container at a given temperature.
• The vapour pressure of a solution is affected by the addition of solutes. According to Raoult's Law, the addition of a non-volatile solute lowers the vapour pressure of a solution by reducing the number of solvent molecules available for evaporation.
• Colligative properties such as vapour pressure lowering depend on the number of solute particles in the solution, not the nature of the solute. These properties include freezing point depression, boiling point elevation, and osmotic pressure.
• Adding different species to a solution can affect its vapour pressure in different ways:
  • Adding water changes the concentration of KI, which in turn changes the vapour pressure.
  • Adding HgI₂ reacts with KI, reducing the number of free ions, thereby affecting the vapour pressure.
  • Adding I₂ does not change the number of ions in the solution and therefore does not affect the vapour pressure.
  • Adding Na₂SO₄ increases the number of ions in the solution, thereby reducing the vapour pressure.

EXPLANATION:

• When water is added to 1 M KI solution, the concentration of KI decreases, which causes a reduction in the vapour pressure. This is because the total number of solvent molecules increases, resulting in fewer solute molecules in the solution, reducing its ability to vaporize. Hence, the vapour pressure changes.
• When HgI₂ is added, it reacts with KI to form a complex, reducing the number of ions in the solution. This reduces the number of particles present for evaporation, which decreases the vapour pressure.

• When I₂ is added to the solution, it does not affect the number of ions in the solution, so there is no change in the vapour pressure.

• When Na₂SO₄ is added, it dissociates into ions, thereby increasing the number of particles in the solution. According to Raoult's Law, this leads to a decrease in vapour pressure due to the greater number of particles in the solution.

Therefore, the correct answer is water and I₂ will cause a change in the vapour pressure of the KI solution.

Concept:

Trends in Properties of Noble Gases

• Solubility in Water: Noble gases are only slightly soluble in water, and their solubility increases down the group due to increased polarizability. The correct trend is:

  He < Ne < Ar < Kr < Xe

• Boiling Points: Boiling points increase down the group due to stronger van der Waals forces as atomic size and polarizability increase. The correct trend is:

  He < Ne < Ar < Kr < Xe

• Chemical Reactivity: Noble gases are generally inert, but heavier noble gases like xenon can form compounds due to lower ionization energy. The correct trend is:

  He < Ne < Ar < Kr < Xe

  (Statement C is incorrect as it reverses the order.)

• Enthalpy of Vaporization: The enthalpy of vaporization increases with atomic size due to stronger intermolecular forces. The correct trend is:

  He < Ne < Ar < Kr < Xe

  (Statement D is incorrect as it reverses the order.)

Explanation:

• Statement A: "Solubility in water He < Ne < Ar < Kr < Xe."
  • Correct, as solubility increases with increasing atomic size.
• Statement B: "Boiling points He < Ne < Ar < Kr < Xe."
  • Correct, as boiling points increase with increasing atomic size due to stronger van der Waals forces.
• Statement C: "Chemical reactivity He > Ne > Ar > Kr > Xe."
  • Incorrect, as the trend is reversed. The correct order is He < Ne < Ar < Kr < Xe.
• Statement D: "Enthalpy of vaporization He > Ne > Ar > Kr > Xe."
  • Incorrect, as the correct trend is He < Ne < Ar < Kr < Xe.

Analysis of Answer Choices:

• A and B only: Correct, as both trends are accurate.
• B and C only: Incorrect, as statement C is incorrect.
• B and D only: Incorrect, as statement D is incorrect.
• A and D only: Incorrect, as statement D is incorrect.

Therefore, the correct answer is A and B only.

Concept:

Gibbs Free Energy (ΔG) and Heat of Reaction

ΔG = ΔH - TΔS

• Gibbs free energy (ΔG) is a thermodynamic quantity used to predict the spontaneity of a reaction at constant temperature and pressure.

Therefore, the heat of the reaction is approx -184.20 kJ.