JEE Main 2024 - FIITJEE Full Test-I

The potential energy of a body in a conservative force field is given by U = x – y, where x and y are the position coordinates. The acceleration of the body can be calculated using the negative gradient of the potential energy.

In an elastic collision, both momentum and kinetic energy are conserved. The ball's initial vertical velocity is canceled by the horizontal velocity of the cart, resulting in a rebound speed determined by the Pythagorean theorem.

When the drum rolls forward, the point of contact between the drum and the ground is momentarily stationary. The distance moved by the man is equal to the distance the drum rolls plus the length of the board.

The surface charge density (σ) of a sheet is directly proportional to the sine of the angle (θ) between the electric field and the surface normal. This means as the angle increases, the surface charge density also increases.

The ratio of the circumference of the first Bohr orbit to the de Broglie wavelength of an electron with the same velocity is 2:1. This arises from the quantization of angular momentum in the Bohr model.

When light travels from a denser medium (glass, refractive index 1.5) to a rarer medium (water, refractive index 4/3), total internal reflection occurs if the angle of incidence exceeds the critical angle. The critical angle is determined by the ratio of refractive indices.

For a glass prism immersed in water, total internal reflection occurs when the angle of incidence (θ) at the BC face is greater than or equal to the critical angle. This condition is based on Snell's law and the refractive indices of glass and water.

The angular momentum of an electron in a hydrogen atom is quantized, meaning it can only take on discrete values. This quantization leads to the existence of specific energy levels for the electron.

The de Broglie wavelength of a particle is inversely proportional to its momentum. This means lighter particles or particles with higher velocities have shorter wavelengths.

The first Bohr orbit of a hydrogen atom represents the ground state, the lowest possible energy level for the electron. The electron can only exist in specific energy levels.

The refractive index of a medium is a measure of how much light bends when it passes from one medium to another. It represents the ratio of the speed of light in vacuum to the speed of light in the medium.

The force required to maintain a constant velocity on the wire is determined by the interaction of the magnetic field and the current flowing through the wire. The magnitude of this force is given by F = I l B, where I is the current, l is the length of the wire, and B is the magnetic field strength.

The impendence of a circuit is a measure of its resistance to the flow of alternating current. It combines the effects of resistance, capacitance, and inductance.

The center of mass of a system rises to a maximum height, which is directly proportional to the square of the initial velocity and inversely proportional to the acceleration due to gravity. This is consistent with the principles of projectile motion.

The position of a point on a rod that maintains equal rates of ice melting and water evaporation is determined by the balance of heat flow. The thermal conductivity of the rod, along with the latent heat of fusion and vaporization, determine the temperature gradient and the position of the point.

The charge density of a dielectric disc in a specific region can be used to calculate the electric field within the disc. The electric field is dependent on the geometry of the charge distribution.

The coefficient of restitution (e) is a measure of how much kinetic energy is conserved in a collision. It is defined as the ratio of the relative velocity of separation to the relative velocity of approach.

In this scenario, the charged particle enters a uniform magnetic field perpendicular to its motion. This causes the particle to move in a circular path. The inelastic collision with the wall means kinetic energy is lost.

Resonance occurs when the frequency of the driving force (tuning fork) matches a natural frequency of the system (air column in the tube). This causes the amplitude of the vibrations to increase significantly.

For resonance in a closed tube (one end closed, one end open), the length of the air column must be an odd multiple of a quarter wavelength of the sound wave. This is because the closed end must be a displacement node (no movement) and the open end a displacement antinode (maximum movement).

The charge flowing to earth is equal to the difference in the initial charges of the plates. Due to charge redistribution, the final charge on each plate will be the same.

When the switch is closed, charge will flow from the plates with higher initial charge to the plates with lower charge until the potential difference between them becomes equal. The charge flowing to earth is that which is released from the plates with higher initial charge.

The force acting on a particle is given by F = (12 – 2t) Newton, where t is time. This force causes the particle to move along the x-axis starting from rest at the origin. The problem asks to find the time it takes for the particle to return to the origin.

An alpha particle, which is a helium nucleus, is accelerated by a potential difference of 10^4 V. This acceleration causes the alpha particle to enter a region with a uniform magnetic field, where it experiences a change in its direction of motion due to the magnetic force. The problem requires finding the angle of this change in direction.

A glass rod is inserted into a glass capillary, creating a narrow space. This arrangement is then brought into contact with water. The water rises inside the capillary due to surface tension and the cohesive forces between water molecules and the glass. The problem aims to determine the height the water will rise in the capillary.

The given AC circuit has a switch that can be placed in two positions. In position 1, the source emf leads the current by an angle of pi/6. When the switch is moved to position 2, the source emf leads the current by an unknown angle. The task is to determine this angle, denoted as 'x'.

The electric field intensity at a point P is determined by the distribution of charges around it. In this case, the problem provides the charge density (delta) and distances, and you need to calculate the electric field intensity at point P.

The problem presents an alpha particle entering a magnetic field and experiencing a change in direction. The problem asks find the angle of deflection.

The force on a particle due to a magnetic field is given by F = q * v * B, where q is the charge, v is the velocity, and B is the magnetic field. This force is perpendicular to both the velocity and the magnetic field, causing the particle to move in a circular path.

The surface tension of water causes it to rise in a narrow tube, such as a capillary, due to the cohesive forces between water molecules and the adhesive forces between water molecules and the tube's walls. This phenomenon is called capillary action.

The electronic configuration of a metal ion describes the arrangement of electrons in its orbitals. For Mn, Fe, and Co ions in complexes, the electrons occupy the t2g and eg orbitals based on crystal field theory, which explains how metal ions interact with ligands.

The complementary strand of a single-stranded DNA molecule has a sequence that pairs with the original strand according to base pairing rules: Adenine (A) pairs with Thymine (T), and Guanine (G) pairs with Cytosine (C).

pKa values indicate the acidity of a compound. A lower pKa value signifies a stronger acid. In this case, the pKa values are ranked based on the stability of the conjugate bases formed after deprotonation.

Radioactive decay follows first-order kinetics, meaning its rate is proportional to the amount of radioactive material present. The rate constant is related to the half-life, which is the time it takes for half of the radioactive material to decay.

Osmosis is the movement of water molecules across a semipermeable membrane from an area of high water concentration to an area of low water concentration. To prevent osmosis, a minimum pressure, called osmotic pressure, needs to be applied on the side with higher water concentration. The osmotic pressure can be calculated using the van't Hoff equation: π = iMRT, where π is the osmotic pressure, i is the van't Hoff factor, M is the molarity of the solution, R is the ideal gas constant, and T is the temperature.

The correct order of pKa values for the given compounds (I - IV) depends on the stability of the conjugate bases formed after deprotonation. This stability is influenced by factors such as resonance and the electronegativity of the atoms involved. In general, more stable conjugate bases correspond to stronger acids and lower pKa values.

The electronic configuration of a transition metal ion in a complex is influenced by the crystal field theory, which explains the splitting of d orbitals in the presence of ligands. The energy levels of these orbitals are then filled according to Hund's rule and the Aufbau principle.

Complementary DNA strands are formed by base pairing, where adenine (A) pairs with thymine (T), and guanine (G) pairs with cytosine (C). The complementary strand is arranged antiparallel to the original strand, meaning that the 5' end corresponds to the 3' end of the original strand and vice versa.