JEE Main Exam January 2024

The formula for calculating the nth term of an arithmetic progression (AP) is given by: Tn = a + (n - 1)d, where 'a' is the first term, 'd' is the common difference, and 'n' is the number of terms.

The formula for calculating the sum of an arithmetic progression (AP) is given by: Sn = (n/2) [2a + (n-1)d], where 'a' is the first term, 'd' is the common difference, and 'n' is the number of terms.

A combination is a way of selecting items from a set where the order doesn't matter. The formula for calculating combinations is given by: nCr = n! / (r! * (n-r)!), where 'n' is the total number of items and 'r' is the number of items being selected.

The formula for calculating the distance between a point and a line in 3D space is given by: d = |(a - b) · (d1 x d2)| / |d1 x d2|, where 'a' is the point, 'b' is a point on the line, 'd1' and 'd2' are direction vectors of the line, and '·' and 'x' represent dot and cross products respectively.

The orthocenter of a triangle is the point of intersection of its altitudes. An altitude is a line segment drawn from a vertex perpendicular to the opposite side.

The differential equation (2x + 3y – 2) dx + (4x + 6y – 7) dy = 0 can be solved using the method of integrating factors. The integrating factor is a function that multiplies the differential equation to make it exact, allowing for straightforward integration. The solution is then found by integrating both sides. Using the initial condition y(0) = 3, the value of the constant of integration can be determined. This leads to the explicit solution of the given differential equation.

The principle of conservation of momentum states that the total momentum of a closed system remains constant in the absence of external forces. In simpler terms, the total momentum of a system before a collision or interaction is equal to the total momentum after the interaction. This principle is a fundamental law of physics and is applicable to various scenarios, including collisions, explosions, and rocket propulsion.

The intensity of an electromagnetic wave is defined as the average power transmitted per unit area perpendicular to the direction of propagation. It is directly proportional to the square of the electric field amplitude and the speed of light. The intensity can be calculated using the formula I = (1/2) ε0E0^2c, where I is intensity, ε0 is permittivity of free space, E0 is the amplitude of the electric field, and c is the speed of light.

The radius of a stationary orbit in the Bohr model of an atom is directly proportional to the square of the principal quantum number (n) and inversely proportional to the atomic number (Z). This relationship is given by the formula r = n^2 * (h^2 / (4π^2 * me * k * e^2 * Z)), where r is the radius, n is the principal quantum number, h is Planck's constant, me is the mass of the electron, k is Coulomb's constant, and e is the charge of an electron.

The average kinetic energy of a molecule is related to the temperature of the system through the equation Kavg = (3/2)kBT, where Kavg is the average kinetic energy, kB is the Boltzmann constant, and T is the absolute temperature.

The change in magnetic flux through a loop is the cause of electromotive force (emf), which is the potential difference induced in a circuit. Faraday's law of electromagnetic induction states that the magnitude of the induced emf is directly proportional to the rate of change of the magnetic flux through the coil. The emf is given by the equation emf = -N(Δ�/Δt), where N is the number of turns in the coil, Δ� is the change in magnetic flux, and Δt is the time interval over which the change occurs.

The potential difference between two points in an electric field is the work done per unit charge in moving a charge from one point to another. The potential difference is given by the equation V = W/q, where V is potential difference, W is work done, and q is charge.

In a meter bridge setup, the unknown resistance is balanced against a known resistance. The ratio of the resistances is equal to the ratio of the corresponding lengths on the meter bridge wire, as per the principle of Wheatstone bridge. This is given by the equation R1/R2 = L1/L2, where R1 is the known resistance, R2 is the unknown resistance, L1 is the length on the meter bridge wire corresponding to R1, and L2 is the length corresponding to R2.

The tension in a wire is a force that is transmitted along the length of the wire, caused by an external pull. In a circular ring, the tension is essentially the force exerted by the wire on the ring, preventing it from expanding or contracting. In the context of electric charge, the tension in a ring is influenced by the electric forces between the charges distributed along the ring and any charges present at the center of it.

Nuclear fission is a reaction where a heavy nucleus splits into two or more lighter nuclei, releasing a tremendous amount of energy. The process is initiated by the absorption of a neutron by the heavy nucleus. This nuclear reaction is the basis of nuclear power plants and atomic bombs.

Moment of inertia is a measure of an object's resistance to changes in its rotation. It depends on the mass distribution and the axis of rotation. For a system of point masses, the moment of inertia is calculated as the sum of the product of each mass with the square of its distance from the axis of rotation. Moment of inertia is analogous to mass in linear motion. A larger moment of inertia indicates a greater resistance to changes in rotation.

Simple harmonic motion (SHM) is a periodic motion where the restoring force is directly proportional to the displacement from the equilibrium position and is always directed towards the equilibrium position. The displacement, velocity, and acceleration of the object executing SHM can be described using sinusoidal functions.

For a uniform magnetic field, the magnetic field lines are parallel and equidistant. A straight wire carrying a current experiences a force due to the magnetic field, which is perpendicular to both the current direction and the magnetic field direction. The force is given by F = BIl, where F is the force, B is the magnetic field strength, I is the current, and l is the length of the wire.

The work done by an applied force is stored as potential energy in a system. This potential energy has the potential to convert back into kinetic energy, causing motion in the system. The potential energy is given by the equation U = mgh, where U is potential energy, m is mass, g is acceleration due to gravity, and h is height.

The magnitude of the electric field is related to the potential difference and the distance between two points by the equation E = ΔV/d, where E is the electric field, ΔV is the potential difference, and d is the distance.

The concept of refractive index is related to the bending of light as it passes from one medium to another. It is defined as the ratio of the speed of light in vacuum to the speed of light in the medium. A higher refractive index indicates a slower speed of light in the medium and a greater bending of light. The formula is n = c/v, where n is the refractive index, c is the speed of light in vacuum, and v is the speed of light in the medium.

The angle of minimum deviation for a prism is defined as the angle of incidence where the angle of deviation is the smallest. At this angle, the light ray travels symmetrically through the prism, entering and exiting at equal angles. The formula for the angle of minimum deviation is given by δm = 2i - A, where δm is the angle of minimum deviation, i is the angle of incidence, and A is the angle of the prism.

The specific rotation of a liquid is a measure of its ability to rotate the plane of polarized light. It is defined as the angle of rotation per unit length per unit concentration. The specific rotation is a characteristic property of a chiral molecule, which refers to a molecule that has a non-superimposable mirror image. The specific rotation is often used in chemistry to identify and quantify chiral compounds, such as sugars, amino acids, and pharmaceuticals.

Viscosity is a measure of a fluid's resistance to flow. High-viscosity fluids flow slowly, while low-viscosity fluids flow easily. Viscosity is influenced by temperature, the size and shape of the molecules, and the strength of the intermolecular forces.

Surface tension is a property of liquids that allows them to resist an external force. Surface tension arises from the cohesive forces between the liquid molecules. A higher surface tension indicates stronger cohesive forces and a greater resistance to external forces. It is measured as the force per unit length acting along the surface of the liquid.

The concept of binding energy in nuclear physics is related to the stability of an atom's nucleus. It represents the energy required to completely separate the nucleons (protons and neutrons) in a nucleus. A higher binding energy per nucleon indicates a more stable nucleus. The formula is BE = (ZmH + NmN - MA)c^2, where BE is binding energy, Z is the number of protons, mH is the mass of a hydrogen atom, N is the number of neutrons, mN is the mass of a neutron, MA is atomic mass, and c is the speed of light.

A spherometer is a device specifically designed for measuring the radius of curvature of spherical surfaces. It operates based on the principle of finding the height difference between the center of curvature and a point on the surface. The spherometer can measure the thickness of thin plates, primarily through the use of a micrometer screw that precisely measures the distance between the tip of a screw and the surface of the object.

The total energy of a system is the sum of its kinetic energy and potential energy. Kinetic energy is the energy an object possesses due to its motion, while potential energy is the energy an object possesses due to its position or configuration. The total energy of the system remains constant in the absence of external forces.

Linear momentum is a measure of the mass in motion of an object. It is calculated by multiplying the object's mass by its velocity. The formula is p = mv, where p is linear momentum, m is mass, and v is velocity.

Moment of force, often referred to as torque, is a measure of the ability of a force to cause rotation around an axis. It is calculated by multiplying the force by the perpendicular distance from the axis of rotation to the line of action of the force. The formula is  = r × F, where  is torque, r is the distance from the axis of rotation, and F is the force.

The total charge stored in a capacitor is directly proportional to the potential difference across its plates. This relationship is given by the equation Q = CV, where Q is the charge, C is the capacitance, and V is the potential difference. The capacitance of a capacitor is a measure of its ability to store charge at a given potential difference.

Electric potential is a scalar quantity that describes the potential energy per unit charge at a given point in an electric field. It is related to the work done in moving a charge from one point to another against the electric forces. The electric potential difference between two points is the work done per unit charge in moving a charge from one point to another.

A reverse-biased diode is a semiconductor device where the positive terminal of the voltage source is connected to the n-type semiconductor and the negative terminal is connected to the p-type semiconductor. In this configuration, the majority carriers are pulled away from the junction, creating a depletion region. The diode's resistance becomes very high, and it essentially blocks the flow of current.

The equation x^2 + x + 1 = 0 represents a quadratic equation with complex roots. The roots of this equation are complex numbers, specifically the complex cube roots of unity, denoted by ω and ω^2. These roots have special properties, including that their sum is -1 and their product is 1. Understanding the properties of these complex roots is essential when dealing with certain mathematical and physical situations.