JEE worked solutions

Learn how to calculate the angular momentum of a particle about a point using the formula L = m(r × v), focusing on choosing the correct reference point and vector cross product.

Derive and understand the relationship between half-life ($T_{1/2}$) and mean-life ($\tau$) for radioactive decay, a key concept in Modern Physics.

Learn the step-by-step derivation of the moment of inertia of a solid sphere about its diameter using integration and the parallel-axis theorem.

Understand why a spinning skater's angular velocity increases when they pull their arms inward, explained through the principle of conservation of angular momentum.

Master Kirchhoff's Voltage Law (KVL) with clear sign conventions for EMFs and resistors. Understand how traversal direction impacts loop analysis in electrical circuits.

Learn how to calculate the effective EMF and internal resistance when multiple cells are connected in parallel, a key concept in current electricity for JEE.

Understand Lenz's Law and why induced EMF opposes the change in magnetic flux, explained through energy conservation.

Derive the self-inductance of a solenoid step-by-step. Understand the concepts of magnetic flux, flux linkage, and the formula L = μ₀n²Al.

Explore how energy oscillates between electric and magnetic fields in an LC circuit, analogous to Simple Harmonic Motion. Understand the frequency of oscillation.

Understand why the electric field inside a conductor is zero in electrostatic equilibrium, explained with free electron redistribution and Gauss's Law.

Understand how inserting a dielectric slab reduces the electric field between capacitor plates due to polarization and bound charges.

Understand Total Internal Reflection (TIR) and derive the critical angle using Snell's Law. Essential for JEE Physics.

Step-by-step JEE Main solution: Electrostatics — V = U/q; scalar vs energy quantity; superposition.

Learn how to calculate the de Broglie wavelength of an electron accelerated through a potential difference V using the formula λ = h/√(2meV).

Understand why temperature remains constant during phase changes like melting or boiling, explained by latent heat and intermolecular forces.

Learn to calculate equivalent resistance in an unbalanced Wheatstone bridge using Kirchhoff's laws and delta-star transformations.

Understand why transition metal complexes are coloured or colourless, focusing on d-d transitions, crystal field splitting, and visible light absorption.

Understand why ortho and para products are favored over meta in electrophilic aromatic substitution due to resonance stabilization of carbocation intermediates by activating groups.

Learn how electron-withdrawing and donating groups affect carboxylic acid acidity by stabilizing or destabilizing the carboxylate anion.

Step-by-step mechanism of aldol condensation for acetaldehyde, detailing enolate formation, nucleophilic addition, and dehydration to form an α,β-unsaturated aldehyde.

Understand why benzene is more stable than its hypothetical cyclohexatriene structure, exploring aromaticity and delocalization energy.

Understand the key differences between E2 and E1cb elimination mechanisms, including the conditions that favor each, base strength, leaving group ability, and carbanion stability.

Learn to determine the coordination number and oxidation state of cobalt in coordination complexes like [Co(en)2Cl2]+, a key concept in Inorganic Chemistry for JEE.

Learn to calculate cell potential ($E$) at non-standard conditions using the Nernst equation. Understand the impact of concentration on electrochemical cell performance.

Understand why the acid dissociation constant (Ka) of a weak acid increases with temperature, explained using Le Chatelier's principle and the endothermic nature of ionization.

Step-by-step JEE Main solution: Physical Chemistry (Solutions) — i = 3 for K2SO4; ΔTb = i·Kb·m.

Learn how to calculate activation energy ($E_a$) using the Arrhenius equation with rate constants ($k_1, k_2$) at two different temperatures ($T_1, T_2$).

Understand why entropy increases when ionic compounds dissolve in water, exploring lattice energy, hydration, and microstates.

Step-by-step JEE Main solution: Organic Chemistry — ionic vs free-radical mechanism; stability of radical intermediates.

Step-by-step JEE Main solution: Physical Chemistry (Electrochemistry) — spontaneous vs non-spontaneous; cathode/anode conventions.

Step-by-step JEE Main solution: Physical Chemistry (Equilibrium) — buffer action; pH = pKa + log([A-]/[HA]).

Master evaluating definite integrals involving absolute value functions by strategically splitting the integration interval at points where the expression inside the modulus changes sign.

Step-by-step JEE Main solution: Complex Numbers — ω = e^(2πi/3); 1 + ω + ω² = 0; ω³ = 1; equilateral triangle.

Master L'Hôpital's Rule for evaluating limits of indeterminate forms 0/0 and ∞/∞. Learn when and how to apply this powerful calculus technique with clear examples.

Learn how to find the area enclosed between two parabolas using definite integration. This guide covers finding intersection points and setting up the integral.

Learn how to find the angle between two planes using their normal vectors and direction cosines with this JEE Main 3D Geometry solution.

Master the JEE Advanced concept of finding the shortest distance between skew lines in 3D. Learn the formula and its derivation using vector methods.

Learn the method of variable separation to find the general solution of first-order separable differential equations, including integration and the constant of integration.

Master De Moivre's theorem to efficiently calculate the nth power of complex numbers in polar form, a key JEE Advanced concept.

Master integrating rational functions with repeated roots using partial fraction decomposition. Learn the systematic approach for JEE Main calculus problems.

Learn how to calculate permutations with identical objects and extend this to the multinomial theorem for arrangements with multiple groups of identical items.

Master Bayes' Theorem for conditional probability. Learn how to update beliefs using prior probabilities and likelihoods with this JEE Main level explanation.

Step-by-step JEE Main solution: Coordinate Geometry (Circles) — 2 tangents; length = √(d²−r²) where d = distance from centre.

Understand the relationship between semi-major axis (a), semi-minor axis (b), and focal distance (c) in an ellipse. Learn how eccentricity (e) defines its shape.

Learn to find the characteristic polynomial of a 2x2 matrix and verify the Cayley-Hamilton theorem with this detailed JEE Advanced solution.

Prove that the sum of squares of binomial coefficients, C(n,0)² + ... + C(n,n)², equals C(2n,n) using the Binomial Theorem.

Master finding matrix rank using row echelon form and determining system consistency with the Rouché-Capelli theorem. Solved JEE Main problem.