Verified JEE & NEET solutions
293 step-by-step solutions across Physics, Chemistry and Maths — every final answer machine-verified, so you're never learning from a confident wrong answer.
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293 shownA projectile is launched from the foot of a fixed incline that rises at 30 degrees to the…
neet mechanics · NEET + JEE
On a horizontal table, block A (2 kg) and block B (3 kg) rest in line, joined by a light string…
neet mechanics · NEET + JEE
A ball of mass 0.5 kg is tied to a light string of length 2 m and whirled in a complete…
neet mechanics · NEET + JEE
A block starts from rest and slides down a smooth incline, descending a vertical height of 2.5…
neet mechanics · NEET + JEE
A uniform solid sphere is released from rest at the top of a rough incline and rolls without…
neet mechanics · NEET + JEE
A particle executes simple harmonic motion of period 12 s about a mean position, starting from…
neet mechanics · NEET + JEE
A satellite of mass 200 kg moves in a circular orbit around the Earth at a height equal to the…
neet mechanics · NEET + JEE
A bullet of mass 0.02 kg moving horizontally at 400 m/s strikes and embeds itself in a…
neet mechanics · NEET + JEE
Car B moves along a straight road at a constant 20 m/s. At the instant B passes a stationary…
neet mechanics · NEET + JEE
A uniform rigid rod of length 2 m is pivoted at one end about a horizontal frictionless axis…
neet mechanics · NEET + JEE
A circular road of radius 20 m is banked at an angle whose tangent is 3/4. The coefficient of…
neet mechanics · NEET + JEE
A body of mass 1 kg, initially at rest on a smooth horizontal surface, is acted upon by a net…
neet mechanics · NEET + JEE
A block of mass 2 kg rests on top of a larger block of mass 8 kg, which itself rests on a…
neet mechanics · NEET + JEE
A light, inextensible string is wound around the rim of a uniform disc of mass 4 kg that is…
neet mechanics · NEET + JEE
A point charge q1 = +2 microcoulomb is fixed at the origin and a point charge q2 = -3…
neet electrostatics current · NEET + JEE
A Wheatstone-type capacitor bridge connects node A to node B. From A two branches start. Branch…
neet electrostatics current · NEET + JEE
Two cells are connected in parallel and drive an external resistor. Cell 1 has EMF 10 V and…
neet electrostatics current · NEET + JEE
A capacitor C1 = 4 microfarad is charged to 100 V and a capacitor C2 = 6 microfarad is charged…
neet electrostatics current · NEET + JEE
An electric dipole has dipole moment p = 2 x 10^-9 C·m. Find the magnitude of the electric…
neet electrostatics current · NEET + JEE
An unbalanced Wheatstone bridge of resistors connects terminals A and B. The arms are: P = 2…
neet electrostatics current · NEET + JEE
A parallel-plate capacitor has capacitance 2 pF in vacuum with plate separation d. A dielectric…
neet electrostatics current · NEET + JEE
An electron enters the region between two horizontal parallel plates moving horizontally with…
neet electrostatics current · NEET + JEE
A 60 V battery is connected to a capacitor network. Capacitor C1 = 3 microfarad is in series…
neet electrostatics current · NEET + JEE
Two identical small balls, each of mass 0.1 g, are suspended from a common point by silk…
neet electrostatics current · NEET + JEE
Four equal point charges, each +1 microcoulomb, are fixed at the four corners of a square of…
neet electrostatics current · NEET + JEE
Two mesh loops share a common middle branch. The left loop contains a 6 V cell and a 1 ohm…
neet electrostatics current · NEET + JEE
Two isolated conducting spheres of radii 0.02 m and 0.04 m are far apart. The smaller sphere…
neet electrostatics current · NEET + JEE
A potentiometer has a uniform wire of length 5 m and total resistance 5 ohm. Its primary…
neet electrostatics current · NEET + JEE
A cyclotron accelerates protons (mass 1.67 x 10^-27 kg, charge 1.6 x 10^-19 C) using a uniform…
neet magnetism emi · NEET + JEE
A charged particle with charge-to-mass ratio q/m = 1.0 x 10^8 C/kg enters a region of uniform…
neet magnetism emi · NEET + JEE
Two long straight parallel wires carry currents 10 A and 15 A in opposite directions and are…
neet magnetism emi · NEET + JEE
A square coil of side 0.10 m has 50 turns and carries a current of 2.0 A. It is placed in a…
neet magnetism emi · NEET + JEE
A conducting rod of length 0.50 m rotates in a horizontal plane about a vertical axis through…
neet magnetism emi · NEET + JEE
A long solenoid has 1000 turns per metre. A short secondary coil of 200 turns and…
neet magnetism emi · NEET + JEE
A series LCR circuit has resistance 30 ohm, inductive reactance 50 ohm and capacitive reactance…
neet magnetism emi · NEET + JEE
A charged particle with charge-to-mass ratio q/m = 5.0 x 10^7 C/kg moves with velocity having…
neet magnetism emi · NEET + JEE
A conducting rod of length 0.40 m slides without friction on two parallel horizontal rails…
neet magnetism emi · NEET + JEE
An alternating current is given by i(t) = 5 sin(100*pi*t) + 3 cos(100*pi*t), where i is in…
neet magnetism emi · NEET + JEE
A circular conducting loop of radius 0.10 m lies in a region of uniform magnetic field…
neet magnetism emi · NEET + JEE
In a mass spectrometer, ions first pass through a velocity selector with crossed fields:…
neet magnetism emi · NEET + JEE
A toroid has a mean radius of 0.20 m and 500 turns carrying a current of 4.0 A. Using mu_0 =…
neet magnetism emi · NEET + JEE
An LC circuit consists of an inductor of 2.0 mH and a capacitor of 8.0 microfarad oscillating…
neet magnetism emi · NEET + JEE
A plano-convex lens is made of glass of refractive index 1.5; its convex face has radius of…
neet optics waves · NEET + JEE
A plano-convex lens of glass (refractive index 1.5) has its convex surface (radius of curvature…
neet optics waves · NEET + JEE
Using the displacement method, a convex lens placed between a fixed object and a screen 100 cm…
neet optics waves · NEET + JEE
A solid glass sphere of refractive index 1.5 and radius 10 cm is placed in air. A point object…
neet optics waves · NEET + JEE
In Young's double-slit experiment with light of wavelength 600 nm, a thin glass sheet of…
neet optics waves · NEET + JEE
A transparent coating of refractive index 1.25 is deposited on a glass plate of higher…
neet optics waves · NEET + JEE
A vibrating tuning fork held over a resonance tube produces its first resonance when the air…
neet optics waves · NEET + JEE
A car moving at 20 m/s sounds a horn of frequency 320 Hz. A stationary observer stands behind…
neet optics waves · NEET + JEE
A prism of refracting angle 60 degrees gives an angle of minimum deviation of 30 degrees for a…
neet optics waves · NEET + JEE
An astronomical telescope has an objective of focal length 100 cm and an eyepiece of focal…
neet optics waves · NEET + JEE
A concave mirror has a focal length of 20 cm. A point object lies on the principal axis 30 cm…
neet optics waves · NEET + JEE
In a two-slit interference experiment the two slits have intensities in the ratio 4:1. At a…
neet optics waves · NEET + JEE
Light of wavelength 600 nm passes through a single slit of width 0.2 mm and forms a diffraction…
neet optics waves · NEET + JEE
Light of a single wavelength falls on two different metal surfaces. For a metal of threshold…
neet modern thermo · NEET + JEE
In a singly-ionized helium atom (He+, atomic number Z = 2) the electron makes a transition from…
neet modern thermo · NEET + JEE
A hypothetical photoelectric experiment uses the same monochromatic light on a metal in two…
neet modern thermo · NEET + JEE
For the n = 2 stationary orbit of a hydrogen atom (Bohr radius a0 = 0.529 angstrom), the de…
neet modern thermo · NEET + JEE
A radioactive sample has an initial activity of 8000 disintegrations per second and a half-life…
neet modern thermo · NEET + JEE
Two deuterium nuclei fuse to form a helium-4 nucleus. Given atomic masses m(2H) = 2.014102 u…
neet modern thermo · NEET + JEE
A Zener diode of breakdown voltage 6 V is used as a voltage regulator. An unregulated 12 V…
neet modern thermo · NEET + JEE
An n-p-n transistor in common-emitter configuration has a current gain (beta) of 100. The base…
neet modern thermo · NEET + JEE
At what temperature (in kelvin) would the rms speed of oxygen molecules (molar mass 32 g/mol)…
neet modern thermo · NEET + JEE
A gas mixture contains 2 moles of helium (monatomic) and 3 moles of oxygen (diatomic, treated…
neet modern thermo · NEET + JEE
A diatomic ideal gas (gamma = 7/5) initially at 300 K is compressed adiabatically until its…
neet modern thermo · NEET + JEE
One mole of an ideal gas is taken around a closed rectangular cycle in the pressure-volume…
neet modern thermo · NEET + JEE
Three moles of a monatomic ideal gas are heated at constant pressure so that the temperature…
neet modern thermo · NEET + JEE
A laser beam of power 10 W is completely absorbed by a black surface. Taking the speed of light…
neet modern thermo · NEET + JEE
A 5.00 g mixture of calcium carbonate (CaCO3, molar mass 100 g/mol) and magnesium carbonate…
phys chem mole · NEET + JEE
A hydrated salt MgSO4.xH2O (anhydrous MgSO4 molar mass 120 g/mol, water 18 g/mol) is heated. A…
phys chem mole · NEET + JEE
A 4.60 g sample of a pure organic compound containing only carbon, hydrogen and oxygen is…
phys chem mole · NEET + JEE
In the Haber process, N2 + 3H2 -> 2NH3. A reaction vessel is charged with 28.0 g of N2 (molar…
phys chem mole · NEET + JEE
Concentrated sulphuric acid is 98.0% H2SO4 by mass (molar mass 98 g/mol) and has density 1.84…
phys chem mole · NEET + JEE
Boron occurs as two isotopes of masses exactly 10 u and 11 u. The observed average atomic mass…
phys chem mole · NEET + JEE
6.5 g of zinc (molar mass 65 g/mol) is added to excess dilute hydrochloric acid: Zn + 2HCl ->…
phys chem mole · NEET + JEE
A 25.0 mL sample of phosphoric acid (H3PO4) solution is exactly neutralized (all three acidic…
phys chem mole · NEET + JEE
A farmer needs to deliver exactly 4.20 kg of nitrogen to a field using ammonium sulphate,…
phys chem mole · NEET + JEE
A sealed vessel contains a mixture of carbon monoxide (CO, molar mass 28) and carbon dioxide…
phys chem mole · NEET + JEE
Washing soda crystals have the formula Na2CO3.xH2O (anhydrous Na2CO3 molar mass 106 g/mol,…
phys chem mole · NEET + JEE
An organic compound contains carbon, hydrogen and oxygen in the mass percentages 40.0% C, 6.67%…
phys chem mole · NEET + JEE
A mixture consists of 4.40 g of CO2 (molar mass 44) and 3.40 g of NH3 (molar mass 17). Using…
phys chem mole · NEET + JEE
Iron pyrite is roasted in air: 4FeS2 + 11O2 -> 2Fe2O3 + 8SO2. All the SO2 produced is then…
phys chem mole · NEET + JEE
2.0 mol of PCl5 is taken in a 4.0 L rigid vessel and allowed to reach equilibrium for the…
phys chem equilibrium · NEET + JEE
At a certain temperature, 1.0 mol of N2O4 is allowed to dissociate as N2O4(g) <=> 2NO2(g). At…
phys chem equilibrium · NEET + JEE
A buffer is prepared containing 0.20 M acetic acid and 0.10 M sodium acetate. The ionization…
phys chem equilibrium · NEET + JEE
The solubility product of silver chromate, Ag2CrO4, is Ksp = 4.0 x 10^-12 at 298 K. Calculate…
phys chem equilibrium · NEET + JEE
The solubility product of AgCl is Ksp = 1.8 x 10^-10 at 298 K. Calculate the molar solubility…
phys chem equilibrium · NEET + JEE
A 0.010 M solution of a monobasic weak acid HA has ionization constant Ka = 1.0 x 10^-5.…
phys chem equilibrium · NEET + JEE
For the equilibrium N2(g) + 3H2(g) <=> 2NH3(g), the partial pressures at equilibrium are found…
phys chem equilibrium · NEET + JEE
For the reaction H2(g) + I2(g) <=> 2HI(g), Kc = 64 at the given temperature. Starting with 1.0…
phys chem equilibrium · NEET + JEE
Calculate the pH of a 0.10 M aqueous solution of sodium acetate (CH3COONa). The ionization…
phys chem equilibrium · NEET + JEE
The molar solubility of Mg(OH)2 in pure water is found to be 1.0 x 10^-4 mol L^-1 at 298 K.…
phys chem equilibrium · NEET + JEE
For the isomerization equilibrium A(g) <=> B(g) the equilibrium constant Kc = 4.0 at a certain…
phys chem equilibrium · NEET + JEE
50 mL of 0.10 M HCl is mixed with 50 mL of 0.20 M NaOH at 298 K. Assuming both are strong…
phys chem equilibrium · NEET + JEE
The solubility product of Mg(OH)2 is Ksp = 1.0 x 10^-11 at 298 K. Calculate the molar…
phys chem equilibrium · NEET + JEE
For the gas-phase dissociation A(g) <=> B(g) + C(g), the equilibrium constant Kp = 3.0 atm.…
phys chem equilibrium · NEET + JEE
For the decomposition of a solid carbonate, the standard enthalpy change is +178.3 kJ per mole…
phys chem thermo electro kinetics · NEET + JEE
A Daniell cell operates with standard cell EMF 1.10 V at 298 K (use the factor 0.0591 V for…
phys chem thermo electro kinetics · NEET + JEE
A first-order reaction is 75% complete in 60 minutes. Determine the time (in minutes) required…
phys chem thermo electro kinetics · NEET + JEE
2 moles of an ideal gas undergo isothermal reversible expansion at 300 K from volume V1 to 3V1.…
phys chem thermo electro kinetics · NEET + JEE
For a reaction, the rate constant increases by a factor of 4 when temperature rises from 300 K…
phys chem thermo electro kinetics · NEET + JEE
A reaction is second order in reactant A with rate constant k = 0.05 L/mol/min and initial…
phys chem thermo electro kinetics · NEET + JEE
For a reaction at 298 K, the standard Gibbs free energy change is -13.6 kJ/mol. Using R = 8.314…
phys chem thermo electro kinetics · NEET + JEE
A zero-order reaction has rate constant k = 0.02 mol/L/s and initial reactant concentration…
phys chem thermo electro kinetics · NEET + JEE
For the ammonia synthesis reaction, the standard enthalpy change is -92.4 kJ and the standard…
phys chem thermo electro kinetics · NEET + JEE
Find the remainder when the power tower 7^(7^7) is divided by 1000. (Here 7^7 = 823543, so you…
number theory · JEE
How many trailing zeros does 2026! have when it is written in base 12? (A trailing zero in base…
number theory · JEE
A positive integer is called 'rough' if it is divisible by the square of some prime…
number theory · JEE
Let D be the set of all positive divisors of 2025. Compute the sum of those divisors d in D…
number theory · JEE
Find the number of ordered pairs of positive integers (a, b) such that lcm(a, b) = 600.
number theory · JEE
Consider repdigit numbers made entirely of the digit 8, i.e. 8, 88, 888, 8888, and so on. Find…
number theory · JEE
Let sigma(n) denote the sum of all positive divisors of n. Count the number of integers n with…
number theory · JEE
Find the number of integers x with 0 ≤ x < 18000 satisfying x^2 ≡ 1 (mod 18000).
number theory · JEE
Compute the value of the sum S = sum_{k=1}^{100} k · gcd(k, 100).
number theory · JEE
Let phi denote Euler's totient function. Find the number of positive integers n for which…
number theory · JEE
Write out 50! in full and delete all of its trailing zeros. The resulting number ends in some…
number theory · JEE
Find the number of integers n with 1 ≤ n ≤ 5000 such that n divides 2^n + 1.
number theory · JEE
Find the number of ordered triples (a, b, c) of positive integers such that a · b · c = 10800.…
number theory · JEE
Let x be the smallest positive integer satisfying simultaneously x ≡ 2 (mod 3), x ≡ 3 (mod 5),…
number theory · JEE
Determine the largest integer k such that 5^k divides the binomial coefficient C(2025, 405).
number theory · JEE
Let N be the smallest positive integer such that N/2 is a perfect square, N/3 is a perfect…
number theory · JEE
Let a, b, c be the (possibly complex) roots of the cubic x^3 - 5x^2 + 6x - 1 = 0. Compute the…
algebra · JEE
A function f defined for all nonzero real x satisfies f(x) + 2 f(1/x) = 3x for every x ≠ 0. Let…
algebra · JEE
Let a, b, c be positive real numbers with abc = 8. Find the minimum possible value of (1 + a)(1…
algebra · JEE
A polynomial P(x) leaves remainder 3 on division by (x - 1), remainder 5 on division by (x -…
algebra · JEE
The cubic x^3 - a x^2 + (a - 1)x - 5 = 0 (with a a real parameter) has roots whose squares sum…
algebra · JEE
Let w be the real number w = cbrt(20 + 14√2) + cbrt(20 - 14√2), where cbrt denotes the real…
algebra · JEE
Evaluate the sum S = 1·1! + 2·2! + 3·3! + 4·4! + 5·5! + 6·6!, where n! denotes factorial. Give…
algebra · JEE
Real numbers a, b, c satisfy a + b + c = 6, a^2 + b^2 + c^2 = 14, and a^3 + b^3 + c^3 = 36.…
algebra · JEE
Find the greatest integer not exceeding (2 + √5)^5; that is, compute ⌊(2 + √5)^5⌋.
algebra · JEE
How many ordered pairs (a, b) of positive integers satisfy 1/a + 1/b = 1/6?
algebra · JEE
Let g(x) = x^2 + 250/x for x > 0. Find the minimum value of g(x).
algebra · JEE
The quartic x^4 - 8x^3 + a x^2 - b x + 16 = 0 has four positive real roots (counted with…
algebra · JEE
A sequence is defined by a_1 = 1 and a_{n+1} = 3 a_n + 2^n for all n ≥ 1. Find a_8.
algebra · JEE
Let a, b, c be positive real numbers with a + b + c = 12. Find the maximum possible value of…
algebra · JEE
Real numbers x and y satisfy x + y = 4 and x^3 + y^3 = 28. Compute x^5 + y^5.
algebra · JEE
Compute floor(1000 * T), where T = sum_{n=0}^{infinity} (n^2 + 1)/n! and floor denotes the…
sequences series · JEE
The sum S = sum_{n=1}^{80} 1/((n+1)*sqrt(n) + n*sqrt(n+1)) equals m/n in lowest terms, where m…
sequences series · JEE
The infinite sum sum_{n=1}^{infinity} n/(n^4 + n^2 + 1) equals m/n in lowest terms. Find m + n.
sequences series · JEE
Let F_1 = 1, F_2 = 1 and F_{n+2} = F_{n+1} + F_n be the Fibonacci numbers. The infinite sum…
sequences series · JEE
The infinite sum sum_{n=1}^{infinity} 1/(n(n+2)(n+4)) equals m/n in lowest terms, where m and n…
sequences series · JEE
Compute sum_{n=1}^{1023} floor(log_2 n), where floor is the greatest integer function and log_2…
sequences series · JEE
How many functions f from {1,2,3,4,5,6,7,8} onto {1,2,3,4} (i.e. surjective functions, every…
combinatorics · JEE
A token moves on the integer grid from (0,0) to (7,5) using only unit steps Right (+1 in x) and…
combinatorics · JEE
Find the number of integer solutions (x1, x2, x3, x4) to x1 + x2 + x3 + x4 = 20 with 0 <= xi <=…
combinatorics · JEE
How many permutations of the numbers 1,2,...,9 have exactly two fixed points (a fixed point is…
combinatorics · JEE
Consider all distinguishable arrangements of the multiset of letters {M, I, I, I, I, S, S, S,…
combinatorics · JEE
Ten distinct balls are distributed into four distinct boxes labelled 1, 2, 3, 4 so that no box…
combinatorics · JEE
How many subsets S of {1, 2, 3, ..., 15} contain no two consecutive integers AND have a sum of…
combinatorics · JEE
How many permutations p of (1, 2, 3, 4, 5, 6, 7, 8) satisfy p(i) is not equal to i for every i…
combinatorics · JEE
Find the number of monomials of total degree exactly 12 in five variables x1, x2, x3, x4, x5 in…
combinatorics · JEE
A sequence of 12 steps is formed, each step being +1 or -1. The sequence starts at height 0,…
combinatorics · JEE
How many integers n with 1 <= n <= 10000 are divisible by at least two of the four numbers 4,…
combinatorics · JEE
How many permutations p of (1, 2, 3, 4, 5, 6, 7) satisfy all of the following: the values in…
combinatorics · JEE
A 5-element subset is chosen uniformly at random from {1, 2, 3, ..., 30}. The probability that…
combinatorics · JEE
In how many ways can the set {1, 2, 3, 4, 5, 6, 7, 8, 9} be partitioned into exactly 3…
combinatorics · JEE
Eight people, including two specific people Anil and Bharti, are to be seated around a circular…
combinatorics · JEE
Count the ordered 5-tuples of positive integers (a, b, c, d, e) such that a + b + c + d + e =…
combinatorics · JEE
A Polya urn starts with 2 red and 3 blue balls. A ball is drawn uniformly at random; its colour…
probability · JEE
A permutation of the numbers 1,2,...,8 is chosen uniformly at random. Let p be the probability…
probability · JEE
A fair coin is flipped repeatedly until the pattern Heads-Tails-Heads (HTH) first appears as…
probability · JEE
Player A repeatedly rolls a fair eight-sided die (faces 1-8) and player B repeatedly rolls a…
probability · JEE
Eight distinct people, three of whom are named A, B, and C, are seated in a random order in a…
probability · JEE
A gambler starts with 3 rupees. In each round he wins 1 rupee with probability 2/5 and loses 1…
probability · JEE
A fair six-sided die is rolled 5 times. Let p be the probability that the maximum value…
probability · JEE
The numbers 1,2,...,9 are arranged in a uniformly random permutation a_1, a_2, ..., a_9. Call a…
probability · JEE
Two players A and B take turns rolling a single fair six-sided die, with A rolling first and…
probability · JEE
A fair coin is flipped repeatedly. Let X be the number of flips up to and including the flip on…
probability · JEE
There are three coins. Coin 1 is fair (P(Heads)=1/2), coin 2 has P(Heads)=1/3, and coin 3 has…
probability · JEE
A function f from {1,2,3,4,5} to {1,2,3,4,5} is chosen uniformly at random among all 5^5 such…
probability · JEE
A fair six-sided die is rolled 6 times. Let E be the expected number of DISTINCT faces (values…
probability · JEE
Seven distinguishable balls are placed independently and uniformly at random into 4…
probability · JEE
Let L = lim_{x->0} ( (1+x)^(1/x) - e + (e x)/2 ) / x^2. The value of L can be written as L =…
limits · JEE
Let L = lim_{n->infinity} n·( e - (1 + 1/n)^n ). The value can be written L = (p/q)·e with p/q…
limits · JEE
Let L = lim_{x->0} ( cosh(x)·cos(x) - 1 ) / x^4. Writing L = m/n in lowest terms (m an integer,…
limits · JEE
Let L = lim_{x->0} ( (1+x)^(1/x) / e )^(1/x). The value is L = e^a for some rational a = m/n in…
limits · JEE
Let L = lim_{x->0} ( arcsin(x) / x )^(1/x^2). The value is L = e^a for a rational a = m/n in…
limits · JEE
Let L = lim_{x->0} ( (1 + x^2)^(1/3) - cos(x) ) / ( x·sin(x) ). Writing L = m/n in lowest terms…
limits · JEE
Let L = lim_{x->0} ( (1 + x + x^2)^(1/x) - e·(1 + x/2) ) / ( e·x^2 ). Writing L = m/n in lowest…
limits · JEE
Compute L = lim_{x->infinity} ( (x^3 + 3 x^2)^(1/3) - (x^2 - 2 x)^(1/2) ). Give L.
limits · JEE
Let L = lim_{x->0} ( (1 - cos x)(1 - cos 2x)(1 - cos 3x) ) / x^6. Writing L = m/n in lowest…
limits · JEE
Let L = lim_{x->0} ( ln(1+x)·ln(1-x) + x^2 ) / x^4. Writing L = m/n in lowest terms (m an…
limits · JEE
Let L = lim_{x->0} ( tan(sin x) - sin(tan x) ) / x^7. Writing L = m/n in lowest terms with…
limits · JEE
Let L = lim_{n->infinity} (1/n)·( (n+1)(n+2)...(2n) )^(1/n). Compute the integer value of…
limits · JEE
Evaluate I = \int_0^1 \frac{x^4(1-x)^4}{1+x^2}\,dx. It can be written as I = \frac{a}{b} - \pi,…
definite integrals · JEE
Evaluate I = \int_0^{\pi/2} \ln^2(\sin x)\,dx and give its value rounded to 4 decimal places.
definite integrals · JEE
Using the symmetry x\to\pi-x, evaluate I = \int_0^{\pi} x\,\sin^4 x\,dx. It equals…
definite integrals · JEE
Evaluate I = \int_0^{2\pi} \frac{dx}{(5-4\cos x)^2}. The result is \frac{m}{n}\pi with m/n in…
definite integrals · JEE
By differentiating under the integral sign, evaluate I = \int_0^{\pi/2} \frac{1}{\cos…
definite integrals · JEE
Using the reduction formula for powers of tangent, evaluate I = \int_0^{\pi/4} \tan^5 x\,dx and…
definite integrals · JEE
Using the Beta function (or Wallis reduction), evaluate I = \int_0^{\pi/2} \sin^6 x\,\cos^4…
definite integrals · JEE
Evaluate I = \int_0^{1} \frac{(\ln x)^2}{1+x^2}\,dx. It equals \frac{m}{n}\pi^3 with m/n in…
definite integrals · JEE
Evaluate I = \int_0^{\pi} \ln\!\big(5 - 4\cos x\big)\,dx and give its value rounded to 4…
definite integrals · JEE
Evaluate I = \int_0^{\pi/2} x\,\cot x\,dx and give its value rounded to 4 decimal places.
definite integrals · JEE
Let x, y, z be positive real numbers with x + y + z = 12. Find the minimum possible value of…
maxima minima · JEE
A straight line of negative slope passes through the point (3, 4) and meets the positive x-axis…
maxima minima · JEE
Let x, y, z be positive real numbers with x + y + z = 6. Find the maximum value of x·y^2·z^3.
maxima minima · JEE
Find the shortest distance from the point (0, 17/4) to the parabola y = x^2.
maxima minima · JEE
A point P moves along the x-axis. Let L(P) = PA + PB where A = (0, 2) and B = (10, 3). Minimize…
maxima minima · JEE
A rectangle has its base on the x-axis and its two upper vertices on the parabola y = 12 - x^2…
maxima minima · JEE
A right circular cone is inscribed in a sphere of radius R so that its apex and base circle lie…
maxima minima · JEE
Find the minimum value of f(x) = 1·|x - 1| + 2·|x - 2| + 3·|x - 3| + 4·|x - 4| + 5·|x - 5| over…
maxima minima · JEE
Let m(a) = max over x in [-1, 1] of |x^3 - a·x|. As a ranges over all real numbers, m(a)…
maxima minima · JEE
A vertical picture is mounted on a wall so that its lower edge is 2 m above the observer's eye…
maxima minima · JEE
A closed cylindrical can must have volume 54π cubic units. The material for the top and bottom…
maxima minima · JEE
Two vertical posts of heights 6 m and 12 m stand 30 m apart on level ground. A single wire runs…
maxima minima · JEE
Let x, y, z be positive real numbers with x + y + z = 1. Find the minimum value of (1 + 1/x)(1…
maxima minima · JEE
Let x and y be positive real numbers with x^2 + y^2 = 1. Find the minimum value of 4/x^2 +…
maxima minima · JEE
Find the number of values of x in the interval [0, 2π) that satisfy the equation sin(5x) =…
trigonometry · JEE
Let S = Σ (from k = 1 to 89) of 1/(sin(k°)·sin((k+1)°)), where the angles are in degrees.…
trigonometry · JEE
Evaluate the product P = (1 + tan 1°)(1 + tan 2°)(1 + tan 3°) ⋯ (1 + tan 44°). Report the value…
trigonometry · JEE
Let S = Σ (from k = 1 to 20) of arctan(1/(k² + k + 1)). The value of tan S can be written as…
trigonometry · JEE
In a triangle, the three angles A, B, C satisfy tan A = 2 and tan B = 3. Then cot A + cot B +…
trigonometry · JEE
Find the number of solutions in [0, 2π) of the equation cos x + cos 2x + cos 3x = 0.
trigonometry · JEE
Find the number of solutions in [0, 2π) of the equation sin⁴x + cos⁴x = sin x · cos x.
trigonometry · JEE
Define S_n = Σ (from k = 1 to n) of 3^(k−1)·sin³(π/3^k). It is known that S_n converges as n →…
trigonometry · JEE
Compute the exact value of tan 9° − tan 27° − tan 63° + tan 81°. Give a single integer.
trigonometry · JEE
Find the number of solutions in [0, 2π) of the equation cos 3x = cos 2x.
trigonometry · JEE
Let M be the maximum value of the function f(x) = sin x·(1 + cos x) over all real x. Compute…
trigonometry · JEE
Compute the exact value of cot²(π/13) + cot²(2π/13) + cot²(3π/13) + cot²(4π/13) + cot²(5π/13) +…
trigonometry · JEE
Compute the exact value of 16·sin 20°·sin 40°·sin 60°·sin 80°. Give a single integer.
trigonometry · JEE
A ray of light starts at the point A(2,3). It first strikes the x-axis, reflects off it, then…
coordinate geometry · JEE
A point P moves in the plane so that the length of the tangent drawn from P to the circle x^2 +…
coordinate geometry · JEE
In triangle ABC the vertices are A(0,0), B(14,0), C(5,12). Consider the triangle whose three…
coordinate geometry · JEE
Consider the ellipse x^2/9 + y^2/4 = 1 with foci S and S'. Let P be the point of the ellipse…
coordinate geometry · JEE
A circle passes through the two points (1,0) and (5,0) and is tangent to the line y = x. There…
coordinate geometry · JEE
PQ is a focal chord of the parabola y^2 = 8x with focus S. The length of the chord PQ is 32/3,…
coordinate geometry · JEE
A single straight line is a common tangent to both parabolas y^2 = 4x and x^2 = 4y. It touches…
coordinate geometry · JEE
For the hyperbola x^2/16 - y^2/9 = 1, the tangent at a variable point P meets the two…
coordinate geometry · JEE
Consider the family of circles passing through the two intersection points of the circle x^2 +…
coordinate geometry · JEE
An ellipse is centred at the origin with its axes along the coordinate axes. It passes through…
coordinate geometry · JEE
A triangle has vertices A(0,0), B(8,0) and C(2,6). Its nine-point circle (the circle through…
coordinate geometry · JEE
A circle is tangent to the x-axis (the line y = 0) and to the line 3x - 4y = 0, lies below the…
coordinate geometry · JEE
The conic 6x^2 - 4xy + 9y^2 - 20x + 20y - 20 = 0 is an ellipse. If the square of its…
coordinate geometry · JEE
From the external point P(7,1) the two tangents are drawn to the circle x^2 + y^2 = 25,…
coordinate geometry · JEE
A point P moves in the plane so that the sum of the squares of its distances from the three…
coordinate geometry · JEE
On the parabola y^2 = 4x, a normal chord is drawn at a point A so that this chord subtends a…
coordinate geometry · JEE
Let w_1, w_2, ..., w_6 denote the six seventh roots of unity other than 1 (i.e. the non-trivial…
complex numbers · JEE
A complex number z satisfies |z| = 1. Determine the maximum possible value of |z^3 - 3z + 2|.…
complex numbers · JEE
Compute the product (2 - 2cos(2*pi*1/13)) * (2 - 2cos(2*pi*2/13)) * ... * (2 -…
complex numbers · JEE
For a complex number z with |z| = 2, find the maximum value of the product |z^2 - z + 1| * |z^2…
complex numbers · JEE
Let w = e^{2*pi*i/10} be a primitive 10th root of unity. Evaluate the sum S = (1 + w^0)^25 + (1…
complex numbers · JEE
Consider the thirty 30th roots of unity z = e^{2*pi*i*k/30} for k = 0, 1, ..., 29. How many of…
complex numbers · JEE
Three complex numbers z_1, z_2, z_3 are the vertices of an equilateral triangle in the complex…
complex numbers · JEE
Let w = e^{i*pi/3} = cos(60 degrees) + i sin(60 degrees). Evaluate the real part of the sum…
complex numbers · JEE
Let w_0, w_1, ..., w_6 be the seven 7th roots of unity (the roots of z^7 = 1). Compute the sum…
complex numbers · JEE
Let w = e^{2*pi*i/17}. Define S = w^1 + w^2 + w^4 + w^8 + w^9 + w^13 + w^15 + w^16, where the…
complex numbers · JEE
Let zeta_1, zeta_2, ..., zeta_8 be the eight primitive 15th roots of unity (the complex numbers…
complex numbers · JEE
Let A = 2 + 3i and B = 8 + 11i. A complex number z varies so that the argument of (z - A)/(z -…
complex numbers · JEE
Find the smallest positive integer n such that ((sqrt(3) + i)/(1 + i))^n is a real number.…
complex numbers · JEE
Let w be a primitive cube root of unity (so w^2 + w + 1 = 0). Define a = 2 + 3w + 5w^2 and b =…
complex numbers · JEE
A non-real complex number z has modulus |z| = 2, and the three points z, conjugate(z), and 1…
complex numbers · JEE
Let a = (1, 2, -1), b = (2, -1, 3) and c = (t, 1, 2), where t is a real parameter. The…
vectors 3d · JEE
Let a, b, c be unit vectors with a·b = 1/4, b·c = 1/3 and c·a = 1/5. Let S denote the square of…
vectors 3d · JEE
Consider the two skew lines L1: r = (1, 2, 3) + s(2, 1, -1) and L2: r = (2, -1, 1) + u(1, 2,…
vectors 3d · JEE
Let a = (3, -1, 2) and b = (1, 5, 1). There is a unique vector r satisfying both r × a = b and…
vectors 3d · JEE
Let Π be the plane through the three points A = (1, 0, 2), B = (3, -1, 1) and C = (2, 2, 4).…
vectors 3d · JEE
Among all planes that contain the line L: (x - 1)/2 = (y - 2)/3 = (z - 3)/1, let Π be the one…
vectors 3d · JEE
Let a = (1, 1, 0), b = (0, 1, 1), c = (1, 0, 1). The vector d = (2, 3, 5) is written uniquely…
vectors 3d · JEE
Let a = (1, -2, 1), b = (2, 1, -1), c = (1, 1, 2), d = (3, -1, 2). Compute the squared…
vectors 3d · JEE
A point Q moves along the line r = (1, 2, 0) + t(1, 1, 1). At the unique position where Q is…
vectors 3d · JEE
Let L be the line of intersection of the planes x + 2y - z = 1 and 2x - y + z = 3. Let θ be the…
vectors 3d · JEE
The vectors a = (1, 2, 2), b = (2, -2, 1) and c = (2, 1, -2) are mutually perpendicular, each…
vectors 3d · JEE
Let a = (2, 1, -2) and b = (1, 3, 2). Over all unit vectors c that are perpendicular to w = (1,…
vectors 3d · JEE
The point P = (4, 2, 7) is reflected in the line L: r = (1, 0, 2) + t(2, 1, 2) to obtain the…
vectors 3d · JEE
A wedge of mass M = 4 kg with a smooth inclined face making 45 degrees with the horizontal…
physics mechanics · JEE
A block of mass 2 kg moving at 10 m/s slides onto the left end of a long plank of mass 8 kg…
physics mechanics · JEE
A particle is projected from the foot of a fixed inclined plane whose surface makes 30 degrees…
physics mechanics · JEE
A thin uniform rod of mass M and length L is hinged at one end about a fixed frictionless…
physics mechanics · JEE
A small particle is placed at rest on the very top of a fixed smooth (frictionless) sphere of…
physics mechanics · JEE
On a frictionless horizontal line, a ball A of mass m moves with speed u and collides head-on…
physics mechanics · JEE
A solid uniform sphere (moment of inertia 2/5 M R^2 about its centre) is released from rest and…
physics mechanics · JEE
A uniform flexible chain of total length 2 m lies on a frictionless horizontal table with a…
physics mechanics · JEE
A car travels on a curved road of radius 50 m that is banked at an angle whose tangent is 0.75.…
physics mechanics · JEE
A particle of mass 2 kg moves along the x-axis under a position-dependent net force F(x) = (6x…
physics mechanics · JEE
From the same point on level ground, two particles are launched simultaneously: the first…
physics mechanics · JEE
A uniform thin rod of mass 3 kg and length 2 m lies at rest on a frictionless horizontal table.…
physics mechanics · JEE
A block of mass 2 kg is released from rest on a rough incline of angle 37 degrees (use sin 37 =…
physics mechanics · JEE
Twelve identical resistors, each of resistance 12 ohm, are soldered along the twelve edges of a…
physics modern em · JEE
An infinite resistor ladder is built from identical repeating sections. In each section a…
physics modern em · JEE
A resistor bridge has four nodes A, B, C, D. The resistors are: A-C = 2 ohm, A-D = 4 ohm, C-B =…
physics modern em · JEE
A bridge of capacitors has four nodes A, B, C, D with capacitances: A-C = 2 uF, A-D = 4 uF, C-B…
physics modern em · JEE
Two batteries are connected in parallel across an external resistor. Battery 1 has EMF 10 V and…
physics modern em · JEE
Capacitor X (2 uF) is charged to 120 V and capacitor Y (3 uF) is charged to 30 V. They are then…
physics modern em · JEE
A parallel-plate capacitor of capacitance 2 uF is connected to a 100 V battery and remains…
physics modern em · JEE
A parallel-plate capacitor has plate separation d. A dielectric slab of dielectric constant 4…
physics modern em · JEE
A singly excited gas of hydrogen-like ions has a ground-state ionization energy of 122.4 eV.…
physics modern em · JEE
An electron with kinetic energy 12.5 eV makes a single head-on inelastic collision with a…
physics modern em · JEE
In a singly ionized helium ion (He+, nuclear charge Z = 2), the electron makes a transition…
physics modern em · JEE
Hydrogen atoms emit light in the Balmer transition from n = 4 to n = 2. This light falls on a…
physics modern em · JEE
When light of wavelength 248 nm strikes a metal, the stopping potential is V0. When light of…
physics modern em · JEE
For a certain metal, monochromatic light of wavelength 200 nm produces a stopping potential of…
physics modern em · JEE