JEE Main - Mathematics (2025)

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is

Let the position vectors of the vertices A, B and C of a tetrahedron ABCD be i + 2j + k, i + 3j - 2k and 2i + j - k respectively. The altitude from the vertex D to the opposite face ABC meets the median line segment through A of the triangle ABC at the point E. If the length of AD is sqrt(110)/3 and the volume of the tetrahedron is sqrt(805)/(6*sqrt(2)), then the position vector of E is

If A, B, and (adj(A^-1) + adj(B^-1)) are non-singular matrices of same order, then the inverse of A(adj(A^-1) + adj(B^-1))^-1 B, is equal to

Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is

Let a curve y = f(x) pass through the points (0,5) and (log_e 2, k). If the curve satisfies the differential equation 2(3+y)e^(2x) dx - (7 + e^(2x)) dy = 0, then k is equal to

If the function f(x) = (2/x){sin((k1+1)x) + sin((k2-1)x)} for x<0, f(0)=4, f(x) = (2/x) log_e((2+k1*x)/(2+k2*x)) for x>0, is continuous at x = 0, then k1^2 + k2^2 is equal to

If the line 3x - 2y + 12 = 0 intersects the parabola 4y = 3x^2 at the points A and B, then at the vertex of the parabola, the line segment AB subtends an angle equal to

Let P be the foot of the perpendicular from the point Q(10,-3,-1) on the line (x-3)/7 = (y-2)/(-1) = (z+1)/(-2). Then the area of the right angled triangle PQR, where R is the point (3,-2,1), is

Let the arc AC of a circle subtend a right angle at the centre O. If the point B on the arc AC, divides the arc AC such that (length of arc AB)/(length of arc BC) = 1/5, and OC = alpha*OA + beta*OB, then alpha + sqrt(2)(sqrt(3)-1)*beta is equal to

Let f(x) = log_e x and g(x) = (x^4 - 2x^3 + 3x^2 - 2x + 2)/(2x^2 - 2x + 1). Then the domain of f o g is

If the system of equations (lambda-1)x + (lambda-4)y + lambda*z = 5; lambda*x + (lambda-1)y + (lambda-4)z = 7; (lambda+1)x + (lambda+2)y - (lambda+2)z = 9 has infinitely many solutions, then lambda^2 + lambda is equal to

The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is

Let R = {(1,2),(2,3),(3,3)} be a relation defined on the set {1,2,3,4}. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:

Let the area of a triangle PQR with vertices P(5,4), Q(-2,4) and R(a,b) be 35 square units. If its orthocenter and centroid are O(2, 14/5) and C(c,d) respectively, then c + 2d is equal to

The value of the integral from e^2 to e^4 of (1/x) * [ e^(((log_e x)^2+1)^-1) / ( e^(((log_e x)^2+1)^-1) + e^(((6-log_e x)^2+1)^-1) ) ] dx is

Let |(z-bar - i)/(2*z-bar + i)| = 1/3, z in C, be the equation of a circle with center at C. If the area of the triangle, whose vertices are at the points (0,0), C and (alpha, 0) is 11 square units, then alpha^2 equals:

The value of (sin 70 deg)(cot 10 deg * cot 70 deg - 1) is

Let I(x) = Integral of dx / [(x-11)^(11/13) * (x+15)^(15/13)]. If I(37) - I(24) = (1/4)*(1/b^(1/13) - 1/c^(1/13)), b,c in N, then 3(b+c) is equal to

If pi/2 <= x <= 3*pi/4, then cos^-1((12/13)cos x + (5/13)sin x) is equal to

Let the circle C touch the line x - y + 1 = 0, have the centre on the positive x-axis, and cut off a chord of length 4/sqrt(13) along the line -3x + 2y = 1. Let H be the hyperbola x^2/alpha^2 - y^2/beta^2 = 1, whose one of the foci is the centre of C and the length of the transverse axis is the diameter of C. Then 2*alpha^2 + 3*beta^2 is equal to

If the equation a(b-c)x^2 + b(c-a)x + c(a-b) = 0 has equal roots, where a + c = 15 and b = 36/5, then a^2 + c^2 is equal to

If the set of all values of a, for which the equation 5x^3 - 15x - a = 0 has three distinct real roots, is the interval (alpha, beta), then beta - 2*alpha is equal to

The sum of all rational terms in the expansion of (1 + 2^(1/2) + 3^(1/2))^6 is equal to

If the area of the larger portion bounded between the curves x^2 + y^2 = 25 and y = |x-1| is (1/4)(b*pi + c), b,c in N, then b + c is equal to

A point particle of charge Q is located at P along the axis of an electric dipole 1 at a distance r as shown in the figure. The point P is also on the equatorial plane of a second electric dipole 2 at a distance r. The dipoles are made of opposite charge q separated by a distance 2a. For the charge particle at P not to experience any net force, which of the following correctly describes the situation?

A spherical surface of radius of curvature R, separates air from glass (refractive index = 1.5). The centre of curvature is in the glass medium. A point object 'O' placed in air on the optic axis of the surface, so that its real image is formed at 'I' inside glass. The line OI intersects the spherical surface at P and PO = PI. The distance PO equals to

The position of a particle moving on x-axis is given by x(t) = A sin t + B cos^2 t + C t^2 + D, where t is time. The dimension of ABC/D is

Given a thin convex lens (refractive index mu2), kept in a liquid (refractive index mu1, mu1 < mu2) having radii of curvatures |R1| and |R2|. Its second surface is silver polished. Where should an object be placed on the optic axis so that a real and inverted image is formed at the same place?

Refer to the circuit diagram given in the figure. Which of the following observations are correct? A. Total resistance of circuit is 6 ohm B. Current in Ammeter is 1 A C. Potential across AB is 4 Volts D. Potential across CD is 4 Volts E. Total resistance of the circuit is 8 ohm. Choose the correct answer from the options given below:

Given below are two statements: Statement I: The hot water flows faster than cold water. Statement II: Soap water has higher surface tension as compared to fresh water. In the light of above statements, choose the correct answer from the options given below

Consider a circular disc of radius 20 cm with centre located at the origin. A circular hole of radius 5 cm is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of centre of mass of residual or remaining disc from the origin will be

The electric flux is phi = alpha*sigma + beta*lambda, where lambda and sigma are linear and surface charge density, respectively. (alpha/beta) represents

A sub-atomic particle of mass 10^-30 kg is moving with a velocity 2.21 x 10^6 m/s. Under the matter wave consideration, the particle will behave closely like (h = 6.63 x 10^-34 J.s)

Consider a moving coil galvanometer (MCG): A. The torsional constant in moving coil galvanometer has dimensions [ML^2T^-2] B. Increasing the current sensitivity may not necessarily increase the voltage sensitivity. C. If we increase number of turns to its double (2N), then the voltage sensitivity doubles. D. MCG can be converted into an ammeter by introducing a shunt resistance of large value in parallel with galvanometer. E. Current sensitivity of MCG depends inversely on number of turns of coil. Choose the correct answer from the options given below:

The electric field of an electromagnetic wave in free space is E = 57 cos[7.5x10^6 t - 5x10^-3(3x+4y)](4i - 3j) N/C. The associated magnetic field in Tesla is

A gun fires a lead bullet of temperature 300 K into a wooden block. The bullet having melting temperature of 600 K penetrates into the block and melts down. If the total heat required for the process is 625 J, then the mass of the bullet is ___ grams. (Latent heat of fusion of lead = 2.5x10^4 J/Kg and specific heat capacity of lead = 125 J/Kg K)

What is the lateral shift of a ray refracted through a parallel-sided glass slab of thickness 'h' in terms of the angle of incidence 'i' and angle of refraction 'r', if the glass slab is placed in air medium?

A radioactive nucleus n2 has 3 times the decay constant as compared to the decay constant of another radioactive nucleus n1. If initial number of both nuclei are the same, what is the ratio of number of nuclei of n2 to the number of nuclei of n1, after one half-life of n1?

A light hollow cube of side length 10 cm and mass 10 g, is floating in water. It is pushed down and released to execute simple harmonic oscillations. The time period of oscillations is y*pi x 10^-2 s, where the value of y is (g = 10 m/s^2, density of water = 10^3 kg/m^3)

Regarding self-inductance: A. The self-inductance of the coil depends on its geometry. B. Self-inductance does not depend on the permeability of the medium. C. Self-induced e.m.f. opposes any change in the current in a circuit. D. Self-inductance is electromagnetic analogue of mass in mechanics. E. Work needs to be done against self-induced e.m.f. in establishing the current. Choose the correct answer from the options given below:

The motion of an airplane is represented by velocity-time graph as shown below. The distance covered by airplane in the first 30.5 second is _______ km.

Identify the valid statements relevant to the given circuit at the instant when the key is closed. A. There will be no current through resistor R. B. There will be maximum current in the connecting wires. C. Potential difference between the capacitor plates A and B is minimum. D. Charge on the capacitor plates is minimum. Choose the correct answer from the options given below:

A solid sphere of mass 'm' and radius 'r' is allowed to roll without slipping from the highest point of an inclined plane of length 'L' and makes an angle 30 deg with the horizontal. The speed of the particle at the bottom of the plane is v1. If the angle of inclination is increased to 45 deg while keeping L constant, then the new speed of the sphere at the bottom of the plane is v2. The ratio v1^2 : v2^2 is

A positive ion A and a negative ion B have charges 6.67x10^-19 C and 9.6x10^-10 C, and masses 19.2x10^-27 kg and 9x10^-27 kg respectively. At an instant, the ions are separated by a certain distance r. At that instant the ratio of the magnitudes of electrostatic force to gravitational force is P x 10^45, where the value of 10P is (Take 1/(4*pi*eps0) = 9x10^9 Nm^2C^-2 and universal gravitational constant as 6.67x10^-11 Nm^2kg^-2). Assume that charge may not be an integral multiple of electron charge.

In the given circuit the sliding contact is pulled outwards such that electric current in the circuit changes at the rate of 8 A/s. At an instant when R is 12 ohm, the value of the current in the circuit will be ______ A.

Two particles are located at equal distance from origin. The position vectors of those are represented by A_vec = 2i + 3nj + 2k and B_vec = 2i - 2j + 4pk, respectively. If both the vectors are at right angle to each other, the value of n^-1 is _____.

An ideal gas initially at temperature 0 deg C is compressed suddenly to one fourth of its volume. If the ratio of specific heat at constant pressure to that at constant volume is 3/2, the change in temperature due to the thermodynamic process is _____ K.

A force f = x^2*y*i + y^2*j acts on a particle in a plane x + y = 10. The work done by this force during a displacement from (0,0) to (4 m, 2 m) is ______ Joule (round off to the nearest integer)