List of top Questions asked in JEE Advanced

Let the function \(f:[1,\infin)→\R\) be defined by
\(f(t) = \begin{cases}     (-1)^{n+1}2, & \text{if } t=2n-1,n\in\N, \\     \frac{(2n+1-t)}{2}f(2n-1)+\frac{(t-(2n-1))}{2}f(2n+1) & \text{if } 2n-1<t<2n+1,n\in\N. \end{cases}\)
Define \(g(x)=\int\limits_{1}^{x}f(t)dt,x\in(1,\infin).\) Let α denote the number of solutions of the equation g(x) = 0 in the interval (1, 8] and \(β=\lim\limits_{x→1+}\frac{g(x)}{x-1}\). Then the value of α + β is equal to _____.

A dimensionless quantity is constructed in terms of electronic charge \(e\), permittivity of free space \(\epsilon_0\) , Planck’s constant ℎ, and speed of light c. If the dimensionless quantity is written as \(e^\alpha\epsilon_0^\beta h^\gamma c^\delta\)and n is a non-zero integer, then\((\alpha, \beta,\gamma,\delta)\) is given by

A block of mass \(5 kg\) moves along the \(x-\)direction subject to the force \(F = (−20x + 10) N,\) with the value of  \(x \) in metre. At time \(t = 0 s,\) it is at rest at position \(x = 1 m\). The position and momentum of the block at \(t = (\pi/4)\) s are

A region in the form of an equilateral triangle (in x-y plane) of height L has a uniform magnetic field 𝐵⃗ pointing in the +z-direction. A conducting loop PQR, in the form of an equilateral triangle of the same height 𝐿, is placed in the x-y plane with its vertex P at x = 0 in the orientation shown in the figure. At 𝑡 = 0, the loop starts entering the region of the magnetic field with a uniform velocity 𝑣 along the +x-direction. The plane of the loop and its orientation remain unchanged throughout its motion.

Which of the following graph best depicts the variation of the induced emf (E) in the loop as a function of the distance (𝑥) starting from 𝑥 = 0?

Two beads, each with charge q and mass m, are on a horizontal, frictionless, non-conducting, circular hoop of radius R. One of the beads is glued to the hoop at some point, while the other one performs small oscillations about its equilibrium position along the hoop. The square of the angular frequency of the small oscillations is given by [ \(\epsilon_0 \)is the permittivity of free space.]

A group of 9 students, s₁, s₂,…., s₉, is to be divided to form three teams X, Y and Z of sizes 2, 3, and 4, respectively. Suppose that s₁ cannot be selected for the team X and s₂ cannot be selected for the team Y. Then the number of ways to form such teams, is _______.

Let \(\vec{p}=2\hat{i}+\hat{j}+3\hat{k}\) and \(\vec{q}=\hat{i}-\hat{j}+\hat{k}\). If for some real numbers α, β and γ we have
\(15\hat{i}+10\hat{j}+6\hat{k}=α(2\vec{p}+\vec{q})+β(\vec{p}-2\vec{q})+γ(\vec{p}\times\vec{q})\),
then the value of γ is ________.

Let X be a random variable, and let P(X = x) denote the probability that X takes the value x. Suppose that the points (x, P(X = x)), x = 0,1,2,3,4, lie on a fixed straight line in the xy -plane, and P(X = x) = 0 for all x ∈ R - {0,1,2,3,4}. If the mean of X is \(\frac{5}{2}\) , and the variance of X is α, then the value of 24α is ______.

A particle of mass 𝑚 is under the influence of the gravitational field of a body of mass 𝑀 (≫ 𝑚). The particle is moving in a circular orbit of radius \(𝑟_0\) with time period \(𝑇_0\) around the mass 𝑀. Then, the particle is subjected to an additional central force, corresponding to the potential energy 𝑉c(𝑟) = 𝑚𝛼/𝑟 3 , where 𝛼 is a positive constant of suitable dimensions and 𝑟 is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius\( 𝑟_0\) in the combined gravitational potential due to 𝑀 and 𝑉c(𝑟), but with a new time period \(𝑇_1\), then\( (𝑇_1^2  − 𝑇_0^ 2 )/𝑇_1^ 2\) is given by [𝐺 is the gravitational constant.]

Two equilateral-triangular prisms \(P_1 \)and \(P_2\) are kept with their sides parallel to each other, in vacuum, as shown in the figure. A light ray enters prism \(P_1\) at an angle of incidence 𝜃 such that the outgoing ray undergoes minimum deviation in prism \(P_2\). If the respective refractive indices of \(P_1\) and\( P_2\) are \(√ 3 /2\) and \(√3\), then \(\theta  = sin{−1}[\sqrt \frac{ 3}{ 2} sin ( \frac{\pi}{B} )],\) where the value of \(\beta\) is ______.

Let R² denote R × R. Let
S = {(a, b, c) : a, b, c ∈ R and ax² + 2bxy + cy² > 0 for all (x, y) ∈ R² - {(0, 0}}.
Then which of the following statements is (are) TRUE ?

A disc of mass \(M\) and radius \(R\) is free to rotate about its vertical axis as shown in the figure. A battery operated motor of negligible mass is fixed to this disc at a point on its circumference. Another disc of the same mass \(M\) and radius \(R/2\) is fixed to the motor’s thin shaft. Initially, both the discs are at rest. The motor is switched on so that the smaller disc rotates at a uniform angular speed \(\omega\). If the angular speed at which the large disc rotates is \(\omega/n\), then the value of \(n\) is _____

A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct answer. Assume that the probability of the student giving the correct answer for a question, given that he has guessed it, is \(\frac{1}{2}\) . Also assume that the probability of the answer for a question being guessed, given that the student’s answer is correct, is \(\frac{1}{6}\) . Then the probability that the student knows the answer of a randomly chosen question is

An infinitely long thin wire, having a uniform charge density per unit length of \(5 nC/m,\) is passing through a spherical shell of radius \(1 m\), as shown in the figure. A \(10 nC\) charge is distributed uniformly over the spherical shell. If the configuration of the charges remains static, the magnitude of the potential difference between points P and R, in Volt, is ______. [Given: In SI units \(\frac{1}{ 4πϵ0} = 9 × 10^9 , ln \ 2 = 0.7\). Ignore the area pierced by the wire.]

According to Bohr’s model, the highest kinetic energy is associated with the electron in the

Let \(\overrightarrow{OP}=\frac{\alpha-1}{\alpha}\hat{i}+\hat{j}+\hat{k},\overrightarrow{OQ}=\hat{i}+\frac{\beta-1}{\beta}\hat{j}+\hat{k}\) and \(\overrightarrow{OR}=\hat{i}+\hat{j}+\frac{1}{2}\hat{k}\) be three vector where α, β ∈ R - {0} and 0 denotes the origin. If \((\overrightarrow{OP}\times\overrightarrow{OQ}).\overrightarrow{OR}=0\) and the point (α, β, 2) lies on the plane 3x + 3y - z + l = 0, then the value of l is _______.

Let A₁, B₁, C₁ be three points in the xy -plane. Suppose that the lines A₁C₁ and B₁C₁ are tangents to the curve y² = 8x at A₁ and B₁, respectively. If O = (0,0) and C₁ = −( 4, 0), then which of the following statements is (are) TRUE ?

Consider the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\). Let S (p, q) be a point in the first quadrant such that \(\frac{p^2}{9}+\frac{q^2}{4}>1\). Two tangents are drawn from S to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point T in the fourth quadrant. Let R be the vertex of the ellipse with positive x-coordinate and Obe the center of the ellipse. If the area of the triangle △ORT is \(\frac{3}{2}\) , then which of the following options is correct ?

Let \(S=\left\{a+b\sqrt2:a,b\in Z\right\},T_1=\left\{(-1+\sqrt2)^n:n\in \N\right\},\ \text{and}\ T_2=\left\{(1+\sqrt2)^n:n\in\N\right\}\). Then which of the following statements is (are) TRUE ?

A closed vessel contains 10 g of an ideal gas X at 300 K, which exerts 2 atm pressure. At the same temperature, 80 g of another ideal gas Y is added to it and the pressure becomes 6 atm. The ratio of root mean square velocities of X and Y at 300 K is

Let \(S=\left\{\begin{pmatrix} 0 & 1 & c \\ 1 & a & d\\ 1 & b & e \end{pmatrix}:a,b,c,d,e\in\left\{0,1\right\}\ \text{and} |A|\in \left\{-1,1\right\}\right\}\), where |A| denotes the determinant of A. Then the number of elements in S is _______.

Let \(k\in R\). If \(\)\(\)\(\)\(\lim\limits_{x→0+}(\sin(\sin kx)+\cos x+x)^{\frac{2}{x}}=e^6\), then the value of k is

The option(s) with correct sequence of reagents for the conversion of P to Q is(are)

In the following reaction sequence, the major product Q is
\(\text{L-Glucose}\xrightarrow[ii)Cr_2,775K,10-20 \text{atm}]{i)HI, \Delta}P\xrightarrow[UV]{Cl_2(excess)}Q\)

Let \(a=3\sqrt2\) and \(b=\frac{1}{5^{1/6}\sqrt5}\). If x, y ∈ R are such that
\(3x+2y=\log_a(18)^{\frac{5}{4}}\) and
\(2x-y=\log_b(\sqrt{1080})\), 
then 4x + 5y is equal to _______.