JEE Advanced 2021 Question Paper: Download Question Paper with Answer Key PDFs

1.

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 _____.

2.

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

• \((2n,-n,-n,-n)\)
• \((n,-n,-2n,-n)\)
• \((n,-n,-n,-2n)\)
• \((2n,-n,-2n,-2n)\)

3.

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

• \(-0.5m,5kg \ \frac{m}{s}\)
• \(0.5m,0kg \ \frac{m}{s}\)
• \(0.5m,0kg \ \frac{m}{s}\)
• \(0.5m,0kg \ \frac{m}{s}\)

4.

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?

5.

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.]

• \(\frac{q^2}{4\pi\epsilon_0R^3m}\)
• \(\frac{q^2}{32\pi\epsilon_0R^3m}\)
• \(\frac{q^2}{8\pi\epsilon_0R^3m}\)
• \(\frac{q^2}{16\pi\epsilon_0R^3m}\)