List of top Questions asked in JEE Advanced- 2021

The major product formed in the following reaction is

Let 
$S_{1}=\{( i , j , k ): i , j , k \in\{1,2, \ldots , 10\}\},$ 
$ S_{2}=\{( i , j ): 1 \leq i < j +2 \leq 10, i , j \in\{1,2, \ldots 10\}\},$ 
$ S_{3}=\{( i , j , k , l ): 1 \leq i < j < k < l , i , j , k , l \in\{1,2, \ldots , 10\}\}$ and 
$ S_{4}=\{( i , j , k , l ): i , j , k $ and $ l$ are distinct elements in $\{1,2, \ldots , 10\}\}$.
 If the total number of elements in the set $S_{ r }$ is $n_{ r }, r =1,2,3,4$, then which of the following statements is(are) TRUE?

Let $f: R \rightarrow R$ be defined by $f(x)=\frac{x^{2}-3 x-6}{x^{2}+2 x+4}$. Then which of the following statements is(are) TRUE?

The area of the region \(\{ {(x, y): 0 ≤ x ≤ \frac{9}{4}, 0 ≤ y ≤ 1, x ≥ 3y, x + y ≥ 2}\}\)

For the reaction, $X (s) \rightleftharpoons Y (s)+ Z (g)$, the plot of $\ln \frac{p_{ Z }}{p^{0}}$ versus $\frac{10^{4}}{T}$ is given below (in solid line), where $p_{ z }$ is the pressure (in bar) of the gas $Z$ at temperature $T$ and $p^0=1$ bar 

(Given, $\frac{ d (\ln K)}{ d \left(\frac{1}{T}\right)}=-\frac{\Delta H^{0}}{R}$, where the equilibrium constant, $K=\frac{p_{z}}{p^{0}}$ and the gas constant, $R=8.314 \,J \,K ^{-1} mol ^{-1}$ ) 
The value of $\Delta S^{0}$ (in $J\, K ^{-1} mol ^{-1}$ ) for the given reaction, at $1000\, K$ is ______

For a prism of prism angle 𝜃 = 60°, the refractive indices of the left half and the right half are, respectively, 𝑛₁ and 𝑛₂ (𝑛₂ ≥ 𝑛₁) as shown in the figure. The angle of incidence 𝑖 is chosen such that the incident light rays will have minimum deviation if 𝑛₁  = 𝑛₂  = 𝑛 = 1.5. For the case of unequal refractive indices, 𝑛₁  = 𝑛 and 𝑛₂ = 𝑛 +∆𝑛 (where∆𝑛≪𝑛), the angle of emergence \(𝑒 =𝑖+∆𝑒\). Which of the following statement(s) is (are) correct?

A wide slab consisting of two media of refractive indices $n_{1}$ and $n_{2}$ is placed in air as shown in the figure. A ray of light is incident from medium $n _{1}$ to $n _{2}$ at an angle $\theta$, where $\sin \theta$ is slightly larger than $\frac1 { n _{1}}$. Take refractive index of air as $1$.Which of the following statement(s) is(are) correct?

Correct option(s) for the following sequence of reactions is(are)

A horizontal force $F$ is applied at the centre of mass of a cylindrical object of mass $m$ and radius $R$, perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is $\mu$. The center of mass of the object has an acceleration $a$. The acceleration due to gravity is $g$. Given that the object rolls without slipping, which of the following statement(s) is(are) correct?

A source, approaching with speed $u$ towards the open end of a stationary pipe of length $L$, is emitting a sound of frequency $f _{ s }$. The farther end of the pipe is closed .The speed of sound in air is $v$ and $f _{0}$ is the fundamental frequency of the pipe. For which of the following combination(s) of $u$ and $f_{s}$, will the sound reaching the pipe lead to a resonance?

An ideal gas undergoes a reversible isothermal expansion from the state I to state II followed by a reversible adiabatic expansion from state II to state III. The correct plot(s) representing the changes from the state I to state III is(are) 
(p: pressure, V: volume, T: temperature, H: enthalpy, S: entropy)

Which of the following statement(s) is(are) correct about the spectrum of hydrogen atom?

Let \( \theta_1,  \theta_2, ….,  \theta_{10}\) be positive valued angles (in radian) such that \( \theta_1+  \theta_2+ ….+  \theta_{10} = 2\pi\). Define the complex numbers \(z_1 = 𝑒^{ 𝑖\theta_1}, 𝑧_𝑘 = 𝑧_{𝑘−1}𝑒^ {𝑖\theta_k} \text{for}\  k = 2, 3, …, 10,\) where \(i = \sqrt{-1}\). Consider the statement P and Q given below:
 \(P: \left|z_2 - z_1\right| + \left|z_3 - z_2\right| + \ldots + \left|z_{10} - z_9\right| + \left|z_1 - z_{10}\right| \leq 2\pi\)
\(Q: \left|z_{22} - z_{12}\right| + \left|z_{32} - z_{22}\right| + \ldots + \left|z_{102} - z_{92}\right| + \left|z_{12} - z_{102}\right| \leq 4\pi\)
Then,

The calculated spin only magnetic moments of $\left[ Cr \left( NH _{3}\right)_{6}\right]^{3+}$ and $\left[ CuF _{6}\right]^{3-}$ in BM, respectively, are
(Atomic number of $Cr$ and $Cu$ are $24$ and $29$ , respectively)

For any $3 \times 3$ matrix $M$, let $| M |$ denote the determinant of $M$. Let
$E=\begin{bmatrix}1 & 2 & 3 \\2 & 3 & 4 \\8 & 13 & 18\end{bmatrix}, P=\begin{bmatrix} 1 & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0\end{bmatrix}$ and $F=\begin{bmatrix} 1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3 \end{bmatrix}$
If $Q$ is a non-singular matrix of order $3 \times 3$, then which of the following statements is(are) TRUE?

For any positive integer $n$, let $S_{n}:(0, \infty) \rightarrow R$ be defined by
$S_{n}(x)=\displaystyle\sum_{k=1}^{n} \cot ^{-1}\left(\frac{1+k(k+1) x^{2}}{x}\right),$
where for any $x \in R, \cot ^{-1}(x) \in(0, \pi)$ and $\tan ^{-1}(x) \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Then which of the following statements is(are) TRUE?

Consider a triangle \(Δ\) whose two sides lie on the x-axis and the line \(x + y + 1 = 0\). If the orthocenter of \(Δ\) is (1, 1), then the equation of the circle passing through the vertices of the triangle \(Δ\) is;

Consider a helium (He) atom that absorbs a photon of wavelength 330 nm. The change in the velocity (in cm s^-1) of the He atom after the photon absorption is_____.
(Assume: Momentum is conserved when the photon is absorbed.
Use: Planck constant = 6.6 × 10^-34 J s, Avogadro number = 6 × 1023 mol^-1, Molar mass of He = 4 g mol^-1)

A projectile is thrown from a point $O$ on the ground at an angle $45^{\circ}$ from the vertical and with a speed $5 \sqrt{2} \,m / s$. The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground, $05\, s$ after the splitting. The other part, $t$ seconds after the splitting, falls to the ground at a distance $x$ meters from the point $O$. The acceleration due to gravity $g =10 \,m / s ^{2}$. The value of $t$ is ____

A long straight wire carries a current, $I =2$ ampere A semi-circular conducting rod is placed beside it on two conducting parallel rails of negligible resistance Both the rails are parallel to the wire The wire, the rod and the rails lie in the same horizontal plane, as shown in the figure Two ends of the semi-circular rod are at distances $1 \,cm$ and $4 \,cm$ from the wire At time $t =0$, the rod starts moving on the rails with a speed $v =30\, m / s$ (see the figure)
 A resistor $R =14 \,\Omega$ and a capacitor $C _{0}=50\, \mu F$ are connected in series between the rails At time $t =0, C _{0}$ is uncharged. Which of the following statement(s) is(are) correct? \([\mu_0 = 4 \pi \times 10^{-7}\) SI units. Take \(ln_2=0.7]\)

A horizontal force $F$ is applied at the centre of mass of a cylindrical object of mass $m$ and radius $R$, perpendicular to its axis as shown in the figure The coefficient of friction between the object and the ground is $\mu$ The center of mass of the object has an acceleration $a$ The acceleration due to gravity is $g$ Given that the object rolls without slipping, which of the following statement(s) is(are) correct?

Given:

The compound(s), which on reaction with HNO₃ will give the product having a degree of rotation, [α]D = –52.7º is(are);

A physical quantity $\vec{ S }$ is defined as $\vec{ S }=(\vec{ E } \times \vec{ B }) / \mu_{0}$, where $\vec{ E }$ is electric field, $\vec{ B }$ is magnetic field and $\mu_{0}$ is the permeability of free space. The dimensions of $\vec{ S }$ are the same as the dimensions of which of the following quantity(ies)?

Two point charges $- Q$ and $+ Q / \sqrt{3}$ are placed in the $xy$-plane at the origin $(0,0)$ and a point $(2,0)$, respectively, as shown in the figure This results in an equipotential circle of radius $R$ and potential $V=0$ in the $x y$-plane with its center at $(b, 0)$ All lengths are measured in meters The value of $b$ is _______ meter