List of top Questions asked in JEE Advanced

Consider a configuration of $n$ identical units, each consisting of three layers The first layer is a column of air of height $h=\frac{1}{3} cm$, and the second and third layers are of equal thickness $d=\frac{\sqrt{3}-1}{2} cm$, and refractive indices $\mu_1=\sqrt{\frac{3}{2}}$ and $\mu_2=\sqrt{3}$, respectively. A light source $O$ is placed on the top of the first unit, as shown in the figure. A ray of light from $O$ is incident on the second layer of the first unit at an angle of $\theta=60^{\circ}$ to the normal. For a specific value of $n$, the ray of light emerges from the bottom of the configuration at a distance $l=\frac{8}{\sqrt{3}} cm$, as shown in the figure The value of $n$ is ______

Let $\hat{i}, \hat{j}$ and $\hat{k}$ be the unit vectors along the three positive coordinate axes Let $ \overrightarrow{ a }=3 \hat{ i }+\hat{ j }-\hat{ k }, $ $\overrightarrow{ b }=\hat{ i }+ b _2 \hat{ j }+ b _3 \hat{ k }, b _2, b _3 \in R , $ $ \overrightarrow{ c }= c _1 \hat{ i }+ c _2 \hat{ j }+ c _3 \hat{ k }, c _1, c _2, c _3 \in R$ be three vectors such that $b_2 b_3>0, \vec{a} \cdot \vec{b}=0$ and $\begin{pmatrix}0 & -c_3 & c_2 \\c_3 & 0 & -c_1 \\-c_2 & c_1 & 0\end{pmatrix}\begin{pmatrix} 1 \\b_2 \\b_3\end{pmatrix}=\begin{pmatrix}3-c_1 \\1-c_2 \\-1-c_3\end{pmatrix} $. Then, which of the following is/are TRUE?

Let $\beta$ be a real number. Consider the matrix
$A=\begin{pmatrix}\beta & 0 & 1 \\2 & 1 & -2 \\3 & 1 & -2\end{pmatrix}$
If $A^7-(\beta-1) A^6-\beta A^5$ is a singular matrix, then the value of $9 \beta$ is __.

Let $G$ be a circle of radius $R>0$. Let $G _1, G _2, \ldots, G _{ n }$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G _1, G _2, \ldots, G _{ n }$ touches the circle $G$ externally. Also, for $i =1,2, \ldots, n -1$, the circle $G _{ i }$ touches $G _{ i +1}$ externally, and $G _{ n }$ touches $G _1$ externally. Then, which of the following statements is/are TRUE?

Considering the following reaction sequence, the correct statement(s) is(are)

The compound(s) which react(s) with $NH _3$ to give boron nitride (BN) is(are)

The correct option(s) related to the extraction of iron from its ore in the blast furnace operating in the temperature range $900-1500 K$ is(are)

Consider the hyperbola $\frac{x^2}{100}-\frac{y^2}{64}=1$ with foci at $S$ and $S_1$, where $S$ lies on the positive $x$-axis Let $P$ be a point on the hyperbola, in the first quadrant Let $\angle \operatorname{SPS}_1=\alpha$, with $\alpha<\frac{\pi}{2}$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola, intersects the straight line $S _1 P$ at $P _1$ Let $\delta$ be the distance of $P$ from the straight line $SP _1$, and $\beta= S _1 P$. Then the greatest integer less than or equal to $\frac{\beta \delta}{9} \sin \frac{\alpha}{2}$ is _____.

A particle of mass $1\, kg$ is subjected to a force which depends on the position as $\vec{F}=-k(x \hat{i}+y \hat{j}) kg\, ms ^{-2}$ with $k=1 \, kgs ^{-2}$ . At time $t=0$, the particle's position $\vec{r}=\left(\frac{1}{\sqrt{2}} \hat{i}+\sqrt{2} \hat{j}\right) m$ and its velocity $\vec{v}=\left(-\sqrt{2} \hat{i}+\sqrt{2} \hat{j}+\frac{2}{\pi} \hat{k}\right) m s^{-1}$ . Let $v_x$ and $v_y$ denote the $x$ and the $y$ components of the particle's velocity, respectively. Ignore gravity. When $z=05\, m$, the value of (xv_y – yv_x) is _______ m²s^–1.

The number of $- CH _2$ - (methylene) groups in the product formed from the following reaction sequence is_____

In the given $P-V$ diagram, a monoatomic gas $\left(\gamma=\frac{5}{3}\right)$ is first compressed adiabatically from state $A$ to state $B$. Then it expands isothermally from state $B$ to state $C$ [Given: $(\frac{1}{3})^{0.6}=05, \ln 2 \simeq 07$].

Which of the following statement(s) is(are) correct?

A bubble has surface tension $S$. The ideal gas inside the bubble has ratio of specific heats $\gamma=\frac{5}{3}$. The bubble is exposed to the atmosphere and it always retains its spherical shape. When the atmospheric pressure is $P_{a 1}$, the radius of the bubble is found to be $r_1$ and the temperature of the enclosed gas is $T_1$. When the atmospheric pressure is $P_{a 2}$, the radius of the bubble and the temperature of the enclosed gas are $r_2$ and $T_2$, respectively.Which of the following statement(s) is(are) correct?

Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of $(\bar{z})^2+\frac{1}{ z ^2}$ are integers, then which of the following is/are possible value(s) of $|z|$?

In a radioactive decay chain reaction, ${ }_{90}^{230} Th$ nucleus decays into ${ }_{84}^{214} Po$ nucleus. The ratio of the number of $\alpha$ to number of $\beta^{-}$particles emitted in this process is _____

Among the following, the correct statement(s) about polymers is(are)

Consider the functions $f, g: R \rightarrow R$ defined by
$f(x)=x^2+\frac{5}{12} \text { and } g(x)= \begin{cases} 2\left(1-\frac{4|x|}{3}\right), & |x| \leq \frac{3}{4}, \\0, & |x|>\frac{3}{4}. \end{cases}$
If $\alpha$ is the area of the region $\left\{( x , y ) \in R \times R :| x | \leq \frac{3}{4}, 0 \leq y \leq \min \{f( x ), g( x )\}\right\},$ then the value of $9 \alpha$ is ____.

Treatment of D-glucose with aqueous NaOH results in a mixture of monosaccharides, which are

Let $\bar{z}$ denote the complex conjugate of a complex number $z$ and let $i=\sqrt{-1}$ In the set of complex numbers, the number of distinct roots of the equation $\bar{z}-z^2=i\left(\bar{z}+z^2\right)$ is ___

Let $l_1, l_2, \ldots, l_{100}$ be consecutive terms of an arithmetic progression with common difference $d_1$, and let $w_1, w_2, \ldots, w_{100}$ be consecutive terms of another arithmetic progression with common difference $d_2$, where $d_1 d_2=10$ For each $i=1,2, \ldots, 100$, let $R_l$ be a rectangle with length $l_i$, width $w_i$ and area $A i$ If $A_{51}-A_{50}=1000$, then the value of $A_{100}-A_{90}$ is ___

Dissolving $1.24 \,g$ of white phosphorous in boiling $NaOH$ solution in an inert atmosphere gives a gas $Q$ The amount of $CuSO _4$ (in g) required to completely consume the gas $Q$ is ___ [Given : Atomic mass of $H =1, O =16, Na =23, P =31, S =32, Cu =63$ ]

The reaction of HClO₃ with HCl gives a paramagnetic gas, which upon reaction with O₃ produces

Two spherical stars $A$ and $B$ have densities $\rho_A$ and $\rho_B$, respectively $A$ and $B$ have the same radius, and their masses $M_A$ and $M_B$ are related by $M_B=2 M_A$ Due to an interaction process, star $A$ loses some of its mass, so that its radius is halved, while its spherical shape is retained, and its density remains $\rho_A$ The entire mass lost by $A$ is deposited as a thick spherical shell on $B$ with the density of the shell being $\rho_A$ If $v_A$ and $v_B$ are the escape velocities from $A$ and $B$ after the interaction process, the ratio $\frac{v_B}{v_A}=\sqrt{\frac{10 n}{15^{1 / 3}}}$ The value of $n$ is ___

Consider the following reaction

On estimation of bromine in $100 \,g$ of $R$ using Carius method, the amount of $AgBr$ formed (in $g$ ) is __ [Given : Atomic mass of $H =1, C =12, O =16, P =31, Br =80, Ag =108$ ]

Dissolving $1.24 \,g$ of white phosphorous in boiling $NaOH$ solution in an inert atmosphere gives a gas $Q$ The amount of $CuSO _4$ (in g) required to completely consume the gas $Q$ is ___ [Given : Atomic mass of $H =1, O =16, Na =23, P =31, S =32, Cu =63$ ]