List of top Questions asked in JEE Advanced

Figure 1 shows the configuration of main scale and Vernier scale before measurement. Fig. 2 shows the configuration corresponding to the measurement of diameter $ D $ of a tube. The measured value of $ D $ is:

Let $ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $ and $ P = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} $. Let $ Q = \begin{pmatrix} x & y \\ z & 4 \end{pmatrix} $ for some non-zero real numbers $ x, y, z $, for which there is a $ 2 \times 2 $ matrix $ R $ with all entries being non-zero real numbers, such that $$ QR = RP $$ Then which of the following statements is (are) TRUE?

Consider the vectors $$ \vec{x} = \hat{i} + 2\hat{j} + 3\hat{k},\quad \vec{y} = 2\hat{i} + 3\hat{j} + \hat{k},\quad \vec{z} = 3\hat{i} + \hat{j} + 2\hat{k}. $$ For two distinct positive real numbers $ \alpha $ and $ \beta $, define $$ \vec{X} = \alpha \vec{x} + \beta \vec{y} - \vec{z},\quad \vec{Y} = \alpha \vec{y} + \beta \vec{z} - \vec{x},\quad \vec{Z} = \alpha \vec{z} + \beta \vec{x} - \vec{y}. $$ If the vectors $ \vec{X}, \vec{Y}, \vec{Z} $ lie in a plane, then the value of $ \alpha + \beta - 3 $ is ________.

The center of a disk of radius $ r $ and mass $ m $ is attached to a spring of spring constant $ k $, inside a ring of radius $ R>r $ as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following Hooke’s law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as $ T = \frac{2\pi}{\omega} $. The correct expression for $ \omega $ is ( $ g $ is the acceleration due to gravity):

A positive point charge of $10^{-8}$ C is kept at a distance of 20 cm from the center of a neutral conducting sphere of radius 10 cm. The sphere is then grounded and the charge on the sphere is measured. The grounding is then removed and subsequently the point charge is moved by a distance of 10 cm further away from the center of the sphere along the radial direction. Taking $ \frac{1}{4\pi\varepsilon_0} = 9 \times 10^9 \, \text{Nm}^2/\text{C}^2 $, which of the following statements is/are correct:

Consider a star of mass $ m_2 $ kg revolving in a circular orbit around another star of mass $ m_1 $ kg with $ m_1 \gg m_2 $. The heavier star slowly acquires mass from the lighter star at a constant rate of $ \gamma $ kg/s. In this transfer process, there is no other loss of mass. If the separation between the centers of the stars is $ r $, then its relative rate of change $ \frac{1}{r} \frac{dr}{dt} $ (in s$^{-1}$) is given by:

Let $$ \alpha = \frac{1}{\sin 60^\circ \sin 61^\circ} + \frac{1}{\sin 62^\circ \sin 63^\circ} + \cdots + \frac{1}{\sin 118^\circ \sin 119^\circ}. $$ Then the value of $$ \left( \frac{\csc 1^\circ}{\alpha} \right)^2 $$ is \rule{1cm}{0.15mm}.

The solubility of barium iodate in an aqueous solution prepared by mixing 200 mL of 0.010 M barium nitrate with 100 mL of 0.10 M sodium iodate is $X \times 10^{-6} \, \text{mol dm}^{-3}$. The value of $X$ is ------.

Use: Solubility product constant $(K_{sp})$ of barium iodate = $1.58 \times 10^{-9}$

At 25$^\circ$C, the concentration of H$^+$ ions in $ 1.00 \times 10^{-3} \, \text{M} $ aqueous solution of a weak monobasic acid having acid dissociation constant $ K_a = 4.00 \times 10^{-11} $ is $ X \times 10^{-7} \, \text{M} $. The value of $ X $ is ____. {Use: Ionic product of water $ K_w = 1.00 \times 10^{-14} $ at 25$^\circ$C

Let $ x_0 $ be the real number such that $ e^{x_0} + x_0 = 0 $. For a given real number $ \alpha $, define $$ g(x) = \frac{3xe^x + 3x - \alpha e^x - \alpha x}{3(e^x + 1)} $$ for all real numbers $ x $. Then which one of the following statements is TRUE?

A projectile of mass 200 g is launched in a viscous medium at an angle $60^\circ$ with the horizontal, with an initial velocity of 270 m/s. It experiences a viscous drag force $\vec{F} = -c \vec{v}$ where the drag coefficient $c = 0.1 \, \text{kg/s}$ and $\vec{v}$ is the instantaneous velocity of the projectile. The projectile hits a vertical wall after 2 s. Taking $e = 2.7$, the horizontal distance of the wall from the point of projection (in m) is ____

Two identical concave mirrors each of focal length $ f $ are facing each other as shown. A glass slab of thickness $ t $ and refractive index $ n_0 $ is placed equidistant from both mirrors on the principal axis. A monochromatic point source $ S $ is placed at the center of the slab. For the image to be formed on $ S $ itself, which of the following distances between the two mirrors is/are correct:

A geostationary satellite above the equator is orbiting around the earth at a fixed distance $r_1$ from the center of the earth. A second satellite is orbiting in the equatorial plane in the opposite direction to the earth’s rotation, at a distance $r_2$ from the center of the earth, such that $r_1 = 1.21 \, r_2$. The time period of the second satellite as measured from the geostationary satellite is $\frac{24}{p}$ hours. The value of $p$ is ___

The major products obtained from the reactions in List-II are the reactants for the named reactions mentioned in List-I. Match each entry in List-I with the appropriate entry in List-II and choose the correct option.

As shown in the figures, a uniform rod $ OO' $ of length $ l $ is hinged at the point $ O $ and held in place vertically between two walls using two massless springs of the same spring constant. The springs are connected at the midpoint and at the top-end $ (O') $ of the rod, as shown in Fig. 1, and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is $ f_1 $. On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2, and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is $ f_2 $. Ignoring gravity and assuming motion only in the plane of the diagram, the value of $\frac{f_1}{f_2}$ is:

An electrochemical cell is fueled by the combustion of butane at 1 bar and 298 K. Its cell potential is $ \frac{X}{F} \times 10^3 $ volts, where $ F $ is the Faraday constant. The value of $ X $ is ____.

Use: Standard Gibbs energies of formation at 298 K are:

$ \Delta_f G^\circ_{CO_2} = -394 \, \text{kJ mol}^{-1}; \quad \Delta_f G^\circ_{water} = -237 \, \text{kJ mol}^{-1}; \quad \Delta_f G^\circ_{butane} = -18 \, \text{kJ mol}^{-1} $

An ideal monatomic gas of $ n $ moles is taken through a cycle $ WXYZW $ consisting of consecutive adiabatic and isobaric quasi-static processes, as shown in the schematic $ V-T $ diagram. The volume of the gas at $ W, X $ and $ Y $ points are, $ 64 \, \text{cm}^3 $, $ 125 \, \text{cm}^3 $ and $ 250 \, \text{cm}^3 $, respectively. If the absolute temperature of the gas $ T_W $ at the point $ W $ is such that $ n R T_W = 1 \, J $ ($ R $ is the universal gas constant), then the amount of heat absorbed (in J) by the gas along the path $ XY $ is

In a Young’s double slit experiment, a combination of two glass wedges $ A $ and $ B $, having refractive indices 1.7 and 1.5, respectively, are placed in front of the slits, as shown in the figure. The separation between the slits is $ d = 2 \text{ mm} $ and the shortest distance between the slits and the screen is $ D = 2 \text{ m} $. Thickness of the combination of the wedges is $ t = 12 \, \mu\text{m} $. The value of $ l $ as shown in the figure is 1 mm. Neglect any refraction effect at the slanted interface of the wedges. Due to the combination of the wedges, the central maximum shifts (in mm) with respect to 0 by ____

The efficiency of a Carnot engine operating with a hot reservoir kept at a temperature of 1000 K is 0.4. It extracts 150 J of heat per cycle from the hot reservoir. The work extracted from this engine is being fully used to run a heat pump which has a coefficient of performance 10. The hot reservoir of the heat pump is at a temperature of 300 K. Which of the following statements is/are correct:

The sum of the spin-only magnetic moment values (in B.M.) of $[\text{Mn}(\text{Br})_6]^{3-}$ and $[\text{Mn}(\text{CN})_6]^{3-}$ is ____.

A linear octasaccharide (molar mass = 1024 g mol$^{-1}$) on complete hydrolysis produces three monosaccharides: ribose, 2-deoxyribose and glucose. The amount of 2-deoxyribose formed is 58.26 % (w/w) of the total amount of the monosaccharides produced in the hydrolyzed products. The number of ribose unit(s) present in one molecule of octasaccharide is _____.
Use: Molar mass (in g mol$^{-1}$): ribose = 150, 2-deoxyribose = 134, glucose = 180; Atomic mass (in amu): H = 1, O = 16

In neutral medium, permanganate reacts with iodide to give:

The left and right compartments of a thermally isolated container of length $L$ are separated by a thermally conducting, movable piston of area $A$. The left and right compartments are filled with $\frac{3}{2}$ and 1 moles of an ideal gas, respectively. In the left compartment the piston is attached by a spring with spring constant $k$ and natural length $\frac{2L}{5}$. In thermodynamic equilibrium, the piston is at a distance $\frac{L}{2}$ from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is $P = \frac{kL}{A} \alpha$, then the value of $\alpha$ is ____

A solid glass sphere of refractive index $ n = \sqrt{3} $ and radius $ R $ contains a spherical air cavity of radius $ \dfrac{R}{2} $, as shown in the figure. A very thin glass layer is present at the point $ O $ so that the air cavity (refractive index $ n = 1 $) remains inside the glass sphere. An unpolarized, unidirectional and monochromatic light source $ S $ emits a light ray from a point inside the glass sphere towards the periphery of the glass sphere. If the light is reflected from the point $ O $ and is fully polarized, then the angle of incidence at the inner surface of the glass sphere is $ \theta $. The value of $ \sin \theta $ is ____

The reaction sequence given below is carried out with 16 moles of X. The yield of the major product in each step is given below the product in parentheses. The amount (in grams) of S produced is ____.

Use: Atomic mass (in amu): H = 1, C = 12, O = 16, Br = 80