List of top Questions asked in JEE Advanced

For the reaction sequence given below, the correct statement(s) is(are):

The complete hydrolysis of $ \text{ICl} $, $ \text{ClF}_3 $, and $ \text{BrF}_5 $, respectively, gives

Monocyclic compounds $ P, Q, R $ and $ S $ are the major products formed in the reaction sequences given below.
The product having the highest number of unsaturated carbon atom(s) is:

During sodium nitroprusside test of sulphide ion in an aqueous solution, one of the ligands coordinated to the metal ion is converted to

The correct reaction/reaction sequence that would produce a dicarboxylic acid as the major product is

The correct statement(s) about intermolecular forces is (are)

An audio transmitter (T) and a receiver (R) are hung vertically from two identical massless strings of length 8 m with their pivots well separated along the $X$ axis. They are pulled from the equilibrium position in opposite directions along the $X$ axis by a small angular amplitude $\theta_0 = \cos^{-1}(0.9)$ and released simultaneously. If the natural frequency of the transmitter is 660 Hz and the speed of sound in air is 330 m/s, the maximum variation in the frequency (in Hz) as measured by the receiver (Take the acceleration due to gravity $g = 10\, \text{m/s}^2$) is ___

A conducting solid sphere of radius $ R $ and mass $ M $ carries a charge $ Q $. The sphere is rotating about an axis passing through its center with a uniform angular speed $\omega$. The ratio of the magnitudes of the magnetic dipole moment to the angular momentum about the same axis is given as $\alpha \frac{Q}{2M}$. The value of $\alpha$ is ___

A hydrogen atom, initially at rest in its ground state, absorbs a photon of frequency $\nu_1$ and ejects the electron with a kinetic energy of 10 eV. The electron then combines with a positron at rest to form a positronium atom in its ground state and simultaneously emits a photon of frequency $\nu_2$. The center of mass of the resulting positronium atom moves with a kinetic energy of 5 eV. It is given that the positron has the same mass as that of electron and the positronium atom can be considered as a Bohr atom, in which the electron and the positron orbit around their center of mass. Considering no other energy loss during the whole process, the difference between the two photon energies (in eV) is ___

Two co-axial conducting cylinders of same length $ \ell $ with radii $ \sqrt{2}R $ and $ 2R $ are kept, as shown in Fig. 1. The charge on the inner cylinder is $ Q $ and the outer cylinder is grounded. The annular region between the cylinders is filled with a material of dielectric constant $ \kappa = 5 $. Consider an imaginary plane of the same length $ \ell $ at a distance $ R $ from the common axis of the cylinders. This plane is parallel to the axis of the cylinders. Ignoring edge effects, the flux of the electric field through the plane is $ (\varepsilon_0 \text{ is the permittivity of free space}) $:

Six infinitely large and thin non-conducting sheets are fixed in configurations I and II. As shown in the figure, the sheets carry uniform surface charge densities which are indicated in terms of $\sigma_0$. The separation between any two consecutive sheets is $1 \, \mu m$. The various regions between the sheets are denoted as 1, 2, 3, 4 and 5. If $\sigma_0 = 9 \, \mu C/m^2$, then which of the following statements is/are correct? (Take permittivity of free space $\epsilon_0 = 9 \times 10^{-12} \, \text{F/m}$)

For a non-zero complex number $ z $, let $\arg(z)$ denote the principal argument of $ z $, with $-\pi < \arg(z) \leq \pi$. Let $\omega$ be the cube root of unity for which $0 < \arg(\omega) < \pi$. Let $$ \alpha = \arg \left( \sum_{n=1}^{2025} (-\omega)^n \right). $$ Then the value of $\frac{3 \alpha}{\pi}$ is _____.

If $$ \alpha = \int_{\frac{1}{2}}^{2} \frac{\tan^{-1} x}{2x^2 - 3x + 2} \, dx, $$ then the value of $ \sqrt{7} \tan \left( \frac{2\alpha \sqrt{7}}{\pi} \right) $ is.
(Here, the inverse trigonometric function $ \tan^{-1} x $ assumes values in $ \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) $.)

Let $\mathbb{R}$ denote the set of all real numbers. Let $f : \mathbb{R} \to \mathbb{R}$ and $g : \mathbb{R} \to (0,4)$ be functions defined by $$ f(x) = \log_e (x^2 + 2x + 4), \quad \text{and} \quad g(x) = \frac{4}{1 + e^{-2x}}. $$ Define the composite function $f \circ g^{-1}$ by $(f \circ g^{-1})(x) = f(g^{-1}(x))$, where $g^{-1}$ is the inverse of the function $g$. Then the value of the derivative of the composite function $f \circ g^{-1}$ at $x=2$ is _____.

A factory has a total of three manufacturing units, $ M_1, M_2, M_3 $, which produce bulbs independently of each other. The units $ M_1, M_2, M_3 $ produce bulbs in the proportions $ 2 : 2 : 1 $, respectively. It is known that 20% of the bulbs produced in the factory are defective. It is also known that, of all the bulbs produced by $ M_1 $, 15% are defective. Suppose that, if a randomly chosen bulb produced in the factory is found to be defective, the probability that it was produced by $ M_2 $ is $ \frac{2}{5} $. If a bulb is chosen randomly from the bulbs produced by $ M_3 $, then the probability that it is defective is ________.

A temperature difference can generate e.m.f. in some materials. Let $ S $ be the e.m.f. produced per unit temperature difference between the ends of a wire, $ \sigma $ the electrical conductivity and $ \kappa $ the thermal conductivity of the material of the wire. Taking $ M, L, T, I $ and $ K $ as dimensions of mass, length, time, current and temperature, respectively, the dimensional formula of the quantity $ Z = \frac{S^2 \sigma}{\kappa} $ is:

Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.

Let $ P(x_1, y_1) $ and $ Q(x_2, y_2) $ be two distinct points on the ellipse $$ \frac{x^2}{9} + \frac{y^2}{4} = 1 $$ such that $ y_1 > 0 $, and $ y_2 > 0 $. Let $ C $ denote the circle $ x^2 + y^2 = 9 $, and $ M $ be the point $ (3, 0) $. Suppose the line $ x = x_1 $ intersects $ C $ at $ R $, and the line $ x = x_2 $ intersects $ C $ at $ S $, such that the $ y $-coordinates of $ R $ and $ S $ are positive. Let $ \angle ROM = \frac{\pi}{6} $ and $ \angle SOM = \frac{\pi}{3} $, where $ O $ denotes the origin $ (0, 0) $. Let $ |XY| $ denote the length of the line segment $ XY $. Then which of the following statements is (are) TRUE?

Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.

Let $ S $ denote the locus of the midpoints of those chords of the parabola $ y^2 = x $, such that the area of the region enclosed between the parabola and the chord is $ \frac{4}{3} $. Let $ \mathcal{R} $ denote the region lying in the first quadrant, enclosed by the parabola $ y^2 = x $, the curve $ S $, and the lines $ x = 1 $ and $ x = 4 $.
Then which of the following statements is (are) TRUE?

Let $ \mathbb{R} $ denote the set of all real numbers. Then the area of the region $$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x},\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0 \right\} $$ is

Let $ S $ denote the locus of the point of intersection of the pair of lines $$ 4x - 3y = 12\alpha,\quad 4\alpha x + 3\alpha y = 12, $$ where $ \alpha $ varies over the set of non-zero real numbers. Let $ T $ be the tangent to $ S $ passing through the points $ (p, 0) $ and $ (0, q) $, $ q > 0 $, and parallel to the line $ 4x - \frac{3}{\sqrt{2}} y = 0 $.
Then the value of $ pq $ is

Let $ \vec{w} = \hat{i} + \hat{j} - 2\hat{k} $, and $ \vec{u} $ and $ \vec{v} $ be two vectors, such that $ \vec{u} \times \vec{v} = \vec{w} $ and $ \vec{v} \times \vec{w} = \vec{u} $. Let $ \alpha, \beta, \gamma $, and $ t $ be real numbers such that: $$ \vec{u} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}, $$ and the system of equations is: $$ -t\alpha + \beta + \gamma = 0 \quad \cdots (1) $$ $$ \alpha - t\beta + \gamma = 0 \quad \cdots (2) $$ $$ \alpha + \beta - t\gamma = 0 \quad \cdots (3) $$ Match each entry in List-I to the correct entry in List-II and choose the correct option.
List-I

[(P)] $ |\vec{v}|^2 $ is equal to
[(Q)] If $ \alpha = \sqrt{3} $, then $ \gamma^2 $ is equal to
[(R)] If $ \alpha = \sqrt{3} $, then $ (\beta + \gamma)^2 $ is equal to
[(S)] If $ \alpha = \sqrt{2} $, then $ t + 3 $ is equal to

List-II

[(1)] 0
[(2)] 1
[(3)] 2
[(4)] 3
[(5)]

Consider the following frequency distribution:

Value	4	5	8	9	6	12	11
Frequency	5	$ f_1 $	$ f_2 $	2	1	1	3

Suppose that the sum of the frequencies is 19 and the median of this frequency distribution is 6.

For the given frequency distribution, let:

$ \alpha $ denote the mean deviation about the mean
$ \beta $ denote the mean deviation about the median
$ \sigma^2 $ denote the variance

Match each entry in List-I to the correct entry in List-II and choose the correct option.

List-I

(P) $ 7f_1 + 9f_2 $ is equal to
(Q) $ 19\alpha $ is equal to
(R) $ 19\beta $ is equal to
(S) $ 19\sigma^2 $ is equal to

List-II

(1) 146
(2) 47
(3) 48
(4) 145
(5) 55

Let $ \mathbb{R} $ denote the set of all real numbers. Let $ z_1 = 1 + 2i $ and $ z_2 = 3i $ be two complex numbers, where $ i = \sqrt{-1} $. Let $$ S = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + iy - z_1| = 2|x + iy - z_2|\}. $$ Then which of the following statements is (are) TRUE?