List of top Questions asked in JEE Advanced

Let $ \mathbb{N} $ denote the set of all natural numbers, and $ \mathbb{Z} $ denote the set of all integers. Consider the functions $ f : \mathbb{N} \to \mathbb{Z} $ and $ g : \mathbb{Z} \to \mathbb{N} $ defined by $$ f(n) = \begin{cases} \frac{n + 1}{2}, & \text{if } n \text{ is odd} \\ \frac{4 - n}{2}, & \text{if } n \text{ is even} \end{cases} \quad \text{and} \quad g(n) = \begin{cases} 3 + 2n, & \text{if } n \ge 0 \\ -2n, & \text{if } n<0 \end{cases} $$ Define $ (g \circ f)(n) = g(f(n)) $ for all $ n \in \mathbb{N} $, and $ (f \circ g)(n) = f(g(n)) $ for all $ n \in \mathbb{Z} $. Then which of the following statements is (are) TRUE?

Consider the matrix $$ P = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}. $$ Let the transpose of a matrix $ X $ be denoted by $ X^T $. Then the number of $ 3 \times 3 $ invertible matrices $ Q $ with integer entries, such that $$ Q^{-1} = Q^T \quad \text{and} \quad PQ = QP, $$ is:

Let $ \mathbb{R} $ denote the set of all real numbers. Define the function $ f: \mathbb{R} \to \mathbb{R} $ by $$ f(x) = \begin{cases} 2 - 2x^2 - x^2 \sin\left(\frac{1}{x}\right), & \text{if } x \ne 0, \\ 2, & \text{if } x = 0. \end{cases} $$ Then which one of the following statements is TRUE?

One of the products formed from the reaction of permanganate ion with iodide ion in neutral aqueous medium is:

The correct order of the wavelength maxima of the absorption band in the ultraviolet-visible region for the given complexes is:

List-I shows various functional dependencies of energy $ E $ on the atomic number $ Z $. Energies associated with certain phenomena are given in List-II. Choose the option that describes the correct match between the entries in List-I to those in List-II.

Consider the depicted hydrogen (H) in the hydrocarbons given below. The most acidic hydrogen (H) is:

Considering ideal gas behavior, the expansion work done (in kJ) when 144 g of water is electrolyzed completely under constant pressure at 300 K is ____. Use: Universal gas constant $ R = 8.3 \, \text{J K}^{-1} \text{mol}^{-1} $; Atomic mass (in amu): H = 1, O = 16

The monomer (X) involved in the synthesis of Nylon 6,6 gives positive carbylamine test. If 10 moles of X are analyzed using Dumas method, the amount (in grams) of nitrogen gas evolved is ____. Use: Atomic mass of N (in amu) = 14

The correct match of the group reagents in List-I for precipitating the metal ion given in List-II from solutions is:

List-I	List-II
(P) Passing H₂S in the presence of NH₄OH	(1) Cu²⁺
(Q) (NH₄)₂CO₃ in the presence of NH₄OH	(2) Al³⁺
(R) NH₄OH in the presence of NH₄Cl	(3) Mn²⁺
(S) Passing H₂S in the presence of dilute HCl	(4) Ba²⁺ (5) Mg²⁺

Match the compounds in List-I with the appropriate observations in List-II and choose the correct option.

Molar volume ($ V_m $) of a van der Waals gas can be calculated by expressing the van der Waals equation as a cubic equation with $ V_m $ as the variable. The ratio (in mol dm$^{-3}$) of the coefficient of $ V_m^2 $ to the coefficient of $ V_m $ for a gas having van der Waals constants $ a = 6.0 \, \text{dm}^6 \, \text{atm mol}^{-2} $ and $ b = 0.060 \, \text{dm}^3 \, \text{mol}^{-1} $ at 300 K and 300 atm is ____. Use: Universal gas constant $ R = 0.082 \, \text{dm}^3 \, \text{atm mol}^{-1} \, \text{K}^{-1} $

In an electrochemical cell, dichromate ions in aqueous acidic medium are reduced to $ \text{Cr}^{3+} $. The current (in amperes) that flows through the cell for 48.25 minutes to produce 1 mole of $ \text{Cr}^{3+} $ is ____. {Use: 1 Faraday $ = 96500 \, \text{C mol}^{-1} $

Consider an electron in the $ n = 3 $ orbit of a hydrogen-like atom with atomic number $ Z $. At absolute temperature $ T $, a neutron having thermal energy $ k_B T $ has the same de Broglie wavelength as that of this electron. If this temperature is given by $$ T = \frac{Z^2 h^2}{\alpha \pi^2 a_0^2 m_N k_B} $$ (where $ h $ is Planck’s constant, $ k_B $ is Boltzmann’s constant, $ m_N $ is the mass of the neutron, and $ a_0 $ is the Bohr radius), then the value of $ \alpha $ is ___.

A single slit diffraction experiment is performed to determine the slit width using the equation, $ \dfrac{b d}{D} = m \lambda $, where $ b $ is the slit width, $ D $ the shortest distance between the slit and the screen, $ d $ the distance between the $ m^{\text{th}} $ diffraction maximum and the central maximum, and $ \lambda $ is the wavelength. $ D $ and $ d $ are measured with scales of least count of 1 cm and 1 mm, respectively. The values of $ \lambda $ and $ m $ are known precisely to be $ 600 \, \text{nm} $ and 3, respectively. The absolute error (in $ \mu\text{m} $) in the value of $ b $ estimated using the diffraction maximum that occurs for $ m = 3 $ with $ d = 5 \, \text{mm} $ and $ D = 1 \, \text{m} $ is ___.

A cube of unit volume contains $ 35 \times 10^7 $ photons of frequency $ 10^{15} \, \text{Hz} $. If the energy of all the photons is viewed as the average energy being contained in the electromagnetic waves within the same volume, then the amplitude of the magnetic field is $ \alpha \times 10^{-9} \, \text{T} $. Taking permeability of free space $ \mu_0 = 4\pi \times 10^{-7} \, \text{Tm/A} $, Planck’s constant $ h = 6 \times 10^{-34} \, \text{Js} $ and $ \pi = \frac{22}{7} $, the value of $ \alpha $ is ____

Let $ \mathbb{R} $ denote the set of all real numbers. For a real number $ x $, let $ \lfloor x \rfloor $ denote the greatest integer less than or equal to $ x $. Let $ n $ denote a natural number. Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I

[(P)] The minimum value of $ n $ for which the function $$ f(x) = \left\lfloor \frac{10x^3 - 45x^2 + 60x + 35}{n} \right\rfloor $$ is continuous on the interval $ [1, 2] $, is
[(Q)] The minimum value of $ n $ for which $$ g(x) = (2n^2 - 13n - 15)(x^3 + 3x), $$ $ x \in \mathbb{R} $, is an increasing function on $ \mathbb{R} $, is
[(R)] The smallest natural number $ n $ which is greater than 5, such that $ x = 3 $ is a point of local minima of $$ h(x) = (x^2 - 9)^n (x^2 + 2x + 3), $$ is
[(S)] Number of $ x_0 \in \mathbb{R} $ such that $$ l(x) = \sum_{k=0}^{4} \left( \sin |x - k| + \cos \left| x - k + \frac{1}{2} \right| \right) $$ is not differentiable at $ x_0 $, is

List-II

[(1)] 8
[(2)] 9
[(3)] 5
[(4)] 6
[(5)] 10

Options:

Let $ S $ be the set of all seven-digit numbers that can be formed using the digits 0, 1 and 2. For example, 2210222 is in $ S $, but 0210222 is NOT in $ S $.
Then the number of elements $ x $ in $ S $ such that at least one of the digits 0 and 1 appears exactly twice in $ x $, is equal to __________.

Let $ \mathbb{R} $ denote the set of all real numbers. Let $ f: \mathbb{R} \to \mathbb{R} $ be a function such that $ f(x)>0 $ for all $ x \in \mathbb{R} $, and $ f(x + y) = f(x)f(y) $ for all $ x, y \in \mathbb{R} $.
Let the real numbers $ a_1, a_2, \ldots, a_{50} $ be in an arithmetic progression. If $ f(a_{31}) = 64f(a_{25}) $, and $$ \sum_{i=1}^{50} f(a_i) = 3(2^{25} + 1), $$ then the value of $$ \sum_{i=6}^{30} f(a_i) $$ is __________.

For any two points $ M $ and $ N $ in the $ XY $-plane, let $ \overrightarrow{MN} $ denote the vector from $ M $ to $ N $, and $ \vec{0} $ denote the zero vector. Let $ P, Q $, and $ R $ be three distinct points in the $ XY $-plane. Let $ S $ be a point inside the triangle $ \Delta PQR $ such that $$ \overrightarrow{SP} + 5\overrightarrow{SQ} + 6\overrightarrow{SR} = \vec{0}. $$ Let $ E $ and $ F $ be the mid-points of the sides $ PR $ and $ QR $, respectively. Then the value of $$ \frac{\text{length of the line segment } EF}{\text{length of the line segment } ES} $$ is __________.

For all $ x>0 $, let $ y_1(x), y_2(x), y_3(x) $ be the functions satisfying $$ \frac{dy_1}{dx} - (\sin x)^2 y_1 = 0, \quad y_1(1) = 5, $$ $$ \frac{dy_2}{dx} - (\cos x)^2 y_2 = 0, \quad y_2(1) = \frac{1}{3}, $$ $$ \frac{dy_3}{dx} - \left(\frac{2 - x^3}{x^3}\right) y_3 = 0, \quad y_3(1) = \frac{3}{5e}, $$ respectively. Then $$ \lim_{x \to 0^+} \frac{y_1(x)y_2(x)y_3(x) + 2x}{e^{3x} \sin x} $$ is equal to __________.

Let the set of all relations $ R $ on the set $ \{a, b, c, d, e, f\} $, such that $ R $ is reflexive and symmetric, and $ R $ contains exactly 10 elements, be denoted by $ S $. Then the number of elements in $ S $ is ___.

Let $ \alpha $ and $ \beta $ be the real numbers such that $$ \lim_{x \to 0} \frac{1}{x^3} \left( \frac{\alpha}{2} \int_{0}^{x} \frac{1}{1 - t^2} \, dt + \beta x \cos x \right) = 2. $$ Then the value of $ \alpha + \beta $ is __________.

Evaluate the definite integral: \[ I = \int_{\frac{1}{2}}^{2} \frac{\tan^{-1} x \, dx}{2x^2 - 3x + 2} \]

What are the gases released when $\mathrm{NH_4NO_2}$ and $\mathrm{NH_4NO_3}$ are heated?