List of top Questions asked in JEE Advanced

Let \(f:\R→\R\) be a function defined by
\(f(x) = \begin{cases}     x^2\sin(\frac{\pi}{x^2}), & \text{if } x \ne 0, \\     0, & \text{if } x = 0. \end{cases}\)
Then which of the following statements is TRUE ?

Based on VSEPR model, match the xenon compounds given in List-I with the corresponding geometries and the number of lone pairs on xenon given in List-II and choose the correct option.

List I		List II
(P)	XeF2	(1)	Trigonal bipyramidal and two lone pair of electrons
(Q)	XeF4	(2)	Tetrahedral and one lone pair of electrons
(R)	XeO3	(3)	Octahedral and two lone pair of electrons
(S)	XeO3F2	(4)	Trigonal bipyramidal and no lone pair of electrons
		(5)	Trigonal bipyramidal and three lone pair of electrons

In a conductometric titration, small volume of titrant of higher concentration is added stepwise to a larger volume of titrate of much lower concentration, and the conductance is measured after each addition.
The limiting ionic conductivity (Λ0) values (in mS m²mol⁻¹ ) for different ions in aqueous solutions are given below:

Ions	Ag+	K+	Na+	H+	\(\text{NO}_{3}^{-}\)	Cl-	\(\text{SO}^{2-}_{4}\)	OH-	CH3COO-
\(\Lambda_0\)	6.2	7.4	5.0	35.0	7.2	7.6	16.0	19.9	4.1

For different combinations of titrates and titrants given in List-I, the graphs of 'conductance' versus ‘volume of titrant’ are given in List-II.
Match each entry in List-I with the appropriate entry in List-II and choose the correct option.
 

List I		List II
(P)	Titrate: KCl Titrant: AgNO3	(1)
(Q)	Titrate: AgNO3 Titrant: KCl	(2)
(R)	Titrate: NaOH Titrant: HCl	(3)
(S)	Titrate: NaOH Titrant: CH3COOH	(4)
		(5)

List-I contains various reaction sequences and List-II contains different phenolic compounds. Match each entry in List-I with the appropriate entry in List-II and choose the correct option.

List I		List II
(P)		(1)
(Q)		(2)
(R)		(3)
(S)		(4)
		(5)

Let \(S=\left\{(x,y)\in\R\times\R:x\ge0,y\ge0,y^2\le12-2x\ \text{and}\ 3y+\sqrt8\ x\le5\sqrt8\right\}\). If the area of the region S is \(\alpha\sqrt2\), then α is equal to

List-I contains various reaction sequences and List-II contains the possible products. Match each entry in List-I with the appropriate entry in List-II and choose the correct option.

List I		List II
(P)		(1)
(Q)		(2)
(R)		(3)
(S)		(4)
		(5)

Considering only the principal values of the inverse trigonometric functions, the value of
\(\tan(\sin^{-1}(\frac{3}{5})-2\cos^{-1}(\frac{2}{\sqrt5}))\)
is

The option(s) in which at least three molecules follow Octet Rule is(are)

Reaction of iso-propylbenzene with O₂ followed by the treatment with H₃O⁺ forms phenol and a by-product P. Reaction of P with 3 equivalents of Cl₂ gives compound Q. Treatment of Q with Ca(OH)₂ produces compound R and calcium salt S.
The correct statement(s) regarding P, Q, R and S is(are)

Aspartame, an artificial sweetener, is a dipeptide aspartyl phenylalanine methyl ester. The structure of aspartame is

The circuit shown in the figure contains an inductor L, a capacitor \(C_0\), a resistor\( R_0\) and an ideal battery. The circuit also contains two keys \(K_1\) and \(K_2\). Initially, both the keys are open and there is no charge on the capacitor. At an instant, key \(K_1\) is closed and immediately after this the current in \(R_0\) is found to be \(I_1\). After a long time, the current attains a steady state value \(I_2\). Thereafter,\( K_2\) is closed and simultaneously \(K_1\) is opened and the voltage across \(C_0\) oscillates with amplitude \(C_0\) and angular frequency \(\omega_0\).

Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

List-I		List-II
P	The value of \(I1\) in Ampere is	I	\(0\)
Q	The value of I2 in Ampere is	II	\(2\)
R	The value of \(\omega_0\) in kilo-radians/s is	III	\(4\)
S	The value of \(V_0\) in Volt is	IV	\(20\)
			200

At room temperature, disproportionation of an aqueous solution of in situ generated nitrous acid (HNO₂) gives the species

Four identical thin, square metal sheets, S1, S2, S3 and S4, each of side a are kept parallel to each other with equal distance \(d (≪ a)\) between them, as shown in the figure. Let \(C_0 = \epsilon_0\frac{a^2}{d}, \)where \(\epsilon_0\) is the permittivity of free space.

Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

List-I		List-II
P	The capacitance between S1 and S4, with S2 and S3 not connected, is	I	\(3C_0\)
Q	The capacitance between S1 and S4, with S2 shorted to S3, is	II	\(\frac{C_0}{2}\)
R	The capacitance between S1 and S3, with S2 shorted to S4, is	III	\(\frac{C_0}{3}\)
S	The capacitance between S1 and S2, with S3 shorted to S1, and S2 shorted to S4, is	IV	\(2\frac{C_0}{3}\)
			\[2C_0\]

One mole of a monatomic ideal gas undergoes the cyclic process J→ K→ L→ M→ J, as shown in the P-T diagram.Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

Match the quantities mentioned in List-I with their values in List-II and choose the correct option. [ℛ is the gas constant.]

List-I		List-II
P	Work done in the complete cyclic process	I	\(ℛT_0 − 4ℛT_0 ln 2\)
Q	Change in the internal energy of the gas in the process JK	II	\(0\)
R	Heat given to the gas in the process KL	III	\(3ℛT_0\)
S	Change in the internal energy of the gas in the process MJ	IV	\(−2ℛT_0 ln 2\)
			\[−3ℛT_0 ln 2\]

 

A light ray is incident on the surface of a sphere of refractive index n at an angle of incidence \(\theta_0\). The ray partially refracts into the sphere with angle of refraction \(\phi\) and then partly reflects from the back surface. The reflected ray then emerges out of the sphere after a partial refraction. The total angle of deviation of the emergent ray with respect to the incident ray is \(a\). Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

List-I		List-II
P	If \(n = 2\) and \(\alpha = 180°\), then all the possible values of \(\theta_0\) will be	I	\(30\degree\) or \(0\degree\)
Q	If \(n = √3\) and \(\alpha= 180°\), then all the possible values of \(\theta_0\) will be	II	\(60\degree\) or \(0\degree\)
R	If \(n = √3\) and \(\alpha= 180°\), then all the possible values of \(\phi_0\) will be	III	\(45\degree\) or \( 0\degree\)
S	If \(n = \sqrt2\) and \(\theta_0 = 45°\), then all the possible values of  \(\alpha\) will be	IV	\(150\degree\)
			\[0\degree\]

A particle of mass m is moving in a circular orbit under the influence of the central force\( f(r) = −kr\), corresponding to the potential energy \(v(r) = \frac{kr^2}{2}\), where \(k\) is a positive force constant and \( r\) is the radial distance from the origin. According to the Bohr’s quantization rule, the angular momentum of the particle is given by \(L= nℏ\), where\( ℏ = ℎ/(2pi), ℎ\) is the Planck’s constant, and \(n\) a positive integer. If \(v\) and \(E\) are the speed and total energy of the particle, respectively, then which of the following expression(s) is(are) correct?

Let y ∈ R be such that the lines \(L_1:\frac{x+11}{1}=\frac{y+21}{2}=\frac{z+29}{3}\) and \(L_2:\frac{x+16}{3}=\frac{y+11}{2}=\frac{z+4}{\gamma}\) intersect. Let R₁ be the point of intersection of L₁ and L₂. Let O = (0, 0 ,0), and \(\hat{n}\) denote a unit normal vector to the plane containing both the lines L₁ and L₂.
Match each entry in List-I to the correct entry in List-II.

List - I		List - II
(P)	γ equals	(1)	\(-\hat{i}-\hat{j}+\hat{k}\)
(Q)	A possible choice for \(\hat{n}\) is	(2)	\(\sqrt{\frac{3}{2}}\)
(R)	\(\overrightarrow{OR_1}\) equals	(3)	1
(S)	A possible value of \(\overrightarrow{OR_1}.\hat{n}\) is	(4)	\(\frac{1}{\sqrt6}\hat{i}-\frac{2}{\sqrt6}\hat{j}+\frac{1}{\sqrt6}\hat{k}\)
		(5)	\(\sqrt{\frac{2}{3}}\)

The correct option is

Let \(f:\R \rightarrow \R\) and \(g:\R \rightarrow\R\) be functions defined by
\(f(x)=\left\{ \begin{array}{ll}     x|x|\sin(\frac{1}{x}), &  x\ne0 \\     0, &  x = 0, \end{array} \right.\text{and} \ g(x)=\left\{ \begin{array}{ll}     1-2x, & 0\leq x\leq \frac{1}{2}, \\     0, & \text{otherwise.} \end{array} \right.\)
Let \(a,b,c,d \in \R\). Define the function \(h:\R\rightarrow\R\) by
\(h(x)=af(x)+b(g(x)+g(\frac{1}{2}-x))+c(x-g(x))+d\ g(x), x\in\R\).
Match each entry in List-I to the correct entry in List-II.

List - I		List - II
(P)	If a = 0, b = 1, c = 0 and d = 0, then	(1)	h is one-one.
(Q)	If a = 1, b = 0, c = 0 and d = 0, then	(2)	h is onto.
(R)	If a = 0, b = 0, c = 1 and d = 0, then	(3)	h is differentiable on \(\R\)
(S)	If a = 0, b = 0, c = 0 and d = 1, then	(4)	the range of h is [0, 1].
		(5)	the range of h is {0, 1}.

The correct option is

Let R³ denote the three-dimensional space. Take two points P = (1, 2, 3) and Q = (4, 2 ,7). Let
dist(X, Y) denote the distance between two points X and Y in R3. Let
\(S=\left\{X\in\R^3:(dist(X,P)^2)-(dist(X,Q))^2=50\right\}\ and\)
\(T=\left\{Y\in \R^3:(dist(Y,Q))^2-(dist(Y,P))^2=50\right\}\)
Then which of the following statements is (are) TRUE ?

Let α and β be the distinct roots of the equation x² + x- 1 = 0. Consider the set T = {1, α, β}. For a 3 × 3 matrix M = ( a_ij )_3×3, define R_i = a_i1 + a_i2 + a_i3 and C_j = a_1j + a_2j + a_3j for i = 1, 2, 3 and j = 1, 2, 3.
Match each entry in List-I to the corret entry in List-II.

List - I		List - II
(P)	The number of matrices M = ( aij )3×3 with all entries in T such that Ri = Cj = 0 for all i, j, is	(1)	1
(Q)	The number of symmetric matrices M = ( aij )3×3 with all entries in T such that Cj = 0 for all j, is	(2)	12
(R)	Let M = ( aij )3×3 be a skew symmetric matrix such that aij ∈ T for i > j. Then the number of elements in the set \(\left\{\begin{pmatrix} x \\ y \\ z \end{pmatrix}:x,y,z\in \R, M\begin{pmatrix} x \\ y \\ z \end{pmatrix}=\begin{pmatrix} a_{12} \\ 0 \\ -a_{23} \end{pmatrix}\right\}\) is	(3)	infinite
(S)	Let M = ( aij )3×3 be a matrix with all entries in T such that Ri = 0 for all i. Then the absolute value of the determinant of M is	(4)	6
		(5)	0

The correct option is

Let the straight line y = 2x touch a circle with center (0, α), α > 0, and radius r at a point A₁. Let B₁ be the point on the circle such that the line segment A₁B₁ is a diameter of the circle. Let \(α+r=5+\sqrt5\).
Match each entry in List-I to the correct entry in List-II.

List - I		List - II
(P)	α equals	(1)	(-2, 4)
(Q)	r equals	(2)	\(\sqrt5\)
(R)	A1 equals	(3)	(-2, 6)
(S)	B1 equals	(4)	5
		(5)	(2, 4)

The correct option is

A monochromatic light wave is incident normally on a glass slab of thickness as shown in the figure. The refractive index of the slab increases linearly from n₁ to n₂ over the height of h. Which of the following statements is/are true about the light wave emerging out of the slab?

Monochromatic light wave
 

For He⁺, a transition takes place from the orbit of radius 105.8 pm to the orbit of radius 26.45 pm. The wavelength (in nm) of the emitted photon during the transition is ___. 
[Use: Bohr radius, a = 52.9 pm, Rydberg constant, 𝑅H = 2.2 × 10⁻¹⁸ J, Planck’s constant, h = 6.6 × 10⁻³⁴ J s, Speed of light, c = 3 × 10⁸ m s⁻¹]

Let  \(X = \{ {(x,y) ∈ Z \times Z : \frac{x^2}{8} + \frac {y^2}{20} < 1 \,and \,\,y^2 < 5x} \}\). Three distinct points P, Q and R are randomly chosen from X . Then the probability that P, Q and R form a triangle whose area is a positive integer, is