List of top Questions asked in JEE Advanced

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 _____

One mole of an ideal gas undergoes two different cyclic processes I and I, as shown in the P-V diagrams below. In cycle, I processes a, b, c and d are isobaric, isothermal, isobaric, and isochoric, respectively. In cycle II, processes a',b',c', and d' are isothermal, isochoric, isobaric, and isochoric, respectively. The total work done during cycle I is 𝑊_I and that during cycle II is 𝑊_II. The ratio W_I/W_II is

An electric dipole is formed by two charges +q and -q located in the x-y plane at (0,2) mm and (0,-2) mm as shown in the figure. The electric potential at point P(100, 100) mm due to dipole is V₀. Now the charges +q and -q are moved to (-1,2)mm and (1,-2)mm respectively. What is the value of the electric potential at point P due to this new dipole?

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 any y∈R, let cot\(^{-1}\)(y) ∈ (0,π) and tan\(^{-1}\)(y) ∈ (\(-\frac{\pi}{2},\frac{\pi}{2}\)). Then the sum of all the solutions of the equation\(tan^{-1}(\frac{6y}{9-y^2})+cot^{-1}(\frac{9-y^2}{6y})=\frac{2\pi}{3}\, for\,0<|y|<3,\) is equal to

H₂S (5 moles) reacts completely with acidified aqueous potassium permanganate solution. In this reaction, the number of moles of water produced is x, and the number of moles of electrons involved is y. The value of (x + y) is ____.

A series LCR circuit is connected to a 45 sin (ωt) volt source. The resonant angular frequency of the circuit is 10⁵ rad/sec and the current amplitude at resonance is I₀. When the angular frequency of the source is ω = 8 x 10⁴ rad/sec, the current amplitude in the circuit is 0.05 I₀. If m = 50 mH, match each entry in the list - I with an approximate value from list - II and choose the option.

	List - I		List - II
(P)	I₀ in mA	(1)	44.4
(Q)	The quality factor of the circuit	(2)	18
(R)	The bandwidth of the circuit in rad/sec	(3)	400
(S)	The peak power dissipated at resonance in watt	(4)	2250
		(5)	500

Let f : (0,1) → R be the function defined as f(x) = √n if x ∈ [\(\frac{1}{n+1},\frac{1}{n}\)] where n ∈ N. Let g : (0,1) → R be a function such that \(\int_{x^2}^{x}\sqrt{\frac{1-t}{t}}dt<g(x)<2\sqrt x\) for all x ∈ (0,1).
Then \(\lim_{x\rightarrow0}f(x)g(x)\)

Let \(\alpha\ and\ \beta\) be real numbers such that \(-\frac{\pi}{4}<\beta<0<\alpha<\frac{\pi}{4}\). If \(\sin (\alpha+\beta)=\frac{1}{3}\ and\ \cos (\alpha-\beta)=\frac{2}{3}\), then the greatest integer less than or equal to
\(\left(\frac{\sin \alpha}{\cos \beta}+\frac{\cos \beta}{\sin \alpha}+\frac{\cos \alpha}{\sin \beta}+\frac{\sin \beta}{\cos \alpha}\right)^2\) is ____

Let R=\(\begin{pmatrix}a&3&b\\c&2&d\\0&5&0\end{pmatrix}\): a,b,c,d ∈ {0,3,5,7,11,13,17,19}. Then the number of invertible matrices in R is

Young’s modulus of elasticity 𝑌 is expressed in terms of three derived quantities, namely, the gravitational constant 𝐺, Planck’s constant ℎ and the speed of light 𝑐, as 𝑌 = 𝑐^𝛼ℎ^𝛽𝐺^𝛾. Which of the following is the correct option?

Let tan^-1 x ∈(\(-\frac π2\) \(\frac π2\)) for x ∈ R. Then the number of real solutions of the equation \(\sqrt{1 + \cos (2x)} = \sqrt2 \tan^{-1}(\tan x)\) in the set \((-\frac{3π}{2}, -\frac{π}{2}) ∪ (-\frac{π}{2},\frac{π}{2}) ∪ (\frac{π}{2},\frac{3π}{2})\) is equal to

A person of height 1.6 m is walking away from a lamp post of height 4 m along a straight path on the flat ground. The lamp post and the person are always perpendicular to the ground. If the speed of the person is 60 cm s^−1, the speed of the tip of the person’s shadow on the ground with respect to the person is _______ cm s^−1.

In an experiment for determination of the focal length of a thin convex lens, the distance of the object from the lens is 10 ± 0.1 cm and the distance of its real image from the lens is 20 ± 0.2 cm. The error in the determination of focal length of the lens is 𝑛 %. The value of 𝑛 is ____ .

A particle of mass m is moving such that its velocity at a point (x,y) is given by v = α (y\(\hat{i}\) + 2x\(\hat{j}\)), where α is a non-zero constant. What is the resultant force acting on the particle?

Let R=\(\begin{pmatrix}a&3&b\\c&2&d\\0&5&0\end{pmatrix}\): a,b,c,d ∈ {0,3,5,7,11,13,17,19}. Then the number of invertible matrices in R is

Let S = (0,1) ∪ (1,2) ∪ (3,4) and T = {0,1, 2,3} . Then which of the following statements is(are) true?

Let P be a point on the parabola y² = 4ax, where a > 0. The normal to the parabola at P meets the x -axis at a point Q. The area of the triangle PFQ where F is the focus of the parabola, is 120. If the slope m of the normal and a are both positive integers, then the pair (a, m) is

The stoichiometric reaction of 516 g of dimethyldichlorosilane with water results in a tetrameric cyclic product X in 75% yield. The weight (in g) of X obtained is___. [Use, molar mass (g mol⁻¹): H = 1, C = 12, O = 16, Si = 28, Cl = 35.5]

Let \( \alpha, \beta, \) and \( \gamma \) be real numbers. Consider the following system of linear equations:
\( x + 2y + z = 7 \)
\( x + \alpha z = 11 \)
\( 2x - 3y + \beta z = \gamma \)
Match each entry in List I to the correct entries in List II

List I		List II
(P)	If \( \beta = \frac{1}{2}(7\alpha - 3) \) and \( \gamma = 28 \), then the system has	(1)	a unique solution
(Q)	If \( \beta = \frac{1}{2}(7\alpha - 3) \) and \( \gamma \neq 28 \), then the system has	(2)	no solution
(R)	If \( \beta \neq \frac{1}{2}(7\alpha - 3) \) where \( \alpha = 1 \) and \( \gamma \neq 28 \), then the system has	(3)	infinitely many solutions
(S)	If \( \beta \neq \frac{1}{2}(7\alpha - 3) \) where \( \alpha = 1 \) and \( \gamma = 28 \), then the system has	(4)	\( x = 11, y = -2 \) and \( z = 0 \) as a solution
		(5)	\( x = -15, y = 4 \) and \( z = 0 \) as a solution

50 mL of 0.2 molal urea solution (density = 1.012 g mL⁻¹ at 300 K) is mixed with 250 mL of a solution containing 0.06 g of urea. Both solutions were prepared in the same solvent. The osmotic pressure (in Torr) of the resulting solution at 300 K is ___.
[Use: Molar mass of urea = 60 g mol⁻¹ ; gas constant, R = 62 L Torr K⁻¹ mol⁻¹; Assume, Δ_mixH = 0, Δ_mixV = 0]

In the given reaction scheme, P is a phenyl alkyl ether, Q is an aromatic compound; R and S are the major products.

The correct statement about S is

List-I shows different radioactive decay processes and List-II provides possible emitted particles. Match each entry in List-I with an appropriate entry from List-II, and choose the correct option.

	List-I		List-II
(P)	\(^{238}_{92}U → ^{234}_{91}Pa\)	(1)	one 𝛼 particle and one 𝛽+ particle
(Q)	\(^{214}_{82}Pb → ^{210}_{82}Pb\)	(2)	three 𝛽− particles and one 𝛼 particle
(R)	\(^{210}_{81}Tl → ^{206}_{82}Pb\)	(3)	two 𝛽− particles and one 𝛼 particle
(S)	\(^{228}_{91}Pa → ^{224}_{88}Ra\)	(4)	one 𝛼 particle and one 𝛽− particle
		(5)	one 𝛼 particle and two 𝛽+ particles

The plot of log 𝑘_𝑓 versus \(\frac{1}{T}\) for a reversible reaction, A (g) ⇌ P (g) is shown.

Pre-exponential factors for the forward and backward reactions are 10¹⁵s^-1 and 10¹¹s^-1 respectively. If the value of log K for the reaction at 500 K is 6, the value of |log kb| at 250K is_________[K = equilibrium constant of the reaction, 𝑘_𝑓 = rate constant of forward reaction, 𝑘_𝑏 = rate constant of backward reaction]

For \(x ∈ R\), let \(tan^{-1} (x) ∈ \) (\(-\frac{\pi}{2},\frac{\pi}{2}\)). Then the minimum value of function \(f: R \rightarrow R\) defined by \(f(x) =\) \(\int_{0}^{x\,tan^{-1}x}\frac{e^{(t-cos\,t)}}{1+t^{2023}}dt\) is