List of top Questions asked in JEE Advanced

For positive integer $n$, define $f(n)=n+\frac{16+5 n-3 n^2}{4 n+3 n^2}+\frac{32+n-3 n^2}{8 n+3 n^2}+\frac{48-3 n-3 n^2}{12 n+3 n^2}+\ldots+\frac{25 n-7 n^2}{7 n^2}$.
Then, the value of $\displaystyle\lim _{n \rightarrow \infty} f(n)$ is equal to

The total number of chiral molecules formed from one molecule of $P$ on complete ozonolysis $\left( O _3, Zn / H _2 O \right)$ is_____

Let $z$ be a complex number with non-zero imaginary part If $\frac{2+3 z+4 z^2}{2-3 z+4 z^2}$ is a real number, then the value of $|z|^2$ is ___ .

Let p, q, r be non-zero real numbers that are, respectively, the 10^th, 100^th and 1000^th terms of a harmonic progression. Consider the system of linear equations
x + y + z = 1
10x + 100y + 1000z = 0
qr x + pr y + pq z = 0
 

List-I		List-II
(I)	If \(\frac{q}{r}=10\), then the system of linear equations has	(P)	x = 0, \(y=\frac{10}{9},z=-\frac{1}{9}\) as a solution
(II)	If \(\frac{p}{r}≠100\), then the system of linear equations has	(Q)	\(x=\frac{10}{9},y=\frac{-1}{9},z=0\) as a solution
(III)	If \(\frac{p}{q}≠10,\) then the system of linear equations has	(R)	infinitely many solutions
(IV)	If \(\frac{p}{q}=10,\) then the system of linear equations has	(S)	no solution
		(T)	at least one solution

Concentration of $H _2 SO _4$ and $Na _2 SO _4$ in a solution is $1 M$ and $1.8 \times 10^{-2} M$, respectively. Molar solubility of $PbSO _4$ in the same solution is $X \times 10^{- Y } M$ (expressed in scientific notation). The value of $Y$ is ________. [Given: Solubility product of $PbSO _4\left(K_{s p}\right)=1.6 \times 10^{-8}$ For $H _2 SO _4, K_{a l}$ is very large and $\left[K_{a 2}=1.2 \times 10^{-2}\right]$

The product of all positive real values of $x$ satisfying the equation $x^{\left(16\left(\log _5 x\right)^3-68 \log _5 x\right)}=5^{-16}$ is _____.

Let $\alpha=\displaystyle\sum_{ k =1}^{\infty} \sin ^{2 k }\left(\frac{\pi}{6}\right)$ Let $g:[0,1] \rightarrow R$ be the function defined by $g(x)=2^{\alpha x}+2^{\alpha(1-x)}$. Then, which of the following statements is/are TRUE?

Consider the ellipse
 \(\frac{x^2}{4} + \frac{y^2}{3} = 1\).
 Let \(H (α, 0), 0 < α < 2\) , be a point. A straight line drawn through H parallel to y-axis crosses the ellipse and its auxiliary circle at points E and F respectively, in the first quadrant. The tangents to the ellipse at the point E intersects the positive x-axis at a point G. Suppose the straight line joining F and the origin makes an angle \(\phi\) with the positive x-axis.
 

List-I		List-II
(I)	If \(\Phi = \frac{\pi}{4}\), then the area of the triangle FGH is	P	\(\frac{(\sqrt{3}-1)^4}{8}\)
(II)	If \(\Phi = \frac{\pi}{3}\), then the area of the triangle FGH is	Q	1
(III)	If \(\Phi = \frac{\pi}{6}\), then the area of the triangle FGH is	R	\(\frac{3}{4}\)
(IV)	If \(\Phi = \frac{\pi}{12}\), then the area of the triangle FGH is	S	\(\frac{1}{2\sqrt{3}}\)
		T	\(\frac{3\sqrt{3}}{2}\)

The treatment of galena with $HNO _3$ produces a gas that is

A projectile is fired from horizontal ground with speed $v$ and projection angle $\theta$ When the acceleration due to gravity is $g$, the range of the projectile is $d$ If at the highest point in its trajectory, the projectile enters a different region where the effective acceleration due to gravity is $g^{\prime}=\frac{g}{081}$, then the new range is $d^{\prime}=n d$ The value of $n$ is ___

The reduction potential \(\left(E^0, in\ V\right)\) of \(MnO _4^{-}( aq ) / Mn ( s )\) is\(\left[ Given : E_{\left( MnO _4^-( aq ) / MnO _2( s )\right)}^0=1.68 V ; E_{\left( MnO _2( s )/ Mn ^{2+}( aq )\right)}^0=1.21 V ; E_{\left( Mn ^2( aq ) / Mn ( s )\right)}^0=-1.03 V \right]\)

If $M=\begin{pmatrix}\frac{5}{2} & \frac{3}{2} \\ -\frac{3}{2} & -\frac{1}{2}\end{pmatrix}$, then which of the following matrices is equal to $M^{2022}$ ?

Consider the parabola $y^2=4 x$ Let $S$ be the focus of the parabola A pair of tangents drawn to the parabola from the point $P=(-2,1)$ meet the parabola at $P_1$ and $P_2$ Let $Q_1$ and $Q_2$ be points on the lines $S P_1$ and $S P_2$ respectively such that $P Q_1$ is perpendicular to $S P_1$ and $P Q_2$ is perpendicular to $S P_2$ Then, which of the following is/are TRUE ?

Consider the equation $\int\limits_1^e \frac{\left(\log _{ e } x \right)^{1 / 2}}{x\left(a-\left(\log _{ e } x \right)^{3 / 2}\right)^2} dx =1, \quad a \in(-\infty, 0) \cup(1, \infty) $ Which of the following statements is/are TRUE ?

Consider the following lists:
 

List I		List II
I	$\left\{x \in\left[-\frac{2 \pi}{3}, \frac{2 \pi}{3}\right]: \cos x+\sin x=1\right\}$	P	has two elements
II	$\left\{x \in\left[-\frac{5 \pi}{18}, \frac{5 \pi}{18}\right]: \sqrt{3} \tan 3 x=1\right\} $	Q	has three elements
III	$ \left\{x \in\left[-\frac{6 \pi}{5}, \frac{6 \pi}{5}\right]: 2 \cos (2 x)=\sqrt{3}\right\} $	R	has four elements
IV	$ \left\{x  \in\left[-\frac{7 \pi}{4}, \frac{7 \pi}{4}\right]: \sin x-\cos x=1\right\}$	S	has five elements
		T	has six elements

The correct option is:

For diatomic molecules, the correct statement(s) about the molecular orbitals formed by the overlap to two $2 p_z$ orbitals is(are)

Let $|M|$ denote the determinant of a square matrix $M$ Let $g:\left[0, \frac{\pi}{2}\right] \rightarrow R$ be the function defined by
 $g (\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1}$ 
where 
$f(\theta)=\frac{1}{2}\begin{vmatrix}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{vmatrix}+\begin{vmatrix}\sin \pi & \cos \left(\theta+\frac{\pi}{4}\right) & \tan \left(\theta-\frac{\pi}{4}\right) \\ \sin \left(\theta-\frac{\pi}{4}\right) & -\cos \frac{\pi}{2} & \log _{ e }\left(\frac{4}{\pi}\right) \\ \cot \left(\theta+\frac{\pi}{4}\right) & \log _{ e }\left(\frac{\pi}{4}\right) & \tan \pi\end{vmatrix}$ 
Let $p (x)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g (\theta)$, and $p (2)=2-\sqrt{2}$ Then, which of the following is/are TRUE?

Considering only the principal values of the inverse trigonometric functions, the value of $\frac{3}{2} \cos ^{-1} \sqrt{\frac{2}{2+\pi^2}}+\frac{1}{4} \sin ^{-1} \frac{2 \sqrt{2} \pi}{2+\pi^2}+\tan ^{-1} \frac{\sqrt{2}}{\pi}$ is ___

Which one of the following options represents the magnetic field $\vec{B}$ at $O$ due to the current flowing in the given wire segments lying on the $x y$ plane?

 

An ideal gas of density $\rho=02 \,kg \, m ^{-3}$ enters a chimney of height $h$ at the rate of $\alpha=08 \,kg \,s ^{-1}$ from its lower end, and escapes through the upper end as shown in the figure The cross-sectional area of the lower end is $A_1=01 \, m ^2$ and the upper end is $A_2=04 \, m ^2$ The pressure and the temperature of the gas at the lower end are $600 \,Pa$ and $300 \,K$, respectively, while its temperature at the upper end is $150 \,K$ The chimney is heat insulated so that the gas undergoes adiabatic expansion Take $g=10 \,ms ^{-2}$ and the ratio of specific heats of the gas $\gamma=2$ Ignore atmospheric pressure Which of the following statement(s) is(are) correct?

Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is 0.5 mm. The circular scale has 100 divisions and for one full rotation of the circular scale, the main scale shifts by two divisions. The measured readings are listed below.
 

Measurement condition	Main scale reading	Circular scale reading
Two arms of gauge touching each other without wire	0 division	4 divisions
Attempt-1: With wire	4 divisions	20 divisions
Attempt-2: With wire	4 divisions	16 divisions

What are the diameter and cross-sectional area of the wire measured using the screw gauge?
 

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

 

Let $z$ be a complex number with non-zero imaginary part If $\frac{2+3 z+4 z^2}{2-3 z+4 z^2}$ is a real number, then the value of $|z|^2$ is ___ .