List of top Questions asked in JEE Main- 2021

The value of

\(\lim_{n\to\infty}\frac{[r]+[2r]+\dots+[nr]}{n^2}\)

, where \(r\) is a non-zero real number and \([r]\) denotes the greatest integer less than or equal to \(r\), is equal to :

Let \(y=y(x)\) be the solution of the differential equation

\(\cos x\,(3\sin x+\cos x+3)\,dy=\left(1+y\sin x\,(3\sin x+\cos x+3)\right)dx,\ 0\le x\le\frac{\pi}{2},\ y(0)=0\)

. Then,

\(y\left(\frac{\pi}{3}\right)\)

is equal to :

Consider the function \(f:\mathbb{R}\to\mathbb{R}\) defined by

\(f(x)=\begin{cases}\left(2-\sin\left(\frac{1}{x}\right)\right)|x|, & x\ne 0\\ 0, & x=0\end{cases}\)

. Then \(f\) is :

Let \(L\) be a tangent line to the parabola \(y^2=4x-20\) at \((6,2)\). If \(L\) is also a tangent to the ellipse

\(\frac{x^2}{2}+\frac{y^2}{b}=1\)

, then the value of \(b\) is equal to :

Let \(S_1, S_2\) and \(S_3\) be three sets defined as

\(S_1=\{z\in\mathbb{C} : |z-1|\le\sqrt{2}\}\)

\(S_2=\{z\in\mathbb{C} : \mathrm{Re}((1-i)z)\ge 1\}\)

\(S_3=\{z\in\mathbb{C} : \mathrm{Im}(z)\le 1\}\)

Then the set \(S_1\cap S_2\cap S_3\) :

The value of the limit

\(\lim_{\theta\to 0}\frac{\tan(\pi\cos^2\theta)}{\sin(2\pi\sin^2\theta)}\)

is equal to :

If the curve \(y=y(x)\) is the solution of the differential equation

\(2(x^2+x^{5/4})\,dy-y(x+x^{1/4})\,dx=2x^{9/4}\,dx\), \(x\gt 0\) which

passes through the point \(\left(1,\,1-\frac{4}{3}\log_e 2\right)\), then the value of \(y(16)\) is equal to :

Let O be the origin. Let

\(\overrightarrow{OP}=x\hat{i}+y\hat{j}-\hat{k}\) and \(\overrightarrow{OQ}=-\hat{i}+2\hat{j}+3x\hat{k}\), \(x,y\in\mathbb{R}\), \(x\gt 0\), be such that \(\left|\overrightarrow{PQ}\right|=\sqrt{20}\) and the vector \(\overrightarrow{OP}\) is perpendicular to \(\overrightarrow{OQ}\). If \(\overrightarrow{OR}=3\hat{i}+z\hat{j}-7\hat{k}\), \(z\in\mathbb{R}\), is coplanar with \(\overrightarrow{OP}\) and \(\overrightarrow{OQ}\), then the value of \(x^2+y^2+z^2\) is equal to :

The number of solutions of the equation

\(\sin^{-1}\left[x^2+\frac{1}{3}\right]+\cos^{-1}\left[x^2-\frac{2}{3}\right]=x^2\), for \(x\in[-1,1]\), and \([x]\) denotes the greatest integer less than or equal to \(x\), is :

If the integral

\(\int_0^{10}\frac{[\sin 2\pi x]}{e^{x-[x]}}\,dx=\alpha e^{-1}+\beta e^{-\frac{1}{2}}+\gamma\)

, where \(\alpha,\beta,\gamma\) are integers and \([x]\) denotes the greatest integer less than or equal to \(x\), then the value of \(\alpha+\beta+\gamma\) is equal to :

Let \(f : \mathbb{R}\to\mathbb{R}\) be defined as \(f(x)=e^{-x}\sin x\). If \(F:[0,1]\to\mathbb{R}\) is a differentiable function with that \(F(x)=\int_0^x f(t)\,dt\), then the value of \(\int_0^1 \left(F'(x)+f(x)\right)e^{x}\,dx\) lies in the interval

Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability of occurrence of 0 at even places be \(\frac{1}{2}\) and probability of occurrence of 0 at the odd place be \(\frac{1}{3}\). Then the probability that '10' is followed by '01' is equal to :

The pressure exerted by a non-reactive gaseous mixture of 6.4g of methane and 8.8 g of carbon dioxide in a 10L vessel at 27\(^\circ\)C is __________ kPa. (Round off to the Nearest Integer).

[Assume gases are ideal, R = 8.314 J \(\mathrm{mol^{-1}\ K^{-1}}\)

Atomic masses : C : 12.0 u, H : 1.0 u, O : 16.0 u]

In the above reaction, 3.9 g of benzene on nitration gives 4.92 g of nitrobenzene. The percentage yield of nitrobenzene in the above reaction is ________\(\%\). (Round off to the Nearest Integer).

(Given atomic mass : C : 12.0 u, H : 1.0 u, O : 16.0 u, N : 14.0 u)

The standard enthalpies of formation of \(\mathrm{Al_2O_3}\) and \(\mathrm{CaO}\) are \(-1675\) kJ \(\mathrm{mol^{-1}}\) and \(-635\) kJ \(\mathrm{mol^{-1}}\) respectively.

For the reaction

\(\mathrm{3CaO + 2Al \to 3Ca + Al_2O_3}\) the standard reaction enthalpy \(\Delta_rH^0\) = _________ kJ.

(Round off to the Nearest Integer)

The reaction of white phosphorus on boiling with alkali in inert atmosphere resulted in the formation of product 'A'. The reaction of 1 mol of 'A' with excess of \(\mathrm{AgNO_3}\) in aqueous medium gives ___________ mol(s) of Ag. (Round off to the Nearest Integer).

The mole fraction of a solute in a 100 molal aqueous solution is ___________ \(\times\) \(10^{-2}\). (Round off to the Nearest Integer).

[Given : Atomic masses : H : 1.0 u, O : 16.0 u ]

The oxygen dissolved in water exerts a partial pressure of 20 kPa in the vapour above water. The molar solubility of oxygen in water is __________ \(\times\) \(10^{-5}\) \(\mathrm{mol\ dm^{-3}}\). (Round off to the Nearest Integer).

[Given : Henry's law constant = \(K_H\) = 8.0 \(\times\) \(10^4\) kPa for \(\mathrm{O_2}\). Density of water with dissolved oxygen = 1.0 kg \(\mathrm{dm^{-3}}\) ]

For a certain first order reaction 32% of the reactant is left after 570s. The rate constant of this reaction is _________ \(\times 10^{-3}\ \mathrm{s^{-1}}\). (Round off to the Nearest Integer). [Given : \(\log_{10}2 = 0.301\), \(\ln 10 = 2.303\)]

Reducing smog is a mixture of :

The point of intersection and sudden increase in the slope, in the diagram given below, respectively, indicates :

A certain orbital has \(n = 4\) and \(m_L = -3\). The number of radial nodes in this orbital is ____________. (Round off to the Nearest Integer).

A colloidal system consisting of a gas dispersed in a solid is called a/an :

The correct order of conductivity of ions in water is :

The INCORRECT statement(s) about heavy water is (are)

(A) used as a moderator in nuclear reactor

(B) obtained as a by-product in fertilizer industry

(C) used for the study of reaction mechanism

(D) has a higher dielectric constant than water

Choose the correct answer from the options given below :