List of top Questions asked in JEE Main- 2021

The number of ordered pairs \((a,b)\), such that the function \(f(x)=4x^{3}-(3a+2b)x^{2}+2abx\) is decreasing in \(\left[\frac32,\frac53\right]\) and increasing in \(\mathbb{R}-\left[\frac32,\frac53\right]\), is ________.

Number of real roots of the equation
\(\sqrt{x+2}-\sqrt{x-2}=\sqrt{4 x-1}\) is ________.

Three fair dice are thrown. The number of ways to obtain a total of 11 is ________.

Which of the following is equivalent to \(p \leftrightarrow q\) ?

The sum of absolute maximum and absolute minimum values of the function \(f(x)=\left|4x^{2}+3x-10\right|+x^{2}-2x+5,\ x\in[-3,1]\) is ________.

If a plane, that contains the line \(\overrightarrow{\mathrm{r}}=(1,-1,1)+\lambda(2,3,-6), \lambda \in \mathbf{R}\), passes through the point \((2,0,2)\) and is at a distance \(\mathrm{d}\) from the origin, then the value of \(73 \mathrm{~d}^{2}\) is ________.

Between 1 and 31, \(\mathrm{n}\) numbers have been inserted in such a way that the resulting sequence is an A.P. If the ratio of \((\mathrm{n}-1)^{\text{th}}\) and \(7^{\text{th}}\) inserted numbers is \(9: 5\), then \(\mathrm{n}\) is equal to ________.

For \(0<x<\pi\), the sum of the solutions of the equation \(4^{\left(\frac{1}{2} \sin x+\frac{1}{4} \sin ^{2} x+\frac{1}{8} \sin ^{3} x+\cdots\right)}=2^{\frac{2}{3}}\) is :

The domain of the function
\(f(x)=\tan ^{-1}\left(\sqrt{x^{2}+x-2}\right)+\sec ^{-1}\left(x^{2}-5 x+3\right)\) is :

Let the distinct numbers \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) be the \(\mathrm{p}^{\text{th}}, \mathrm{q}^{\text{th}}\) and \(\mathrm{r}^{\text{th}}\) terms of a geometric progression of positive terms. Then the angle between the vectors \(\overrightarrow{\mathrm{u}}=\left(\log_{\mathrm{e}} \mathrm{a}^{2}\right) \hat{i}+\left(\log_{\mathrm{e}} \mathrm{b}^{2}\right) \hat{j}+\left(\log_{\mathrm{e}} \mathrm{c}^{2}\right) \hat{k}\) and \(\overrightarrow{\mathrm{v}}=(\mathrm{q}-\mathrm{r}) \hat{i}+(\mathrm{r}-\mathrm{p}) \hat{j}+(\mathrm{p}-\mathrm{q}) \hat{k}\) is :

If the angle between the line \(\frac{x+1}{1}=\frac{y-1}{2}=\frac{z+2}{-1}\) and the plane \(\mathrm{P}: 2x-y-\lambda z+4=0\), \(\lambda > 0\), is \(\sin^{-1}\left(\frac{1}{9\sqrt6}\right)\), then the distance of the point \((-1,1,-2)\) from the plane \(\mathrm{P}\) is :

If a line \(\mathrm{p}x+\mathrm{q}y=12\) is a tangent to the ellipse \(4x^2+9y^2=16\), then \(\mathrm{pq}\) can NOT be equal to :

A rod AB of length 4 units rests against a vertical wall with A on the horizontal floor. P is a point on AB such that AP : PB = 2 : 1. If the rod slides down along the wall in a vertical plane, then the point \(P(x,y)\) moves on the curve :

Let \(y=y(x)\) be the solution of the differential equation \(x\frac{dy}{dx}=(2x+3)e^{x}+xy,\ x > 0\). If \(y(e)=(2e+3)e^{e}\), then \(\frac{d^2y}{dx^2}-\frac{dy}{dx}\) at \(x=1\) is equal to :

Let \(\mathrm{A}(1,2)\) and \(\mathrm{B}(3,6)\) be two points on a circle of radius \(\mathrm{R}\), whose one of the diameters is along the line \(2x-y+4=0\). Then \(\mathrm{R}^2\) is equal to :

If \(\mathrm{I}=\int_{0}^{1} \tan^{-1}(\sqrt{x}+1)\,dx\), \(\mathrm{J}=\int_{0}^{1} \frac{\sqrt{x}\,dx}{x+2\sqrt{x}+2}\), then the value of \(2\mathrm{I}+\mathrm{J}\) is equal to :

A line perpendicular to the line \(4x=3y\) is a tangent to the parabola \(y^2=3x+1\) at a point \(\mathrm{P}\) on it. If \(\mathrm{S}\) is the focus of the parabola, then the equation of the line \(\mathrm{SP}\) is :

Let \(f\) be a non-zero polynomial function such that \(f(3x)=f'(x)f''(x)\). Then the value of \(f(6)\) is :

Consider yourself a fish in the aquarium that is in a room and a girl is standing near the aquarium and playing with the fish. Create the view of the room according the eye of fish.
OR
Create picture with fish/fishes, a girl, boat, farmers, tractor, trees and plants.

Draw a proportionate sketch of given reference image with black and white rendering techniques of your choice. Picture should fit in given space for this answer.

OR
Create a wall mosaic for a metro station using triangles of any size and overall panel size of your choice. The theme of mosaic is "Environment". Use colours to make the mosaic attractive and meaningful.

Monsoon first reach to which state of India ?

One of the following answer figure is hidden in the problem figure in same size and direction. Select correct one.

One of the following answer figure is hidden in the problem figure in same size and direction. Select correct one.

The 3D figure shows the view of an object. Identify the correct view looking in the direction of arrow, from the answer figure's.

The 3D figures shows the view of an object. Identify the correct view looking in the direction of arrow, from the given answer figures.