List of top Questions asked in JEE Main

A bullet of mass 10 g and speed 500 m/s is fired into a door and gets embedded exactly at the centre of the door. The door is 1.0 m wide and weighs 12 kg. It is hinged at one end and rotates about a vertical axis practically without friction. The angular speed of the door just after the bullet embeds into it will be :

This question has Statement-1 and Statement-2. Of the four choices given after the Statements, choose the one that best describes the two Statements.
Statement-1: A capillary is dipped in a liquid and liquid rises to a height \(h\) in it. As the temperature of the liquid is raised, the height \(h\) increases (if the density of the liquid and the angle of contact remain the same).
Statement-2: Surface tension of a liquid decreases with the rise in its temperature.

From the following, the quantity (constructed from the basic constants of nature), that has the dimensions, as well as correct order of magnitude, vis-a-vis typical atomic size, is:

If the ratio of lengths, radii and Young's moduli of steel and brass wires in the figure are \(a\), \(b\) and \(c\) respectively,

then the corresponding ratio of increase in their lengths is :

Two springs of force constants 300 N/m (Spring A) and 400 N/m (Spring B) are joined together in series. The combination is compressed by 8.75 cm. The ratio of energy stored in A and B is \(\frac{E_A}{E_B}\). Then \(\frac{E_A}{E_B}\) is equal to:

A vector \(\vec{n}\) is inclined to \(x\)-axis at \(45^\circ\), to \(y\)-axis at \(60^\circ\) and at an acute angle to \(z\)-axis. If \(\vec{n}\) is a normal to a plane passing through the point \((\sqrt{2}, -1, 1)\) then the equation of the plane is :

If the lines \(\frac{x+1}{2} = \frac{y-1}{1} = \frac{z+1}{3}\) and \(\frac{x+2}{2} = \frac{y-k}{3} = \frac{z}{4}\) are coplanar, then the value of \(k\) is :

If \(x=\int_0^y\frac{dt}{\sqrt{1+t^2}}\), then \(\frac{d^2y}{dx^2}\) is equal to :

The area bounded by the curve \(y = \ln(x)\) and the lines \(y = 0\), \(y = \ln(3)\) and \(x = 0\) is equal to:

Let \(\vec{a} = 2\hat{i} - \hat{j} + \hat{k}\), \(\vec{b} = \hat{i} + 2\hat{j} - \hat{k}\) and \(\vec{c} = \hat{i} + \hat{j} - 2\hat{k}\) be three vectors. A vector of the type \(\vec{b} + \lambda\vec{c}\) for some scalar \(\lambda\), whose projection on \(\vec{a}\) is of magnitude \(\sqrt{\frac{2}{3}}\) is :

The vector \((\hat{i} \times \vec{a} \cdot \vec{b})\hat{i} + (\hat{j} \times \vec{a}\vec{b})\hat{j} + (\hat{k} \times \vec{a} \cdot \vec{b})\hat{k}\) is equal to:

The mean of a data set consisting of 20 observations is 40. If one observation 53 was wrongly recorded as 33, then the correct mean will be :

If \(a,b,c\) are sides of a scalene triangle, then the value of \(\begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix}\) is :

Let \(f(x)=\frac{x^2-x}{x^2+2x}\), \(x\ne0,-2\). Then \(\frac{d}{dx}\left[f^{-1}(x)\right]\) (wherever it is defined) is equal to :

The matrix \(A^2+4A-5I\), where \(I\) is identity matrix and \(A=\begin{bmatrix}1&2\\4&-3\end{bmatrix}\), equals :

Let \(A=\{1,2,3,4\}\) and \(R:A\to A\) be the relation defined by \(R=\{(1,1),(2,3),(3,4),(4,2)\}\). The correct statement is :

Statement-1: The statement \(A\to(B\to A)\) is equivalent to \(A\to(A\vee B)\).
Statement-2: The statement \(\sim[(A\wedge B)\to(\sim A\vee B)]\) is a Tautology.

If each of the lines \(5x+8y=13\) and \(4x-y=3\) contains a diameter of the circle \(x^2+y^2-2(a^2-7a+11)x-2(a^2-6a+6)y+b^3+1=0\), then :

If \(a\) and \(c\) are positive real numbers and the ellipse \(\frac{x^2}{4c^2}+\frac{y^2}{c^2}=1\) has four distinct points in common with the circle \(x^2+y^2=9a^2\), then

Equation of the line passing through the points of intersection of the parabola \(x^2=8y\) and the ellipse \(\frac{x^2}{3}+y^2=1\) is :

Statement-1: The slope of the tangent at any point \(P\) on a parabola, whose axis is the axis of \(x\) and vertex is at the origin, is inversely proportional to the ordinate of the point \(P\).
Statement-2: The system of parabolas \(y^2=4ax\) satisfies a differential equation of degree 1 and order 1.

The value of \(\lim_{x\to0}\frac1x\left[\tan^{-1}\left(\frac{x+1}{2x+1}\right)-\frac\pi4\right]\) is :

A committee of 4 persons is to be formed from 2 ladies, 2 old men and 4 young men such that it includes at least 1 lady, at least 1 old man and at most 2 young men. Then the total number of ways in which this committee can be formed is :

The sum of the series: \(1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots\ldots\) upto 10 terms, is :

A value of \(x\) for which \(\sin\left(\cot^{-1}(1+x)\right)=\cos\left(\tan^{-1}x\right)\), is :