List of top Questions asked in JEE Main

The density of a solid metal sphere is determined by measuring its mass and its diameter. The maximum error in the density of the sphere is \(\left(\frac{x}{100}\right)\%\). If the relative errors in measuring the mass and the diameter are 6.0% and 1.5% respectively, the value of x is_______.

An insect is at the bottom of a hemispherical ditch of radius 1 m. It crawls up the ditch but starts slipping after it is at height h from the bottom. If the coefficient of friction between the ground and the insect is 0.75, then h is :
(g = 10 \(\mathrm{ms^{-2}}\))

A point like object is placed at a distance of 1 m in front of a convex lens of focal length 0.5 m. A plane mirror is placed at a distance of 2 m behind the lens. The position and nature of the final image formed by the system is :

An electron, a doubly ionized helium ion (\(\mathrm{He^{++}}\)) and a proton are having the same kinetic energy. The relation between their respective de-Broglie wavelengths \(\lambda_e\), \(\lambda_{He^{++}}\) and \(\lambda_p\) is :

Charges \(Q_1\) and \(Q_2\) are at points A and B of a right angle triangle OAB (see figure). The resultant electric field at point O is perpendicular to the hypotenuse, then \(\frac{Q_1}{Q_2}\) is proportional to :

A satellite is in an elliptical orbit around a planet P. It is observed that the velocity of the satellite when it is farthest from the planet is 6 times less than that when it is closest to the planet. The ratio of distances between the satellite and the planet at closest and farthest points is:

Four point masses, each of mass m, are fixed at the corners of a square of side \(l\). The square is rotating with angular frequency \(\omega\), about an axis passing through one of the corners of the square and parallel to its diagonal, as shown in the figure. The angular momentum of the square about this axis is :

A clock has a continuously moving second's hand of 0.1 m length. The average acceleration of the tip of the hand (in units of \(\mathrm{ms^{-2}}\)) is of the order of :

An object of mass m is suspended at the end of a massless wire of length L and area of crosssection A. Young modulus of the material of the wire is Y. If the mass is pulled down slightly its frequency of oscillation along the vertical direction is :

In the figure below, P and Q are two equally intense coherent sources emitting radiation of wavelength 20 m. The separation between P and Q is 5 m and the phase of P is ahead of that of Q by \(90^{\circ}\). A, B and C are three distinct points of observation, each equidistant from the midpoint of PQ. The intensities of radiation at A, B, C will be in the ratio :

An electron is moving along +x direction with a velocity of \(6 \times 10^{6}\ \mathrm{ms^{-1}}\). It enters a region of uniform electric field of 300 V/cm pointing along +y direction. The magnitude and direction of the magnetic field set up in this region such that the electron keeps moving along the x direction will be :

Let AD and BC be two vertical poles at A and B respectively on a horizontal ground. If AD = 8 m, BC = 11 m and AB = 10 m; then the distance (in meters) of a point M on AB from the point A such that \(MD^2+MC^2\) is minimum is _______.

The angle of elevation of the top of a hill from a point on the horizontal plane passing through the foot of the hill is found to be \(45^\circ\). After walking a distance of 80 meters towards the top, up a slope inclined at an angle of \(30^\circ\) to the horizontal plane, the angle of elevation of the top of the hill becomes \(75^\circ\). Then the height of the hill (in meters) is _______.

You are given that Mass of \(^{7}_{3}\mathrm{Li}\) = 7.0160u,
Mass of \(^{4}_{2}\mathrm{He}\) = 4.0026u
and Mass of \(^{1}_{1}\mathrm{H}\) = 1.0079u.
When 20 g of \(^{7}_{3}\mathrm{Li}\) is converted into \(^{4}_{2}\mathrm{He}\) by proton capture, the energy liberated, (in kWh), is :
[Mass of nucleon = 1 GeV/c²]

A particle of charge q and mass m is moving with a velocity \(-v\hat{i}\) (\(v \ne 0\)) towards a large screen placed in the Y - Z plane at a distance d. If there is a magnetic field \(\vec{B} = B_0\hat{k}\), the maximum value of v for which the particle will not hit the screen is :

If \(\vec{a}\) and \(\vec{b}\) are unit vectors, then the greatest value of
\(\sqrt{3}\,|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|\)
is _______.

The negation of the Boolean expression \(p\vee(\sim p\wedge q)\) is equivalent to :

The shortest distance between the lines
\(\frac{x-1}{0}=\frac{y+1}{-1}=\frac{z}{1}\)
and \(x+y+z+1=0,\ 2x-y+z+3=0\) is :

Which of the following points lies on the locus of the foot of perpedicular drawn upon any tangent to the ellipse,
\(\frac{x^2}{4}+\frac{y^2}{2}=1\)
from any of its foci?

Let \(f:R\to R\) be defined as
\(f(x)=\begin{cases}x^5\sin\left(\frac{1}{x}\right)+5x^2, & x\lt 0\\ 0, & x=0\\ x^5\cos\left(\frac{1}{x}\right)+\lambda x^2, & x\gt 0\end{cases}\)
The value of \(\lambda\) for which \(f''(0)\) exists, is _______.

Let \(L_1\) be a tangent to the parabola \(y^2=4(x+1)\) and \(L_2\) be a tangent to the parabola \(y^2=8(x+2)\) such that \(L_1\) and \(L_2\) intersect at right angles. Then \(L_1\) and \(L_2\) meet on the straight line :

\(\displaystyle\lim_{x\to 1}\left(\frac{\int_{0}^{(x-1)^2} t\cos\left(t^2\right)dt}{(x-1)\sin(x-1)}\right)\)

A ray of light coming from the point \((2,\,2\sqrt{3})\) is incident at an angle \(30^\circ\) on the line \(x=1\) at the point A. The ray gets reflected on the line \(x=1\) and meets x-axis at the point B. Then, the line AB passes through the point :

If \(I_1=\displaystyle\int_0^1(1-x^{50})^{100}dx\) and \(I_2=\displaystyle\int_0^1(1-x^{50})^{101}dx\) such that \(I_2=\alpha I_1\) then \(\alpha\) equals to :

The general solution of the differential equation
\(\sqrt{1+x^2+y^2+x^2y^2}+xy\dfrac{dy}{dx}=0\) is :
(where C is a constant of integration)