List of top Questions asked in JEE Main

Bob of a simple pendulum of length \(l\) is made of iron. The pendulum is oscillating over a horizontal coil carrying direct current. If the time period of the pendulum is T then :

Consider a finite insulated, uncharged conductor placed near a finite positively charged conductor. The uncharged body must have a potential :

Wax is coated on the inner wall of a capillary tube and the tube is then dipped in water. Then, compared to the unwaxed capillary, the angle of contact \(\theta\) and the height \(h\) upto which water rises change. These changes are :

The ratio of the coefficient of volume expansion of a glass container to that of a viscous liquid kept inside the container is 1 : 4. What fraction of the inner volume of the container should the liquid occupy so that the volume of the remaining vacant space will be same at all temperatures ?

A thin tube sealed at both ends is 100 cm long. It lies horizontally, the middle 20 cm containing mercury and two equal ends containing air at standard atmospheric pressure. If the tube is now turned to a vertical position,

by what amount will the mercury be displaced ?
(Given : cross-section of the tube can be assumed to be uniform)

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: The internal energy of a perfect gas is entirely kinetic and depends only on absolute temperature of the gas and not on its pressure or volume.
Statement 2: A perfect gas is heated keeping pressure constant and later at constant volume. For the same amount of heat the temperature of the gas at constant pressure is lower than that at constant volume.

A particle of mass 2 kg is moving such that at time \(t\), its position, in meter, is given by \(\vec{r}(t)=5\hat{i}-2t^2\hat{j}\). The angular momentum of the particle at \(t=2\,\mathrm{s}\) about the origin in \(\mathrm{kgm^{-2}\,s^{-1}}\) is :

A body of mass ' \(m\) ' is tied to one end of a spring and whirled round in a horizontal plane with a constant angular velocity. The elongation in the spring is 1 cm. If the angular velocity is doubled, the elongation in the spring is 5 cm. The original length of the spring is :

A copper wire of length 1.0 m and a steel wire of length 0.5 m having equal cross-sectional areas are joined end to end. The composite wire is stretched by a certain load which stretches the copper wire by 1 mm. If the Young's modulii of copper and steel are respectively \(1.0\times10^{11}\ \mathrm{Nm^{-2}}\) and \(2.0\times10^{11}\ \mathrm{Nm^{-2}}\), the total extension of the composite wire is :

A, B, C try to hit a target simultaneously but independently. Their respective probabilities of hitting the targets are \(\frac{3}{4},\frac{1}{2},\frac{5}{8}\). The probability that the target is hit by A or B but not by C is :

A 70 kg man leaps vertically into the air from a crouching position. To take the leap the man pushes the ground with a constant force \(F\) to raise himself. The center of gravity rises by 0.5 m before he leaps. After the leap the c.g. rises by another 1 m. The maximum power delivered by the muscles is : (Take \(g=10\ \mathrm{ms^{-2}}\))

A boy of mass 20 kg is standing on a 80 kg free to move long cart. There is negligible friction between cart and ground. Initially, the boy is standing 25 m from a wall. If he walks 10 m on the cart towards the wall, then the final distance of the boy from the wall will be

The value of \(\displaystyle\int_{-\pi/2}^{\pi/2}\dfrac{\sin^2 x}{1+2^x}\,dx\) is :

The area under the curve \(y=|\cos x-\sin x|,\ 0\le x\le\frac{\pi}{2}\), and above \(x\)-axis is :

If the projections of a line segment on the \(x\), \(y\) and \(z\)-axes in 3-dimensional space are 2, 3 and 6 respectively, then the length of the line segment is :

If the curves \(\frac{x^2}{\alpha}+\frac{y^2}{4}=1\) and \(y^3=16x\) intersect at right angles, then a value of \(\alpha\) is :

If a curve passes through the point \(\left(2, \frac{7}{2}\right)\) and has slope \(\left(1-\frac{1}{x^2}\right)\) at any point \((x, y)\) on it, then the ordinate of the point on the curve whose abscissa is \(-2\) is :

Statement-1: The system of linear equations
\(x+(\sin\alpha)y+(\cos\alpha)z=0\)
\(x+(\cos\alpha)y+(\sin\alpha)z=0\)
\(x-(\sin\alpha)y-(\cos\alpha)z=0\)
has a non-trivial solution for only one value of \(\alpha\) lying in the interval \(\left(0, \frac{\pi}{2}\right)\).
Statement-2: The equation in \(\alpha\)
\(\begin{vmatrix}\cos\alpha & \sin\alpha & \cos\alpha\\ \sin\alpha & \cos\alpha & \sin\alpha\\ \cos\alpha & -\sin\alpha & -\cos\alpha\end{vmatrix}=0\)
has only one solution lying in the interval \(\left(0, \frac{\pi}{2}\right)\)

\(S=\tan^{-1}\left(\frac{1}{n^2+n+1}\right)+\tan^{-1}\left(\frac{1}{n^2+3n+3}\right)+\ldots+\tan^{-1}\left(\frac{1}{1+(n+19)(n+20)}\right)\), then \(\tan S\) is equal to :

Let \(f\) be a composite function of \(x\) defined by \(f(u)=\frac{1}{u^2+u-2},\ u(x)=\frac{1}{x-1}\). Then the number of points \(x\) where \(f\) is discontinuous is :

If the median and the range of four numbers \(\{x, y, 2x+y, x-y\}\), where \(0 \lt y \lt x \lt 2y\), are 10 and 28 respectively, then the mean of the numbers is :

Let \(R=\{(x, y) : x, y \in N \text{ and } x^2-4xy+3y^2=0\}\), where \(N\) is the set of all natural numbers. Then the relation \(R\) is :

Let A, other than I or \(-I\), be a \(2 \times 2\) real matrix such that \(A^2=I\), I being the unit matrix. Let \(\mathrm{Tr}(A)\) be the sum of diagonal elements of A.
Statement-1: \(\mathrm{Tr}(A)=0\)
Statement-2: \(\det(A)=-1\)

For integers \(m\) and \(n\), both greater than 1, consider the following three statements : \(P : m\) divides \(n\) \(Q : m\) divides \(n^2\) \(R : m\) is prime, then

If the circle \(x^2 + y^2 - 6x - 8y + (25 - a^2) = 0\) touches the axis of \(x\), then \(a\) equals.