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The horizontal range and the maximum height of a projectile are equal. The angle of projection of the projectile is

The diameter of a cylinder is measured using a Vernier calipers with no zero error. It is found that the zero of the Vernier scale lies between $5.10\text{cm}$ and $5.15\text{cm}$ of the main scale. The Vernier scale has $50$ divisions equivalent to $2.45\text{cm}$. The ${24}^{th}$ division of the Vernier scale excatly coincides with one of the main scale divisions. The diameter of the cylinder is :-

The current in the given circuit is increasing at a rate of $4{\text{As}}^{-1}.$ The charge on the capacitor at an instant when the current in the circuit is $2\text{A}$ will be-

In $L{i}^{++}$ electron in first Bohr orbit is excited to a level by a radiation of wavelength $X$. When the ion gets deexcited to the ground state in all possible ways (including intermediate emissions), a total of six spectral lines are observed. What is the value of $\lambda$ ?

The wave equation of a pulse is given by $y=\frac{3}{2+(x-4t{)}^2}$ where y is in metre and t in second. The wave velocity of the pulse is

A satellite is moving in a low, nearly circular orbit around the earth. Its radius is roughly equal to that of the earth's radius $R_e$. By firing rockets attached to it, Its speed is instantaneously increased in the direction of its motion so that it become $\sqrt{\frac{3}{2}}$ times larger. Due to this, the farthest distance from the centre of the earth that the satellite reaches is $R.$ The value of $R$ is-

A particle is projected at an angle of ${30}^{\circ }$ w.r.t horizontal with speed $20m/s.$ Take the point of projection as the origin of the coordinate axes and find the angle between velocity vector and position at $t=1s$

A machine is blowing spherical soap bubbles of different radii filled with helium gas. It is found that if the bubbles have a radius smaller than 1 cm, then they sink to the floor in still air. Larger bubbles float in the air. Assume that the thickness of the soap film in all bubbles is uniform and equal. Assume that the density of soap solution is same as that of water (= 1000 $kg{m}^{-3})$. The density of helium inside the bubbles and air are 0.18 kg $m^{-3}$ and 1.23 kg $m^{-3}$, respectively. Then the thickness of the soap film of the bubbles is (note 1 $\mu$ m = ${10}^{-6}$ m)

A planet of mass $m$ moves around the sun of mass $M$ in an elliptical orbit. The maximum and minimum distances of the planet from the sun are $r_1$ and $r_2$ respectively. The time period of revolution of planet is proportional to

A copper wire of diameter $2.4\text{mm}$ carries a current $i$. The maximum magnetic field due to this wire $5\times {10}^{-4}\text{T}$, then the value of $i$ is

A mass of $5\text{kg}$ is suspended from a spring of stiffness constant $4000\text{N/m}$. The system is fitted with a damper with a damping coefficient of $0.2$. The mass is pulled down by $50\text{mm}$ and then released. Calculate the damped frequency.

For the expression $2x^2–7x-5$, the roots are α and β, then the value of α+β is

In the combination of following gates, the output $Y$ can be written in terms of the inputs $A\text{and}B$ as :

Two resistors have their resistances in the ratio $2:3.$ If these two resistors are connected in parallel, then the ratio of respective power dissipated through the resistors will be equal to



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A ball of mass $0.25\text{kg}$ attached to the string of length $1.96\text{m}$ is moving in a horizontal circle. The string will break if the tension in the string is more than $25\text{N}$. What is the maximum angular velocity with which the ball can be moved?

If $\overrightarrow{a}=\hat{i}+2\hat{j}+4\hat{k},\overrightarrow{b}=\hat{i}+\lambda \hat{j}+4\hat{k},\overrightarrow{c}=2\hat{i}+4\hat{j}+({\lambda }^2-1)\hat{k},$ are coplanar vectors and $\overrightarrow{a},\overrightarrow{c}$ are not parallel, then $|\overrightarrow{a}\times \overrightarrow{c}|$ is equal to

Two identical glass $({\mu }_g=3/2)$ equiconvex lenses of focal length $f$ are kept in contact. The space between the two lenses is filled with water $({\mu }_w=4/3)$. The focal length of the combination is

A series LCR circuit containing a resistance of $120\Omega$ has resonance frequency $4\times {10}^5\text{rad/s}$. At resonance, the voltage across resistor and inductor are $60\text{V}$ and $40\text{V}$, respectively. The value of capacitance is -

A conducting circular loop is placed in a uniform magnetic field $0.04\text{T}$ with its plane perpendicular to the magnetic field. The radius of the loop starts shrinking at a uniform rate of $2\text{mm/s}$. The induced emf in the loop, when the radius is $2\text{cm}$, is:

Abhishek decides he knows enough about how stuff works to fix his TV. He then proceeds to turn the switch on and put his finger in the socket. Before the resulting electric shock causes any serious consequential damage, Himanshu decides to violently smash Abhishek’s hand away to save his life. Which of the following objects must he use to ensure he doesn't put himself in any kind of danger while maximizing Abhishek's chances of surviving?

Electric field in a plane electromagnetic wave is given by

$E=50\sin (500x-10\times {10}^{10}t)\text{V/m}$.

The velocity of electromagnetic wave in this medium is :

(Given $c=$ speed of light in vacuum)

Water is flowing through a horizontal tube of non - uniform cross - section. At a place the radius of the tube is $1.0\text{cm}$ and the velocity of water is $2\text{m/s}$. Then, what will be the velocity of water at a point where the radius of the pipe is $2.0\text{cm}$?

The radioactive decay of uranium into thorium is represented by the equation
${}_{92}^{238}U\to {}_{90}^{234}Th+x$
What is x?

Find the derivative of $y=e^{\sin \theta }$ with respect to $\theta$

Consider a spherical gaseous cloud of mass density $\rho \left(r\right)$ in free space, where $r$ is the radial distance from its centre. The gaseous cloud is made of particles of equal mass $m$ moving in circular orbits about the common center with the same kinetic energy $K$. The force acting on the particles is their mutual gravitational force. If $\rho \left(r\right)$ is constant in time, the particle number density $n\left(r\right)=\rho \left(r\right)/m$ is [$G$ is universal gravitational constant]

Activities of three radioactive substances $A,B$ and $C$ are represented by the curves $A,B$ and $C$, in the figure. Then their half-lives $T_{\frac{1}{2}}\left(A\right):T_{\frac{1}{2}}\left(B\right):T_{\frac{1}{2}}\left(C\right)$ are in the ratio

Which of the following is not correct for the condition for work not to be done:

Two blocks $A$ and $B$ of equal masses are released from an inclined plane of inclination ${45}^{\circ }$ at $t=0$ . Both the blocks are initially at rest. The coefficient of kinetic friction between the block $A$ and the inclined plane is $0.2$ while it is $0.3$ for block $B$. Initially the block $A$ is $\sqrt{2}\text{m}$ behind the block $B$. When and where from the position of A, front faces of both the blocks will come in a line? (Take $g=10m/s^2)$