Two objects are projected with same velocity ' \(u\) ' however at different angles \(\alpha\) and \(\beta\) with the horizontal. If \(\alpha+\beta=90^{\circ}\), the ratio of horizontal range of the first object to the \(2^{\text {nd }}\) object will be :

A Carnot freezer takes heat from water at \(0\,^oC\) inside it and rejects it to the room at a temperature of \(27\,^oC\). The latent heat of ice is \(336 \times 10^3\, J\,kg^{-1}\).  lf \(5\, kg\) of water at \(0\,^oC\) is converted into ice at  \(0\,^oC\) by the freezer, then the energy consumed by the freezer is close to

In a Young's double slit experiment, the distance between the two identical slits is \(6.1\) times larger than the slit width. Then the number of intensity maxima observed with in the central maximum of the single slit diffraction pattern is

A radioactive sample disintegrates via two independent decay processes having half lives \(T _{1 / 2}^{(1)}\) and \(T _{1 / 2}^{(2)}\) respectively. The effective half- life \(T _{1 / 2}\) of the nuclei is

The \(SI\) unit of a physical quantity is pascal-second. The dimensional formula of this quantity will be ..............

In a Vernier Calipers. \(10\) divisions of Vernier scale is equal to the \(9\) divisions of main scale. When both jaws of Vernier calipers touch each other, the zero of the Vernier scale is shifted to the left of \(zero\) of the main scale and \(4^{\text {th }}\) Vernier scale division exactly coincides with the main scale reading. One main scale division is equal to \(1\,mm\). While measuring diameter of a spherical body, the body is held between two jaws. It is now observed that zero of the Vernier scale lies between \(30\) and \(31\) divisions of main scale reading and \(6^{\text {th }}\) Vernier scale division exactly. coincides with the main scale reading. The diameter of the spherical body will be \(.......cm\)

A wire of density \(9 \times 10^{-3} \,kg\, cm ^{-3}\) is stretched between two clamps \(1\, m\) apart. The resulting strain in the wire is \(4.9 \times 10^{-4}\). The lowest frequency of the transverse vibrations in the wire is......\(HZ\) (Young's modulus of wire \(Y =9 \times 10^{10}\, Nm ^{-2}\) ), (to the nearest integer),

A ray of light passing through an equilateral prism is having velocity \(2.12 \times 10^8\) m/s in the prism material, then the minimum angle of deviation is _________ degrees.

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Knowing initial position \(x_0\) and initial momentum \(p_0\) is enough to determine the position and momentum at any time \(t\) for a simple harmonic motion with a given angular frequency \(\omega\).
Reason (R): The amplitude and phase can be expressed in terms of \(x_0\) and \(\mathrm{p}_0\).
In the light of the above statements, choose the correct answer from the options given below :

In the reported figure, there is a cyclic process \(ABCDA\) on a sample of \(1\, {mol}\) of a diatomic gas. The temperature of the gas during the process \({A} \rightarrow {B}\) and \({C} \rightarrow {D}\) are \({T}_{1}\) and \({T}_{2}\left({T}_{1}\,>\,{T}_{2}\right)\) respectively. Choose the correct option out of the following for work done if processes \(B C\) and \(D A\) are adiabatic.

A body of mass \(200\,g\) is tied to a spring of spring constant \(12.5\,N / m\), while the other end of spring is fixed at point \(O\). If the body moves about \(O\) in a circular path on a smooth horizontal surface with constant angular speed \(5\,rad / s\), then the ratio of extension in the spring to its natural length will be :

A bullet of mass \(0.1\, kg\) is fired on a wooden block to pierce through it, but it stops after moving a distance of \(50\, cm\) into it. If the velocity of bullet before hitting the wood is \(10\) \(m / s\) and it slows down with uniform deceleration, then the magnitude of effective retarding force on the bullet is \('x' N.\) The value of \('x'\) to the nearest integer is................. \(N\)

A convex lens is put \(10\, cm\) from a light source and it makes a sharp image on a screen, kept \(10\, cm\) from the lens. Now a glass block (refractive index \(1.5\)) of \(1.5\, cm\) thickness is placed in contact with the light source. To get the sharp image again, the screen is shifted by a distance \(d\). Then \(d\) is