One mole of an ideal diatomic gas expands from volume V to 2 V isothermally at a temperature \(27^{\circ} C\) and does W joule of work. If the gas undergoes same magnitude of expansion adiabatically from \(27^{\circ} C\) doing the same amount of work W , then its final temperature will be (close to) ___________ \({ }^{\circ} C\).

Hot water cools from \(60\,^oC\) to \(50\,^oC\) in the first \(10\,minutes\) and to \(42\,^oC\) in the next \(10\,minutes.\) The temperature of the surroundings is ...... \(^oC\)

With reference to the observations in photo-electric effect, identify the correct statements from below: \(A.\) The square of maximum velocity of photoelectrons varies linearly with frequency of incident light. \(B.\) The value of saturation current increases on moving the source of light away from the metal surface. \(C.\) The maximum kinetic energy of photo-electrons decreases on decreasing the power of LED (light emitting diode) source of light. \(D.\) The immediate emission of photo-electrons out of metal surface can not be explained by particle nature of light/electromagnetic waves. \(E.\) Existence of threshold wavelength can not be explained by wave nature of light/electromagnetic waves. Choose the correct answer from the options given below:

The relation between root mean square speed \(\left( v _{ rms }\right)\) and most probable speed \(\left( v _{ p }\right)\) for the molar mass \(M\) of oxygen gas molecule at the temperature of \(300\, K\) will be

A rod of length \(50\,cm\) is pivoted at one end. It is raised such that if makes an angle of \(30^o\) fro the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in \(rad\,s^{-1}\) ) will be \((g = 10\,ms^{-2})\)

A particle is doing simple harmonic motion of amplitude \(0.06 \mathrm{~m}\) and time period \(3.14 \mathrm{~s}\). The maximum velocity of the particle is _______ \(\mathrm{cm} / \mathrm{s}\).

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 ........ \(m\)

If the domain of the function \(f(x)=\sin ^{-1}\left(\frac{5-x}{3+2 x}\right)+\frac{1}{log _e (10-x)}\) is \((-\infty, \alpha]\)\(\cup[\beta, \gamma)-\{\delta\}\), then \(6(\alpha+\beta+\gamma+\delta)\) is equal to

A particle executes simple harmonic motion. Its amplitude is \(8 \,cm\) and time period is \(6 \,s\). The time it will take to travel from its position of maximum displacement to the point corresponding to half of its amplitude, is ............. \(s\)

When a rubber-band is stretched by a distance \(x\), it exerts restoring force of magnitude  \(F=ax+bx^2\) where \(a\) and \(b\) areconstants. The work done in stretching the unstretched rubber-band by \(L\) is

Let the circle \(x^{2}+y^{2}=4\) intersect x-axis at the points \(A(a,0), a>0\) and \(B(b,0)\). Let \(P(2 \cos\alpha, 2 \sin\alpha), 0<\alpha<\frac{\pi}{2}\) and \(Q(2 \cos\beta, 2 \sin\beta)\) be two points such that \((\alpha-\beta)=\frac{\pi}{2}.\) Then the point of intersection of AQ and BP lies on:

A copper block of mass \(5.0\, kg\) is heated to a temperature of \(500^{\circ} C\) and is placed on a large ice block. What is the maximum amount of ice (ઇન \(kg\)) that can melt? [Specific heat of copper: \(0.39\, Jg ^{-1 \circ} C ^{-1}\) and latent heat of fusion of water : \(335 \,J g ^{-1}\) ]

The displacement time graph of a particle executing \(S.H.M.\) is given in figure: (sketch is schematic and not to scale) Which of the following statements is are true for this motion? \((A)\) The force is zero \(t=\frac{3 T}{4}\) \((B)\) The acceleration is maximum at \(t=T\) \((C)\) The speed is maximum at \(t =\frac{ T }{4}\) \((D)\) The \(P.E.\) is equal to \(K.E.\) of the oscillation at \(t=\frac{T}{2}\)