The following bodies, \((1)\) a ring \((2)\) a disc \((3)\) a solid cylinder \((4)\) a solid sphere, of same mass \(m\) and radius \(R\) are allowed to roll down without slipping simultaneously from the top of the inclined plane. The body which will reach first at the bottom of the inclined plane is ........... [Mark the body as per their respective numbering given in the question]

In a Young's double slit experiment, two slits are located 1.5 mm apart. The distance of screen from slits is 2 m and the wavelength of the source is 400 nm. If the 20 maxima of the double slit pattern are contained within the centre maximum of the single slit diffraction pattern, then the width of each slit is \(\mathrm{x} \times 10^{-3} \mathrm{~cm}\), where x -value is ________.

A hot body, obeying Newton's law of cooling is cooling down from its peak value \(80\,^oC\) to an ambient temperature of \(30\,^oC\) . It takes \(5\, minutes\) in cooling down from \(80\,^oC\) to \(40\,^oC\).  ........  \(\min.\) will it take to cool down from \(62\,^oC\) to \(32\,^oC\) ? (Given \(ln\, 2\, = 0.693, ln\, 5\, = 1.609\))

A straight wire carrying a current of \(14\,A\) is bent into a semicircular are of radius \(2.2\,cm\) as shown in the figure. The magnetic field produced by the current at the centre \((O)\) of the arc. is \(.........\,\times 10^{-4}\, T\)

A wire \(X\) of length \(50\; cm\) carrying a current of \(2\,A\) is placed parallel to a long wire \(Y\) of length \(5\,m\). The wire \(Y\) carries a current of \(3\,A\). The distance between two wires is \(5\,cm\) and currents flow in the same direction. The force acting on the wire \(Y\) is.

Three rods of Copper, Brass and Steel are welded together to form a \(Y\) shaped structure. Area of cross - section of each rod \(= 4\ cm^2\) . End of copper rod is maintained at \(100^o C \) where as ends ofbrass and steel are kept at \(0^o C\). Lengths of the copper, brass and steel rods are \(46, 13\) and \(12\ cms\) respectively. The rods are thermally insulated from surroundings excepts at ends. Thermal conductivities of copper, brass and steel are \(0.92, 0.26\) and \(0.12\ CGS\) units respectively. Rate ofheat flow through copper rod is ....... \(cal\, s^{-1}\)

Two stars of masses \(3\times10^{31}\, kg\) each, and at distance \(2\times10^{11}\, m\) rotate in a plane about their common centre of mass \(O. A\) meteorite passes through \(O\) moving perpendicular to the star’s rotation plane. In order to escape from the gravitational field of this double star, the minimum speed that meteorite should have at \(O\) is : (Take Gravitational constant \(G = 6.67\times10^{-11}\, Nm^2\, kg^{-2}\))

A capillary tube is immersed vertically in water and the height of the water column is \(x\). When this arrangement is taken into a mine of depth \(d\), the height of the water column is \(y\). If \(R\) is the radius of earth, the ratio \(\frac{x}{y}\) is

The gravitational field in a region is given by \(\overrightarrow g  = 5\,N/kg\hat i\, + \,12\,N/kg\hat j\). The change in the gravitational potential energy of a particle of mass \(1\, kg\) when it is taken from the origin to a point \((7\, m, - 3\, m)\) is .......  \(J\)

Diameter of a steel ball is measured using a Vernier callipers which has divisions of \(0. 1\,cm\) on its main scale \((MS)\) and \(10\) divisions of its vernier scale \((VS)\) match \(9\) divisions on the main scale. Three such measurements for a ball are given as

S.No.	\(MS\;(cm)\)	\(VS\) divisions
\((1)\)	\(0.5\)	\(8\)
\((2)\)	\(0.5\)	\(4\)
\((3)\)	\(0.5\)	\(6\)

If the zero error is \(- 0.03\,cm,\) then mean corrected diameter is  ........... \(cm\)

Given below are two statements: one is labelled as Assertion \(A\) and the other is labelled as Reason \(R\). \(Assertion\) \(A\) : Number of photons increases with increase in frequency of light. \(Reason\) \(R\) : Maximum kinetic energy of emitted electrons increases with the frequency of incident radiation. In the light of the above statements, choose the most appropriate answer from the options given below :

A body moves on a frictionless plane starting from rest. If \(\mathrm{S}_{\mathrm{n}}\) is distance moved between \(\mathrm{t}=\mathrm{n}-1\) and \(\mathrm{t}\) \(=\mathrm{n}\) and \(\mathrm{S}_{\mathrm{n}-1}\) is distance moved between \(\mathrm{t}=\mathrm{n}-2\) and \(t=n-1\), then the ratio \(\frac{S_{n-1}}{S_n}\) is \(\left(1-\frac{2}{x}\right)\) for \(n\) \(=10\). The value of \(x\) is _______.

If one mole of an ideal gas at \(\left( P _{1}, V _{1}\right)\) is allowed to expand reversibly and isothermally (\(A\) to \(B\) ) its pressure is reduced to one-half of the original pressure (see \(figure\)). This is followed by a constant volume cooling till its pressure is reduced to one-fourth of the initial value \(( B \rightarrow C ) .\) Then it is restored to its initial state by a reversible adiabatic compression (\(C\) to \(A\)). The net workdone by the gas is equal to  ...... .