A force of \(- P \hat{k}\) acts on the origin of the coordinate system. The torque about the point \((2,-3)\) is \(P(a \hat{i}+b \hat{j})\), the ratio of \(\frac{a}{b}\) is \(\frac{x}{2}\). The value of \(x\) is

Two different wires having lengths \(L _{1}\) and \(L _{2}\) and respective temperature coefficient of linear expansion \(\alpha_{1}\) and \(\alpha_{2},\) are joined end-to-end. Then the effective temperature coefficient of linear expansion is

Two masses m and 2m are connected by a light string going over a pulley (disc) of mass 30m with radius r = 0.1 m. The pulley is mounted in a vertical plane and it is free to rotate about its axis. The 2m mass is released from rest and its speed when it has descended through a height of 3.6 m is __________ m/s. (Assume string does not slip and \( g=10 m/s^{2} \))

The co-ordinates of a particle moving in \(x-y\) plane are given by :  \(\mathrm{x}=2+4 \mathrm{t}, \mathrm{y}=3 \mathrm{t}+8 \mathrm{t}^2 .\) The motion of the particle is _______.

Voltage rating of a parallel plate capacitor is \(500\,V\). Its dielectric can withstand a maximum electric field of \({10^6}\,\frac{V}{m}\). The plate area is \(10^{-4}\, m^2\) . What is the dielectric constant if the capacitance is \(15\, pF\) ? (given \({ \in _0} = 8.86 \times {10^{ - 12}}\,{C^2}\,/N{m^2}\))

The temperature of \(3.00\, {mol}\) of an ideal diatomic gas is increased by \(40.0^{\circ} {C}\) without changing the pressure of the gas. The molecules in the gas rotate but do not oscillate. If the ratio of change in internal energy of the gas to the amount of workdone by the gas is \(\frac{{x}}{10} .\) Then the value of \({x}\) (round off to the nearest integer) is ..... . \(\left(\right.\) Given \(\left.{R}=8.31\, {J} {mol}^{-1} {K}^{-1}\right)\)

A radioactive nucleus \(A\) with a half life \(T\), decays into a nucleus \(B.\) At \(t = 0\), there is no nucleus \(B\). At sometime \(t\), the ratio of the number of \(B\) to that of \(A\) is \(0.3\). Then, \(t\) is given by

The trajectory of projectile, projected from the ground is given by \(y=x-\frac{x^2}{20}\). Where \(x\) and \(y\) are measured in meter. The maximum height attained by the projectile will be \(...........\,m\)

An inductor of self inductance 1 H connected in series with a resistor of \(100 \pi \mathrm{ohm}\) and an ac supply of \(100 \pi\) volt, 50 Hz. Maximum current flowing in the circuit is ________ A.

A radar sends an electromagnetic signal of electric field \(\left( E _{0}\right)=2.25\,V / m\) and magnetic field \(\left( B _{0}\right)=1.5 \times 10^{-8}\,T\) which strikes a target on line of sight at a distance of \(3\,km\) in a medium After that, a pail of signal \((echo)\) reflects back towards the radar vitli same velocity and by same path. If the signal was transmitted at time \(t_{0}\) from radar. then after how much time (in  \(\times 10^{-5}\,s\)) echo will reach to the radar?

In an experiment, a set of reading are obtained \(-1.24 mm, 1.25 mm, 1.23 mm, 1.21 mm\). The expected least count of the instrument used in recording these readings is ___________ mm.

Match List\(-I\) with List\(-II.\)

List\(-I\)	List\(-II\)
\((a)\) Torque	\((i)\) \({MLT}^{-1}\)
\((b)\) Impulse	\((ii)\) \({MT}^{-2}\)
\((c)\) Tension	\((iii)\) \({ML}^{2} {T}^{-2}\)
\((d)\) Surface Tension	\((iv)\) \({MI} {T}^{-2}\)

Choose the most appropriate answer from the option given below :

Consider a sphere of radius \(\mathrm{R}\) which carries a uniform charge density \(\rho .\) If a sphere of radius \(\frac{\mathrm{R}}{2}\) is carved out of it, as shown, the ratio \(\frac{\left|\overrightarrow{\mathrm{E}}_{\mathrm{A}}\right|}{\left|\overrightarrow{\mathrm{E}}_{\mathrm{B}}\right|}\) of magnitude of electric field \(\overrightarrow{\mathrm{E}}_{\mathrm{A}}\) and \(\overrightarrow{\mathrm{E}}_{\mathrm{B}}\) respectively, at points \(\mathrm{A}\) and \(\mathrm{B}\) due to the remaining portion is