A particle executes \(S.H.M.\) of amplitude A along \(x\)-axis. At \(t =0\), the position of the particle is \(x=\frac{A}{2}\) and it moves along positive \(x\)-axis the displacement of particle in time \(t\) is \(x=A \sin (\omega t+\delta)\), then the value \(\delta\) will be

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : In an insulated container, a gas is adiabatically shrunk to half of its initial volume. The temperature of the gas decreases.
Reason (R): Free expansion of an ideal gas is an irreversible and an adiabatic process.
In the light of the above statements, choose the correct answer from the options given below :

In the given circuit, the charge on \(4\, \mu F\) capacitor will be.....\(\mu C\)

In the experimental setup of meter bridge shown in the figure, the null point is obtained at a distance of \(40\,cm\) from \(A\). If a \(10\,\Omega \) resistor is connected in series with \(R_1\), the null point shifts by \(10\,cm\). The resistance that should be connected in parallel with \(\left( {{R_1} + 10} \right)\,\Omega \) such that the null point shifts back to its initial position is .............. \(\Omega\)

Water falls from a height of 200 m into a pool. Calculate the rise in temperature of the water assuming no heat dissipation from the water in the pool.
\(\left(\right.\)Take \(g=10 \mathrm{~m} / \mathrm{s}^2\), specific heat of water \(\left.=4200 \mathrm{~J} /(\mathrm{kg} \mathrm{K})\right)\)

If the r.m.s. speed of chlorine molecule is \(490\,m / s\) at \(27^{\circ}\,C\), the r.m.s. speed of argon molecules at the same temperature will be \(......\,m/s\) (Atomic mass of argon \(=39.9\,u\), molecular mass of chlorine \(=70.9\,u )\)

The equilateral triangle \(ABC\) is cut from a thin solid sheet of wood. (See figure) \(D, E\) and \(F\) are the mid points of its sides as shown and \(G\) is the centre of the triangle. The moment of inertia of the triangle about an axis passing through \(G\) and perpendicular to the plane of the triangle is \(I_0.\) If the smaller triangle \(DEF\) is removed from \(ABC,\) the moment of inertia of the remaining figure about the same axis is \(I.\) Then

A sample of an ideal gas is taken through the cyclic process \(abca\) as shown in the figure. The change in the internal energy of the gas along the path \(ca\) is \(-180\, J\). The gas absorbs \(250\, J\) of heat along the path \(ab\) and \(60\, J\) along the path \(bc\). The work done by the gas along the path \(abc\) is ..... \(J\)

The distance \(x\) covered by a particle in one dimensional motion varies with time \(t\) as \(\mathrm{x}^{2}=\mathrm{at}^{2}+2 \mathrm{bt}+\mathrm{c.}\) If the acceleration of the particle depends on \(\mathrm{x}\) as \(\mathrm{x}^{-\mathrm{n}},\) where \(\mathrm{n}\) is an integer, the value of \(\mathrm{n}\) is

A thermally insulated vessel contains an ideal gas of molecular mass \(M\) and ratio of specific heats \(1.4\). Vessel is moving with speed \(v\) and is suddenly brought to rest. Assuming no heat is lost to the surrounding and vessel temperature of the gas increases by ... ( \(R =\) universal gas constant )

A loop ABCDA , carrying current \(\mathrm{I}=12 \mathrm{~A}\), is placed in a plane, consists of two semi-circular segments of radius \(R_1=6 \pi \mathrm{~m}\) and \(\mathrm{R}_2=4 \pi \mathrm{~m}\). The magnitude of the resultant magnetic field at center O is \(\mathrm{k} \times 10^{-7} \mathrm{~T}\). The value of k is ______ (Given \(\mu_0=4 \pi \times 10^{-7} \mathrm{Tm} \mathrm{A}^{-1}\))

Two thin convex lenses of focal length 30 cm and 10 cm are placed coaxially, 10 cm apart. The power of this combination is :