The volume of an ideal gas \((\gamma=1.5)\) is changed adiabatically from \(5\) litres to \(4\) litres. The ratio of initial pressure to final pressure is _______.

A cubical volume is bounded by the surfaces \(x =0, x = a , y =0, y = a , z =0, z = a\). The electric field in the region is given by \(\overrightarrow{ E }= E _0 \times \hat{ i }\). Where \(E _0=4 \times 10^4 NC ^{-1} m ^{-1}\). If \(a =2 cm\), the charge contained in the cubical volume is \(Q \times 10^{-14} C\). The value of \(Q\) is \(...........\) Take \(\left.\varepsilon_0=9 \times 10^{-12} C ^2 / Nm ^2\right)\) 

A uniform chain of \(6\, m\) length is placed on a table such that a part of its length is hanging over the edge of the table. The system is at rest. The co-efficient of static friction between the chain and the surface of the table is \(0.5\), the maximum length of the chain hanging from the table is.......\(m.\)

Read the following statements: \(A.\) When small temperature difference between a liquid and its surrounding is doubled the rate of loss of heat of the liquid becomes twice. \(B.\) Two bodies \(P\) and \(Q\) having equal surface areas are maintained at temperature \(10^{\circ}\,C\) and \(20^{\circ}\,C\). The thermal radiation emitted in a given time by \(P\) and \(Q\) are in the ratio \(1: 1.15\) \(C.\) A carnot Engine working between \(100\,K\) and \(400\,K\) has an efficiency of \(75 \%\) \(D.\) When small temperature difference between a liquid and its surrounding is quadrupled, the rate of loss of heat of the liquid becomes twice. Choose the correct answer from the options given below :

Two objects of equal masses placed at certain distance from each other attracts each other with a force of \(F\). If one-third mass of one object is transferred to the other object, then the new force will be

Assume that the earth is a solid sphere of uniform density and a tunnel is dug along its diameter throughout the earth. It is found that when a particle is released in this tunnel, it executes a simple harmonic motion. The mass of the particle is \(100 g\). The time period of the motion of the particle will be (approximately) (take \(g =10\,ms ^{-2}\), radius of earth \(=6400\,km\) )

Given below are two statements: one is labelled a Assertion \((A)\) and the other is labelled as Reason\((R)\) \(Assertion\) \((A)\) : Work done by electric field on moving a positive charge on an equipotential surface is always zero. \(Reason\) \((R)\) : Electric lines of forces are always perpendicular to equipotential surfaces. In the light of the above statements, choose the most appropriate answer from the options given below 

In a meter bridge experiment resistances are connected as shown in the figure. Initially resistance \(P\, = 4\,\Omega \) and the neutral point \(N\) is at \(60\,cm\) from \(A\). Now an unknown resistance \(R\) is connected in series to \(P\) and the new position of the neutral point is at \(80\,cm\) from \(A\) . The value of unknown resistance \(R\) is

Let the sum of the first \(n\) terms of a non-constant \(A.P., a_1, a_2, a_3, ……\) be \(50\,n\, + \,\frac{{n\,(n\, - 7)}}{2}A,\) where \(A\) is a constant. If \(d\) is the common difference of this \(A.P.,\) then the ordered pair \((d,a_{50})\) is equal to

Two gases-argon (atomic radius \(0.07 \;\mathrm{nm}\),atomic weight \(40\) ) and xenon (atomic radius \(0.1\; \mathrm{nm},\) atomic weight \(140\) ) have the same number density and are at the same temperature. The raito of their respective mean free times is closest to

The sum \(\dfrac{1^3}{1} + \dfrac{1^3 + 2^3}{1 + 3} + \dfrac{1^3 + 2^3 + 3^3}{1 + 3 + 5} + \cdots\) up to 8 terms, is:

If two tangents drawn from a point \(\mathrm{P}\) to the parabola \(\mathrm{y}^{2}=16(\mathrm{x}-3)\) are at right angles, then the locus of point \(\mathrm{P}\) is :

The value of the integral \(\displaystyle\int_{0}^{2} \dfrac{\sqrt{x(x^2+x+1)}}{(\sqrt{x+1})(\sqrt{x^4+x^2+1})} \, dx\) is equal to: