What is the dimensional formula of \(a b^{-1}\) in the equation? \(\left(\mathrm{P}+\frac{\mathrm{a}}{\mathrm{V}^2}\right)(\mathrm{V}-\mathrm{b})=\mathrm{RT}\), where letters have their usual meaning.

A particle of mass \(m\) is moving along the \(x\) -axis with initial velocity \(u \hat i\). It collides elastically with a particle of mass \(10\, m\) at rest and then moves with half its initial kinetic energy (see figure). If \(\sin \theta_{1}=\sqrt{n} \sin \theta_{2}\) then value of \(n\) is.....

A body of mass \(M\) thrown horizontally with velocity \(v\) from the top of the tower of height \(\mathrm{H}\) touches the ground at a distance of \(100 \mathrm{~m}\) from the foot of the tower. A body of mass \(2 \mathrm{M}\) thrown at a velocity \(\frac{v}{2}\) from the top of the tower of height \(4 \mathrm{H}\) will touch the ground at a distance of _______.

A plane is inclined at an angle \(\alpha  = 30^o\) with respect to the horizontal. A particle is projected with a speed \(u = 2\,ms^{-1},\) from the base of the plant, making an angle \(\theta = 15^o\) with respect to the plane as shown in the figure. The distance from the base at which the particle hits the plane is close to  ........ \(cm\) (Take \(g = 10\,ms^2\) )

A force \(F=\left(5+3 y^2\right)\) acts on a particle in the \(y\) direction, where \(F\) is newton and \(y\) is in meter. The work done by the force during a displacement from \(y =2\,m\) to \(y =5\,m\) is \(.............\,j\).

An ideal gas occupies a volume of \(2\, m^3\) at a pressure of \(3\times10^6\, Pa\). The energy of  the gas is

The figure shows a square loop \(L\) of side \(5\, cm\) which is connected to a network of resistances. The whole set up is moving towards right with a constant speed of \(1\, cms^{-1}\). At some instant, a part of \(L\) is in a uniform magnetic field of \(1\, T\), perpendicular to the plane of the loop. If the resistance of \(L\) is \(1.7\,\Omega \), the current in the loop at that instant will be close to.....\(\mu A\)

A wedge \(Y\) with mass of \(10\) kg and all frictionless surfaces and the inclined surface making \(37°\) with horizontal. A block \(X\) with mass \(2\) kg is placed at the highest point of the wedge as shown in figure is at rest. At \(t = 0\) wedge \((Y)\) is pulled toward right with constant force \((f)\) of \(24\) N. Taking the block \(X\) at rest at \(t = 0\), the time taken by it to slide down \(8.8\) m on the slope, while \(Y\) is on the move, is _____ s.
(take \(\tan(37°) = 3/4\) and \(g = 10\) m/s\(^2\))

Match List I with List \(II\) Choose the correct answer from the options given below:

A wire of resistance \(R_{1}\) is drawn out so that its length is increased by twice of its original length.The ratio of new resistance to original resistance is.

Figure shows a rod \({AB}\), which is bent in a \(120^{\circ}\) circular arc of radius \(R\). A charge \((-Q)\) is uniformly distributed over rod \({AB}\). What is the electric field \(\overrightarrow{{E}}\) at the centre of curvature \({O}\) ?

Boolean relation at the output stage\(-Y\) for the following circuit is

A new unit \((\alpha)\) of length is chosen such that it is equal to the speed of light in vacuum. What is the distance between Venus and Earth in terms of \(\alpha\) units if light takes \(6\) min. \(40\) s to cover this distance?