At time \(t =0\) a particle starts travelling from a height \(7\,\hat{z} cm\) in a plane keeping \(z\) coordinate constant. At any instant of time it's position along the \(x\) and \(y\) directions are defined as \(3\,t\) and \(5\,t^{3}\) respectively. At \(t =1\,s\) acceleration of the particle will be.

Match List \(I\) with List \(II\) :

List \(I\)	List \(II\)
\(A\) Isothermal Process	\(I\) Work done by the gas decreases internal energy
\(B\) Adiabatic Process	\(II\) No change in internal energy
\(C\) Isochoric Process	\(III\) The heat absorbed goes partly to increase internal energy and partly to do work
\(D\) Isobaric Process	\(IV\) No work is done on or by the gas

Choose the correct answer from the options given below :

The source that illuminates the double - slit in 'double - slit interference experiment' emits two distinct monochromatic waves of wavelength \(500\,nm\) and \(600\, nm\), each of them producing its own pattern on the screen. At the central point of the pattern when path difference is zero, maxima of both the patterns coincide and the resulting interference pattern is most distinct at the region of zero path difference. But as one moves out of this central region, the two fringe systems are gradually out of step such that maximum due to on wavelength coincides with the minimum due to the other and the combined fringe system becomes completely indistinct. This may happen when path difference in \(nm\) is

Two parallel, long wires are kept \(0.20\,m\) apart in vacuum, each carrying current of \(x\) in the same direction. If the force of attraction per meter of each wire is \(2 \times 10^{-6}\,N\), then the value of \(x\) is approximately

Let the moment of inertia of a hollow cylinder of length \(30\, cm\) (inner radius \(10\, cm\) and outer radius \(20\, cm\)), about its axis be \(I\). The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also \(I\), is ......... \(cm\).

A particle of mass \(20\,g\) is released with an initial velocity \(5\,m/s\) along the curve from the point \(A,\) as shown in the figure. The point \(A\) is at height \(h\) from point \(B.\) The particle slides along the frictionless surface. When the particle reaches point \(B,\) its angular momentum about \(O\) will be ......... \(kg - m^2/s\). [Take \(g = 10\,m/s^2\) ]

A bi-convex lens has radius of curvature of both the surfaces same as \(1 / 6 \mathrm{~cm}\). If this lens is required to be replaced by another convex lens having different radii of curvatures on both sides \(\left(R_1 \neq R_2\right)\), without any change in lens power then possible combination of \(R_1\) and \(R_2\) is :

If \(m\) and \(M\) are the minimum and the maximum values of \(4 + \frac{1}{2}\,{\sin ^2}\,2x - 2\,{\cos ^4}\,x\,,x\, \in R,\)  then \(M - m\) is equal to

particle moves with simple harmonic motion in a straight line. In first \(\tau\ s\), after starting from rest it travels a distance \(a\), and in next \(\tau\ s\) it travels \(2a\), in same direction, then

If a directrix of a hyperbola centered at the origin and passing through the point \((4, -2\sqrt 3)\) is \(5x = 4\sqrt 5\) and its eccentricity is \(e\), then

A uniform disc of radius \(R\) and mass \(M\) is free to rotate only about its axis. A string is wrapped over its rim and a body of mass \(m\) is tied to the free end of the string as shown in the figure. The body is released from rest. Then the acceleration of the body is

Let PQ be a focal chord of the parabola \(y^2=36 x\) of length \(100\), making an acute angle with the positive \(x\)-axis. Let the ordinate of \(P\) be positive and \(M\) be the point on the line segment \(P Q\) such that \(P M: M Q=3: 1\). Then which of the following points does NOT lie on the line passing through \(M\) and perpendicular to the line \(PQ\) ?

Two long, straight wires carry equal currents in opposite directions as shown in figure. The separation between the wires is \(5.0 \mathrm{~cm}\). The magnitude of the magnetic field at a point \(P\) midway between the wires is __________\(\mu \mathrm{T}\) (Given : \(\mu_0=4 \pi \times 10^{-7} \mathrm{TmA}^{-1}\) )