A longitudinal wave is represented by \(x =10 \sin 2 \pi\left( nt -\frac{ x }{\lambda}\right) \,cm\). The maximum particle velocity will be four times the wave velocity if the determined value of wavelength is equal to .....................

A thin biconvex lens is prepared from the glass (\(\mu=1.5\)) both curved surfaces of which have equal radii of \(20\) cm each. Left side surface of the lens is silvered from outside to make it reflecting. To have the position of image and object at the same place, the object should be placed, from the lens at a distance of ________ cm.

A rod of length \(50\,cm\) is pivoted at one end. It is raised such that if makes an angle of \(30^o\) fro the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in \(rad\,s^{-1}\) ) will be \((g = 10\,ms^{-2})\)

As shown in the figure a block of mass \(10\,kg\) lying on a horizontal surface is pulled by a force \(F\) acting at an angle \(30^{\circ}\), with horizontal. For \(\mu_{ s }=0.25\), the block will just start to move for the value of \(F..........\,N\) : \(\left[\right.\) Given \(\left.g =10\,ms ^{-2}\right]\)

The parallel combination of two air filled parallel plate capacitors of capacitance \(C\) and \(nC\) is connected to a battery of voltage, \(V\). When the capacitor are fully charged, the battery is removed and after that a dielectric material of dielectric constant \(K\) is placed between the two plates of the first capacitor. The new potential difference of the combined system is

Two charges \(7 \mu \mathrm{c}\) and \(-4 \mu \mathrm{c}\) are placed at \((-7 \mathrm{~cm}, 0,0)\) and \((7 \mathrm{~cm}, 0,0)\) respectively. Given, \(\epsilon_0=8.85 \times 10^{-12} \mathrm{C}^2 \mathrm{~N}^{-1} \mathrm{~m}^{-2}\), the electrostatic potential energy of the charge configuration is :

Given below in the left column are different modes of communication using the kinds of waves given in the right column

\(A.\) Optical Fiber Communication	\(P.\) Ultrasound
\(B.\) Radar	\(Q.\) Infrared Light
\(C.\) Sonar	\(R.\) Microwaves
\(D.\) Mobile Phones	\(S.\) Radio Waves

From the options given below, find the most appropriate match between entries in the left and the right column

A simple harmonic oscillator of angular frequency \(2\,rad\,s^{-1}\) is acted upon by an external force \(F = sin\,t\,N .\) If the oscillator is at rest in its equilibrium position at \(t = 0,\) its position at later times is proportional to

An object of uniform density rolls up the curved path with the initial velocity \(v_0\) as shown in the figure. If the maximum height attained by an object is \(\dfrac{7v_0^2}{10g}\) (\(g\) = acceleration due to gravity), the object is a _______.

In a compound microscope the focal length of objective lens is \(1.2\, cm\) and focal length of eye piece is \(3.0\, cm\). When object is kept at \(1.25\, cm\) in front of objective, final image is formed at infinity. Magnifying power of the compound microscope should be

A solid cylinder and a solid sphere, having same mass \(M\) and radius \(R\), roll down the same inclined plane from top without slipping. They start from rest. The ratio of velocity of the solid cylinder to that of the solid sphere, with which they reach the ground, will be.