The density and breaking stress of a wire are \(6 \times\) \(10^4 \mathrm{~kg} / \mathrm{m}^3\) and \(1.2 \times 10^8 \mathrm{~N} / \mathrm{m}^2\) respectively. The wire is suspended from a rigid support on a planet where acceleration due to gravity is \(\frac{1^{\text {rd }}}{3}\) of the value on the surface of earth. The maximum length of the wire with breaking is _______ \(\mathrm{m}\) (take, \(\mathrm{g}=\) \(\left.10 \mathrm{~m} / \mathrm{s}^2\right)\)

A person observes two moving trains, '\(A\)' reaching the station and '\(B\)' leaving the station with equal speed of \(30\,m / s\). If both trains emit sounds with frequency \(300\,Hz\), (Speed of sound : \(330\,m / s\) ) approximate difference of frequencies heard by the person will be \(..........Hz\)

Angle of minimum deviation is equal to the half of the angle of prism in an equilateral prism. The refractive index of the prism is __________.

Identify the correct statements from the following descriptions of various properties of electromagnetic waves. \(A\). In a plane electromagnetic wave electric field and magnetic field must be perpendicular to each other and direction of propagation of wave should be along electric field or magnetic field. \(B.\) The energy in electromagnetic wave is divided equally between electric and magnetic fields. \(C.\) Both electric field and magnetic field are parallel to each other and perpendicular to the direction of propagation of wave. \(D.\) The electric field, magnetic field and direction of propagation of wave must be perpendicular to each other. \(E.\) The ratio of amplitude of magnetic field to the amplitude of electric field is equal to speed of light. Choose the most appropriate answer from the options given below:

A light cylindrical vessel is kept on a horizontal surface. Area of base is A. A hole of crosssectional area \('a'\) is made just at its bottom side. The minimum coefficient of friction necessary to prevent sliding the vessel due to the impact force of the emerging liquid is \((a\,<\,<\,A)\)

A potentiometer wire of length \(10 \,m\) and resistance\(20 \,\Omega\) is connected in series with a \(25 \,V\) battery and an external resistance \(30\, \Omega\). A cell of emf \(E\) in secondary circuit is balanced by \(250\, cm\) long potentiometer wire. The value of \(E\) (in volt) is \(\frac{x}{10}\). The value of \(x\) is.......

The displacement of a damped harmonic oscillator is given by \(x\left( t \right) = {e^{ - 0.1\,t}}\,\cos \left( {10\pi t + \varphi } \right)\)  The time taken for its amplitude of vibration to drop to half of its initial value is close to .... \(s\)

A particle is moving in one dimension (along the x-axis) under the action of a variable force. Its position \( x \) (in meters) varies with time \( t \) (in seconds) according to the equation \( x = -3t^3 + 18t^2 + 16t + 16 \). The velocity of the particle when its acceleration becomes zero is _________ \( \text{m/s} \).

A ball of mass \(10\, kg\) moving with a velocity \(10 \sqrt{3}\, ms ^{-1}\) along \(X-\)axis, hits another ball of mass \(20\, kg\) which is at rest. After collision, the first ball comes to rest and the second one disintegrates into two equal pieces. One of the pieces starts moving along \(Y-\) axis at a speed of \(10\, m / s\). The second piece starts moving at a speed of \(20\, m / s\) at an angle \(\theta\) (degree) with respect to the \(X-\)axis. The configuration of pieces after collision is shown in the figure. The value of \(\theta\) to the nearest integer is

A telescope with objective diameter \(R\) is used to observe a distant star emitting light of wavelength \(500\) nm, at a resolution of \(5 \times 10^{-7}\) radian. The value of \(R\) is _____ cm.

Two beams of light having intensities \(I\) and \(4\,I\) interfere to produce a fringe pattern on a screen. The phase difference between the two beams are \(\pi / 2\) and \(\pi / 3\) at points \(A\) and \(B\) respectively. The difference between the resultant intensities at the two points is \(x I\). The value of \(x\) will be.