The kinetic energy of an electron, \(\alpha\)-particle and a proton are given as \(4K,2K\) and \(K\) respectively. The de-Broglie wavelength associated with electron ( \(\lambda e), \alpha\)-particle \((\lambda \alpha)\) and the proton \((\lambda p)\) are as follows:

Two blocks (\(P\) and \(Q\)) with respectively masses \(2\) kg and \(1.5\) kg are joined by a massless thread. These blocks are mounted on a frictionless pulley which is fixed on the edge of a cube (\(S\)), as shown in the figure below. Block \(P\) is positioned on the top surface which has no friction and block \(Q\) is in contact with side-surface, having coefficient friction \(\mu\). The cube (\(S\)) moves towards the right with acceleration of \(\dfrac{g}{2}\), where \(g\) is gravitational acceleration. During this movement the block \(P\) and \(Q\) remain stationary. The value of \(\mu\) is _______. (take \(g = 10\) m/s\(^2\))

\(A\) power transmission line feeds input power at \(2\).\(3\) \(\mathrm{kV}\) to a step down transformer with its primary winding having \(3000\) turns. The output power is delivered at \(230 \mathrm{~V}\) by the transformer. The current in the primary of the transformer is \(5 \mathrm{~A}\) and its efficiency is \(90 \%\). The winding of transformer is made of copper. The output current of transformer is _______ \(A.\)

A metal sample carrying a current along \(X-\) axis with density \(J_x\) is subjected to a magnetic field \(B_z\) ( along \(z-\) axis ). The electric field \(E_y\) developed along \(Y-\) axis is directly proportional io \(J_x\) as well as \(B_z\) . The constant of proportionality has \(SI\, unit\).

The threshold wavelength for photoelectric emission from a material is \(5500\,\mathring A\). Photoelectrons will be emitted, when this material is illuminated with monochromatic radiation from a \(A.\) \(75\,W\) infra-red lamp \(B.\) \(10\,W\) infra-red lamp \(C.\) \(75\,W\) ultra-violet lamp \(D.\) \(10\,W\) ultra-violet lamp Choose the correct answer from the options given below :

A monochromatic neon lamp with wavelength of \(670.5\, {nm}\) illuminates a photo-sensitive material which has a stopping voltage of \(0.48\, {V}\). What will be the stopping voltage if the source light is changed with another source of wavelength of \(474.6 \,{nm} ?\) (In \({V}\))

Given below are two statement  Statement \(-I\) : What \(\mu\) amount of an ideal gas undergoes adiabatic change from state \(\left( P _{1}, V _{1}, T _{1}\right)\) to state \(\left( P _{2}, V _{2}, T _{2}\right)\), the work done is \(W =\frac{1 R \left( T _{2}- T _{1}\right)}{1-\gamma}\), where \(\gamma=\frac{ C _{ P }}{ C _{ V }}\) and \(R =\) universal gas constant, Statement \(-II\) : In the above case. when work is done on the gas. the temperature of the gas would rise. Choose the correct answer from the options given below

Two satellites \(A\) and \(B\) having masses in the ratio \(4: 3\) are revolving in circular orbits of radii \(3\,r\) and \(4\,r\) respectively around the earth. The ratio of total mechanical energy of \(A\) to \(B\) is.

Which of the following are true? \(A.\) Speed of light in vacuum is dependent on the direction of propagation. \(B.\) Speed of light in a medium in independent of the wavelength of light. \(C.\) The speed of light is independent of the motion of the source. \(D.\) The speed of light in a medium is independent of intensity. Choose the correct answer from the option given below :

A cyclist starts from the point \(P\) of a circular ground of radius \(2 \mathrm{~km}\) and travels along its circumference to the point \(\mathrm{S}\). The displacement of a cyclist is _______.

Let a function \(f:\left( {0,\infty } \right) \to \left( {0,\infty } \right)\) be defined by \(f\left( x \right) = \left| {1 - \frac{1}{x}} \right|\). Then \(f\) is

Let \(f: R \rightarrow R\) be defined as \(f(\mathrm{x})= -\frac{4}{3} x^{3}+2 x^{2}+3 x ,\quad x>0\) \(\quad\quad\quad\quad 3 x e^{x}, \quad\quad\quad\quad\quad\quad\mathrm{x} \leq 0\) Then \(\mathrm{f}\) is increasing function in the interval.

Four fair dice are thrown independently \(27\) times. Then the expected number of times, at least two dice show up a three or a five, is