A particle is moving with speed \(v= b\sqrt x\) along positive \(x-\) axis. Calculate the speed of the particle at time \(t = \tau\) (assume that the particle is at origin at \(t = 0\) ).

Two light beams of intensities \(4\,I\) and \(9\,I\) interfere on a screen. The phase difference between these beams on the screen at point \(A\) is \(zero\) and at point \(B\) is \(\pi\). The difference of resultant intensities, at the point \(A\) and \(B\), will be \(....I\).

Consider a solid sphere of radius \(\mathrm{R}\) and mass density \(\rho(\mathrm{r})=\rho_{0}\left(1-\frac{\mathrm{r}^{2}}{\mathrm{R}^{2}}\right),  0<\mathrm{r} \leq \mathrm{R} .\) The minimum density of a liquid in which it will float is

A 400 g solid cube having an edge of length 10 cm floats in water. How much volume of the cube is outside the water ?
(Given : density of water \(=1000 \mathrm{~kg} \mathrm{~m}^{-3}\) )

A thin lens made of glass (refractive index \(= 1.5\)) of focal length \(f=16\; \mathrm{cm}\) is immersed in a liquid of refractive index \(1.42 .\) If its focal length in liquid is \(f_{l},\) then the ratio \(f_{l} /f\) is closest to the integer

Given below are two statements: one is labelled as Assertion \(A\) and the other is labelled as Reason \(R\) Assertion \(A:\) When you squeeze one end of a tube to get toothpaste out from the other end, Pascal's principle is observed. Reason \(R:\) A change in the pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of its container. In the light of the above statements, choose the most appropriate answer from the options given below

A circular coil of \(1000 \,\,turns\) each with area \(1\, m ^{2}\) is rotated about its vertical diameter at the rate of one revolution per second in a uniform horizontal magnetic field of \(0.07 \,T\). The maximum voltage generation will be......... \(V\).

If the ratio of lengths, radii and Young's moduli of steel and brass wires in the figure are \(a, b\) and \(c\) respectively, then the corresponding ratio of increase in their lengths is

The percentage decrease in the weight of a rocket, when taken to a height of \(32 km\) above the surface of earth will, be\(.....\%\) (Radius of earth \(=6400\,km\) )

A triangular shaped wire carrying \(10 A\) current is placed in a uniform magnetic field of \(0.5\,T\), as shown in figure. The magnetic force on segment \(CD\) is \(....N\) \((\) Given \(BC = CD = BD =5\,cm )\).