The percentage errors in quantities \(P, Q, R\)  and \(S\)  are \(0.5\%,\,1\%,\,3\%\)  and  \(1 .5\%\) respectively in the  measurement of a physical quantity \(A\, = \,\frac{{{P^3}{Q^2}}}{{\sqrt {R}\,S }}\) . the maximum percentage error in the value of \(A\)  will be ........... \(\%\)

For the determination of refractive index of glass slab, a travelling microscope is used whose main scale contains 300 equal divisions equals to 15 cm. The vernier scale attached to the microscope has 25 divisions equals to 24 divisions of main scale. The least count (LC) of the travelling microscope is (in cm) :

A passenger train of length \(60\, m\) travels at a speed of \(80\, km/hr\). Another freight train of length \(120\, m\) travels at a speed of \(30\, km/hr\). The ratio of times taken by the passenger train to completely cross the freight train when: \((i)\) they are moving in the same direction, and \((ii)\) in the opposite directions is

A cricket ball of mass \(0.15\, kg\) is thrown vertically up by a bowling machine so that it rises to a maximum height of \(20 \;m\) after leaving the machine. If the part pushing the ball applies a constant force \(F\) on the ball and moves horizontally a distance of \(0.2\, m\) while launching the ball, the value of \(F(\) in \(N)\) is  \(\left(g=10\, m s^{-2}\right)\)

In an experiment, a set of reading are obtained \(-1.24 mm, 1.25 mm, 1.23 mm, 1.21 mm\). The expected least count of the instrument used in recording these readings is ___________ mm.

Which of the following are correct expression for torque acting on a body?
A. \(\vec{\tau}=\vec{r}\times\vec{L}\)
B. \(\vec{\tau}=\frac{\mathrm{d}}{\mathrm{dt}}(\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{p}})\)
C. \(\vec{\tau}=\overrightarrow{\mathrm{r}} \times \frac{\mathrm{d} \mathrm{p}}{\mathrm{dt}}\)
D. \(\vec{\tau}=\mathrm{I} \vec{\alpha}\)
E. \(\vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}\)
( \(\overrightarrow{\mathrm{r}}=\) position vector; \(\overrightarrow{\mathrm{p}}=\) linear momentum;
\(\overrightarrow{\mathrm{L}}=\) angular momentum; \(\vec{\alpha}=\) angular acceleration;
\(I=\) moment of inertia; \(\vec{F}=\) force; \(t=\) time)
Choose the correct answer from the options given below :

Choose the correct option:

An air bubble of radius \(0 .1\, cm\) is in a liquid having surface tension \(0.06\, N/m\) and density \(10^3\, kg/m^3\). The pressure inside the bubble is \(1100\, Nm^{-2}\) greater than the atmospheric pressure. At ....... \(m\) depth is the bubble below the surface of the liquid ? \((g\, = 9.8\, ms^{- 2})\)

Eight copper wire of length \(l\) and diameter \(d\) are joined in parallel to form a single composite conductor of resistance \(R\). If a single copper wire of length \(2\,l\) have the same resistance \((R)\) then its diameter will be \(.....d\).

A body of mass \(1 kg\) begins to move under the action of a time dependent force \(\vec{F}=\left(t \hat{i}+3 t^2 \hat{ j }\right) N\). where \(\hat{i}\) and \(\hat{j}\) are the unit vectors along \(x\) and \(y\) axis. The 10.Power developed by above force, at the time \(t=2 s\). will be \(..............W\).

The electric field of a plane electromagnetic wave propagating along the \(x\) direction in vacuum is \(\overrightarrow{ E }= E _{0} \hat{ j } \cos (\omega t - kx )\). The magnetic field \(\overrightarrow{ B },\) at the moment \(t =0\) is :