In two different experiments, an object of mass \(5\; kg\) moving with a speed of \(25\; ms ^{-1}\) hits two different walls and comes to rest within \((i)\) \(3\) second, \((ii)\) \(5\) seconds, respectively. Choose the correct option out of the following :

When an air bubble of radius \(r\) rises from the bottom to the surface of a lake, its radius becomes \(\frac{{5r}}{4}\).Taking the atmospheric pressure to be equal to \(10\,m\) height of water column, the depth of the lake would approximately be ....... \(m\) (ignore the surface tension and the effect of temperature)

In an ideal gas at temperature \(T,\)  the average force that a molecule applies on the walls of a closed container depends on \(T\) as \(T^q\) . A good estimate for \(q\) is

A metallic ring is uniformly charged as shown in figure. AC and BD are two mutually perpendicular diameters. Electric field due to \(\operatorname{arc} \mathrm{AB}\) to ' O ' is ' E ' is magnitude. What would be the magnitude of electric field at ' O ' due to arc ABC ?

A star has \(100 \%\) helium composition. It starts to convert three \({ }^4 \mathrm{He}\) into one \({ }^{12} \mathrm{C}\) via triple alpha process as \({ }^4 \mathrm{He}+{ }^4 \mathrm{He}+{ }^4 \mathrm{He} \rightarrow{ }^{12} \mathrm{C}+\mathrm{Q}\). The mass of the star is \(2.0 \times 10^{32} \mathrm{~kg}\) and it generates energy at the rate of \(5.808 \times 10^{30} \mathrm{~W}\). The rate of converting these \({ }^4 \mathrm{He}\) to \({ }^{12} \mathrm{C}\) is \(\mathrm{n} \times 10^{42} \mathrm{~s}^{-1}\), where \(\mathrm{n}\) is _______. [Take, mass of \({ }^4 \mathrm{He}=4.0026 \mathrm{u}\), mass of \({ }^{12} \mathrm{C}=12 \mathrm{u}\)]

A paramagnetic substance in the form of a cube with sides \(1\, cm\) has a magnetic dipole moment of \(20 \times 10^{-6}\, J/T\) when a magnetic intensity of \(60 \times 10^3\, A/m\) is applied. Its magnetic susceptibility is

An electron is moving along the positive \(x\)-axis. If the uniform magnetic field is applied parallel to the negative \(z\)-axis. then \(A.\) The electron will experience magnetic force along positive \(y\)-axis \(B.\) The electron will experience magnetic force along negative \(y\)-axis \(C.\) The electron will not experience any force in magnetic field \(D.\) The electron will continue to move along the positive \(x\)-axis \(E.\) The electron will move along circular path in magnetic field Choose the correct answer from the options given below:

The area of the region given by \(\left\{(x, y): x y \leq 8,1, \leq y \leq x^2\right\}\) is :

Let \(y = y(x)\) be the solution of the differential equation \(\frac{{dy}}{{dx}} + y\,\tan \,x = 2x\, + \,{x^2}\,\tan \,x\,,\,x\, \in \,\left( { - \frac{\pi }{2},\frac{\pi }{2}} \right),\) such that \(y(0) = 1.\) Then

For a moving coil galvanometer, the deflection in the coil is \(0.05\,rad\) when a current of \(10\,mA\) is passed through it. If the torsional constant of suspension wire is \(4.0 \times 10^{-5}\,Nm\,rad ^{-1}\), the magnetic field is \(0.01\,T\) and the number of turns in the coil is \(200\),the area of each turn (in \(cm ^2\) ) is :

A monochromatic light of wavelength \(6000\)A is incident on the single slit of width \(0.01 \mathrm{~mm}\). If the diffraction pattern is formed at the focus of the convex lens of focal length \(20 \mathrm{~cm}\), the linear width of the central maximum is _______.

A diode has Zener voltage of \(10\) V and maximum power dissipation of \(0.5\) W, then the minimum resistance to be used in series with this diode for safety when it is connected to a \(25\) V power supply is ______ \(\Omega\).

An object ' \(o\) ' is placed at a distance of \(100\,cm\) in front of a concave mirror of radius of curvature \(200\,cm\) as shown in the figure. The object starts moving towards the mirror at a speed \(2\,cm / s\). The position of the image from the mirror after \(10\,s\) will be at ...... \(cm\).