In an experiment of determine the Young's modulus of wire of a length exactly \(1\; m\), the extension in the length of the wire is measured as \(0.4\,mm\) with an uncertainty of \(\pm 0.02\,mm\) when a load of \(1\,kg\) is applied. The diameter of the wire is measured as \(0.4\,mm\) with an uncertainty of \(\pm 0.01\,mm\). The error in the measurement of Young's modulus \((\Delta Y)\) is found to be \(x \times 10^{10}\,Nm ^{-2}\). The value of \(x\) is \(\left[\right.\) Take \(\left.g =10\,m / s ^{2}\right]\)

Given below are two statements : one is labelled as Assertion \(A\) and the other is labelled as Reason \(R.\) Assertion \(A :\) Moment of inertia of a circular disc of mass \('M'\) and radius \('R'\) about \(X, Y\) axes (passing through its plane) and \(Z-\)axis which is perpendicular to its plane were found to be \(I_{x}, I_{y}\) and \({I}_{z}\) respectively. The respective radii of gyration about all the three axes will be the same. Reason \(R\) : A rigid body making rotational motion has fixed mass and shape. In the light of the above statements, choose the most appropriate answer from the options given below :

A voltage regulating circuit consisting of Zener diode, having break-down voltage of 10 V and maximum power dissipation of 0.4 W , is operated at 15 V . The approximate value of protective resistance in this circuit is _________ \(\Omega\).

An ideal gas undergoes a process maintaining relation between pressure \((P)\) and volume \((V)\) as \(P = P_o\left(1 + \left(\dfrac{V_o}{V}\right)^2\right)^{-1}\), where \(P_o\) and \(V_o\) are constants. If two samples \(A\) and \(B\) (two moles each) with initial volumes \(V_o\) and \(3V_o\) respectively undergo above mentioned process and attain same pressure, then the difference at the temperatures of these samples, \(T_B - T_A\) is _____. (\(R = \) gas constant)

The electric current through a wire varies with time as \(I=I_0+\beta t\). where \(I_0=20 \mathrm{~A}\) and \(\beta=3 \mathrm{~A} / \mathrm{s}\). The amount of electric charge crossed through a section of the wire in \(20 \mathrm{~s}\) is :

The position vectors of two 1 kg particles, (A) and (B), are given by
\(\overrightarrow{\mathrm{r}}_{\mathrm{A}}=\left(\alpha_1 \mathrm{t}^2 \hat{i}+\alpha_2 \mathrm{t} \hat{j}+\alpha_3 \mathrm{t} \hat{k}\right) \mathrm{m}\) and \(\overrightarrow{\mathrm{r}}_{\mathrm{B}}=(\beta_1 \mathrm{t} \hat{i}+\beta_2 \mathrm{t}^2 \hat{j}\) \(+\beta_3 \mathrm{t} \hat{k}) \mathrm{m}\), respectively; \((\alpha_1=1 \mathrm{~m} / \mathrm{s}^2, \alpha_2=3 \mathrm{n} \mathrm{m} / \mathrm{s},\) \(\alpha_3=2 \mathrm{~m} / \mathrm{s},\) \(\beta_1=2 \mathrm{~m} / \mathrm{s}, \beta_2=-1 \mathrm{~m} / \mathrm{s}^2, \beta_3=4 \mathrm{pm} / \mathrm{s})\), where t is time, n and p are constants. At \(t=1 \mathrm{~s},\left|\overrightarrow{V_A}\right|=\left|\vec{V}_B\right|\) and velocities \(\vec{V}_A\) and \(\vec{V}_B\) of the particles are orthogonal to each other. At \(t=1 \mathrm{~s}\), the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is \(\sqrt{\mathrm{L}} \mathrm{kgm}^2 \mathrm{~s}^{-1}\). The value of L is _______ .

The orbital angular momentum of a satellite is \(L\), when it is revolving in a circular orbit at height \(h\) from earth surface. If the distance of satellite from the earth centre is increased by eight times to its initial value, then the new angular momentum will be \(............\,L\)

Given below are two statements: One is labelled as Assertion \((A)\) and other is labelled as Reason \((R)\). Assertion \((A)\) : Time period of oscillation of a liquid drop depends on surface tension \((S)\), if density of the liquid is \(p\) and radius of the drop is \(r\), then \(T = k \sqrt{ pr ^{3} / s ^{3 / 2}}\) is dimensionally correct, where \(K\) is dimensionless. Reason \((R)\): Using dimensional analysis we get \(R.H.S.\) having different dimension than that of time period. In the light of above statements, choose the correct answer from the options given below.

Water from a tap emerges vertically downwards with an initial speed of \(1.0\,ms^{-1}.\) The cross-sectional area of the tap is \(10^{-4}\,m^2.\) Assume that the pressure is constant throughout the stream of water and that flow is streamlined. The cross-sectional area of the stream, \(0.15\,m\) below the tap would be: (take \(g = 10\,ms^{-2}\) )

An \(X-\)ray tube is operated at \(1.24\) million volt. The shortest wavelength of the produced photon will be ..... \(nm\)

A charged particle moves along circular path in a uniform magnetic field in a cyclotron. The kinetic energy of the charged particle increases to \(4\) times its initial value. What will be the ratio of new radius to the original radius of circular path of the charged particle

The electromagnetic waves travel in a medium at a speed of \(2.0 \times 10^{8}\, m / s\). The relative permeability of the medium is \(1.0.\) The relative permittivity of the medium will be