The position-time graphs for two students \(A\) and \(B\) returning from the school to their homes are shown in figure \((A)\) \(A\) lives closer to the school \((B)\) \(B\) lives closer to the school \((C)\) \(A\) takes lesser time to reach home \((D)\) \(A\) travels faster than \(B\) \((E)\) \(B\) travels faster than \(A\) Choose the correct answer from the options given below :

A plane electromagnetic wave is propagating along the direction \(\frac{\hat{i}+\hat{j}}{\sqrt{2}},\) with its polarization along the direction \(\hat{\mathrm{k}}\). The correct form of the magnetic field of the wave would be (here \(\mathrm{B}_{0}\) is an appropriate constant)

Orange light of wavelength \(6000 \times 10^{-10}\, m\) in illuminates a single slit of width \(0.6 \times 10^{-4}\, m\) The maximum possible number of diffraction minima produced on both sides of the central maximum is\(........\)

In the first configuration (1) as shown in the figure, four identical charges \(\left(q_0\right)\) are kept at the corners \(A, B, C\) and \(D\) of square of side length 'a'. In the second configuration (2), the same charges are shifted to mid points \(\mathrm{G}, \mathrm{E}, \mathrm{H}\) and F, of the square, If \(\mathrm{K}=\frac{1}{4 \pi \varepsilon_0}\), the difference between the potential energies of configuration (2) and (1) is given by :

An unknown metal of mass \(192\, g\) heated to a temperature of \(100\,^oC\) was immersed into a brass calorimeter of mass \(128\, g\) containing \(240\, g\) of water at a temperature of \(8.4\,^oC\). Calculate the specific heat of the unknown metal if water temperature stabilizes at \(21.5\,^oC\). (Specific heat of brass is \(394\, J\, kg^{-1}\, K^{-1}\)) ......... \(J\, kg^{-1}\, K^{-1}\)

Let the moment of inertia of a hollow cylinder of length \(30\, cm\) (inner radius \(10\, cm\) and outer radius \(20\, cm\)), about its axis be \(I\). The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also \(I\), is ......... \(cm\).

Refer to the circuit diagram given below. The heat generated across the \(6\,\Omega\) resistance in \(100\) second is \(\dfrac{\alpha}{100}\) J. The value of \(\alpha\) is _______. (Nearest integer)

An object is projected in the air with initial velocity \(u\) at an angle \(\theta\). The projectile motion is such that the horizontal range \(R\), is maximum. Another object is projected in the air with a horizontal range half of the range of first object. The initial velocity remains same in both the case. The value of the angle of projection, at which the second object is projected, will be \(.......\) degree.

A parallel plate capacitor was made with two rectangular plates, each with a length of \(l=3 \mathrm{~cm}\) and breath of \(\mathrm{b}=1 \mathrm{~cm}\). The distance between the plates is \(3 \mu \mathrm{~m}\). Out of the following, which are the ways to increase the capacitance by a factor of 10 ?
A. \(l=30 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=1 \mu \mathrm{~m}\)
B. \(l=3 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=30 \mu \mathrm{~m}\)
C. \(l=6 \mathrm{~cm}, \mathrm{~b}=5 \mathrm{~cm}, \mathrm{~d}=3 \mu \mathrm{~m}\)
D. \(l=1 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=10 \mu \mathrm{~m}\)
E. \(l=5 \mathrm{~cm}, \mathrm{~b}=2 \mathrm{~cm}, \mathrm{~d}=1 \mu \mathrm{~m}\)
Choose the correct answer from the options given below:

A solid sphere (\(A\)) of mass \(5m\) and a spherical shell (\(B\)) of mass \(m\), both having same radius, are placed on a rough surface. When a force of same magnitude is applied tangentially at the highest points of \(A\) and \(B\), they start rolling without slipping with an acceleration of \(a_A\) and \(a_B\), respectively. The ratio of \(a_A\) and \(a_B\) is __________.