An electric dipole is placed at a distance of 2 cm from an infinite plane sheet having positive charge density \(\sigma_{\mathrm{o}}\). Choose the correct option from the following.

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R .
Assertion A : If oxygen ion \(\left(\mathrm{O}^{-2}\right)\) and Hydrogen ion \(\left(\mathrm{H}^{+}\right)\)enter normal to the magnetic field with equal momentum, then the path of \(\mathrm{O}^{-2}\) ion has a smaller curvature than that of \(\mathrm{H}^{+}\).
Reason R : A proton with same linear momentum as an electron will form a path of smaller radius of curvature on entering a uniform magnetic field perpendicularly.
In the light of the above statements, choose the correct answer from the options given below :

The total number of turns and cross-section area in solenoid is fixed. However, its length \(L\) is varied by adjusting the separation between windings. The inductance of solenoid will be proportional to

A uniform wire of length \(l\) and radius \(r\) has a resistance of \(100\, \Omega \). It is recast into a wire of radius \(\frac{r}{2}\). The resistance of new wire will be ............... \(\Omega\)

Consider the combination of \(2\) capacitors \(C _{1}\) and \(C _{2},\) with \(C _{2}> C _{1},\) when connected in parallel, the equivalent capacitance is \(\frac{15}{4}\) time the equivalent capacitance of the same connected in series. Calculate the ratio of capacitors, \(\frac{ C _{2}}{ C _{1}}\)

A uniform cylinder of length \(L\) and mass \(M\) having crosssectional area \(A\) is suspended, with its length vertical, from a fixed point by a massless spring such that it is half submerged in a liquid of density \(\sigma\) at equilibrium position. The extension \(x_0\) of the spring when it is in equilibrium is

If you are provided a set of resistances \(2\, \Omega, 4\, \Omega\) \(6\, \Omega\) and \(8\, \Omega\). Connect these resistances so as to obtain an equivalent resistance of \(\frac{46}{3}\, \Omega\).

If the refractive index of the material of a prism is \(\cot \left(\frac{A}{2}\right)\), where \(A\) is the angle of prism then the angle of minimum deviation will be

Let \(A=\left[a_{i j}\right]\) be a real matrix of order \(3 \times  3\), such that \(a_{i 1}+a_{i 2}+a_{i 3}=1\), for \(i=1,2,3\). Then, the sum of all the entries of the matrix \(A^{3}\) is equal to:

Consider an electrical circuit containing a two way switch \(^{\prime}{S}^{\prime}\). Initially \({S}\) is open and then \({T}_{1}\) is connected to \({T}_{2} .\) As the current in \({R}=6 \,\Omega\) attains a maximum value of steady state level, \({T}_{1}\) is disconnected from \({T}_{2}\) and immediately connected to \({T}_{3} .\) Potential drop across \({r}=3\, \Omega\) resistor immediately after \(T_{1}\) is connected to \(T_{3}\) is \(....\,V.\) (Round off to the Nearest Integer)

Two gases-argon (atomic radius \(0.07 \;\mathrm{nm}\),atomic weight \(40\) ) and xenon (atomic radius \(0.1\; \mathrm{nm},\) atomic weight \(140\) ) have the same number density and are at the same temperature. The raito of their respective mean free times is closest to

The acute angle between the planes \(P_{1}\) and \(P_{2}\), when \(P_{1}\) and \(P_{2}\) are the planes passing through the intersection of the planes \(5 x+8 y+13 z-29=0\) and \(8 x-7 y+z-20=0\) and the points \((2,1,3)\) and \((0,1,2)\), respectively, is

The sum of the digits in the unit's place of all the \(4-\) digit numbers formed by using the numbers \(3, 4 , 5\) and \(6\), without repetition, is