Capacitance of an isolated conducting sphere of radius \(R_{1}\) becomes \(n\) times when it is enclosed by a concentric conducting sphere of radius \(R_{2}\) connected to earth. The ratio of their radii \(\left(\frac{ R _{2}}{ R _{1}}\right)\) is:

Identify the correct statements from the following descriptions of various properties of electromagnetic waves. \(A\). In a plane electromagnetic wave electric field and magnetic field must be perpendicular to each other and direction of propagation of wave should be along electric field or magnetic field. \(B.\) The energy in electromagnetic wave is divided equally between electric and magnetic fields. \(C.\) Both electric field and magnetic field are parallel to each other and perpendicular to the direction of propagation of wave. \(D.\) The electric field, magnetic field and direction of propagation of wave must be perpendicular to each other. \(E.\) The ratio of amplitude of magnetic field to the amplitude of electric field is equal to speed of light. Choose the most appropriate answer from the options given below:

A \(LCR\) circuit behaves like a damped harmonic oscillator. Comparing it with a physical springmass damped oscillator having damping constant \(\mathrm{b}\), the correct equivalence would be:

In a measurement, it is asked to find modulus of elasticity per unit torque applied on the system. The measured quantity has dimension of \(\left[M^a L^b T^c\right]\). If \(b=-3\), the value of \(c\) is ________

When a resistance of \(5\,\Omega\) is shunted with a moving coil galvanometer, it shows a full scale deflection for a current of \(250\,mA\), however when \(1050\,\Omega\) resistance is connected with it in series, it gives full scale deflection for \(25\) volt. The resistance of galvanometer is \(......\,\Omega\).

A body is projected vertically upwards from the surface of earth with a velocity equal to one third of escape velocity. The maximum height attained by the body will be \(......km\) (Take radius of earth \(=6400\,km\) and \(g =10\,ms^{-2}\) )

If \(\vec{A}\) and \(\vec{B}\) are two vectors satisfying the relation \(\vec{A} . \vec{B}=[\vec{A} \times \vec{B}]\). Then the value of \([\vec{A}-\vec{B}]\). will be :

The correct match between the entries in column \(I\) and column \(II\) are              \(I\)                         \((II)\) Radiation                     Wavelength \((a)\) Microwave          \((i)\) \(100 m\) \((b)\) Gamma rays      \((ii)\) \(10^{-15} m\) \((c)\) \(A.M.\) radio waves        \((iii)\) \(10^{-10} m\) \((d)\) \(X-\)rays             \((iv)\) \(10^{-3} m\)

Let a line \(L\) pass through the point of intersection of the lines \(b x+10 y-8=0\) and \(2 x-3 y=0\), \(b \in R -\left\{\frac{4}{3}\right\}\). If the line \(L\) also passes through the point \((1,1)\) and touches the circle \(17\left( x ^{2}+ y ^{2}\right)=16\), then the eccentricity of the ellipse \(\frac{x^{2}}{5}+\frac{y^{2}}{b^{2}}=1\) is.

In a meter bridge experiment null point is obtained at \(40\, cm\) from one end of the wire when resistance \(X\) is balanced against another resistance \(Y\). If \(X < Y\), then the new position of the null point from the same end, if one decides to balance a resistance of \(3X\) against \(Y\), will be close to .............. \(cm\)

If the foot of the perpendicular from point \((4,3,8)\) on the line \(L _{1}: \frac{ x - a }{l}=\frac{ y -2}{3}=\frac{ z - b }{4},\) \(l \neq 0\) is \((3,5,7),\) then the shortest distance between the line \(L _{1}\) and line \(L _{2}: \frac{ x -2}{3}=\frac{ y -4}{4}=\frac{ z -5}{5}\) is equal to:

Let \(N\) be the set of natural numbers and a relation \(R\) on \(N\) be defined by \(R=\left\{(x, y) \in N \times N: x^{3}-3 x^{2} y-x y^{2}+3 y^{3}=0\right\} .\) Then the relation \(R\) is:

Let \(P\) be the plane, passing through the point \((1,-1,-5)\) and perpendicular to the line joining the points \((4,1,-3)\) and \((2,4,3)\). Then the distance of \(P\) from the point \((3,-2,2)\) is