A cell of internal resistance \(r\) drives current through an external resistance \(R\) . The power delivered by the cell to the external resistance will be maximum when:

In the circuit shown, the potential difference between \(A\) and \(B\) is ............. \(V\)

For an object placed at a distance \(2.4\,m\) from a lens, a sharp focused image is observed on a screen placed at a distance \(12\,cm\) from the lens. A glass plate of refractive index \(1.5\) and thickness \(1\,cm\) is introduced between lens and screen such that the glass plate plane faces parallel to the screen. By what distance should the object be shifted so that a sharp focused image is observed again on the screen \(\dots\,m\)

Given below are two statements: One is labelled as Assertion \(A\) and the other is labelled as Reason \(R\). Assertion \(A:\) Two metallic spheres are charged to the same potential. One of them is hollow and another is solid, and both have the same radii. Solid sphere will have lower charge than the hollow one. Reason \(R:\) Capacitance of metallic spheres depend on the radii of spheres. In the light of the above statements, choose the correct answer from the options given below.

An insulated copper wire of \(100\) turns is wrapped around a wooden cylindrical core of the crosssectional area \(24\,cm ^2\). The two ends of the wire are connected to a resistor. The total resistance in the circuit is \(12\,\Omega\). If an externally applied uniform magnetic field in the core along its axis changes from \(1.5 T\) in one direction to \(1.5\,T\) in the opposite direction, the charge flowing through a point in the circuit during the change of magnetic field will be \(.........\,mC\).

The magnetic field vector of an electromagnetic wave is given by \({B}={B}_{o} \frac{\hat{{i}}+\hat{{j}}}{\sqrt{2}} \cos ({kz}-\omega {t})\); where \(\hat{i}, \hat{j}\) represents unit vector along \({x}\) and \({y}\)-axis respectively. At \(t=0\, {s}\), two electric charges \(q_{1}\) of \(4\, \pi\) coulomb and \({q}_{2}\) of \(2 \,\pi\) coulomb located at \(\left(0,0, \frac{\pi}{{k}}\right)\) and \(\left(0,0, \frac{3 \pi}{{k}}\right)\), respectively, have the same velocity of \(0.5 \,{c} \hat{{i}}\), (where \({c}\) is the velocity of light). The ratio of the force acting on charge \({q}_{1}\) to \({q}_{2}\) is :-

The electric field of a plane electromagnetic wave propagating along the \(x\) direction in vacuum is \(\overrightarrow{ E }= E _{0} \hat{ j } \cos (\omega t - kx )\). The magnetic field \(\overrightarrow{ B },\) at the moment \(t =0\) is :

Consider the line \(\mathrm{L}\) passing through the points \((1,2,3)\) and \((2,3,5)\). The distance of the point \(\left(\frac{11}{3}, \frac{11}{3}, \frac{19}{3}\right)\) from the line \(\mathrm{L}\) along the line \(\frac{3 x-11}{2}=\frac{3 y-11}{1}=\frac{3 z-19}{2}\) is equal to :

A physical quantity \(P\) is given as \(P=\frac{a^2 b^3}{c \sqrt{d}}\) The percentage error in the measurement of \(a, b, c\) and \(d\) are \(1 \%, 2 \%, 3 \%\) and \(4 \%\) respectively. The percentage error in the measurement of quantity \(P\) will be \(.......\%\)

The magnetic potential due to a magnetic dipole at a point on its axis situated at a distance of \(20 \mathrm{~cm}\) from its center is \(1.5 \times 10^{-5} \  \mathrm{Tm}\). The magnetic moment of the dipole is _______ \(\mathrm{Am}^2\). (Given : \(\frac{\mu_0}{4 \pi}=10^{-7} \  \mathrm{TmA}^{-1}\) )

Let \(A B\) and \(P Q\) be two vertical poles, \(160\, m\) apart from each other. Let \(C\) be the middle point of \(B\) and \(Q\), which are feet of these two poles. Let \(\frac{\pi}{8}\) and \(\theta\) be the angles of elevation from \(C\) to \(P\) and \(A\), respectively. If the height of pole \(P Q\) is twice the height of pole \(AB\), then \(\tan ^{2} \theta\) is equal to

If the mean and variance of the following data: \(6,10,7,13, a, 12, b, 12\) are 9 and \(\frac{37}{4}\) respectively, then \((a-b)^{2}\) is equal to:

If the point of intersections of the ellipse \(\frac{ x ^{2}}{16}+\frac{ y ^{2}}{ b ^{2}}=1\) and the circle \(x ^{2}+ y ^{2}=4 b , b > 4\) lie on the curve \(y^{2}=3 x^{2},\) then \(b\) is equal to: