Two cells of emfs \(1\) V and \(2\) V and internal resistance \(2\,\Omega\) and \(1\,\Omega\), respectively connected in parallel, gave a current of \(1\) A through an external resistance. If the polarity of one cell is reversed, then value of current through the external resistance will be \(\dfrac{\alpha}{5}\) A. The value of \(\alpha\) is __________.

When a coil is connected across a \(20 \mathrm{~V}\) DC supply, it draws a current of \(5 \mathrm{~A}\). When it is connected across \(20 \mathrm{~V}, 50 \mathrm{~Hz}\) AC supply, it draws a current of \(4 \mathrm{~A}\). The self inductance of the coil is _______ \(\mathrm{mH}\). (Take \(\pi=3\) )

This question has Statement \(-1\) and Statement \(-2.\) Of the four choices given after the Statements, choose the one that best describes the two Statements. Statement \(-1:\) In Young's double slit experiment, the number of fringes observed in the field of view is small With longer wavelength of light and is large with shorter wavelength of light. Statement \(-2:\) In the double slit experiment the fringe width depends directly on the wavelength of light.

A particle of charge \(16\times10^{-16}\, C\) moving with velocity \(10\, ms^{-1}\) along \(x-\) axis enters a region where magnetic field of induction \(\vec B\) is along the \(y-\) axis and an electric field of magnitude \(10^4\, Vm^{-1}\) is along the negative \(z-\) axis. If the charged particle continues moving along \(x-\) axis, the magnitude of \(\vec B\) is

A charge of \(4\,\mu C\) is to be divided into two. The distance between the two divided charges is constant. The magnitude of the divided charges so that the force between them is maximum, will be.

A wire of length \(L\) is hanging from a fixed support. The length changes to \(L _{1}\) and \(L _{2}\) when masses \(1 \,kg\) and \(2 \,kg\) are suspended respectively from its free end. Then the value of \(L\) is equal to ..................

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) :


Three identical spheres of same mass undergo one dimensional motion as shown in figure with initial velocities \(v_{\mathrm{A}}=5 \mathrm{~m} / \mathrm{s}, v_{\mathrm{B}}=2 \mathrm{~m} / \mathrm{s}, v_{\mathrm{C}}=4 \mathrm{~m} / \mathrm{s}\). If we wait sufficiently long for elastic collision to happen, then \(v_{\mathrm{A}}=4 \mathrm{~m} / \mathrm{s}, v_{\mathrm{B}}=2 \mathrm{~m} / \mathrm{s}\), \(v_{\mathrm{C}}=5 \mathrm{~m} / \mathrm{s}\) will be the final velocities.

Reason (R): In an elastic collision between identical masses, two objects exchange their velocities.
In the light of the above statements, choose the correct answer from the options given below :

If the solution curve, of the differential equation \(\frac{d y}{d x}=\frac{x+y-2}{x-y}\) passing through the point \((2,1)\) is \(\tan ^{-1}\left(\frac{y-1}{x-1}\right)-\frac{1}{\beta} \log _e\left(\alpha+\left(\frac{y-1}{x-1}\right)^2\right)=\log _e|x-1|\), then \(5 \beta+\alpha\) is equal to

A tube of length \(50\,cm\) is filled completely with an incompressible liquid of mass \(250\,g\) and closed at both ends. The tube is then rotated in horizontal plane about one of its ends with a uniform angular velocity \(x \sqrt{ F }\) rad \(s ^{-1}\). If \(F\) be the force exerted by the liquid at the other end then the value of \(x\) will be\(....\)

A line with direction ratios \(1, -1, 2\) intersects the lines \(\dfrac{x}{2} = \dfrac{y}{3} = \dfrac{z+1}{3}\) and \(\dfrac{x+1}{-1} = \dfrac{y-2}{1} = \dfrac{z}{4}\) at the points \(P\) and \(Q\), respectively. If the length of the line segment \(PQ\) is \(\alpha\), then \(225\alpha^2\) is equal to:

A bag contains \(8\) balls, whose colours are either white or black. \(4\) balls are drawn at random without replacement and it was found that \(2\) balls are white and other \(2\) balls are black. The probability that the bag contains equal number of white and black balls is :

A galvanometer having a coil resistance of \(100\ \Omega\). gives a full scale deflection, when a currect of \(I\ mA\) is passed through it. The value of the resistance, which can convert this galvanometer into ammeter giving a full scale deflection for a current of \(10\ A\), is :......\(Ω\)

A solid circular disc of mass \(50 \mathrm{~kg}\) rolls along a horizontal floor so that its center of mass has a speed of \(0.4 \mathrm{~m} / \mathrm{s}\). The absolute value of work done on the disc to stop it is _______ \(\mathrm{J}\).