The distance between an object and a screen is \(100\, cm .\) A lens can produce real image of the object on the screen for two different positions between the screen and the object. The distance between these two positions is \(40\, cm .\) If the power of the lens is close to \(\left(\frac{ N }{100}\right) D\) where \(N\) is an integer, the value of \(N\) is.......

Two known resistance of \(R \Omega\) and \(2 R \Omega\) and and one unknown resistance \(X \Omega\) are connected in a circuit as shown in the figure. If the equivalent resistance between points A and B in the circuit is \(X \Omega\), then the value of X is _________ \(\Omega\).

For a nucleus \({ }_Z^A X\) having mass number \(A\) and atomic number \(Z\) \(A.\) The surface energy per nucleon \(\left(b_s\right)=a_1 A^{2 / 3}\) \(B.\) The Coulomb contribution to the binding energy \(b_c=-a_2 \frac{Z(Z-1)}{A^{4 / 3}}\) \(C.\) The volume energy \(b_v=a_3 A\) \(D.\) Decrease in the binding energy is proportional to surface area. \(E.\) While estimating the surface energy, it is assumed that each nucleon interacts with \(12\) nucleons, \(\left(a_1, a_2\right.\) and \(a _3\) are constants) Choose the most appropriate answer from the options given below:

Two concentric circular coils, \(C _{1}\) and \(C _{2}\), are placed in the \(XY\) plane. \(C _{1}\) has \(500\) turns, and a radius of \(1\; cm . C _{2}\) has \(200\) turns and radius of \(20\, cm . C _{2}\) carries a time dependent current \(I(t)=\left(5 t^{2}-2 t+3\right) \;A\) where \(t\) is in \(s\). The \(emf\) nduced in \(C _{1}\) (in \(mV\) ), at the instant \(t =1\; s\) is \(\frac{4}{ x }\). The value of \(x\) is

Consider an electromagnetic wave propagating in vacuum . Choose the correct statement

A convex lens made of glass (refractive index \(=1.5\) ) has focal length 24 cm in air. When it is totally immersed in water (refractive index \(=1.33\) ), its focal length changes to ________.

A parallel plate capacitor is made of two circular plates separated by a distance \(5\ mm\) and with a dielectric of dielectric constant \(2.2\) between them. When the electric field in the dielectric is \(3 \times 10^4\) \( Vm^{-1}\) the charge density of the positive plate will be close to

The magnetic moment of a bar magnet is \(0.5 \mathrm{Am}^2\). It is suspended in a uniform magnetic field of \(8 \times 10^{-2} \mathrm{~T}\). The work done in rotating it from its most stable to most unstable position is _______.

Three identical particle \(A , B\) and \(C\) of mass \(100\, kg\) each are placed in a straight line with \(AB = BC =13 \,m\). The gravitational force on a fourth particle \(P\) of the same mass is \(F\), when placed at a distance \(13 \,m\) from the particle \(B\) on the perpendicular bisector of the line \(AC\). The value of \(F\) will be approximately\(....G\)

The half-life of a radioactive nucleus is \(5\) years, The fraction of the original sample that would decay in \(15\) years is

An aeroplane, with its wings spread \(10\, m ,\) is flying at a speed of \(180 \,km / h\) in a horizontal direction. The total intensity of earth's field at that part is \(2.5 \times 10^{-4}\, Wb / m ^{2}\) and the angle of dip is \(60^{\circ}\). The emf induced between the tips of the plane wings will be ...... \(mV\)

An external pressure \(P\) is applied on a cube at \(0^o C\) so that it is equally compressed from all sides. \(K\) is the bulk modulus of the material of the cube and a is its coefficient of linear expansion. Suppose we want to bring the cube to its original size by heating. The temperature should be raised by

Match List \(I\) with List \(II\) of Electromagnetic waves with corresponding wavelength range :

List \(I\)	List \(II\)
\((A)\) Microwave	\((I)\) \(400\,nm\) to \(1\,nm\)
\((B)\) Ultraviolet	\((II)\) \(1\,nm\) to \(10^{-3}\,nm\)
\((C)\) \(X-\)Ray	\((III)\) \(1\,mn\) to \(700\,nm\)
\((D)\) Infra-red	\((IV)\) \(0.1\,m\) to \(1\,mm\)

Choose the correct answer from the options given below: