The magnitude of the magnetic field at the centre of an equilateral triangular loop of side \(1\,m\) which is carrying a current of \(10\,A\) is:......\(\mu T\) [Take \(\mu _0 = 4\pi  \times 10^{-7}\,NA^{-2}\)]

An oscillator of mass \(M\)  is at rest in its equilibrium position in a potential \(V\, = \,\frac{1}{2}\,k{(x - X)^2}.\) A particle of mass \(m\)  comes from right with speed \(u\)  and collides completely inelastically with \(M\) and sticks to it . This process repeats every time the oscillator crosses its equilibrium position .The amplitude of oscillations after \(13\)  collisions is: \((M = 10,\, m = 5,\, u = 1,\, k = 1 ).\) 

The \(V-I\) characteristic of a diode is shown in the figure. The ratio of forward to reverse bias resistance is

A common transistor radio set requires \(12\,V (D.C.)\) for its operation. The \(D.C.\) source is constructed by using a transformer and a rectifier circuit, which are operated at \(220\, V\) \((A.C.)\) on standard domestic \(A.C.\) supply. The number of turns of secondary coil are \(24,\) then the number of turns of primary are

Given below are two statement : Statement \(I :\) If the number of turns in the coil of a moving coil galvanometer is doubled then the current sensitivity becomes double. Statement \(II :\) Increasing current sensitivity of a moving coil galvanometer by only increasing the number of turns in the coil will also increase its voltage sensitivity in the same ratio: In the light of the above statement, choose the correct answer from the options given below :

\(\mathrm{A}, \mathrm{B}\) and C are disc, solid sphere and spherical shell respectively with same radii and masses. These masses are placed as shown in figure.

The moment of inertia of the given system about \(P Q\) is \(\frac{X}{15} I\), where \(I\) is the moment of inertia of the disc about its diameter. The value of x is _______.

A force \(\overrightarrow{\mathrm{F}}=2 \hat{i}+\mathrm{b} \hat{j}+\hat{k}\) is applied on a particle and it undergoes a displacement \(\hat{i}-2 \hat{j}-\hat{k}\). What will be the value of \(b\), if work done on the particle is zero.

A disc of mass \(1\,kg\) and radius \(R\) is free of rotate about a horizontal axis passing through its centre and perpendicular to the plane of disc. A body of same mass as that of disc is fixed at the highest point of the disc. Now the system is released, when the body comes to the lowest position, its angular speed will be \(4 \sqrt{\frac{x}{3 R}}\) rad s\(^{-1}\) where \(x=\) \(\left( g =10\,ms ^{-2}\right)\)

If a radioactive element having half-life of \(30\,min\) is undergoing beta decay, the fraction of radioactive element remains undecayed after \(90\,min\). will be :

Two bodies A and B of equal mass are suspended from two massless springs of spring constant \(k_1\) and \(k_2\), respectively. If the bodies oscillate vertically such that their amplitudes are equal, the ratio of the maximum velocity of \(A\) to the maximum velocity of \(B\) is _______.

A body of mass \(1\,kg\) falls freely from a height of \(100\,m,\) on a platform of mass \(3\,kg\) which is mounted on a spring having spring constant \(k = 1.25 \times 10^6\, N/m.\) The body sticks to the platform and the spring’s maximum compression is found to be \(x.\) Given that \(g = 10\,ms^{-2},\) the value of \(x\) will be close to ................ \(\mathrm{cm}\)

A uniform disc of radius \(R\) and mass \(M\) is free to oscillate about the axis \(A\) as shown in the figure. For small oscillations the time period is ______.
(\(g\) is acceleration due to gravity)

A concave mirror of focal length \(10\) cm forms an image which is double the size of object when the object is placed at two different positions. The distance between the two positions of the object is __________ cm.