The potential energy \((U)\) of a diatomic molecule is a function dependent on \(r\) (interatomic distance) as \(U =\frac{\alpha}{ r ^{10}}-\frac{\beta}{ r ^{5}}-3\) where, \(\alpha\) and \(\beta\) are positive constants. The equilibrium distance between two atoms will be \(\left(\frac{2 \alpha}{\beta}\right)^{\frac{a}{b}},\) where a \(=\) ..........

Three parallel plate capacitors \(\mathrm{C}_1, \mathrm{C}_2\) and \(\mathrm{C}_3\) each of capacitance \(5 \mu \mathrm{~F}\) are connected as shown in figure. The effective capacitance between points A and B, when the space between the parallel plates of \(C_1\) capacitor is filled with a dielectric medium having dielectric constant of 4, is :

A circuit is arranged as shown in figure. The output voltage \({V}_{0}\) is equal to \(\ldots \ldots . {V}\).

The diffraction pattern of a light of wavelength \(400\) \(\mathrm{nm}\) diffracting from a slit of width \(0.2 \mathrm{~mm}\) is focused on the focal plane of a convex lens of focal length \(100 \mathrm{~cm}\). The width of the \(1^{\text {st }}\) secondary maxima will be _______.

A particle moves along the \(x\)-axis and has its displacement \(x\) varying with time \(t\) according to the equation \(\mathrm{x}=\mathrm{c}_0\left(\mathrm{t}^2-2\right)+\mathrm{c}(\mathrm{t}-2)^2\) where \(c_0\) and \(c\) are constants of appropriate dimensions. Then, which of the following statements is correct?

For the given circuit, comment on the type of transformer used:

In Young's double slit experiment, two slits \(S_1\) and \(S_2\) are ' \(d\) ' distance apart and the separation from slits to screen is \(D\) (as shown in figure). Now if two transparent slabs of equal thickness \(0.1\,mm\) but refractive index \(1.51\) and \(1.55\) are introduced in the path of beam \((\lambda=4000 \mathring A)\) from \(S_1\) and \(S_2\) respectively. The central bright fringe spot will shift by  \(..........\) number of fringes.

In Young's double slit experiment, if the source of light changes from orange to blue then

A transverse wave is represented by  \(y = \frac{{10}}{\pi }\,\sin \,\left( {\frac{{2\pi }}{T}t - \frac{{2\pi }}{\lambda }x} \right)\) For what value of the wavelength the wave velocity is twice the maximum particle velocity ..... \(cm\) ?

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason \(\mathbf{R}\)
Assertion A: A sound wave has higher speed in solids than gases.
Reason R: Gases have higher value of Bulk modulus than solids.
In the light of the above statements, choose the correct answer from the options given below

In the given circuit ' \(a\) ' is an arbitrary constant. The value of \(m\) for which the equivalent circuit resistance is minimum, will be \(\sqrt{\frac{ x }{2}}\). The value of \(x\) is ...........