A source (S) of sound has frequency \(240 \mathrm{~Hz}\). When the observer \((\mathrm{O})\) and the source move towards each other at a speed \(v\) with respect to the ground (as shown in Case 1 in the figure), the observer measures the frequency of the sound to be \(288 \mathrm{~Hz}\). However, when the observer and the source move away from each other at the same speed \(v\) with respect to the ground (as shown in Case 2 in the figure), the observer measures the frequency of sound to be \(n \mathrm{~Hz}\). The value of \(n\) is ______

The binding energy of nucleons in a nucleus can be affected by the pairwise Coulomb repulsion. Assume that all nucleons are uniformly distributed inside the nucleus. Let the binding energy of a proton be $E_b^p$ and the binding energy of a neutron be $E_b^n$ in the nucleus.
Which of the following statement(s) is(are) correct?

The center of a disk of radius \(r\) and mass \(m\) is attached to a spring of spring constant \(k\), inside a ring of radius \(R>r\) as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following the Hooke's law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as \(T=\frac{2 \pi}{\omega}\). The correct expression for \(\omega\) is ( \(g\) is the acceleration due to gravity):

Length, breadth and thickness of a strip having a uniform cross section are measured to be 10.5 cm , 0.05 mm , and \(6.0 \mu \mathrm{~m}\), respectively. Which of the following option(s) give(s) the volume of the strip in \(\mathrm{cm}^3\) with correct significant figures:

Which one of the following statement is WRONG in the context of X-rays generated from X-ray tube?

A Young's double slit interference arrangement with slits ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ is immersed in water (refractive index $=\frac{4}{3}$ ) as shown in the figure. The positions of maxima on the surface of water are given by ${\mathrm{x}}^2={\mathrm{p}}^2{\mathrm{m}}^2{\mathrm{\lambda }}^2-{\mathrm{d}}^2$ , where $\mathrm{\lambda }$ is the wavelength of light in air (refractive index = 1), $2\mathrm{d}$ is the separation between the slits and m is an integer. The value of p is

One mole of a monatomic ideal gas is taken along two cyclic processes $\text{E}\to \text{F}\to \text{G}\to \text{E} and \text{E}\to \text{F}\to \text{H}\to \text{E}$ as shown in the PV diagram. The processes involved are purely isochoric, isobaric, isothermal or adiabatic.

Match the paths in List I with the magnitudes of the work done in List II and select the correct answer using the codes given below the lists.
$\begin{array}{ccccc}& \text{List I}& & & \text{List II}\\ \text{A.}& \text{G}\mathrm{\to }\text{E }& & \mathrm{P}\text{.}& 16\mathrm{0 }{\text{P}}_0{\text{V}}_0\text{ ln}2\\ \text{B.}& \text{G}\mathrm{\to }\text{H }& & \mathrm{Q}\text{.}& 36{\text{ P}}_0{\text{V}}_0\\ \text{C.}& \text{F}\mathrm{\to }\text{H }& & \mathrm{R}\text{.}& 24{\text{ P}}_0{\text{V}}_0\\ \text{D.}& \text{F}\mathrm{\to }\text{G }& & \mathrm{S}\text{.}& 31{\text{ P}}_0{\text{V}}_0\end{array}$

Let $a, b, c$ be three non-zero real numbers such that the equation $\sqrt{3}a\cos x+2b\sin x=c, x\in \left[-\frac{\pi }{2}, \frac{\pi }{2}\right]$ has two distinct real roots $\alpha$ and $\beta$ with $\alpha +\beta =\frac{\pi }{3}$. Then, the value of $\frac{b}{a}$ is

A $10\mathrm{}\mathrm{c}\mathrm{m}$ long perfectly conducting wire $\mathrm{P}\mathrm{Q}$ is moving, with a velocity $1\mathrm{}\mathrm{c}\mathrm{m}/\mathrm{s}$ on a pair of horizontal rails of zero resistance. One side of the rails is connected to an inductor $\mathrm{L}=1\mathrm{}\mathrm{m}\mathrm{H}$ and a resistance $\mathrm{R}=1\mathrm{\Omega }$ as shown in figure. The horizontal rails, $\mathrm{L}$ and $\mathrm{R}$ lie in the same plane with a uniform magnetic field $\mathrm{B}=1\mathrm{}\mathrm{T}$ perpendicular to the plane. If the key $\mathrm{S}$ is closed at certain instant, the current in the circuit after $1$ milli second is $\mathrm{x}\times {10}^{-3}\mathrm{A},$ where the value of $\mathrm{x}$ is_______.
[Assume the velocity of wire $\mathrm{P}\mathrm{Q}$ remains constant $\left(1\mathrm{c}\mathrm{m}/\mathrm{s}\right)$ after key $\mathrm{S}$ is closed. Given: ${\mathrm{e}}^{-1}=0.37,$ where $\mathrm{e}$ is base of the natural logarithm]

Three objects \(A, B\) and \(C\) are kept in a straight line on a frictionless horizontal surface. These have masses \(m, 2 m\) and \(m\), respectively. The object \(A\) moves towards \(B\) with a speed \(9 \mathrm{~ms}^{-1}\) and makes an elastic collision with it. Thereafter, \(B\) makes completely inelastic collision with \(C\). All motions occur on the same straight line. Find the final speed (in \(\mathrm{ms}^{-1}\) ) of the object \(C\).

Statement 1 Two cylinders, one hollow (metal) and the other solid (wood) with the same mass and identical dimensions are simultaneously allowed to roll without slipping down an inclined plane from the same height. The hollow cylinder will reach the bottom of the inclined plane first.
and Statement 2 By the principle of conservation of energy, the total kinetic energies of both the cylinders are identical when they reach the bottom of the incline.

Let \(\mathbb{R}\) denote the set of all real numbers. For a real number \(x\), let \([x]\) denote the greatest integer less than or equal to \(x\). Let \(n\) denote a natural number.
Match each entry in List-I to the correct entry in List-II and choose the correct option.

	LIST - I		LIST - II
(P)	The minimum value of \(n\) for which the function \(f(x)=\left[\frac{10 x^3-45 x^2+60 x+35}{n}\right]\) is continuous on the interval \([1,2]\),is	(1)	8
(Q)	The minimum value of \(n\) for which \(g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right),x \in \mathbb{R}\),is an increasing function on \(\mathbb{R}\),is	(2)	9
(R)	The smallest natural number \(n\) which is greater than 5,such that \(x=3\) is a point of local minima of \(h(x)=\left(x^2-9\right)^{\mathrm{n}}\left(x^2+2 x+3\right)\),is	(3)	5
(S)	Number of \(x_0 \in \mathbb{R}\) such that \(l(x)=\sum_{k=0}^4\left(\sin |x-k|+\cos \left|x-k+\frac{1}{2}\right|\right),x \in \mathbb{R}\),is NOT differentiable at \(x_0\),is	(4)	6
		(5)	10

In the following circuit, the current through the resistor $R(=2\mathrm{\Omega })$ is $\mathrm{I}$ Amperes. The value of $\mathrm{I}\mathrm{}$ is