During Searle's experiment, zero of the Vernier scale lies between $3.20\times {10}^{-2}\mathrm{ }\mathrm{m}$ and $3.25\times {10}^{-2}\mathrm{ }\mathrm{m}$ of the main scale. The ${20}^{\mathrm{t}\mathrm{h}}$ division of the Vernier scale exactly coincides with one of the main scale divisions. When an additional load of 2 kg is applied to the wire, the zero of the Vernier scale still lies between $3.20\times {10}^{-2\mathrm{ }}\mathrm{m}$ and $3.25\times {10}^{-2}\mathrm{ }\mathrm{m}$ of the main scale but now the ${45}^{\mathrm{t}\mathrm{h}}$ division of Vernier scale coincides with one of the main scale divisions. The length of the thin metallic wire is 2 m and its cross-sectional area is $8\times {10}^{-7}\mathrm{ }{\mathrm{m}}^2$ . The least count of the Vernier scale is $1.0\times \mathrm{ }{10}^{-5}\mathrm{ }\mathrm{m}$ . The maximum percentage error in the Young's modulus of the wire is

Statement I If the accelerating potential in an X-ray tube is increased, the wavelengths of the characteristic \(X\)-rays do not change.
Statement II When an electron beam strikes the target in an X-ray tube, part of the kinetic energy is converted into X-ray energy.

Paragraph II: Two particles, 1 and 2 , each of mass \(m\), are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at \(x_0\), are oscillating with amplitude \(a\) and angular frequency \(\omega\). Thus, their positions at time \(t\) are given by \(x_1(t)=\left(x_0+d\right)+a \sin \omega t\) and \(x_2(t)=\left(x_0-d\right)-a \sin \omega t\), respectively, where \(d>2 a\). Particle 3 of mass \(m\) moves towards this system with speed \(u_0=a \omega / 2\), and undergoes instantaneous elastic collision with particle 2 , at time \(t_0\). Finally, particles 1 and 2 acquire a center of mass speed \(v_{\mathrm{cm}}\) and oscillate with amplitude \(b\) and the same angular frequency \(\omega\).

If the collision occurs at time \(t_0=0\), the value of \(v_{\mathrm{cm}} /(a \omega)\) will be ________ .

In the List-I below, four different paths of a particle are given as functions of time. In these functions, $\alpha \mathrm{a}\mathrm{n}\mathrm{d} \beta$ are positive constants of appropriate dimensions and $\alpha \ne \beta$ . In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned: $\overrightarrow{p}$ is the linear momentum, $\overrightarrow{L}$ is the angular momentum about the origin, K is the kinetic energy, U is the potential energy and E is the total energy. Match each path in List-I with those quantities in List-II, which are conserved for that path.

	LIST -I		LIST-II
A)	$\overrightarrow{r}\left(t\right)=\alpha t\widehat{i}+\beta \widehat{j}$	P)	$\overrightarrow{p}$
B)	$\overrightarrow{r}\left(t\right)=\alpha \cos \omega t\widehat{i}+\beta \sin \omega t\widehat{j}$	Q)	$\overrightarrow{L}$
C)	$\overrightarrow{r}\left(t\right)=\alpha \cos \omega t\widehat{i}+\sin \omega t\widehat{j}$	R)	$K$
D)	$\overrightarrow{r}\left(t\right)=\alpha t\widehat{i}+\frac{\beta }{2}{t}^2\widehat{j}$	S)	$U$
		T)	$E$

An extended object is placed at point $O$, $10 \mathrm{cm}$ in front of a convex lens $L_1$ and a concave lens $L_2$ is placed $10 \mathrm{cm}$ behind it, as shown in the figure. The radii of curvature of all the curved surfaces in both the lenses are $20 \mathrm{cm}$. The refractive index of both the lenses is $1.5$. The total magnification of this lens system is:

A drop of liquid of radius $R={10}^{-2}m$ having surface tension $S=\frac{0.1}{4\pi }N{m}^{-1}$ divides itself into K identical drops. In the process the total change in the surface energy $\mathrm{\Delta }U={10}^{-3}J$ . If $K={10}^{\alpha }$ then the value of $\alpha$ is

Paragraph :
A spray gun is shown in the figure where a piston pushes air out of a nozzle. A thin tube of uniform cross section is connected to the nozzle. The other end of the tube is in a small liquid container. As the piston pushes air through the nozzle, the liquid from the container rises into the nozzle and is sprayed out. For the spray gun shown, the radii of the piston and the nozzle are \(20 \mathrm{~mm}\) and \(1 \mathrm{~mm}\), respectively. The upper end of the container is open to the atmosphere.

Question :
If the piston is pushed at a speed of \(5 \mathrm{~mms}^{-1}\), the air comes out of the nozzle with a speed of

Two adjacent sides of a parallelogram \(A B C D\) are given by \(\overrightarrow{\mathbf{A B}}=2 \hat{\mathbf{i}}+10 \hat{\mathbf{j}}+11 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{A D}}=-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\). The side \(A D\) is rotated by an acute angle \(\alpha\) in the plane of the parallelogram so that \(A D\) becomes \(A D^{\prime}\). If \(A D^{\prime}\) makes a right angle with the side \(A B\), then the cosine of the angle \(\alpha\) is given by

The compound (s) having peroxide linkage is(are) :

Three concentric metallic spherical shells of radii \(R, 2 R, 3 R\) are given charges \(Q_1, Q_2, Q_3\), respectively. It is found that the surface charge densities on the outer surfaces of the shells are equal. Then, the ratio of the charges given to the shells, \(Q_1: Q_2: Q_3\), is

Two particles of mass \(m\) each are tied at the ends of a light string of length \(2 a\). The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance \(a\) from the centre \(P\) (as shown in the figure). Now, the mid-point of the string is pulled vertically upwards with a small but constant force \(F\). As a result, the particles move towards each other on the surface.
The magnitude of acceleration, when the separation between them becomes \(2 x\), is

Consider a hydrogen atom with its electron in the ${\mathrm{n}}^{\mathrm{t}\mathrm{h}}$ orbital. An electromagnetic radiation of wavelength 90 nm is used to ionize the atom. If the kinetic energy of the ejected electron is 10.4 eV, then the value of n is $\left(\mathrm{h}\mathrm{c}=1242\mathrm{ }\mathrm{e}\mathrm{V}\mathrm{ }\mathrm{n}\mathrm{m}\right)$

Let $\mathrm{f}\left(\mathrm{x}\right)=\sin \left(\frac{\mathrm{\pi }}{6}\sin \left(\frac{\mathrm{\pi }}{2}\sin \mathrm{x}\right)\right)$, for all $\mathrm{x}\in R$ and $\mathrm{g}\left(\mathrm{x}\right)=\frac{\mathrm{\pi }}{2}\sin \mathrm{x}$, for all $\mathrm{x}\in R$. Let $\left(\mathrm{f}o\mathrm{g}\right)\mathrm{ }\left(\mathrm{x}\right)$ denote $\mathrm{f}\left(\mathrm{g}\left(\mathrm{x}\right)\right)$ and $\left(\mathrm{g}o\mathrm{f}\right)\mathrm{ }\left(\mathrm{x}\right)$ denote $\mathrm{g}\left(\mathrm{f}\left(\mathrm{x}\right)\right)$. Then which of the following is (are) true?