A particle of mass $M=0.2 \mathrm{kg}$ is initially at rest in the $xy$-plane at a point $\left(x=-l, y=-h\right)$, where $l=10 \mathrm{m}$ and $h=1 \mathrm{m}$. The particle is accelerated at time $t=0$ with a constant acceleration $a=10 \mathrm{m} {\mathrm{s}}^{-2}$ along the positive $x$-direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by $\overrightarrow{L}$ and  $\overrightarrow{\tau }$, respectively. $\widehat{\mathrm{i}}, \widehat{\mathrm{j}}$ and $\widehat{\mathrm{k}}$ are unit vectors along the positive $x,y$ and $z$-directions, respectively. If $\widehat{\mathrm{k}}=\widehat{\mathrm{i}}\times \widehat{\mathrm{j}}$ then which of the following statement(s) is (are) correct?

Steel wire of length \(L\) at \(40^{\circ} \mathrm{C}\) is suspended from the ceiling and then a mass \(m\) is hung from its free end. The wire is cooled down from \(40^{\circ} \mathrm{C}\) to \(30^{\circ} \mathrm{C}\) to regain its original length \(L\). The coefficient of linear thermal expansion of the steel is \(10^{-5} /{ }^{\circ} \mathrm{C}\), Young's modulus of steel is \(10^{11} \mathrm{~N} / \mathrm{m}^2\) and radius of the wire is \(1 \mathrm{~mm}\). Assume that \(L>\) diameter of the wire. Then, the value of \(m\) in \(\mathrm{kg}\) is nearly

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A uniform thin cylindrical disk of mass \(M\) and radius \(R\) is attached to two identical massless springs of spring constant \(k\) which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance \(d\) from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is \(L\). The disk is initially at its equilibrium position with its centre of mass \((C M)\) at a distance Lfrom the wall. The disk rolls without slipping with velocity \(\mathbf{v}_0=v_0 \hat{\mathbf{i}}\) The coefficient of friction is \(\mu\).
Question :
The net external force acting on the disk when its centre of mass is at displacement \(x\) with respect to its equilibrium position 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

A rocket is launched normal to the surface of the Earth, away from the Sun, along the line joining the Sun and the Earth. The Sun is $3\times {10}^5$ times heavier than the Earth and is at a distance $2.5\times {10}^4$ times larger than the radius of the Earth. The escape velocity from Earth's gravitational field is $v_e=11.2 km s^{-1}$ . The minimum initial velocity $\left(v_s\right)$ required for the rocket to be able to leave the Sun-Earth system is closest to
(Ignore the rotation and revolution of the Earth and the presence of any other planet)

A metal is heated in a furnace where a sensor is kept above the metal surface to read the power radiated ($P$) by the metal. The sensor has a scale that displays ${\log }_2\left(\frac{P}{P_0}\right)$, where $P_0$ is a constant. When the metal surface is at a temperature of $487 °\mathrm{C}$, the sensor shows a value $1$. Assume that the emissivity of the metallic surface remains constant. What is the value displayed by the sensor when the temperature of the metal surface is raised to $2767 °\mathrm{C}$?

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A special metal \(S\) conducts electricity without any resistance. A closed wire loop, made of \(S\), does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius \(a\), with its center at the origin. A magnetic dipole of moment \(m\) is brought along the axis of this loop from infinity to a point at distance \(r(\gg a)\) from the center of the loop with its north pole always facing the loop, as shown in the figure below.

The magnitude of magnetic field of a dipole \(m\), at a point on its axis at distance \(r\), is \(\frac{\mu_{0}}{2 \pi} \frac{m}{r^{3}}\), where \(\mu_{0}\) is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, \(m_{1}\) and \(m_{2}\), separated by a distance \(r\) on the common axis, with their north poles facing each other, is \(\frac{k m_{1} m_{2}}{r^{4}}\), where \(k\) is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

Question :
When the dipole m is placed at a distance r from the centre of the loop (as shown in the figure), the current induced in the loop will be proportional to:

A dimensionless quantity is constructed in terms of electronic charge \(e\), permittivity of free space \(\varepsilon_0\), Planck's constant \(h\) and speed of light \(c\). If the dimensionless quantity is written as \(e^\alpha \varepsilon_0^\beta h^\gamma c^\delta\) and \(n\) is a non-zero integer, then \((\alpha, \beta, \gamma, \delta)\) is given by

A piece of ice (heat capacity \(=2100 \mathrm{~J} \mathrm{~kg}^{-1}{ }^{\circ} \mathrm{C}^{-1}\) and latent heat \(=3.36 \times 10^5 \mathrm{~J} \mathrm{~kg}^{-1}\) ) of mass \(\mathrm{m}\) gram is at \(-5^{\circ} \mathrm{C}\) at atmospheric pressure. It is given \(420 \mathrm{~J}\) of heat so that the ice starts melting. Finally when the ice-water mixture is in equilibrium, it is found that \(1 \mathrm{~g}\) of ice has melted. Assuming there is no other heat exchange in the process, the value of \(m\) is

Match each set of hybrid orbitals from LIST-I with complex (es) given in LIST-II.

	LIST -I		LIST -II
A)	$ds{p}^2$	P)	${\left[Fe{F}_6\right]}^{4-}$
B)	$s{p}^3$	Q)	$\left[Ti{\left(H_{2}O\right)}_{3}C{l}_3\right]$
C)	$s{p}^3{d}^2$	R)	${\left[Cr{\left(N{H}_3\right)}_6\right]}^{3+}$
D)	$d^{2}s{p}^3$	S)	${\left[FeC{l}_4\right]}^{2-}$
		T)	$Ni{\left(CO\right)}_4$
		U)	${\left[Ni{\left(CN\right)}_4\right]}^{2-}$

Two large, identical water tanks, 1 and 2, kept on the top of a building of height \(H\), are filled with water up to height \(h\) in each tank. Both the tanks contain an identical hole of small radius on their sides, close to their bottom. A pipe of the same internal radius as that of the hole is connected to tank 2 , and the pipe ends at the ground level. When the water flows from the tanks 1 and 2 through the holes, the times taken to empty the tanks are \(t_1\) and \(t_2\), respectively. If \(H=\left(\frac{16}{9}\right) h\), then the ratio \(t_1 / t_2\) is _______ .

In the balanced condition, the values of the resistances of the four arms of a Wheatstone bridge are shown in the figure below. The resistance $R_3$ has temperature coefficient $0.0004°{\mathrm{C}}^{-1}$. If the temperature of $R_3$ is increased by $100°\mathrm{C},$ the voltage developed between $S$ and $T$ will be __________ volt.

A block of mass $2\mathrm{M}$ is attached to a massless spring with spring-constant $\mathrm{k}.$ This block is connected to two other blocks of masses $\mathrm{M}$ and $2\mathrm{M}$ using two massless pulleys and strings. The accelerations of the blocks are ${\mathrm{a}}_1,{\mathrm{a}}_2$ and ${\mathrm{a}}_3$ as shown in figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is ${\mathrm{x}}_0.$ Which of the following option(s) is/are correct?
[ $\mathrm{g}$ is the acceleration due to gravity. Neglect friction]