A large square container with thin transparent vertical walls and filled with water $\left(\text{refractive index}\frac{4}{3}\right)$ is kept on a horizontal table. A student holds a thin straight wire vertically inside the water $12\mathrm{cm}$ from one of its corners, as shown schematically in the figure. Looking at the wire from this corner, another student sees two images of the wire, located symmetrically on each side of the line of sight as shown. The separation (in $\mathrm{cm}$) between these images is ________.

A solid sphere of mass $1 \mathrm{kg}$ and radius $1 \mathrm{m}$ rolls without slipping on a fixed inclined plane with an angle of inclination $\theta =30°$ from the horizontal. Two forces of magnitude $1 \mathrm{N}$ each, parallel to the incline, act on the sphere, both at distance $r=0.5 \mathrm{m}$ from the center of the sphere, as shown in the figure. The acceleration of the sphere down the plane is $\mathrm{m} {\mathrm{s}}^{-2}$. (Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)
If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places.

Paragraph:
Modern trains are based on Maglev technology in which trains are magnetically leviated, which runs its EDS Maglev system.
There are coils on both sides of wheels. Due to motion of train, current induces in the coil of track which levitate it. This is in accordance with Lenz's law. If trains lower down then due to Lenz's law a repulsive force increases due to which train gets uplifted and if it goes much high then there is a net downward force disc to gravity. The advantage of Maglev train is that there is no friction between the train and the track, thereby reducing power consumption and enabling the train to attain very high speeds.
Disadvantage of Maglev train is that as it slows down the electromagnetic forces decreases and it becomes difficult to keep it leviated and as it moves forward according to Lenz law, there is an electromagnetic drag force.
Question:
What is the advantage of this system?

Paragraph:

A point charge \(Q\) is moving in a circular orbit of radius \(R\) in the \(x-y\) plane with an angular velocity \(\omega .\) This can be considered as equivalent to a loop carrying a steady current \(\frac{Q \omega}{2 \pi}\). A uniform magnetic field along the positive \(z\)-axis is now switched on, which increases at a constant rate from 0 to \(B\) in one second. Assume that the radius of the orbit remains constant. The application of the magnetic field induces an emf in the orbit. The induced emf is defined as the work done by an induced electric field in moving a unit positive charge around a closed loop. It is known that, for an orbiting charge, the magnetic dipole moment is proportional to the angular momentum with a proportionality constant \(\gamma\).

Question:

The magnitude of the induced electric field in the orbit at any instant of time during the time interval of the magnetic field change is

Paragraph
The key feature of Bohr's theory of spectrum of hydrogen atom is the quantization of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find quantized rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr's quantization condition.
Question:
It is found that the excitation frequency from ground to the first excited state of rotation for the \(\mathrm{CO}\) molecule is close to \(\frac{4}{\pi} \times 10^{11} \mathrm{~Hz}\). Then the moment of inertia of CO molecule about its centre of mass is close to (Take \(h=2 \pi \times 10^{-34} J-s\) )

A cylindrical furnace has height $\left(H\right)$ and diameter $\left(D\right)$ both $1\mathrm{m}$. It is maintained at temperature $360\mathrm{K}$. The air getsheated inside the furnace at constant pressure$P_a$and its temperature becomes $T=360\mathrm{K}$. The hot air with density $\rho$rises up a vertical chimney of diameter $d=0.1\mathrm{m}$ and height $h=9\mathrm{m}$ above the furnace and exits the chimney (seethe figure). As a result, atmospheric air of density${\rho }_a=1.2\mathrm{kg}{\mathrm{m}}^{-3}$,pressure $P_a$and temperature $T_a=300\mathrm{K}$ entersthe furnace. Assume air as an ideal gas, neglect the variations in$\rho$and $T$ inside the chimney and the furnace. Also ignore the viscous effects.
[Given: The acceleration due to gravity$g=10\mathrm{m}{\mathrm{s}}^{-2}$and$\pi =3.14$]


When the chimney is closed using a cap at the top, a pressure difference $∆P$develops between the top and the bottom surfaces of the cap. If the changes in the temperature and density of the hot air, due to the stoppage of air flow, are negligible then the value of $∆P$is _____ $\mathrm{N}{\mathrm{m}}^{-2}$.

An annular disk of mass $M$, inner radius $a$ and outer radius $b$ is placed on a horizontal surface with coefficient of friction $\mu$, as shown in the figure. At some time, an impulse $J_0\stackrel{\wedge }{x}$ is applied at a height $h$ above the center of the disk. If $h=h_m$ then the disk rolls without slipping along the $x$-axis. Which of the following statement(s) is(are) correct?

The option(s) with only amphoteric oxides is(are)

For the reduction of \(\mathrm{NO}_3^{-}\)ion in an aqueous solution \(E^{\circ}\) is \(+0.96 \mathrm{~V}\). Values of \(E^{\circ}\) for some metal ions are given below.
\(
\begin{aligned}
& \mathrm{V}^{2+}(a q)+2 e^{-} \longrightarrow \mathrm{V} ; E^{\circ}=-1.19 \mathrm{~V} \\
& \mathrm{Fe}^{3+}(a q)+3 e^{-} \longrightarrow \mathrm{Fe} ; \mathrm{E}^{\circ}=-0.04 \mathrm{~V} \\
& \mathrm{Au}^{3+}(a q)+3 e^{-} \longrightarrow \mathrm{Au} ; E^{\circ}=+1.40 \mathrm{~V}
\end{aligned}
\)
\(
\mathrm{Hg}^{2+}(a q)+2 e^{-} \longrightarrow \mathrm{Hg} ; E^{\circ}=+0.86 \mathrm{~V}
\)
The pair(s) of metals that is (are) oxidised by \(\mathrm{NO}_3^{-}\)in aqueous solution is (are)

Paragraph:
If a continuous \(f\) defined on the real line \(R\), assume positive and negative values in \(R\), then the equation \(f(x)=0\) has a root in \(R\). For example, if it is known that a continuous function \(f\) on \(R\) is positive at some point and its minimum values is negative, then the equation \(f(x)=0\) has a root in \(R\).
Consider \(f(x)=k e^x-x\) for all real \(x\), where \(k\) is real constant.Question:
The line \(y=x\) meets \(y=k e^x\) for \(k \leq 0\) at

The nitrogen oxide(s) that contain(s) N.N bond(s) is (are)

Let \(\mathbb{R}\) denote the set of all real numbers. Let \(z_1=1+2 i\) and \(z_2=3 i\) be two complex numbers, where \(i=\sqrt{-1}\).
Let \(S=\{(x, y) \in \mathbb{R} \times \mathbb{R}:\left|x+i y-z_1\right|=2\)\(\left|x+i y-z_2\right|\}\).
Then which of the following statements is (are) TRUE?

A point mass of \(1 \mathrm{~kg}\) collides elastically with a stationary point mass of \(5 \mathrm{~kg}\). After their collision, the \(1 \mathrm{~kg}\) mass reverses its direction and moves with a speed of \(2 \mathrm{~ms}^{-1}\). Which of the following statement(s) is/are correct for the system of these two masses?