One mole of a monoatomic ideal gas goes through a thermodynamic cycle, as shown in the volume versus temperature $\left(\mathrm{V}-\mathrm{T}\right)$ diagram. The correct statement(s) is/are:
[ $\mathrm{R}$ is the gas constant]

A thin flexible wire of length \(L\) is connected to two adjacent fixed points and carries a current \(I\) in the clockwise direction, as shown in the figure. When the system is put in a uniform magnetic field of strength \(B\) going into the plane of the paper, the wire takes the shape of a circle. The tension in the wire is

\(5.6 \mathrm{~L}\) of helium gas at STP is adiabatically compressed to \(0.7 \mathrm{~L}\). Taking the initial temperature to be \(T_1\), the work done in the process is

List I contains four combinations of two lenses ( 1 and 2 ) whose focal lengths (in cm ) are indicated in the figures. In all cases, the object is placed 20 cm from the first lens on the left, and the distance between the two lenses is 5 cm . List II contains the positions of the final images.

	List - I		List - II
(I)		(P)	Final image is formed at 7.5 cm on the right side of lens 2.
(II)		(Q)	Final image is formed at 60.0 cm on the right side of lens 2.
(III)		(R)	Final image is formed at 30.0 cm on the left side of lens 2.
(IV)		(S)	Final image is formed at 6.0 cm on the right side of lens 2.
		(T)	Final image is formed at 30.0 cm on the right side of lens 2.

Which one of the following options is correct?

A temperature difference can generate e.m.f. in some materials. Let \(S\) be the e.m.f. produced per unit temperature difference between the ends of a wire, \(\sigma\) the electrical conductivity and \(\kappa\) the thermal conductivity of the material of the wire. Taking \(M, L, T, I\) and \(K\) as dimensions of mass, length, time, current and temperature, respectively, the dimensional formula of the quantity \(Z=\frac{S^2 \sigma}{\kappa}\) is :-

System shown in figure is in equilibrium and at rest. The spring and string are massless, now the string is cut. The acceleration of mass \(2 m\) and \(m\) just after the string is cut will be

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A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity \(\omega\) is an example of a non-inertial frame of reference.

The relationship between the force \(\vec{F}_{\text {rot }}\) experienced by a particle of mass \(m\) moving on the rotating disc and the force \(\vec{F}_{\text {in }}\) experienced by the particle in an inertial frame of reference is

\(\vec{F}_{\mathrm{rot}}=\vec{F}_{\mathrm{in}}+2 m\left(\vec{v}_{\mathrm{rot}} \times \vec{\omega}\right)+m(\vec{\omega} \times \vec{r}) \times \vec{\omega}\),

where \(\vec{v}_{\text {rot }}\) is the velocity of the particle in the rotating frame of reference and \(\vec{r}\) is the position vector of the particle with respect to the centre of the disc.

Now consider a smooth slot along a diameter of a disc of radius \(R\) rotating counter-clockwise with a constant angular speed \(\omega\) about its vertical axis through its center. We assign a coordinate system with the origin at the center of the disc, the \(x\)-axis along the slot, the \(y\)-axis perpendicular to the slot and the \(z\)-axis along the rotation axis \((\vec{\omega}=\omega \hat{k}) .\) A small block of mass \(m\) is gently placed in the slot at \(\vec{r}=(R / 2) \hat{i}\) at \(t=0\) and is constrained to move only along the slot.

Question :
The net reaction of the disc on the block is

Consider the strong electrolytes ${\mathrm{Z}}_{\mathrm{m}}{\mathrm{X}}_{\mathrm{n}}, {\mathrm{U}}_{\mathrm{m}}{\mathrm{Y}}_{\mathrm{p}}$ and ${\mathrm{V}}_{\mathrm{m}}{\mathrm{X}}_{\mathrm{n}}$. Limiting molar conductivity of ${\mathrm{U}}_{\mathrm{m}}{\mathrm{Y}}_{\mathrm{p}}$ and ${\mathrm{V}}_{\mathrm{m}}{\mathrm{X}}_{\mathrm{n}}$ are $250$ and $440 \mathrm{S} {\mathrm{cm}}^2 {\mathrm{mol}}^{-1}$, respectively. The value of $\left(\mathrm{m}+\mathrm{n}+\mathrm{p}\right)$ is Given:

Ion	\(Z ^{ n +}\)	\(U ^{ p +}\)	\(V ^{ n +}\)	\(X ^{ m }\)	\(Y ^{ m -}\)
\(\lambda^0(\text{S cm} ^2\) \(\text{mol}^{-1})\)	50.0	25.0	100.0	80.0	100.0

${\lambda }^0$ is the limiting molar conductivity of ions The plot of molar conductivity $\left(\Lambda \right)$ of ${\mathrm{Z}}_{\mathrm{m}}{\mathrm{X}}_{\mathrm{n}}$ vs ${\mathrm{c}}^{\frac{1}{2}}$ is given below. If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places.

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p-amino-N, N-dimethylaniline is added to a strongly acidic solution of \(X\). The resulting solution is treated with a few drops of aqueous solution of \(Y\) to yield blue colouration due to the formation of methylene blue. Treatment of the aqueous solution of \(Y\) with the reagent potassium hexacyanoferrate(II) leads to the formation of an intense blue precipitate. The precipitate dissolves an excess addition of the reagent. Similarly, treatment of the solution of \(Y\) with the solution of potassium hexacyanoferrate(III) leads to a brown colouration due to the formation of \(Z\).
Question:
The compound \(Z\), is

Let $\alpha$ and $\beta$ be real numbers such that $-\frac{\pi }{4}<\beta <0<\alpha <\frac{\pi }{4}$. If $\sin \left(\alpha +\beta \right)=\frac{1}{3}$ and $\cos \left(\alpha -\beta \right)=\frac{2}{3}$, then the greatest integer less than or equal to ${\left(\frac{\sin \alpha }{\cos \beta }+\frac{\cos \beta }{\sin \alpha }+\frac{\cos \alpha }{\sin \beta }+\frac{\sin \beta }{\cos \alpha }\right)}^2$ is

If $M=\left(\begin{array}{cc}\frac{5}{2}& \frac{3}{2}\\ -\frac{3}{2}& -\frac{1}{2}\end{array}\right)$, then which of the following matrices is equal to $M^{2022}$ ?

Let $x\in R$ and let $P=\left[\begin{array}{ccc}1& 1& 1\\ 0& 2& 2\\ 0& 0& 3\end{array}\right], Q=\left[\begin{array}{ccc}2& x& x\\ 0& 4& 0\\ x& x& 6\end{array}\right]$ and $R=PQ{P}^{-1}.$
Then which of the following options is/are correct?

The diffusion coefficient of an ideal gas is proportional to its mean free path and mean speed. The absolute temperature of an ideal gas is increased $4$ times and its pressure $2$ times. As a result, the diffusion coefficient of this gas increases $\mathrm{x}$ times. The value of $\mathrm{x}$ is: