A thermodynamic system is taken from an initial state $\mathrm{i}$ with internal energy ${\mathrm{U}}_{\mathrm{i}}\mathrm{}=100\mathrm{}\mathrm{J}$ to the final state f along two different paths iaf and ibf, as schematically shown in the figure. The work done by the system along the paths af, ib and bf are ${\mathrm{w}}_{\mathrm{a}\mathrm{f}}=\mathrm{}200\mathrm{}\mathrm{J},\mathrm{}{\mathrm{W}}_{\mathrm{i}\mathrm{b}}\mathrm{}=\mathrm{}50\mathrm{}\mathrm{J}$ and ${\mathrm{W}}_{\mathrm{b}\mathrm{f}}\mathrm{}=\mathrm{}100\mathrm{}\mathrm{J}$ respectively. The heat supplied to the system along the path iaf, ib and bf are iaf ib ${\mathrm{Q}}_{\mathrm{i}\mathrm{a}\mathrm{f}},\mathrm{}{\mathrm{Q}}_{\mathrm{i}\mathrm{b}}$ and ${\mathrm{Q}}_{\mathrm{b}\mathrm{f}}$ respectively. If the internal energy of the system in the state b is ${\mathrm{U}}_{\mathrm{b}}=200\mathrm{}\mathrm{J}$ and ${\mathrm{Q}}_{\mathrm{i}\mathrm{a}\mathrm{f}}=500\mathrm{}\mathrm{J}$ , the ratio $\frac{{\mathrm{Q}}_{\mathrm{b}\mathrm{f}}}{{\mathrm{Q}}_{\mathrm{i}\mathrm{b}}}\mathrm{}$ is

Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimension of $\mathrm{L},$ which of the following statement (s) is/are correct?

Three identical capacitors $C_1,C_2\mathrm{a}\mathrm{n}\mathrm{d}{C}_3$ have a capacitance of $1.0\mu F$ each and they are uncharged initially. They are connected in a circuit as shown in the figure and $C_1$ is then filled completely with a dielectric material of relative permittivity ${ϵ}_r$ . The cell electromotive force (emf) $V_0=8V$ . First the switch $S_1$ is closed while the switch $S_2$ is kept open. When the capacitor $C_3$ is fully charged, $S_1$ is opened and $S_2$ is closed simultaneously. When all the capacitors reach equilibrium, the charge on $C_3$ is found to be $5\mu C$ . The value of ${ϵ}_r$ =____________.

Statement \(1 \mathrm{In}\) a Meter Bridge experiment, null point for an unknown resistance is measured. Now, the unknown resistance is put inside an enclosure maintained at a higher temperature. The null point can be obtained at the same point as before by decreasing the value of the standard resistance.
and Statement 2 Resistance of a metal increase with increase in temperature.

A point object is placed at distance of \(20 \mathrm{~cm}\) from a thin planoconvex lens of focal length \(15 \mathrm{~cm}\). The plane surface of the lens is now silvered. The image created by the system is at

A conducting loop in the shape of right angled isosceles triangle of height 10 cm is kept such that the ${90}^o$ vertex is very close to an infinitely long conducting wire (see the figure). The wire is electrically insulated from the loop. The hypotenuse of the triangle is parallel to the wire. The current in the triangular loop is in counterclockwise direction and increased at constant rate of $10A{s}^{-1}.$ Which of the following statement(s) is (are) true?

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If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation $z =\frac{x}{y}$. If the errors in $x, y \mathrm{a}\mathrm{n}\mathrm{d} z \mathrm{a}\mathrm{r}\mathrm{e} ∆x, ∆y \mathrm{a}\mathrm{n}\mathrm{d} ∆z$ , respectively, then

$z\pm ∆z=\frac{x\pm ∆x}{y\pm ∆y}=\frac{x}{y}\left(1\pm \frac{∆x}{x}\right){\left(1\pm \frac{∆y}{y}\right)}^{-1}$

The series expansion for ${\left(1\pm \frac{∆y}{y}\right)}^{-1}$, to first power in $∆y/y, \mathrm{i}\mathrm{s} 1\mp \left(∆y/y\right)$. The relative errors in independent variables are always added. So the error in $z$ will be

$\text{Δ}z=z\left(\frac{\text{Δ}x}{x}+\frac{\text{Δ}y}{y}\right).$
Question :
The above derivation makes the assumption that $\frac{\text{Δ}r}{x}\ll 1,\frac{\text{Δ}y}{y}\ll 1$. Therefore, the higher powers of these quantities are neglected.

Vessel-1 contains \(\mathbf{w}_2 \mathrm{~g}\) of a non-volatile solute \(\mathbf{X}\) dissolved in \(\mathbf{w}_1 \mathrm{~g}\) of water. Vessel- 2 contains \(\mathbf{w}_2 \mathrm{~g}\) of another non-volatile solute \(\mathbf{Y}\) dissolved in \(\mathbf{w}_1 \mathrm{~g}\) of water. Both the vessels are at the same temperature and pressure. The molar mass of \(\mathbf{X}\) is \(80 \%\) of that of \(\mathbf{Y}\). The van't Hoff factor for \(\mathbf{X}\) is 1.2 times of that of \(\mathbf{Y}\) for their respective concentrations.
The elevation of boiling point for solution in Vessel-1 is ________. \(\%\) of the solution in Vessel-2.

The freezing point (in \({ }^{\circ} \mathrm{C}\) ) of solution containing \(0.1 \mathrm{~g}\) of \(\mathrm{K}_3\left[\mathrm{Fe}(\mathrm{CN})_6\right]\) (molecular weight 329\()\) in \(100 \mathrm{~g}\) of water \(\left(K_f=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\right)\) is

A trinitro compound, $1,3,5$-tris-($4$-nitrophenyl) benzene, on complete reaction with an excess of $\mathrm{Sn}/\mathrm{HCl}$ gives a major product, which on treatment with an excess of ${\mathrm{NaNO}}_2/\mathrm{HCl}$ at $0°\mathrm{C}$ provides $\mathbf{P}$ as the product. $\mathbf{P}$, upon treatment with excess of ${\mathrm{H}}_2\mathrm{O}$ at room temperature, gives the product $\mathbf{Q}$. Bromination of $\mathbf{Q}$ in aqueous medium furnishes the product $\mathbf{R}$. The compound $\mathbf{P}$ upon treatment with an excess of phenol under basic conditions gives the product $\mathbf{S}$. The molar mass difference between compounds $\mathbf{Q}$ and $\mathbf{R}$ is $474 \mathrm{g} {\mathrm{mol}}^{-1}$ and between compounds $\mathbf{P}$ and $\mathbf{S}$ is $172.5 \mathrm{g} {\mathrm{mol}}^{-1}$.
The number of heteroatoms present in one molecule of \(R\) is [Use: Molar mass (in \(\text{g mol} -1\) ): \(H =1,\) \(C =12, N=14, O =16,\) \(Br =80, Cl =35.5\) Atoms other than C and H are considered as heteroatoms]

In the circuit shown below, the key is pressed at time t = 0. Which of the following statement(s) is(are) true?

A student is performing an experiment using a resonance column and a tuning fork of frequency \(244 s^{-1}\). He is told that the air in the tube has been replaced by another gas (assume that the column remains filled with the gas). If the minimum height at which resonance occurs is \((0.350 \pm 0.005) m\), the gas in the tube is (Useful information: \(\sqrt{167\text{ RT}}=640\text{ J}^{\frac{1}{2}}\text{ mol}^{-\frac{1}{2}} ;\) \(\sqrt{140\text{ RT}}=590\text{ J}^{\frac{1}{2}}\text{ mole}^{-\frac{1}{2}}\) . The molar masses M in grams are given in the options. Take the values of \(\sqrt{\frac{10}{\text M }}\) for each gas as given there.)

Let $f\left(x\right)=\frac{\sin \pi x}{x^2}, x>0.$
Let $x_1<x_2<x_3<\dots <x_n<\dots$ be all the points of local maximum of $f$ and $y_1<y_2<y_3<\dots <y_n<\dots$ be all the points of local minimum of $f.$
Then which of the following options is/are correct?