The plot given below shows $\mathrm{P}-\mathrm{T}$ curves (where, $\mathrm{P}$ is the pressure and $\mathrm{T}$ is the temperature) for two solvents $\mathrm{X}$ and $\mathrm{Y}$ and isomolal solutions of $\mathrm{NaCl}$ in these solvents. $\mathrm{NaCl}$ is completely dissociates in both the solvents.

On the addition of equal number of moles of a non-volatile solute $\mathrm{S}$ in equal amount (in $\mathrm{kg}$) of these solvents, the elevation of boiling point of solvent $\mathrm{X}$ is three times that of solvent $\mathrm{Y}$. The solute $\mathrm{S}$ is known to undergo dimerization in these solvents. If the degree of dimerization is $0.7$ in the solvent $\mathrm{Y}$, the degree of dimerization in the solvent $\mathrm{X}$ is _______.

A positron is emitted from \({ }_{11}^{23} \mathrm{Na}\). The ratio of the atomic mass and atomic number of the resulting nuclide is

Based on the compounds of group 15 elements, the correct statement(s) is (are)

Different possible thermal decomposition pathways for peroxyesters are shown below. Match each pathway from List I with an appropriate structure from List II and select the correct answer using the code given below the lists.

	List – I		List – II
A	Pathway P	P.
B.	Pathway Q	Q.
C.	Pathway R	R.
D.	Pathway S	S.

In the scheme given below, the total number of intramolecular aldol condensation products formed from ' \(Y\) ' is

For the given compound X, the total number of optically active stereoisomers is ____.

The reversible expansion of an ideal gas under adiabatic and isothermal conditions is shown in the figure. Which of the following statement(s) is (are) correct?

Paragraph :
When liquid medicine of density \(\rho\) is to be put in the eye, it is done with the help of a dropper. As the bulb on the top of the dropper is pressed, a drop forms at the opening of the dropper. We wish to estimate the size of the drop.
We first assume that the drop formed at the opening is spherical because that requires a minimum increase in its surface energy. To determine the size, we calculate the net vertical force due to the surface tension \(T\) when the radius of the drop is \(R\). When this force becomes smaller than the weight of the drop, the drop gets detached from the dropper.
Question :
If the radius of the opening of the dropper is \(r\), the vertical force due to the surface tension on the drop of radius \(R\) (assuming \(r< < R\) ) is

A steel wire of diameter 0.5 mm and Young's modulus \(2 \times 10^{11} Nm ^{-2}\) carries a load of mass M . The length of the wire with the load is 1.0 m . A vernier scale with 10 divisions is attached to the end of this wire. Next to the steel wire is a reference wire to which a main scale, of least count 1.0 mm , is attached. The 10 divisions of the vernier scale correspond to 9 divisions of the main scale. Initially, the zero of vernier scale coincides with the zero of main scale. If the load on the steel wire is increased by \(1.2 k g\), the vernier scale division which coincides with a main scale division is _____________ . Take \(g =10 m s^{-1}\) and \(\pi=3.2\).

Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E:\frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P:y^2=12x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the points $A_1$ and $A_2$, respectively and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$, respectively. Then which of the following statements is(are) true?

Let the functions $f:R\to R$ and $g:R\to R$ be defined by
$f\left(x\right)=e^{x-1}-e^{-\left|x-1\right|}$ and $g\left(x\right)=\frac{1}{2}\left(e^{x-1}+e^{1-x}\right)$.
Then the area of the region in the first quadrant bounded by the curves $y=f\left(x\right), y=g\left(x\right)$ and $x=0$ is

A spring - block system is resting on a frictionless floor as shown in the figure. The spring constant is $2.0N{m}^{-1}$ and the mass of the block is $2.0kg$ . Ignore the mass of the spring. Initially the spring is in an unstretched condition. Another block of mass 1.0kg moving with a speed of $2.0m{s}^{-1}$ collides elastically with the first block. The collision is such that the 2.0kg block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is _________.

Let ${\text{S}}_{\text{n}}=\underset{\text{k}=1}{\overset{4\text{n}}{\Sigma }}{\left(-1\right)}^{\frac{\text{k}\left(\text{k}+1\right)}{2}}{\text{k}}^2$. Then S_n can take value(s)