\(\mathrm{MgSO}_4\) on reaction with \(\mathrm{NH}_4 \mathrm{OH}\) and \(\mathrm{Na}_2 \mathrm{HPO}_4\) forms a white crystalline precipitate. What is its formula?

The correct statement(s) about O₃ is(are)

The unbalanced chemical reactions given in List I show missing reagent or condition (?) which are provided in list II. match List I with List II and select the correct answer using the code given below the lists :

List - I	List - II
A. \(PbO _2+ H _2 SO _4 \xrightarrow{?}\) \(PbSO _4 + O _2+\) other product	P. \(\text{NO}\)
B. \(Na _2 S_2 O _3+ H _2 O \xrightarrow{?}\) \(NaHSO _4~+\) other product	Q. \(\text{I}_2\)
C. \(N _2 H _4 \xrightarrow{?} N_2 +\) other product	R. Warm
D. \(XeF _2 \xrightarrow{?} Xe ~+\) other product	S. \(\text{Cl}_2\)

The wave function, ${\mathrm{\psi }}_{\mathrm{n},\mathrm{}\mathrm{l},\mathrm{}{\mathrm{m}}_{\mathrm{l}}}$ is a mathematical function whose value depends upon spherical polar coordinates $\left(\mathrm{r},\mathrm{}\mathrm{\theta },\mathrm{}\mathrm{\varphi }\right)$ of the electron and characterized by the quantum numbers $\mathrm{n},\mathrm{}\mathrm{l}$ and ${\mathrm{m}}_{\mathrm{l}}$ . Here $\mathrm{r}$ is the distance from the nucleus, $\mathrm{\theta }$ is colatitude and $\mathrm{\varphi }$ is azimuth. In the mathematical functions given in the Table, $\mathrm{Z}$is the atomic number and ${\mathrm{a}}_{\mathrm{o}}$ is Bohr radius

Column 1	Column 2	Column 3
(I) 1s orbital	\(\begin{array}{l}\text { (i) } \psi_{ n , 1, m_1} \\ \propto\left(\frac{ Z }{ a _{ o }}\right)^{\frac{3}{2}} e ^{-\left(\frac{ Zr }{ e _{ o }}\right)}\end{array}\)	(P)
(II) 2s orbital	(ii) One radial node	(Q) Probability density at the nucleus \(\propto \frac{1}{ a _{ o }^3}\).
(III) \(2 p _{ z }\) orbital	\(\begin{array}{l}\text { (iii) } \psi_{ n }, 1, m_{ l } \\ \propto\left(\frac{ Z }{ a _{ o }}\right)^{\frac{5}{2}} re ^{-\left(\frac{ Zr }{2 a _{ o }}\right)} \cos \theta\end{array}\)	(R) Probability density is maximum at the nucleus.
(IV) \(3 d_{ z }^2\) orbital	(iv) \(x y\)-plane is a nodal plane	(S) Energy needed to excite an electron from \(n =2\) state to \(n =4\) state is \(\frac{27}{32}\) times the energy needed to excite are electron from \(n =2\) state to \(n =6\) state.

For the given orbital in column 1, the only correct combination for any hydrogen-like species is

In a conductometric titration, small volume of titrant of higher concentration is added stepwise to a larger volume of titrate of much lower concentration, and the conductance is measured after each addition.
The limiting ionic conductivity \(\left(\Lambda_0\right)\) values (in \(\mathrm{mS} \mathrm{m}^2 \mathrm{~mol}^{-1}\)) for different ions in aqueous solutions are given below:

For different combinations of titrates and titrants given in List-I, the graphs of 'conductance' versus 'volume of titrant' are given in List-II.
Match each entry in List-I with the appropriate entry in List-II and choose the correct option.

Using the data provided, calculate the multiple bond energy \(\left(\mathrm{kJ} \mathrm{mol}^{-1}\right)\) of a \(\mathrm{C} \equiv \mathrm{C}\) bond in \(\mathrm{C}_{2} \mathrm{H}_{2}\). That energy is (take the bond energy of a \(\mathrm{C}-\mathrm{H}\) bond as \(350 \mathrm{~kJ} \mathrm{~mol}^{-1}\) ) \(2 \mathrm{C}(\mathrm{s})+\mathrm{H}_{2}(\mathrm{~g}) \longrightarrow \mathrm{HC}=\mathrm{CH}(\mathrm{g}) ; \Delta \mathrm{H}=\) \(225 \mathrm{~kJ} \mathrm{~mol}^{-1}\) \(2 \mathrm{C}(\mathrm{s}) \longrightarrow 2 \mathrm{C}(\mathrm{g}) ; \Delta \mathrm{H}=1410 \mathrm{~kJ} \mathrm{~mol}^{-1}\) \(\mathrm{H}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{H}(\mathrm{g}) ; \Delta \mathrm{H}=330 \mathrm{~kJ} \mathrm{~mol}^{-1}\)

The complex showing a spin-only magnetic moment of \(2.82\) B.M. is

Paragraph :
A thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle, as shown in the figure below. On each side of the partition, there is one mole of an ideal gas, with specific heat at constant volume, \(C_{V}=2 R\). Here, \(R\) is the gas constant. Initially, each side has a volume \(V_{0}\) and temperature \(T_{0}\). The left side has an electric heater, which is turned on at very low power to transfer heat \(Q\) to the gas on the left side. As a result the partition moves slowly towards the right reducing the right side volume to \(V_{0} / 2\). Consequently, the gas temperatures on the left and the right sides become \(T_{L}\) and \(T_{R}\), respectively. Ignore the changes in the temperatures of the cylinder, heater and the partition.

Question :
The value of $\frac{Q}{R{T}_0}$ is:

The synthesis of 3-octyne is achieved by adding a bromoalkane into a mixture of sodium amide and an alkyne. The bromoalkane and alkyne respectively are

Let \(\alpha, \beta\) and \(\gamma\) be real numbers such that the system of linear equations
\[
\begin{array}{c}
x+2 y+3 z=\alpha \\
4 x+5 y+6 z=\beta \\
7 x+8 y+9 z=\gamma-1
\end{array}
\]
is consistent. Let \(|M|\) represent the determinant of the matrix
\[
M=\left[\begin{array}{ccc}
\alpha & 2 & \gamma \\
\beta & 1 & 0 \\
-1 & 0 & 1
\end{array}\right]
\]
Let \(P\) be the plane containing all those \((\alpha, \beta, \gamma)\) for which the above system of linear equations is consistent, and \(D\) be the square of the distance of the point \((0,1,0)\) from the plane \(P\).
The value of $\left|M\right|$ is

LIST-$\mathrm{I}$ contains metal species and LIST-$\mathrm{II}$ contains their properties.

List - I	List - II
(I) \(\left[ Cr ( CN )_6\right]^{4-}\)	(P) \(t _{2 g}\) orbitals contain 4 electrons
(II) \(\left[ RuCl _6\right]^{2-}\)	(Q) \(\mu\) spin - only \(=4.9\) BM
(III) \(\left[ Cr \left( H _2 O \right)_6\right]^{2+}\)	(R) low spin complex ion
(IV) \(\left[ Fe \left( H _2 O \right)_6\right]^{2+}\)	(S) metal ion in 4 + oxidation state
	(T) \(d ^4\) species

[Given: Atomic number of $\mathrm{Cr}=24,\mathrm{Ru}=44,\mathrm{Fe}=26$]
Match each metal species in LIST-$\mathrm{I}$ with their properties in LIST-$\mathrm{II}$, and choose the correct option

The circle passing through the point \((-1,0)\) and touching the \(Y\)-axis at \((0,2)\), also passes through the point

A particle of mass $m$ is moving in the $xy$-plane such that its velocity at a point $(x, y)$ is given as $\overrightarrow{v}=\alpha \left(y\stackrel{\wedge }{x}+2x\stackrel{\wedge }{y}\right)$ where $\alpha$ is a non-zero constant. What is the force $\overrightarrow{F}$ acting on the particle?