Among the following, the paramagnetic compound is

A sample \((5.6 \mathrm{~g})\) containing iron is completely dissolved in cold dilute \(\mathrm{HCl}\) to prepare a \(250 \mathrm{~mL}\) of solution. Titration of \(25.0 \mathrm{~mL}\) of this solution requires \(12.5 \mathrm{~mL}\) of \(0.03 \mathrm{M} \mathrm{KMnO}_{4}\) solution to reach the end point. Number of moles of \(\mathrm{Fe}^{2+}\) present in \(250 \mathrm{~mL}\) solution is \(\mathbf{x} \times 10^{-2}\) (consider complete dissolution of \(\mathrm{FeCl}_{2}\) ). The amount of iron present in the sample is \(\mathbf{y} \%\) by weight.
(Assume: \(\mathrm{KMnO}_{4}\) reacts only with \(\mathrm{Fe}^{2+}\) in the solution Use: Molar mass of iron as \(56 \mathrm{~g} \mathrm{~mol}^{-1}\) )
The value of $\mathrm{x}$ is________ .

The correct match of the group reagents in List-I for precipitating the metal ion given in List-II from solutions, is

List - I	List - II
(P) Passing \(H _2 S\) in the presence of \(NH _4 OH\)	(1) \(Cu ^{2+}\)
(Q) \(\left( NH _4\right)_2 CO _3\) in the presence of \(NH _4 OH\)	(2) \(Al ^{3+}\)
(R) \(NH _4 OH\) in the presence of \(NH _4 Cl\)	(3) \(Mn ^{2+}\)
(S) Passing \(H _2 S\) in the presence of dilute HCl	(4) \(Ba ^{2+}\)
	(5) \(Mg ^{2+}\)

Both \(\left[\mathrm{Ni}(\mathrm{CO})_4\right]\) and \(\left[\mathrm{Ni}(\mathrm{CN})_4\right]^{2-}\) are diamagnetic. The hybridisations of nickel in these complexes, respectively, are

A closed vessel contains \(10 \mathrm{~g}\) of an ideal gas \(\mathbf{X}\) at \(300 \mathrm{~K}\), which exerts 2 atm pressure. At the same temperature, \(80 \mathrm{~g}\) of another ideal gas \(\mathbf{Y}\) is added to it and the pressure becomes \(6 \mathrm{~atm}\). The ratio of root mean square velocities of \(\mathbf{X}\) and \(\mathbf{Y}\) at \(300 \mathrm{~K}\) is

Adsorption of phenol from its aqueous solution on to fly ash obeys Freundlich isotherm. At a given temperature, from \(10 \mathrm{mg} \mathrm{g}^{-1}\) and \(16 \mathrm{mg} \mathrm{g}^{-1}\) aqueous phenol solutions, the concentrations of adsorbed phenol are measured to be \(4 \mathrm{mg} \mathrm{g}^{-1}\) and \(10 \mathrm{mg} \mathrm{g}^{-1}\), respectively. At this temperature, the concentration (in \(\mathrm{mg} \mathrm{g}^{-1}\) ) of adsorbed phenol from \(20 \mathrm{mg} \mathrm{g}^{-1}\) aqueous solution of phenol will be \(\qquad\) .
Use : \(\log _{10} 2=0.3\)

When \(20 \mathrm{~g}\) of naphthoic acid \(\left(\mathrm{C}_{11} \mathrm{H}_8 \mathrm{O}_2\right)\) is dissolved in \(50 \mathrm{~g}\) of benzene \(\left(K_f=1.72 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\right)\), a freezing point depression of \(2 \mathrm{~K}\) is observed. The van't Hoff factor \((i)\) is

Two balls, having linear momenta \(\overrightarrow{\mathbf{p}}_1=p \hat{\mathbf{i}}\) and \(\overrightarrow{\mathbf{p}}_2=-p \hat{\mathbf{i}}\), undergo a collision in free space. There is no external force acting on the balls. Let \(\overrightarrow{\mathbf{p}}_1^{\prime}\) and \(\overrightarrow{\mathbf{p}}_2^{\prime}\) be their final momenta. The following option(s) is/are NOT ALLOWED for any non-zero value of \(p, a_1, a_2, b_1, b_2, c_1\) and \(c_2\)

The number of hydroxyl group(s) in Q is

A transparent slab of thickness d has a refractive index n (z) that increases with z. Here z is the vertical distance inside the slab, measured from the top. The slab is placed between two media with uniform refractive indices $n_{1}and{n}_2\left(>n_1\right),$ as shown in the figure. A ray of light is incident with angle ${\theta }_i$ from medium 1 and emerges in medium 2 with refraction angle ${\theta }_f$ with a lateral displacement $\mathcal{l}$ . Which of the following statement(s) 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?

Answer the following by appropriately matching the lists based on the information given in the paragraph.
A musical instrument is made using four different metal strings, $1,\mathrm{ }2,\mathrm{ }3$ and $4$ with mass per unit length $\mathrm{\mu },2\mathrm{\mu },3\mathrm{\mu }$ and $4\mathrm{\mu }$ respectively. The instrument is played by vibrating the strings by varying the free length in between the range ${\mathrm{L}}_0$ and $2{\mathrm{L}}_0.$ It is found that in string- $1\left(\mathrm{\mu }\right)$ at free length ${\mathrm{L}}_0$ and tension ${\mathrm{T}}_0$ the fundamental mode frequency is ${\mathrm{f}}_0.$
List-I gives the above four strings while list-II lists the magnitude of some quantity.
 

	List I		List II
$\left(\mathrm{a}\right)$	String $-1\left(\mathrm{\mu }\right)$	$\left(\mathrm{p}\right)$	$1$
$\left(\mathrm{b}\right)$	String $-2\left(2\mathrm{\mu }\right)$	$\left(\mathrm{q}\right)$	$\frac{1}{2}$
$\left(\mathrm{c}\right)$	String $-3\left(3\mathrm{\mu }\right)$	$\left(\mathrm{r}\right)$	$\frac{1}{\sqrt{2}}$
$\left(\mathrm{d}\right)$	String $-4\left(4\mathrm{\mu }\right)$	$\left(\mathrm{s}\right)$	$\frac{1}{\sqrt{3}}$
		$\left(\mathrm{t}\right)$	$\frac{3}{16}$
		$\left(\mathrm{u}\right)$	$\frac{1}{16}$

If the tension in each string is ${\mathrm{T}}_0,$ the correct match for the highest fundamental frequency in ${\mathrm{f}}_0$ units will be,

The \(\beta\)-decay process, discovered around 1900 , is basically the decay of a neutron \((n)\). In the laboratory, a proton \((p)\) and an electron \(\left(e^{-}\right)\)are observed as the decay products of the neutron. Therefore, considering the decay of a neutron as a two-body decay process, it was predicted theoretically that the kinetic energy of the electron should be a constant. But experimentally, it was observed that the electron kinetic energy has continuous spectrum. Considering a three-body decay process, i.e.
\(n \rightarrow p+e^{-}+\bar{v}_{e}\), around 1930, Pauli explained the observed electron energy spectrum. Assuming the anti-neutrino \(\left(\bar{v}_{e}\right)\) to be massless and possessing negligible energy, and the neutron to be at rest, momentum and energy conservation principles are applied. From this calculation, the maximum kinetic energy of the electron is \(0.8 \times 10^{6} \mathrm{eV}\). The kinetic energy carried by the proton is only the recoil energy.

Question:
If the anti-neutrino had a mass of \(3 \mathrm{eV} / \mathrm{c}^{2}\) (where \(\mathrm{c}\) is the speed of light) instead of zero mass, what should be the range of the kinetic energy, \(K\), of the electron?