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 $-4\left(4\mathrm{\mu }\right)$	$\left(\mathrm{s}\right)$	$\frac{1}{\sqrt{3}}$
		$\left(\mathrm{t}\right)$	$\frac{3}{16}$
		$\left(\mathrm{u}\right)$	$\frac{1}{16}$

The length of the strings $1,\mathrm{ }2,\mathrm{ }3$ and $4$ are kept fixed at ${\mathrm{L}}_0,\frac{3{\mathrm{L}}_0}{2},\frac{5{\mathrm{L}}_0}{4}$ and $\frac{7{\mathrm{L}}_0}{4},$ respectively. Strings $1,\mathrm{ }2,\mathrm{ }3$ and $4$ are vibrated at their $1^{\mathrm{s}\mathrm{t}},\mathrm{ }{3}^{\mathrm{r}\mathrm{d}},\mathrm{ }{5}^{\mathrm{t}\mathrm{h}}$ and ${14}^{\mathrm{t}\mathrm{h}}$ harmonics, respectively such that all the strings have same frequency. The correct match for the tension in the four strings in the units of ${\mathrm{T}}_0$ will be.

A metal rod \(A B\) of length \(10 x\) has its one end \(A\) in ice at \(0^{\circ} \mathrm{C}\) and the other end \(B\) in water at \(100^{\circ} \mathrm{C}\). If a point \(P\) on the rod is maintained at \(400^{\circ} \mathrm{C}\), then it is found that equal amounts of water and ice evaporate and melt per unit time. The latent heat of evaporation of water is \(540 \mathrm{calg}^{-1}\) and latent heat of melting of ice is \(80 \mathrm{calg}^{-1}\). If the point \(P\) is at a distance of \(\lambda x\) from the ice end \(A\), find the value of \(\lambda\). [Neglect any heat loss to the surrounding.]

The specific heat capacity of a substance is temperature dependent and is given by the formula \(C=k T\), where \(k\) is a constant of suitable dimensions in SI units, and \(T\) is the absolute temperature. If the heat required to raise the temperature of \(1 \mathrm{~kg}\) of the substance from \(-73^{\circ} \mathrm{C}\) to \(27^{\circ} \mathrm{C}\) is \(n k\), the value of \(n\) is _____
[Given: \(0 \mathrm{~K}=-273^{\circ} \mathrm{C}\).]

An electron in an excited state of $\mathrm{L}{\mathrm{i}}^{2+}$ ion has angular momentum $\frac{3\mathrm{h}}{2\mathrm{\pi }}$ . The de Broglie wavelength of the electron in this state is $\mathrm{p}\mathrm{\pi }{\mathrm{\alpha }}_0$ (where ${\mathrm{a}}_0$ is the Bohr radius). The value of $\mathrm{p}$ is

Two large circular discs separated by a distance of $0.01\mathrm{m}$ are connected to a battery via a switch as shown in the figure. Charged oil drops of density $900\mathrm{kg}{\mathrm{m}}^{-3}$ are released through a tiny hole at the centre of the top disc. Once some oil drops achieve terminal velocity, the switch is closed to apply a voltage of $200\mathrm{V}$ across the discs. As a result, an oil drop of radius $8\times {10}^{-7}\mathrm{m}$ stops moving vertically and floats between the discs. The number of electrons present in this oil drop is __________. (neglect the buoyancy force, take acceleration due to gravity $=10\mathrm{m}{\mathrm{s}}^{-2}$ and charge on an electron (e) $=1.6\times {10}^{-19}\mathrm{C}$)

Two particles of mass \(m\) each are tied at the ends of a light string of length \(2 a\). The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance \(a\) from the centre \(P\) (as shown in the figure). Now, the mid-point of the string is pulled vertically upwards with a small but constant force \(F\). As a result, the particles move towards each other on the surface.
The magnitude of acceleration, when the separation between them becomes \(2 x\), is

Paragraph:

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 distance \(r\) of the block at time \(t\) is

In a radioactive decay chain, ${\mathrm{ }}_{90}^{232}\mathrm{Th}$ nucleus decays to ${\mathrm{ }}_{82}^{212}\mathrm{Pb}$ nucleus. Let $N_{\alpha } \text{and} N_{\beta }$ be the number of $\alpha \text{and} {\beta }^{-}$ particles, respectively, emitted in this decay process. Which of the following statements is (are) true?

Let \(z\) be a complex number such that the imaginary part of \(z\) is non-zero and \(a=z^{2}+z+1\) is real. Then a cannot take the value

The work function ( \(\Phi\) ) of some metals is listed below. The number of metals which will show photoelectric effect when light of \(300 \mathrm{~nm}\) wavelength falls on the metal is

Let \(S=\{a+b \sqrt{2}: a, b \in \mathbb{Z}\}, T_1=\)\(\left\{(-1+\sqrt{2})^n: n \in \mathbb{N}\right\}\), and \(T_2=\left\{(1+\sqrt{2})^n: n \in \mathbb{N}\right\}\). Then which of the following statements is (are) TRUE?

The quadratic equation $\mathrm{p}\left(\mathrm{x}\right)\mathrm{ }=\mathrm{ }0$ with real coefficients has purely imaginary roots. Then the equation $\mathrm{p}\left(\mathrm{p}\right(\mathrm{x}\left)\right)\mathrm{ }=\mathrm{ }0$ has

For non-negative integers $n,$ let
$f\left(n\right)=\frac{\sum _{k=0}^n\sin \left(\frac{k+1}{n+2}\pi \right)\sin \left(\frac{k+2}{n+2}\pi \right)}{\sum _{k=0}^n{\sin }^2\left(\frac{k+1}{n+2}\pi \right)}$
Assuming ${\cos }^{-1}x$ takes values in $\left[0, \pi \right],$ which of the following options is/are correct?