In the Young's double slit experiment using a monochromatic light of wavelength $\lambda$, the path difference ( in terms of an integer $n$ ) corresponding to any point having half the peak intensity 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?

Consider one mole of helium gas enclosed in a container at initial pressure $P_1$ and volume $V_1$. It expands isothermally to volume $4V_1$. After this, the gas expands adiabatically and its volume becomes $32V_1$. The work done by the gas during isothermal and adiabatic expansion processes are $W_{\mathrm{iso}}$ and $W_{\text{adia}},$ $\frac{W_{\mathrm{iso}}}{W_{\mathrm{adia}}}=f\ln \left(2\right)$, then $f$ is _________ .

Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is $0.5 \mathrm{mm}$. The circular scale has $100$ divisions and for one full rotation of the circular scale, the main scale shifts by two divisions. The measured readings are listed below.

Measurement condition	Main scale reading	Circular scale reading
Two arms of gauge touching each other without wire	$0$ division	$4$ divisions
Attempt-$1$: With wire	$4$ divisions	$20$ divisions
Attempt-$2$: With wire	$4$ divisions	$16$ divisions

What are the diameter and cross-sectional area of the wire measured using the screw gauge?

A solid sphere is in pure rolling motion on an inclined surface having inclination \(\theta\).

Match List I of the nuclear processes with List II containing parent nucleus and one of the end products of each process and then select the correct answer using the codes given below the lists :

List - i	List - II
A. Alpha decay	P. \({ }_8^{15} O \rightarrow{ }_7^{15} N+\ldots \ldots \ldots\).
B. \(\beta^{+}\)decay	Q. \({ }_{92}^{238} U \rightarrow{ }_{90}^{234} Th +\ldots \ldots \ldots\).
C. Fission	R. \({ }_{83}^{185} Bi \rightarrow{ }_{82}^{184} Pb+\ldots \ldots \ldots\).
D. Proton emission	S. \({ }_{94}^{239} Pu \rightarrow{ }_{57}^{140} La +\ldots \ldots \ldots\).

In the figure below, the switches $S_1\mathrm{a}\mathrm{n}\mathrm{d}{S}_2$ are closed simultaneously at $t=0$ and a current starts to flow in the circuit. Both the batteries have the same magnitude of the electromotive force (emf) and the polarities are as indicated in the figure. Ignore mutual inductance between the inductors. The current I in the middle wire reaches its maximum magnitude $I_{max}$ time $t=\tau$. Which of the following statements is (are) true?

The number of $3\times 3$ matrices $M$ with entries from $\{\mathrm{0,1},2\}$, such that the sum of the diagonal elements of $M^{T}M$ is $5$, are

Statement 1 The stream of water flowing at high speed from a garden hose pipe tends to spread like a fountain when held vertically up, but tends to narrow down when held vertically down.
and Statement 2 In any steady flow of an incompressible fluid, the volume flow rate of the fluid remains constant.

A uniform electric field, $\overrightarrow{E}=-400\sqrt{3}\hat{\mathrm{j}}\mathrm{N}{\mathrm{C}}^{-1}$ is applied in a region. A charged particle of mass $m$ carrying positive charge $q$ is projected in this region with an initial speed of $2\sqrt{10}\times {10}^6\mathrm{m}{\mathrm{s}}^{-1}$. This particle is aimed to hit a target $T$, which is$5\mathrm{m}$ away from its entry point into the field as shown schematically in the figure. Take $\frac{q}{m}={10}^{10}\mathrm{C}{\mathrm{kg}}^{-1}$. Then

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The mass of a nucleus \({ }_{Z}^{A} X\) is less than the sum of the masses of \((A-Z)\) number of neutrons and \(Z\) number of protons in the nucleus. The energy equivalent to the corresponding mass difference is known as the binding energy of the nucleus. A heavy nucleus of mass \(M\) can break into two light nuclei of masses \(m_{1}\) and \(m_{2}\) only if \(\left(m_{1}+m_{2}\right) \lt M\). Also two light nuclei of masses \(m_{3}\) and \(m_{4}\) can undergo complete fusion and form a heavy nucleus of mass \(M^{\prime}\) only if \(\left(m_{3}+m_{4}\right) \gt M^{\prime}\). The masses of some neutral atoms are given in the table below:


\(\left(1 u=932 M e V / c^{2}\right)\)

Question:

The correct statement is

Consider a branch of the hyperbola \(x^2-2 y^2-2 \sqrt{2} x-4 \sqrt{2} y-6=0\) with vertex at the point \(A\). Let \(B\) be one of the end points of its latus rectum. If \(C\) is the focus of the hyperbola nearest to the point \(A\), then the area of the \(\triangle A B C\) is

Statement \(1 \mathrm{~Pb}^{4+}\) compounds are stronger oxidising agents than \(\mathrm{Sn}^{4+}\) compounds.
Statement 2 The higher oxidation states for the group 14 elements are more stable for the heavier members of the group due to 'inert pair effect'.