Two coherent monochromatic point sources $S_1$ and $S_2$ of wavelength $\lambda =600nm$ are placed symmetrically on either side of the center of the circle as shown. The sources are separated by a distance $d=1.8mm.$ This arrangement produces interference fringes visible as alternate bright and dark spots on the circumference of the circle. The angular separation between two consecutive bright spots is $\mathrm{\Delta }\theta$ . Which of the following options is/are correct?

A particle of mass $m$ is projected from the ground with an initial speed $u_{\circ }$ at an angle $\alpha$ with the horizontal. At the highest point of its trajectory, it makes a completely inelastic collision with another identical particle, which was thrown vertically upward from the ground with the same initial speed $u_{\circ }$. The angle that the composite system makes with the horizontal immediately after the collision is

A disk of radius $R$ with uniform positive charge density $\sigma$ is placed on the $xy$ plane with its center at the origin. The Coulomb potential along the $z$-axis is
$V\left(z\right)=\frac{\sigma }{2{ϵ}_0}\left(\sqrt{R^2+z^2}-z\right)$ A particle of positive charge $q$ is placed initially at rest at a point on the $z$-axis with $z=z_0$ and $z_0>0$. In addition to the Coulomb force, the particle experiences a vertical force $\overrightarrow{F}=-c\hat{k}$ with $c>0$. Let $\beta =\frac{2c{\epsilon }_0}{q\sigma }$. Which of the following statement(s) is(are) correct?

List I describes thermodynamic processes in four different systems. List II gives the magnitudes (either exactly or as a close approximation) of possible changes in the internal energy of the system due to the process.

	List-I		List-II
(I)	${10}^{-3} \mathrm{kg}$ of water $100 °\mathrm{C}$ is converted to steam at the same temperature, at a pressure of ${10}^5\mathrm{Pa}$. The volume of the system changes from ${10}^{-6}{\mathrm{m}}^3$ to ${10}^{-3}{\mathrm{m}}^3$ in the process. Latent heat of water $=2250 \mathrm{kJ} {\mathrm{kg}}^{-1}$	(P)	$2 \mathrm{kJ}$
(II)	$0.2$ moles of a rigid diatomic ideal gas with volume $V$ at temperature $500 \mathrm{K}$ undergoes an isobaric expansion to volume $3 V$. Assume $8.0 \mathrm{J} {\mathrm{mol}}^{–1} {\mathrm{K}}^{–1}$	(Q)	$7 \mathrm{kJ}$
(III)	One mole of a monoatomic ideal gas is compressed adiabatically from volume $V=\frac{1}{3}{\mathrm{m}}^3$ and pressure $2 \mathrm{kPa}$ to volume $\frac{V}{8}$.	(R)	$4 \mathrm{kJ}$
(IV)	Three moles of a diatomic ideal gas whose molecules can vibrate, is given $9 \mathrm{kJ}$ of heat and undergoes isobaric expansion.	(S)	$5 \mathrm{kJ}$
		(T)	$3 \mathrm{kJ}$

Which one of the following options is correct?

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Modern trains are based on Maglev technology in which trains are magnetically leviated, which runs its EDS Maglev system.
There are coils on both sides of wheels. Due to motion of train, current induces in the coil of track which levitate it. This is in accordance with Lenz's law. If trains lower down then due to Lenz's law a repulsive force increases due to which train gets uplifted and if it goes much high then there is a net downward force disc to gravity. The advantage of Maglev train is that there is no friction between the train and the track, thereby reducing power consumption and enabling the train to attain very high speeds.
Disadvantage of Maglev train is that as it slows down the electromagnetic forces decreases and it becomes difficult to keep it leviated and as it moves forward according to Lenz law, there is an electromagnetic drag force.
Question:
Which force causes the train to elevate up?

A small object is placed 50 cm to the left of thin convex lens of focal length 30 cm. A convex spherical mirror of radius of curvature 100 cm is placed to the right of the lens at a distance of 50 cm. The mirror is tilted such that the axis of the mirror is at an angle $\theta ={30}^o$ to the axis of the lens, as shown in the figure. If the origin of the coordinate system is taken to be at the centre of the lens, the coordinates (in cm) of the point (x, y) at which the image is formed are:

A string of length $1 \mathrm{m}$ and mass $2\times {10}^{-5} \mathrm{kg}$ is under tension $T$. When the string vibrates, two successive harmonics are found to occur at frequencies $750 \mathrm{Hz}$ and $1000 \mathrm{Hz}$. The value of tension $T$ is _____ newton.

If the distance of the point \(P(1,-2,1)\) from the plane \(x+2 y-2 z=\alpha\), where \(\alpha>0\), is 5 , then the foot of the perpendicular from \(P\) to the plane is

Among the following, the paramagnetic compound is

A sample initially contains only U-238 isotope of uranium. With time, some of the U-238 radioactively decays into \(\mathrm{Pb}-206\) while the rest of it remains undisintegrated.
When the age of the sample is \(\mathbf{P} \times 10^8\) years, the ratio of mass of \(\mathrm{Pb}-206\) to that of \(\mathrm{U}-238\) in the sample is found to be 7 . The value of \(\mathbf{P}\) is ______
[Given: Half-life of U-238 is \(4.5 \times 10^9\) years; \(\log _c 2=0.693\) ]

A vernier calipers has \(1 \mathrm{~mm}\) marks on the main scale. It has 20 equal divisions on the vernier scale which match with 16 main scale divisions. For this vernier calipers, the least count is

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Light guidance in an optical fiber can be understood by considering a structure comprising of thin solid glass cylinder of refractive index \(n_{1}\) surrounded by a medium of lower refractive index \(n_{2}\). The light guidance in the structure takes place due to successive total internal reflections at the interface of the media \(n_{1}\) and \(n_{2}\) as shown in the figure. All rays with the angle of incidence \(i\) less than a particular value \(i_{m}\) are confined in the medium of refractive index \(n_{1}\). The numerical aperture (NA) of the structure is defined as \(\sin i_{m}\).



Question:

If two structures of same cross-sectional area, but different numerical apertures \(N A_{1}\) and \(N A_{2}\left(N A_{2} < N A_{1}\right)\) are joined longitudinally, the numerical aperture of the combined structure is

Let \(S_1=\{(i, j, k): i, j, k \in\{1,2, \ldots, 10\}\}\)
\(S_2=\{(i, j): 1 \leq i < j+2 \leq 10, i, j\) \(\in\{1,2, \ldots, 10\}\} \)
\(S_3=\{(i, j, k, l): 1 \leq i < j < k < l, i,\) \(j, k, l \in\{1,2, \ldots . ., 10\}\}\)
and \(S_4=\{(i, j, k, l): i, j, k\) and \(l\) are distint elements in \(\{1,2, \ldots, 10\}\}\) If the total number of elements in the set \(S_r\) is \(n_r, r=1,2,3,4\) then which of the following statements is (are) TRUE?