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?

Figure 1 shows the configuration of main scale and Vernier scale before measurement. Fig. 2 shows the configuration corresponding to the measurement of diameter \(D\) of a tube. The measured value of \(D\) is:

Paragraph : In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $\left[E\right] \mathrm{a}\mathrm{n}\mathrm{d} \left[B\right]$ stand for dimensions of electric and magnetic fields respectively, while $\left[{\epsilon }_0\right]$ and $\left[{\mu }_0\right]$ stand for dimensions of the permittivity and permeability of free space respectively. $\left[L\right]\mathrm{a}\mathrm{n}\mathrm{d} \left[T\right]$ are dimensions of length and time respectively. All the quantities are given in $SI$ units

Question : The relation between $\left[{ϵ}_0\right] \mathrm{a}\mathrm{n}\mathrm{d} \left[{\mu }_0\right]$ is

A flat plate is moving normal to its plane through a gas under the action of a constant force F. The gas is kept at a very low pressure. The speed of the plate v I much less than the average speed u of the gas molecules. Which of the following options is true?

Steel wire of length \(L\) at \(40^{\circ} \mathrm{C}\) is suspended from the ceiling and then a mass \(m\) is hung from its free end. The wire is cooled down from \(40^{\circ} \mathrm{C}\) to \(30^{\circ} \mathrm{C}\) to regain its original length \(L\). The coefficient of linear thermal expansion of the steel is \(10^{-5} /{ }^{\circ} \mathrm{C}\), Young's modulus of steel is \(10^{11} \mathrm{~N} / \mathrm{m}^2\) and radius of the wire is \(1 \mathrm{~mm}\). Assume that \(L>\) diameter of the wire. Then, the value of \(m\) in \(\mathrm{kg}\) is nearly

Which of the following statement is correct in Infinitely long wire kept perpendicula to the paper carrying current inwards the given figure?

In the determination of Young's modulus \(\left(Y=\frac{4 M L g}{\pi l d^{2}}\right)\) by using Searle's method, a wire of length \(L=2 \mathrm{~m}\) and diameter \(d=0.5 \mathrm{~mm}\) is used. For a load \(M=2.5 \mathrm{~kg}\), an extension \(l=0.25 \mathrm{~mm}\) in the length of the wire is observed. Quantities \(d\) and \(l\) are measured using a screw gauge and a micrometer, respectively. They have the same pitch of \(0.5 \mathrm{~mm}\). The number of divisions on their circular scale is 100 . The contributions to the maximum probable error of the \(Y\) measurement

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.

Paragraph :
A uniform thin cylindrical disk of mass \(M\) and radius \(R\) is attached to two identical massless springs of spring constant \(k\) which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance \(d\) from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is \(L\). The disk is initially at its equilibrium position with its centre of mass \((C M)\) at a distance Lfrom the wall. The disk rolls without slipping with velocity \(\mathbf{v}_0=v_0 \hat{\mathbf{i}}\) The coefficient of friction is \(\mu\).
1
Question :
The centre of mass of the disk undergoes simple harmonic motion with angular frequency \(\omega\) equal to

Consider the lines \(L_{1}\) and \(L_{2}\) defined by
\[
L_{1}: x \sqrt{2}+y-1=0 \text { and } L_{2}: x \sqrt{2}-y+1=0
\]
For a fixed constant \(\lambda\), let \(C\) be the locus of a point \(P\) such that the product of the distance of \(P\) from \(L_{1}\) and the distance of \(P\) from \(L_{2}\) is \(\lambda^{2}\). The line \(y=2 x+1\) meets \(C\) at two points \(R\) and \(S\), where the distance between \(R\) and \(S\) is \(\sqrt{270}\).
Let the perpendicular bisector of \(R S\) meet \(C\) at two distinct points \(R^{\prime}\) and \(S^{\prime}\). Let \(D\) be the square of the distance between \(R^{\prime}\) and \(S^{\prime}\).
The value of $D$ is

Assuming that Hund's rule is violated, the bond order and magnetic nature of the diatomic molecule \(\mathrm{B}_2\) is

Consider all rectangles lying in the region $\left\{\left(x, y\right)\in R\times R:0\le x\le \frac{\pi }{2}and0\le y\le 2\sin \left(2x\right)\right\}$ and having one side on the $x$-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is

Two inductors $L_i$ (inductance $1 \mathrm{mH}$, internal resistance $3 \mathrm{\Omega }$ ) and $L_2$ (inductance $2 \mathrm{mH}$, internal resistance $4 \mathrm{\Omega }$ ), and a resistor $R$(resistance $12 \mathrm{\Omega }$ ) are all connected in parallel across a $5 \mathrm{V}$ battery. The circuit is switched on at time $t=0$. The ratio of the maximum current $\left(\frac{I_{max}}{I_{min}}\right)$ drawn from the battery is: