Two resistances $R_1=X \mathrm{\Omega }$ and $R_2=1 \mathrm{\Omega }$ are connected to a wire $AB$ of uniform resistivity, as shown in the figure. The radius of the wire varies linearly along its axis from $0.2 \mathrm{mm}$ at $A$ to $1 \mathrm{mm}$ at $B$. A galvanometer $\left(G\right)$ connected to the center of the wire, $50 \mathrm{cm}$ from each end along its axis, shows zero deflection when $A$ and $B$ are connected to a battery. The value of $X$ is _______.

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:

Two vectors $\overrightarrow{A} \mathrm{a}\mathrm{n}\mathrm{d} \overrightarrow{B}$ are defined as $\overrightarrow{A}=a\widehat{i} and \overrightarrow{B}=a\left(\cos \omega t \widehat{i}+\sin \omega t\widehat{j}\right)$, where a is a constant and $\omega =\frac{\pi }{6} rad s^{-1}.$ If $\left|\overrightarrow{A}+\overrightarrow{B}\right|=\sqrt{3}\left|\overrightarrow{A}-\overrightarrow{B}\right|$ at time $t=\tau$ for the first time, the value of $\tau$ in seconds, is_______

A piece of wire is bent in the shape of a parabola \(y=k x^2\) (y-axis vertical) with a bead of mass \(m\) on it. The bead can slide on the wire without friction. It stays at the lowest point of the parabola when the wire is at rest. The wire is now accelerated parallel to the \(x\)-axis with the constant acceleration \(a\). The distance of the new equilibrium position of the bead, where the bead can stays at rest with respect to the wire, from the \(y\)-axis is

A spherical body of radius $\mathrm{R}$ consists of a fluid of constant density and is in equilibrium under its own gravity. If $\mathrm{P}\left(\mathrm{r}\right)$ is the pressure at $\mathrm{r}(\mathrm{r}<\mathrm{R})$ , then the correct option(s) is (are)

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A special metal \(S\) conducts electricity without any resistance. A closed wire loop, made of \(S\), does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius \(a\), with its center at the origin. A magnetic dipole of moment \(m\) is brought along the axis of this loop from infinity to a point at distance \(r(\gg a)\) from the center of the loop with its north pole always facing the loop, as shown in the figure below.

The magnitude of magnetic field of a dipole \(m\), at a point on its axis at distance \(r\), is \(\frac{\mu_{0}}{2 \pi} \frac{m}{r^{3}}\), where \(\mu_{0}\) is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, \(m_{1}\) and \(m_{2}\), separated by a distance \(r\) on the common axis, with their north poles facing each other, is \(\frac{k m_{1} m_{2}}{r^{4}}\), where \(k\) is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

Question :
When the dipole m is placed at a distance r from the centre of the loop (as shown in the figure), the current induced in the loop will be proportional to:

A block of mass ${\mathrm{m}}_1=1\mathrm{}\mathrm{k}\mathrm{g}$ another mass ${\mathrm{m}}_2=2\mathrm{}\mathrm{k}\mathrm{g}$ , are placed together (see figure) on an inclined plane with angle of inclination $\mathrm{\theta }$ . Various values of $\mathrm{\theta }$ are given in List I. The coefficient of friction between the block ${\mathrm{m}}_1$ and the plane is always zero. The coefficient of static and dynamic friction between the block ${\mathrm{m}}_2$ and the plane are equal to $\mathrm{\mu }=0.3$ . In List II expressions for the friction on the block ${\mathrm{m}}_2$ are given. Match the correct expression of the friction in List II with the angles given in List I, and choose the correct option. The acceleration due to gravity is denoted by g.
[Useful information: $\tan \left({5.5}^{\mathrm{o}}\right)\approx 0.1;\tan \left({11.5}^{\mathrm{o}}\right)\approx 0.2;\tan \left({16.5}^{\mathrm{o}}\right)\approx 0.3$ ]

	List I		List II
A.	$\mathrm{\theta }=5^{\mathrm{o}}$	$\mathrm{P}.$	${\mathrm{m}}_2\mathrm{g}\sin \mathrm{\theta }$
B.	$\mathrm{\theta }={10}^{\mathrm{o}}$	$\mathrm{Q}.$	$\left({\mathrm{m}}_1+{\mathrm{m}}_2\right)\mathrm{g}\sin \mathrm{\theta }$
C.	$\mathrm{\theta }={15}^{\mathrm{o}}$	$\mathrm{R}.$	$\mathrm{\mu }{\mathrm{m}}_2\mathrm{}\mathrm{g}\cos \mathrm{\theta }$
D.	$\mathrm{\theta }={20}^{\mathrm{o}}$	$\mathrm{S}.$	$\mathrm{\mu }\left({\mathrm{m}}_1+{\mathrm{m}}_2\right)\mathrm{g}\cos \mathrm{\theta }$

Paragraph: There are five students $S_1, S_2, S_3, S_4$ and $S_5$ in a music class and for them there are five seats $R_1, R_2, R_3, R_4$ and $R_5$ arranged in a row, where initially the seat $R_i$ is allotted to the student $S_i, i = \mathrm{1,2},\mathrm{3,4},5.$ But, on the examination day, the five students are randomly allotted the five seats
Question : For $i = 1,2,3,4$ Let $T_i$ denote the event that the students $S_i$ and $S_{i+1}$ do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event $T_1\cap T_2\cap T_3\cap T_4$ is

For a > b > c > 0, the distance between (1, 1) and the point of intersection of the lines ax + by + c = 0 and bx + ay + c = 0 is less than $2\sqrt{2}$.Then

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The concentration of potassium ions inside a biological cell is at least twenty times higher than the outside. The resulting potential difference across the cell is important in several processes such as transmission of nerve impulses and maintaining the ion balance. A simple model for such a concentration cell involving a metal \(\mathrm{M}\) is :
\(M(s) \mid M^{+}(\alpha q ; 0.05 \text { molar }) \| M^{+}(a q ; 1 \text { molar })\) \(\mid M(s)\)
For the above electrolytic cell the magnitude of the cell potential \(\left|E_{\text {cell }}\right|=70 \mathrm{mV}\).
Question
For the above cell

An engineer is required to visit a factory for exactly four days during the first $15$ days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during $1-15$ June $2021$ is_____

Hydrogen peroxide in its reaction with KIO₄ and NH₂OH respectively, is acting as a

Consider the hyperbola $\frac{x^2}{100}-\frac{y^2}{64}=1$ with foci at $S$ and $S_1$, where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $\angle SP{S}_1=\alpha$, with $\alpha <\frac{\pi }{2}$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola, intersects the straight line $S_{1}P$ at $P_1$. Let $\delta$ be the distance of $P$ from the straight line $S{P}_1$, and $\beta =S_{1}P$. Then the greatest integer less than or equal to $\frac{\beta \delta }{9}\sin \frac{\alpha }{2}$ is _____ .