Consider regular polygons with number of sides $n=\mathrm{3,4},5\dots$ as shown in the figure. The center of mass of all the polygons is at height $h$ from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is $\mathrm{\Delta }$ . Then, $\mathrm{\Delta }$ depends on $n$ and $h$ as

In the experiment to determine the speed of sound using a resonance column:

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.

In a Young's double slit experiment, a combination of two glass wedges \(A\) and \(B\), having refractive indices 1.7 and 1.5, respectively, are placed in front of the slits, as shown in the figure. The separation between the slits is \(d=2 \mathrm{~mm}\) and the shortest distance between the slits and the screen is \(D=2 \mathrm{~m}\). Thickness of the combination of the wedges is \(t=12 \mu \mathrm{~m}\). The value of \(l\) as shown in the figure is 1 mm . Neglect any refraction effect at the slanted interface of the wedges. Due to the combination of the wedges, the central maximum shifts (in mm ) with respect to O by ______

In the balanced condition, the values of the resistances of the four arms of a Wheatstone bridge are shown in the figure below. The resistance $R_3$ has temperature coefficient $0.0004°{\mathrm{C}}^{-1}$. If the temperature of $R_3$ is increased by $100°\mathrm{C},$ the voltage developed between $S$ and $T$ will be __________ volt.

A meter bridge is set up as shown in figure, to determine an unknown resistance \(X\) using a standard \(10 \Omega\) resistor. The galvanometer shows null point when tapping key is at \(52 \mathrm{~cm}\) mark. The end-corrections are \(1 \mathrm{~cm}\) and \(2 \mathrm{~cm}\) respectively for the ends \(A\) and \(B\). The determined value of \(X\) is

A uniform capillary tube of inner radius $r$ is dipped vertically into a beaker filled with water. The water rises to a height $h$ in the capillary tube above the water surface in the beaker. The surface tension of water is $\sigma$ . The angle of contact between water and the wall of the capillary tube is $\theta$ . Ignore the mass of water in the meniscus. Which of the following statements is (are) true?

In the following circuit $C_1=12 \mathrm{\mu F},C_2=C_3=4 \mathrm{\mu F}$ and $C_4=C_5=2 \mathrm{\mu F}$. The charge stored in $C_3$ is $\mathrm{\mu C}.$

Paragraph:

A box \(B_{1}\) contains \(1\) white ball, \(3\) red balls and \(2\) black balls. Another box \(B_{2}\) contains \(2\) white balls, \(3\) red balls and \(4\) black balls. A third box \(B_{3}\) contains \(3\) white balls, \(4\) red balls and \(5\) black balls.


Question:

If \(2\) balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these \(2\) balls are drawn from box \(B_{2}\) is

The major products obtained from the reactions in List-II are the reactants for the named reactions mentioned in List-I. Match List-I with List-II and choose the correct option.

	List I		List II
$\left(\mathrm{P}\right)$	Etard reaction	$1$	$\mathrm{Acetophenone} \stackrel{\mathrm{Zn}-\mathrm{Hg},\mathrm{HCl}}{\to }$
$\left(\mathrm{Q}\right)$	Gattermann reaction	$2$	$\mathrm{Toluene} \underset{\left(\mathrm{ii}\right) {\mathrm{SOCl}}_2 }{\overset{\left(\mathrm{i}\right) {\mathrm{KMnO}}_4, \mathrm{KOH},\mathbf{ }\mathrm{\Delta } }{\to }}$
$\left(\mathrm{R}\right)$	Gattermann-Koch reaction	$3$	$\mathrm{Benzene} \underset{\mathrm{Anhyd}. {\mathrm{AlCl}}_3}{\overset{{\mathrm{CH}}_3\mathrm{Cl}}{\to }}$
$\left(\mathrm{S}\right)$	Rosenmund reduction	$4$	$\mathrm{Aniline} \frac{{\mathrm{NaNO}}_2 /\mathrm{HCl}}{273-278 \mathrm{K}}$
		$5$	$\mathrm{Phenol} \stackrel{\mathrm{Zn},\mathrm{\Delta }}{\to }$

According to Bohr's model, the highest kinetic energy is associated with the electron in the

Let the straight line \(\mathrm{y}=2 \mathrm{x}\) touch a circle with center \((0, \alpha), \alpha>0\), and radius \(\mathrm{r}\) at a point \(\mathrm{A}_1\). Let \(\mathrm{B}_1\) be the point on the circle such that the line segment \(A_1 B_1\) is a diameter of the circle. Let \(\alpha+r=5+\sqrt{5}\).
Match each entry in List-I to the correct entry in List-II.

The correct option is

During an experiment with a metre bridge, the galvanometer shows a null point when the jockey is pressed at $40.0\mathrm{cm}$using a standard resistance of $90\mathrm{}\mathrm{\Omega }$ ,as shown in the figure. The least count of the scale used in the metre bridge is $1\mathrm{mm}$. The unknown resistance is