Using the expression $2d \sin \theta =\lambda$, one calculates the values of $d$ by measuring the corresponding angle $\text{θ}$ in the range $0$ to $90°$. The wavelength $\lambda$ is exactly known and the error in $\theta$ is constant for all the values of $\theta$. As $\text{θ}$ increases from $0°$,

Paragraph :
When liquid medicine of density \(\rho\) is to be put in the eye, it is done with the help of a dropper. As the bulb on the top of the dropper is pressed, a drop forms at the opening of the dropper. We wish to estimate the size of the drop.
We first assume that the drop formed at the opening is spherical because that requires a minimum increase in its surface energy. To determine the size, we calculate the net vertical force due to the surface tension \(T\) when the radius of the drop is \(R\). When this force becomes smaller than the weight of the drop, the drop gets detached from the dropper.Question :
After the drop detaches, its surface energy is

A boy is pushing a ring of mass \(2 \mathrm{~kg}\) and radius \(0.5 \mathrm{~m}\) with a stick as shown in the figure. The stick applies a force of \(2 \mathrm{~N}\) on the ring and rolls it without slipping with an acceleration of \(0.3 \mathrm{~m} / \mathrm{s}^2\). The coefficient of friction between the ground and the ring is large enough that rolling always occurs and the coefficient of friction between the stick and the ring is \(\frac{P}{10}\). The value of \(P\) is

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 _______.

Four point charges, each of \(+q\), are rigidly fixed at the four corners of a square planar soap film of side \(a\). The surface tension of the soap film is \(\gamma\). The system of charges and planar film are in equilibrium, and \(a=k\left[\frac{q^2}{\gamma}\right]^{1 / N}\), where \(k\) is a constant. Then, \(N\) is

A series LCR circuit is connected to a $45 \sin \left(\omega t\right)$ Volt source. The resonant angular frequency of the circuit is ${10}^5 \mathrm{rad} {\mathrm{s}}^{-1}$ and current amplitude at resonance is $I_0$. When the angular frequency of the source is $\omega =8\times {10}^4 \mathrm{rad} {\mathrm{s}}^{-1}$, the current amplitude in the circuit is $0.05 I_0$. If $L=50 \mathrm{mH}$, match each entry in List-$\mathrm{I}$ with an appropriate value from List-$\mathrm{II}$ and choose the correct option.

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\mathrm{P}$	$I_0$ in $\mathrm{mA}$	$1$	$44.4$
$\mathrm{Q}$	The quality factor of the circuit	$2$	$18$
$\mathrm{R}$	The bandwidth of the circuit in $\mathrm{rad} {\mathrm{s}}^{-1}$	$3$	$400$
$\mathrm{S}$	The peak power dissipated at resonance in Watt	$4$	$2250$
		$5$	$500$

A bi-convex lens is formed with two thin plano-convex lenses as shown in the figure. Refractive index \(n\) of the first lens is \(1.5\) and that of the second lens is \(1.2\). Both the curved surface are of the same radius of curvature \(R=14\) \(\mathrm{cm}\). For this bi-convex lens, for an object distance of 40 \(\mathrm{cm}\), the image distance will be

Let \(g(x)=\frac{(x-1)^n}{\log \cos ^m(x-1)} ; 0 < x < 2, m\) and \(n\) are integers, \(m \neq 0, n>0\) and let \(p\) be the left hand derivative of \(|x-1|\) at \(x=1\). If \(\lim _{x \rightarrow 1^{+}} g(x)=p\), then

In a triangle $ABC$, let $AB=\sqrt{23},BC=3$ and $CA=4$. Then the value of $\frac{\cot A+\cot C}{\cot B}$ is

A tiny spherical oil drop carrying a net charge \(q\) is balanced in still air with a vertical uniform electric field of strength \(\frac{81 \pi}{7} \times 10^5 \mathrm{Vm}^{-1}\). When the field is switched off, the drop is observed to fall with terminal velocity \(2 \times 10^{-3} \mathrm{~ms}^{-1}\). Given \(g=9.8 \mathrm{~ms}^{-2}\), viscosity of the air \(=1.8 \times 10^{-5} \mathrm{Ns} \mathrm{m}^{-2}\) and the density of oil \(=900 \mathrm{~kg} \mathrm{~m}^{-3}\), the magnitude of \(q\) is

An incompressible liquid is kept in a container having a weightless piston with a hole. A capillary tube of inner radius $0.1\mathrm{mm}$ is dipped vertically into the liquid through the airtight piston hole, as shown in the figure. The air in the container is isothermally compressed from its original volume $V_0$to$\frac{100}{101}{V}_0$with the movable piston.Considering air as an ideal gas, the height $\left(h\right)$ of the liquid column in the capillary above the liquid level in $\mathrm{cm}$is _____.
[Given: Surface tension of the liquid is $0.075\mathrm{N}{\mathrm{m}}^{-1}$, atmospheric pressure is ${10}^5\mathrm{N}{\mathrm{m}}^{-2}$, acceleration due to gravity $\left(g\right)$ is $10\mathrm{m}{\mathrm{s}}^{-2}$, density of the liquid is ${10}^3\mathrm{kg}{\mathrm{m}}^{-3}$and contact angle of capillary surface with the liquid is zero]

Two men are walking along a horizontal straight line in the same direction. The man in front walks at a speed $1.0 m s^{-1}$ and the man behind walks at a speed $2.0 m s^{-1}$ . A third man is standing at a height $12 m$ above the same horizontal line such that all three men are in a vertical plane. The two walking men are blowing identical whistles which emit a sound of frequency $1430 Hz$ . The speed of sound in air is $330 m s^{-1}.$ At the instant, when the moving men are $10 m$ apart, the stationary man is equidistant from them. The frequency of beats in $Hz$ , heard by the stationary man at this instant, is __________.

Consider a hydrogen atom with its electron in the ${\mathrm{n}}^{\mathrm{t}\mathrm{h}}$ orbital. An electromagnetic radiation of wavelength 90 nm is used to ionize the atom. If the kinetic energy of the ejected electron is 10.4 eV, then the value of n is $\left(\mathrm{h}\mathrm{c}=1242\mathrm{ }\mathrm{e}\mathrm{V}\mathrm{ }\mathrm{n}\mathrm{m}\right)$