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If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation $z =\frac{x}{y}$. If the errors in $x, y \mathrm{a}\mathrm{n}\mathrm{d} z \mathrm{a}\mathrm{r}\mathrm{e} ∆x, ∆y \mathrm{a}\mathrm{n}\mathrm{d} ∆z$ , respectively, then

$z\pm ∆z=\frac{x\pm ∆x}{y\pm ∆y}=\frac{x}{y}\left(1\pm \frac{∆x}{x}\right){\left(1\pm \frac{∆y}{y}\right)}^{-1}$

The series expansion for ${\left(1\pm \frac{∆y}{y}\right)}^{-1}$, to first power in $∆y/y, \mathrm{i}\mathrm{s} 1\mp \left(∆y/y\right)$. The relative errors in independent variables are always added. So the error in $z$ will be

$\text{Δ}z=z\left(\frac{\text{Δ}x}{x}+\frac{\text{Δ}y}{y}\right).$
Question :
The above derivation makes the assumption that $\frac{\text{Δ}r}{x}\ll 1,\frac{\text{Δ}y}{y}\ll 1$. Therefore, the higher powers of these quantities are neglected.

A parallel plate capacitor of capacitance $\mathrm{C}$ has spacing d between two plates having area $\mathrm{A}.$ The region between the plates is filled with $\mathrm{N}$ dielectric layers, parallel to its plates, each with thickness $\mathrm{\delta }=\frac{\mathrm{d}}{\mathrm{N}}.$ The dielectric constant of the ${\mathrm{m}}^{\mathrm{t}\mathrm{h}}$ layer is ${\mathrm{K}}_{\mathrm{m}}=\mathrm{K}\left(1+\frac{\mathrm{m}}{\mathrm{N}}\right).$ For a very large $\mathrm{N}\left(>{10}^3\right),$ the capacitance $\mathrm{C}$ is $\mathrm{\alpha }\left(\frac{\mathrm{K}{\in }_0\mathrm{A}}{\mathrm{d}\mathrm{l}\mathrm{n}2}\right).$ The value of $\mathrm{\alpha }$ will be____
$[{\in }_0$ is the permittivity of free space]

A double slit setup is shown in the figure. One of the slits is in medium $2$ of refractive index $n_2$. The other slit is at the interface of this medium with another medium $1$ of refractive index $n_1\left(\ne n_2\right)$. The line joining the slits is perpendicular to the interface and the distance between the slits is $d$. The slit widths are much smaller than $d$. A monochromatic parallel beam of light is incident on the slits from medium $1$. A detector is placed in medium $2$ at a large distance from the slits, and at an angle $\theta$ from the line joining them, so that $\theta$ equals the angle of refraction of the beam. Consider two approximately parallel rays from the slits received by the detector.
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Which of the following statement(s) is(are) correct?

Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields \(\vec{E}=E_{0} \hat{j}\) and \(\vec{B}=B_{0} \hat{j}\). At time \(t=0\), this charge has velocity \(\vec{v}\) in the in the \(x\)-y plane, making an angle \(\theta\) with the \(x\)-axis. Which of the following option(s) is (are) correct for time \(t>0\) ?

Two large vertical and parallel metal plates having a separation of \(1 \mathrm{~cm}\) are connected to a DC voltage source of potential difference \(X\). A proton is released at rest midway between the two plates. It is found to move at \(45^{\circ}\) to the vertical JUST after release. Then \(X\) is nearly

A free hydrogen atom after absorbing a photon of wavelength ${\mathrm{\lambda }}_{\mathrm{a}}$ gets excited from the state $\mathrm{n}=1$ to the state $\mathrm{n}=4.$ Immediately after that the electron jumps to $\mathrm{n}=\mathrm{m}$ state by emitting a photon of wavelength ${\mathrm{\lambda }}_{\mathrm{e}}.$ Let the change in momentum of atom due to the absorption and the emission are $\mathrm{\Delta }{\mathrm{p}}_{\mathrm{a}}$ and $\mathrm{\Delta }{\mathrm{p}}_{\mathrm{e}},$ respectively. If $\frac{{\mathrm{\lambda }}_{\mathrm{a}}}{{\lambda }_e}=\frac{1}{5}.$ Which of the option(s) is/are correct?
[Use $\mathrm{h}\mathrm{c}=1242\mathrm{e}\mathrm{V}\mathrm{n}\mathrm{m},1\mathrm{n}\mathrm{m}={10}^{-9}\mathrm{m},$ $\mathrm{h}$ and $\mathrm{c}$ are Planck's constant and speed of light, respectively]

A source (S) of sound has frequency \(240 \mathrm{~Hz}\). When the observer \((\mathrm{O})\) and the source move towards each other at a speed \(v\) with respect to the ground (as shown in Case 1 in the figure), the observer measures the frequency of the sound to be \(288 \mathrm{~Hz}\). However, when the observer and the source move away from each other at the same speed \(v\) with respect to the ground (as shown in Case 2 in the figure), the observer measures the frequency of sound to be \(n \mathrm{~Hz}\). The value of \(n\) is ______

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A point charge \(Q\) is moving in a circular orbit of radius \(R\) in the \(x-y\) plane with an angular velocity \(\omega .\) This can be considered as equivalent to a loop carrying a steady current \(\frac{Q \omega}{2 \pi}\). A uniform magnetic field along the positive \(z\)-axis is now switched on, which increases at a constant rate from 0 to \(B\) in one second. Assume that the radius of the orbit remains constant. The application of the magnetic field induces an emf in the orbit. The induced emf is defined as the work done by an induced electric field in moving a unit positive charge around a closed loop. It is known that, for an orbiting charge, the magnetic dipole moment is proportional to the angular momentum with a proportionality constant \(\gamma\).

Question:

The magnitude of the induced electric field in the orbit at any instant of time during the time interval of the magnetic field change is

let $f_1 :R\to R, f_2 :\left[0, \infty \right)\to R, f_3 :R\to R and f_4 :R\to [0, \infty )$ be defined by
$f_1\left(x\right)=\left\{\begin{array}{ccc}\left|x\right|& if& x<0\\ e^x& if& x\ge 0\end{array}\right. ;f_2\left(x\right)=x^2;f_3\left(x\right)=\left\{\begin{array}{ccc}\sin x& if& x<0\\ x& if& x\ge 0\end{array}\right. and f_4\left(x\right)=\left\{\begin{array}{ccc}{f}_2\left(f_1\left(x\right)\right)& if& x<0\\ f_2\left(f_1\left(x\right)\right)-1& if& x\ge 0\end{array}\right.$
 

	List – I		List – II
(A)	$f_4$ is	(P)	Onto but not one – one
(B)	$f_3$ is	(Q)	neither continuous nor one-one
(C)	$f_2$ of $f_1$ is	(R)	differentiable but not one-one
(D)	$f_2$ is	(S)	continuous and one-one

Let $f:\left[0,2\right]\to R$ be a function which is continuous on $\left[0,2\right]$ and is differentiable on $\left(0,2\right) f\left(0\right)=1$. Let $F\left(x\right)=\underset{0}{\overset{x^2}{\int }}f\mathrm{ }\left(\sqrt{t}\right)dt$ for $x\in \left[0,2\right].$ If $F^{\text{'}}\left(x\right)=f^{\text{'}}\left(x\right)$ for all $x\in \left(0,2\right),$ then $F\left(2\right)$ equals

Match the extraction processes listed in column I with metals listed in column II.

A uniform rod of length \(L\) and mass \(M\) is pivoted at the centre. Its two ends are attached to two springs of equal spring constants \(k\). The springs are fixed to rigid supports as shown in the figure, and rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle \(\theta\) in one direction and released. The frequency of oscillation is

Newman projections $\mathrm{P},\mathrm{Q},\mathrm{R}$ and $\mathrm{S}$ are shown below:

Which one of the following options represent identical molecules?