A student performed the experiment of determination of focal length of a concave mirror by \(u-v\) method using an optical bench of length \(1.5 \mathrm{~m}\). The focal length of the mirror used is 24 \(\mathrm{cm}\). The maximum error in the location of the image can be \(0.2 \mathrm{~cm}\).
The 5 sets of \((u, v)\) values recorded by the student (in \(\mathrm{cm})\) are : \((42,56),(48\), \(48),(60,40),(66,33),(78,39)\). The data set(s) that cannot come from experiment and is (are) incorrectly recorded, is(are)

In the List-I below, four different paths of a particle are given as functions of time. In these functions, $\alpha \mathrm{a}\mathrm{n}\mathrm{d} \beta$ are positive constants of appropriate dimensions and $\alpha \ne \beta$ . In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned: $\overrightarrow{p}$ is the linear momentum, $\overrightarrow{L}$ is the angular momentum about the origin, K is the kinetic energy, U is the potential energy and E is the total energy. Match each path in List-I with those quantities in List-II, which are conserved for that path.

	LIST -I		LIST-II
A)	$\overrightarrow{r}\left(t\right)=\alpha t\widehat{i}+\beta \widehat{j}$	P)	$\overrightarrow{p}$
B)	$\overrightarrow{r}\left(t\right)=\alpha \cos \omega t\widehat{i}+\beta \sin \omega t\widehat{j}$	Q)	$\overrightarrow{L}$
C)	$\overrightarrow{r}\left(t\right)=\alpha \cos \omega t\widehat{i}+\sin \omega t\widehat{j}$	R)	$K$
D)	$\overrightarrow{r}\left(t\right)=\alpha t\widehat{i}+\frac{\beta }{2}{t}^2\widehat{j}$	S)	$U$
		T)	$E$

Two small particles of equal masses start moving in opposite directions from a point \(A\) in a horizontal circular orbit. Their tangential velocities are \(v\) and \(2 v\) respectively, as shown in the figure. Between collisions, the particles move with constant speeds. After making how many elastic collisions, other than that at \(A\), these two particles will again reach the point \(A\) ?

An electric dipole with dipole moment $\frac{{\mathrm{p}}_0}{\sqrt{2}}\left(\widehat{\mathrm{i}}+\widehat{\mathrm{j}}\right)$ is held fixed at the origin $\mathrm{O}$ in the presence of an uniform electric field of magnitude ${\mathrm{E}}_0.$ If the potential is constant on a circle of radius $\mathrm{R}$ centered at the origin as shown in figure, then the correct statement(s) is/are:
( ${\mathrm{\epsilon }}_0$ is permittivity of free space, $\mathrm{R}>>$ dipole size)

Statement 1 It is easier to pull a heavy object than to push it on a level ground.
and Statement 2 The magnitude of frictional force depends on the nature of the two surfaces in contact.

An optical bench has $1.5\mathrm{ }\mathrm{m}$ long scale having four equal divisions in each $\mathrm{c}\mathrm{m}.$ While measuring the focal length of a convex lens, the lens is kept at $75\mathrm{ }\mathrm{c}\mathrm{m}$ mark of the scale and the object pin is kept at $45\mathrm{ }\mathrm{c}\mathrm{m}$ mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at $135\mathrm{ }\mathrm{c}\mathrm{m}$ mark. In this experiment, the percentage error in the measurement of the focal length of the lens is_____

One end of a horizontal uniform beam of weight $W$ and length $L$ is hinged on a vertical wall at point $O$ and its other end is supported by a light inextensible rope. The other end of the rope is fixed at point $Q$, at a height $L$ above the hinge at point $O$. A block of weight $\alpha W$ is attached at the point $P$ of the beam, as shown in the figure (not to scale). The rope can sustain a maximum tension of  $\left(2\sqrt{2}\right)W$. Which of the following statement(s) is(are) correct?

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Consider the function \(f:(-\infty, \infty) \rightarrow(-\infty, \infty)\) defined by \(f(x)=\frac{x^2-a x+1}{x^2+a x+1} ; 0 < a < 2\)Question:
Which of the following is true ?

In a Young's double slit experiment, each of the two slits A and B, as shown in the figure, are oscillating about their fixed center and with a mean separation of \(0.8 \mathrm{~mm}\). The distance between the slits at time \(t\) is given by \(d=(0.8+0.04 \sin \omega t) \mathrm{mm}\), where \(\omega=0.08 \mathrm{rad} \mathrm{s}^{-1}\). The distance of the screen from the slits is \(1 \mathrm{~m}\) and the wavelength of the light used to illuminate the slits is \(6000 Ã…\). The interference pattern on the screen changes with time, while the central bright fringe (zeroth fringe) remains fixed at point \(O\).

The maximum speed in \(\mu \mathrm{m} / \mathrm{s}\) at which the \(8^{\text {th }}\) bright fringe will move is ________ .

Two co-axial conducting cylinders of same length \(\ell\) with radii \(\sqrt{2} R\) and \(2 R\) are kept, as shown in Fig. 1. The charge on the inner cylinder is \(Q\) and the outer cylinder is grounded. The annular region between the cylinders is filled with a material of dielectric constant \(\kappa=5\). Consider an imaginary plane of the same length \(\ell\) at a distance R from the common axis of the cylinders. This plane is parallel to the axis of the cylinders. The cross-sectional view of this arrangement is shown in Fig. 2. Ignoring edge effects, the flux of the electric field through the plane is ( \(\epsilon_0\) is the permittivity of free space):

Let $E, F$ and $G$ be three events having probabilities $P\left(E\right)=\frac{1}{8},P\left(F\right)=\frac{1}{6}$ and $P\left(G\right)=\frac{1}{4},$ and let $P(E\cap F\cap G)=\frac{1}{10}$
For any event $H$, if $H^c$ denotes its complement, then which of the following statements is (are) TRUE ?

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	$4$	$2250$
		$5$	$500$

For a polynomial $g\left(x\right)$ with real coefficients, let $m_g$ denote the number of distinct real roots of $g\left(x\right)$. Suppose $S$ is the set of polynomials with real coefficients defined by $S=\left\{{\left(x^2-1\right)}^2\left(a_0+a_{1}x+a_2{x}^2+a_3{x}^3\right) :a_0, a_1, a_2, a_3\in R\right\}$. For a polynomial $f$, let $f\text{'}$ and $f^{″}$ denote its first and second order derivatives respectively. Then the minimum possible value of $\left(m_{f^{\prime }}+m_{f^{″}}\right)$, where $f\in S$, is ____