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_______

Consider a disc rotating in the horizontal plane with a constant angular speed \(\omega\) about its centre \(\mathrm{O}\). The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown, two pebbles \(P\) and \(Q\) are simultaneously projected at an angle towards \(R\). The velocity of projection is in the \(y\) - \(z\) plane and is same for both pebbles with respect to the disc. Assume that (i) they land back on the disc before the disc has completed \(1 / 8\) rotation, (ii) their range is less than half the disc radius, and (iii) \(\omega\) remains constant throughout. Then

The wavelength of the first spectral line in the Balmer series of hydrogen atom is \(6561 Å\). The wavelength of the second spectral line in the Balmer series of singly ionized helium atom is

A steady current $I$ flows along an infinitely long hollow cylindrical conductor of radius $R$. This cylinder is placed co-axially inside an infinite solenoid of radius $2R$. The solenoid has $n$ turns per unit length and carries a steady current $I$. Consider a point $P$ at a distance $r$ from the common axis. The correct statement(s) is(are):

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 ______

Paragraph:
Modern trains are based on Maglev technology in which trains are magnetically leviated, which runs its EDS Maglev system.
There are coils on both sides of wheels. Due to motion of train, current induces in the coil of track which levitate it. This is in accordance with Lenz's law. If trains lower down then due to Lenz's law a repulsive force increases due to which train gets uplifted and if it goes much high then there is a net downward force disc to gravity. The advantage of Maglev train is that there is no friction between the train and the track, thereby reducing power consumption and enabling the train to attain very high speeds.
Disadvantage of Maglev train is that as it slows down the electromagnetic forces decreases and it becomes difficult to keep it leviated and as it moves forward according to Lenz law, there is an electromagnetic drag force.
Question:
What is the disadvantage of this system?

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

Let $f\left(x\right)=\sin \left(\pi \cos x\right)$ and $g\left(x\right)=\cos \left(2\pi \sin x\right)$ be two functions defined for $x>0.$ Define the following sets whose elements are written in the increasing order:
$X=\left\{x:f\left(x\right)=0\right\}, Y=\left\{x:f^{\text{'}}\left(x\right)=0\right\}$
$Z=\left\{x:g\left(x\right)=0\right\}, W=\left\{x:g^{\text{'}}\left(x\right)=0\right\}$
List- $I$ contains the sets $X, Y, Z$ and $W.$ List- $\mathrm{I}\mathrm{I}$ contains some information regarding these sets.
 

List- $\mathrm{I}$	List- $\mathrm{I}\mathrm{I}$
$\left(I\right)$	$X$	$\left(P\right)$	$\supseteq \left\{\frac{\pi }{2},\frac{3\pi }{2}, 4\pi , 7\pi \right\}$
$\left(II\right)$	$Y$	$\left(Q\right)$	an arithmetic progression
$\left(III\right)$	$Z$	$\left(R\right)$	NOT an arithmetic progression
$\left(IV\right)$	$W$	$\left(S\right)$	$\supseteq \left\{\frac{\pi }{6},\frac{7\pi }{6},\frac{13\pi }{6}\right\}$
		$\left(T\right)$	$\supseteq \left\{\frac{\pi }{3},\frac{2\pi }{3}, \pi \right\}$
		$\left(U\right)$	$\supseteq \left\{\frac{\pi }{6},\frac{3\pi }{4}\right\}$

Which of the following is the only correct combination?

The distance between two stars of masses $3{\mathrm{M}}_S$ and $6{\mathrm{M}}_S$ is $9\mathrm{R}$. Here $R$ is the mean distance between the centres of the Earth and the Sun, and $M_S$ is the mass of the Sun. Two stars orbit around their common centre of mass in circular orbits with period $nT$, where$T$ is the period of Earth's revolution around the Sun. The value of $n$ is _____.

For the following reaction scheme, percentage yields are given along the arrow:

\(\mathbf{x} g\) and \(\mathbf{y} g\) are mass of \(\mathbf{R}\) and \(\mathbf{U}\), respectively.
(Use: Molar mass (in \(\mathrm{g} \mathrm{mol}^{-1}\) ) of \(\mathrm{H}, \mathrm{C}\) and \(\mathrm{O}\) as 1,12 and 16 , respectively)
The value of$\mathrm{y}$ is ___.

Let the eccentricity of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) be reciprocal to that of the ellipse \(x^2+4 y^2=4\). If the hyperbola passes through a focus of the ellipse, then

The figure shows a system consisting of (i) a ring of outer radius \(3 R\) rolling clockwise without slipping on a horizontal surface with angular speed \(\omega\) and (ii) an inner disc of radius \(2 R\) rotating anti-clockwise with angular speed \(\omega / 2\). The ring and disc are separated by frictionless ball bearings. The point \(P\) on the inner disc is at a distance \(R\) from the origin, where \(O P\) makes an angle of \(30^{\circ}\) with the horizontal. Then with respect to the horizontal surface,

Four charges ${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_2,\mathrm{}{\mathrm{Q}}_3$ and ${\mathrm{Q}}_4$ of same magnitude are fixed along the x axis at $\mathrm{x}=-2\mathrm{a},\mathrm{}-\mathrm{a},\mathrm{}+\mathrm{a}$ and $+2\mathrm{a}$ , respectively. A positive charge q is placed on the positive y axis at a distance $\mathrm{b}\mathrm{}>\mathrm{}0$. Four options of the signs of these charges are given in List I. The direction of the forces on the charge q is given in List II. Match List I with List II and select the correct answer using the code given below the lists.

	List I		List II
A.	${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_2,\mathrm{}{\mathrm{Q}}_3\mathrm{}{\mathrm{Q}}_4$ all positive	$\mathrm{P}.$	$+\mathrm{x}$
B.	${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_2$ positive; ${\mathrm{Q}}_3,\mathrm{}{\mathrm{Q}}_4$ negative	Q.	$-\mathrm{x}$
C.	${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_4$ positive; ${\mathrm{Q}}_2,\mathrm{}{\mathrm{Q}}_3$ negative	$\mathrm{R}.$	$+\mathrm{y}$
D.	${\mathrm{Q}}_1,\mathrm{}{\mathrm{Q}}_3$ positive; ${\mathrm{Q}}_2,\mathrm{}{\mathrm{Q}}_4$ negative	$\mathrm{S}.$	$-\mathrm{y}$