A particle of mass $m$ is moving in the $xy$-plane such that its velocity at a point $(x, y)$ is given as $\overrightarrow{v}=\alpha \left(y\stackrel{\wedge }{x}+2x\stackrel{\wedge }{y}\right)$ where $\alpha$ is a non-zero constant. What is the force $\overrightarrow{F}$ acting on the particle?

A vibrating string of certain length \(l\) under a tension \(T\) resonates with a mode corresponding to the first overtone (third harmonic) of an air column of length \(75 \mathrm{~cm}\) inside a tube closed at one end. The string also generates 4 beat/s when excited along with a tuning fork of frequency \(n\). Now when the tension of the string is slightly increased the number of beats reduces to 2 per second. Assuming the velocity of sound in air to be \(340 \mathrm{~m} / \mathrm{s}\), the frequency \(n\) of the tuning fork (in \(\mathrm{Hz}\) ) is

A block of weight $100\mathrm{}\mathrm{N}$ is suspended by copper and steel wires of same cross-sectional area $0.5\mathrm{}\mathrm{c}{\mathrm{m}}^2$ and, length $\sqrt{3}\mathrm{}\mathrm{m}$ and $1\mathrm{}\mathrm{m},$ respectively. Their other ends are fixed on a ceiling as shown in figure. The angles subtended by copper and steel wires with ceiling are ${30}^{\mathrm{o}}$ and ${60}^{\mathrm{o}},$ respectively. If elongation in copper wire is $\left(\mathrm{\Delta }{\mathrm{l}}_{\mathrm{C}}\right)$ and elongation in steel wire is $\left(\mathrm{\Delta }{\mathrm{l}}_{\mathrm{S}}\right),$ then the ratio $\frac{\mathrm{\Delta }{\mathrm{l}}_{\mathrm{C}}}{\mathrm{\Delta }{\mathrm{l}}_{\mathrm{S}}}$ is _____.
[Young's modulus for copper and steel are $1\times {10}^{11}\mathrm{N}/{\mathrm{m}}^2$ and $2\times {10}^{11}\mathrm{N}/{\mathrm{m}}^2$ respectively]

The densities of two solid spheres $\mathrm{A}$ and $\mathrm{B}$ of the same radii $\mathrm{R}$ vary with radial distance $\mathrm{r}$ as ${\mathrm{\rho }}_{\mathrm{A}}\left(\mathrm{r}\right)=\mathrm{k}\left(\frac{\mathrm{r}}{\mathrm{R}}\right)$ and ${\mathrm{\rho }}_{\mathrm{B}}\left(\mathrm{r}\right)=\mathrm{k}{\left(\frac{\mathrm{r}}{\mathrm{R}}\right)}^5$ , respectively, where $\mathrm{k}$ is a constant. The moments of inertia of the individual spheres about axes passing through their centres are ${\mathrm{I}}_{\mathrm{A}}$ and ${\mathrm{I}}_{\mathrm{B}}$ , respectively. If $\frac{{\mathrm{I}}_{\mathrm{B}}}{{\mathrm{I}}_{\mathrm{A}}}=\frac{\mathrm{n}}{10}$ , the value of $\mathrm{n}$ is

Paragraph :
A uniform thin cylindrical disk of mass \(M\) and radius \(R\) is attached to two identical massless springs of spring constant \(k\) which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance \(d\) from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is \(L\). The disk is initially at its equilibrium position with its centre of mass \((C M)\) at a distance Lfrom the wall. The disk rolls without slipping with velocity \(\mathbf{v}_0=v_0 \hat{\mathbf{i}}\) The coefficient of friction is \(\mu\).
Question :
The net external force acting on the disk when its centre of mass is at displacement \(x\) with respect to its equilibrium position is

A biconvex lens of focal length \(15 \mathrm{~cm}\) is in front of a plane mirror. The distance between the lens and the mirror is 10 \(\mathrm{cm}\). A small object is kept at a distance of \(30 \mathrm{~cm}\) from the lens. The final image is

A monochromatic beam of light is incident at ${60}^{\mathrm{o}}$ on one face of an equilateral prism of refractive index $\mathrm{n}$ and emerges from the opposite face making an angle $\mathrm{\theta }\left(\mathrm{n}\right)$ with the normal (see the figure). For $\mathrm{n}=\sqrt{3}$ the value of $\mathrm{\theta }$ is ${60}^{\mathrm{o}}$ and $\frac{\mathrm{d}\mathrm{\theta }}{\mathrm{d}\mathrm{n}}=\mathrm{m}$ . The value of $\mathrm{m}\mathrm{}$ is

Three identical capacitors $C_1,C_2\mathrm{a}\mathrm{n}\mathrm{d}{C}_3$ have a capacitance of $1.0\mu F$ each and they are uncharged initially. They are connected in a circuit as shown in the figure and $C_1$ is then filled completely with a dielectric material of relative permittivity ${ϵ}_r$ . The cell electromotive force (emf) $V_0=8V$ . First the switch $S_1$ is closed while the switch $S_2$ is kept open. When the capacitor $C_3$ is fully charged, $S_1$ is opened and $S_2$ is closed simultaneously. When all the capacitors reach equilibrium, the charge on $C_3$ is found to be $5\mu C$ . The value of ${ϵ}_r$ =____________.

Answer the following by appropriately matching the lists based on the information given in the paragraph.
A musical instrument is made using four different metal strings, $1,\mathrm{ }2,\mathrm{ }3$ and $4$ with mass per unit length $\mathrm{\mu },2\mathrm{\mu },3\mathrm{\mu }$ and $4\mathrm{\mu }$ respectively. The instrument is played by vibrating the strings by varying the free length in between the range ${\mathrm{L}}_0$ and $2{\mathrm{L}}_0.$ It is found that in string- $1\left(\mathrm{\mu }\right)$ at free length ${\mathrm{L}}_0$ and tension ${\mathrm{T}}_0$ the fundamental mode frequency is ${\mathrm{f}}_0.$
List-I gives the above four strings while list-II lists the magnitude of some quantity.

	List I		List II
$\left(\mathrm{a}\right)$	String $-1\left(\mathrm{\mu }\right)$	$\left(\mathrm{p}\right)$	$1$
$\left(\mathrm{b}\right)$	String $-2\left(2\mathrm{\mu }\right)$	$\left(\mathrm{q}\right)$	$\frac{1}{2}$
$\left(\mathrm{c}\right)$	String $-3\left(3\mathrm{\mu }\right)$	$\left(\mathrm{r}\right)$	$\frac{1}{\sqrt{2}}$
$\left(\mathrm{d}\right)$	String $-4\left(4\mathrm{\mu }\right)$	$\left(\mathrm{s}\right)$	$\frac{1}{\sqrt{3}}$
		$\left(\mathrm{t}\right)$	$\frac{3}{16}$
		$\left(\mathrm{u}\right)$	$\frac{1}{16}$

The length of the strings $1,\mathrm{ }2,\mathrm{ }3$ and $4$ are kept fixed at ${\mathrm{L}}_0,\frac{3{\mathrm{L}}_0}{2},\frac{5{\mathrm{L}}_0}{4}$ and $\frac{7{\mathrm{L}}_0}{4},$ respectively. Strings $1,\mathrm{ }2,\mathrm{ }3$ and $4$ are vibrated at their $1^{\mathrm{s}\mathrm{t}},\mathrm{ }{3}^{\mathrm{r}\mathrm{d}},\mathrm{ }{5}^{\mathrm{t}\mathrm{h}}$ and ${14}^{\mathrm{t}\mathrm{h}}$ harmonics, respectively such that all the strings have same frequency. The correct match for the tension in the four strings in the units of ${\mathrm{T}}_0$ will be.

For every integer \(n\), let \(a_{n}\) and \(b_{n}\) be real numbers. Let function \(f: I R \rightarrow I R\) be given by

\(f(x)=\left\{\begin{array}{lll}

a_{n}+\sin \pi x, & \text { for } & x \in[2 n, 2 n+1] \\

b_{n}+\cos \pi x, & \text { for } & x \in(2 n-1,2 n)

\end{array}\right.\)

for all integers \(n\). If \(f\) is continuous, then which of the following hold \((s)\) for all \(n\) ?

The total number of compounds having at least one bridging oxo group among the molecules given below is
\(\text N _2\text O _5,\)\(\text P _4\text O _6,\)\(\text P _4\text O _8,\)\(\text H _4\text P _2\text O _5,\)\(\text H _5\text P _3\text O _{10},\) \(\text H _2\text S_2\text O _3,\text H _2\text S_2\text O _5\)

Consider the following reactions (unbalanced)
$\mathrm{Z}\mathrm{n}+H\mathrm{o}\mathrm{t}\mathrm{ }\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}.\mathrm{ }{\mathrm{H}}_2\mathrm{S}{\mathrm{O}}_4\to \mathrm{G}+\mathrm{R}+\mathrm{X}$
$\mathrm{Z}\mathrm{n}+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}.\mathrm{ }\mathrm{N}\mathrm{a}\mathrm{O}\mathrm{H}\to \mathrm{T}+\mathrm{Q}$
\(\text G +\text H _2\text S+ \text{NH} _4 \text{OH} \rightarrow\text Z ~(\)a precipitate\()+\text X +\text Y\)
Choose the correct option(s).

The value of the integral $\underset{0}{\overset{\frac{1}{2}}{\int }}\frac{1+\sqrt{3}}{{\left({\left(x+1\right)}^2{\left(1-x\right)}^6\right)}^{\frac{1}{4}}} dx$ is